W. F. BRINKMAN? P. A. GANNON‘ J. S. NOWAK!
H. O. BURTON? T. J. HERR* L. C. SEIFERT®
). CHERNAK! D. M. HILL? W. E. STRICH”
M. F. COCCA! D. HIRSCH? J. W. TIMKO!
B. R. DARNALL' S. HORING' Vv. A. VYSSOTSKY!|
A. FEINER? N. W. NILSON? J. H. WEBER®

‘AT&T Bell Laboratories =? AT&T Information Systems — * Sandia National Laboratories
* AT&T Network Systems >*AT&T Technology Systems °® AT&T Technologies
7 AT&T Communications 8 AT&T

Coherent Lightwave Communications 2153
J. Salz
Cross-Polarization Cancellation and Equalization in Digital 2211

Transmission Over Dually Polarized Multipath Fading
Channels
M. Kavehrad and J. Salz

Cross-Polarization Interference Cancellation and Nonminimum 2247
Phase Fades
M. Kavehrad

Analysis/Simulation Study of Cross-Polarization Cancellation 2261
in Dual-Polarization Digital Radio
L. J. Greenstein

A Laboratory Simulation Facility for Multipath Fading 2281
Microwave Radio Channels

Single-Frame Vowel Recognition Using Vector Quantization 2319
With Several Distance Measures
L. R. Rabiner and F. K. Soong

Traffic Capabilities of Two Rearrangeably Nonblocking 2331
Photonic Switching Modules |
R. A. Thompson

Union Bounds on Viterbi Algorithm Performance 2375
W. Turin
A New File Transfer Protocol 2387

A VCR-Based Access System for Large Pictorial Databases
K. Y. Eng, O. Yue, B. G. Haskell, and C. Grimes

On Binary Differential Detection for Coherent Lightwave
Communication
J. E. Mazo

Performance Signatures for Dual-Polarized Transmission of
M-QAM Signals Over Fading Multipath Channels
M. Kavehrad and C. A. Siller, Jr.

Statistical Model for Amplitude and Delay of Selective Fading
P. Balaban

The chief objective of this paper is to develop a fundamental understanding
of the effects of laser phase noise on the performance of coherent lightwave
communication systems. A comprehensive treatment applicable to a wide
variety of coherent receiver designs under a broad range of conditions is
provided. Our models and analytical tools are developed in sufficient detail to
encompass a broad range of applications. Formulas are derived for the bit
error rate in homodyne and heterodyne Phase Shift Keying (PSK), Differen-
tial Phase Shift Keying (DPSK), Frequency Shift Keying (FSK) and on-off
keying. Estimates are provided of the penalties accrued due to phase noise.
Based on detailed mathematical analysis and estimates, we made several
findings. Near quantum-limited receiver sensitivity can be achieved with PSK
using homodyne detection only at signaling rates 3000 times greater than the
laser linewidth. A receiver sensitivity 3 to 6 decibels poorer than the quantum
limit can be achieved with heterodyne rather than homodyne detection. DPSK,
for example, can operate at rates only 300 times greater than the laser
linewidth. At lower rates, FSK is an attractive candidate. It can be designed
to be extremely tolerant of phase noise by using wide frequency deviations.

Despite the rapid advance of lightwave technology over the past
decade, the basic operation of an optical fiber communications link
has remained essentially unchanged. Direct modulation of the source
(on-off keying) and direct detection at the receiver using a pin diode
or Avalanche Photodiode (APD) have been the mainstays of lightwave

systems since their infancy. Recent advances in lightwave components,
however, now permit significant improvements on this time-honored
approach. For example, external modulation of the laser with an
electro-optic device has recently produced better long-distance per-
formance at very high bit rates (24 Gb/s) than direct modulation of
the laser itself. Another promising technique now being pursued in
research laboratories around the world is the use of coherent light-
wave—the optical analog of superheterodyne radio reception. Here we
provide a comprehensive analytical treatment applicable to a wide
variety of coherent receiver designs under a broad range of conditions.
Recognizing that not all contingencies can be covered explicitly, we
have endeavored to develop our model and analytical tools in sufficient
detail so they can be applied to other cases of interest.

Unlike direct detection, where the optical signal is converted directly
into a demodulated electrical output, the coherent receiver first adds
to the signal a locally generated optical wave and then detects the
sum. The resulting photocurrent is a replica of the original signal,
translated down in frequency from the optical domain (~10° GHz) to
the radio domain (sS few GHz), where conventional electronic tech-
niques can be used for further signal processing, such as filtering and
demodulation. This method offers significant improvements in re-
ceiver sensitivity and wavelength selectivity compared with direct
detection. In the 1.3- to 1.6-um lightwave band, for example, an ideal
coherent receiver requires a signal energy of only 10 to 20 photons per
bit to achieve a bit error rate of 10-°—far less than the roughly 1000
photons required by today’s APDs. And because of its improved
selectivity, a coherent receiver might permit wavelength-division-
multiplexed systems with channel spacings of only, say 100 MHz,
instead of the 100 GHz required with conventional optical multiplexing
technology. A further advantage of coherent reception, not often cited
but potentially very important, is that it allows the use of electronic
equalization to compensate for the effects of optical pulse dispersion
in the fiber. a

The possible advantages of coherent optical communications have
been explored for numerous applications. Much of the earlier work
emphasized space communications, where highly collimated laser
beams could be used to span enormous distances.” More recently, the
use of coherent techniques in optical fiber systems has received con-
siderable attention. Especially at bit rates above 2 Gb/s, where APD
performance begins to deteriorate, the high sensitivity and potentially
broad bandwidth of coherent receivers is a powerful stimulus to further
research. As part of this effort, theroretical analyses of several types
of coherent receivers have been published in the literature.*° (Since
these investigations are generally based on fundamental equations and

hardware designs closely resembling those encountered in the micro-
wave domain, it is perhaps not too surprising that their general
conclusions are also similar. In both cases, for example, the most
energy-efficient binary system uses phase shift keying and coherent
demodulation.) The performance of several experimental receivers has
been compared with theory, and the agreement, is generally good,
especially when HeNe or YAG lasers are used for the optical sources.
In practical coherent lightwave systems, however, itis expected that
semiconductor injection lasers will be used; when these have been
employed in coherent receiver experiments, measured sensitivity is
almost always degraded. The effect is most pronounced in angle-
modulation experiments, where receiver performance is often so poor
that low error rates (<10~°) cannot be achieved at all.°’ The cause of
this degradation has been identified as laser phase noise, an impair-
ment that is particularly severe in semiconductor devices. The effect
of this noise mechanism is to impress random phase modulation on
the otherwise monochromatic output of the laser, thereby impairing
its performance in angle-modulation experiments. A fundamental
understanding of this impairment and its effects on performance is
the primary objective of this paper.

Laser phase noise is usually characterized in terms of the linewidth
of the laser emission spectrum, (a readily measurable quantity that is
directly proportional to the spectral density of the underlying phase
noise process.) As was implied above, the linewidths available with
today’s distributed feedback semiconductor lasers, typically 5 to 50
MHz, are too broad to take full advantage of coherent techniques.
Consequently the realization of a stable, reliable narrow-linewidth
source is an extremely high priority in lightwave research. Several
promising techniques have been demonstrated in the laboratory, but
their usefulness under actual field conditions has yet to be established.
Since reducing laser linewidth appears to be a difficult task, it is
important to understand the effects of this impairment in order to
establish precisely how much reduction is required.

The paper begins with an executive summary. Section II provides a
brief review of direct detection methods and fundamental limits.
Properties of phase noise in lasers are reviewed in Section III. Analysis
of phase-lock technique are presented in Section IV. Frequency Shift
Keying (FSK) is treated in Section V while Differential Phase Shift
Keying (DPSK) and on-off-keying are treated in Sections VI and VII.

A coherent lightwave receiver is the optical analog of a superheter-
odyne radio set. Instead of detecting photons directly, the coherent
receiver first converts the incoming signal from the optical regime

down to the radio regime, and then uses conventional electronic
circuitry to perform various signal processing operations, such as
amplification and demodulation. In principle, this technique can yield
large increases (“20 dB) in receiver sensitivity compared with direct
detection using today’s avalanche photodiodes. Indeed, in the 1.3- to
1.6-um lightwave band, coherent receivers offer the only realistic hope
of approaching the so-called “quantum limit” of receiver sensitivity:
~10 photons/bit at 10~° error rate.'To date, however, the performance
of experimental coherent receivers (especially those employing semi-
conductor lasers) has fallen short of the idealized. theoretical predic-
tions. One of the prime causes of this degradation has been identified
as laser phase noise, a phenomenon that is known to be particularly
serious in semiconductor devices. And since semiconductor lasers are,
at present, the preferred candidates for coherent systems applications,
it is imperative to develop an understanding of their noise properties.
Our goal in this paper is to present a comprehensive treatment of the
deleterious effects of laser phase noise in digital lightwave systems, so
that the resulting degradation can be understood and predicted.

Laser phase noise is a random process driven by spontaneous
emissions within the laser cavity, which cause the phase of the optical
output wave to execute a random walk away from the value it would
have had in the absence of spontaneous emission. This random phase
process manifests itself as a broadening of the laser emission spectrum;
it is the cause of the broad linewidth (typically 5 to 100 MHz) of
today’s InGaAsP Distributed Feedback (DFB) lasers. In communica-
tion systems, phase noise degrades performance because unwanted
phase fluctuations in the received wave impair the demodulation
process, especially when Phase Shift Keying (PSK) is used. At low
signaling rates, the accumulated phase “wander” during a signaling
interval might be so great that PSK cannot be used at all. In general,
however, as the bit rate is increased, the impairment due to phase
noise can be made negligibly small.

The central question addressed in this paper is, How high must the
signaling rate be in order to ensure tolerable system degradation due
to phase noise? The answer, not surprisingly, depends on system
design constraints. For example, if one requires quantum-limited
receiver sensitivity, then PSK with homodyne detection must be used.
Based on detailed mathematical analysis and estimates we conclude
that the degradation or penalty due to phase noise can be kept small
(<1 dB) only if the ratio of signaling rate to laser linewidth, R/B_z, is
greater than 3000. For a laser with B, = 10 MHz, this condition
implies a signaling rate of 30 Gb/s—well outside the range of current

technology. To operate at lower rates, one might use any of several
- techniques available for narrowing laser linewidth, but only at the
price of a substantial increase in complexity. With the theoretical
guidelines presented in this paper, the design engineer can strike a
balance between the cost of linewidth reduction and-the value of
_ improved system performance. |

If a system design can tolerate a receiver sensitivity 3 to 6 dB poorer
than the quantum limit, then considerable robustness against phase
noise can be achieved by using heterodyne, rather than homodyne,
detection. With heterodyne differential detection of PSK, for example,
the phase-noise penalty is less than 1 dB for R/B, 2 300, an order-of-
magnitude improvement over the homodyne case. Finally, we consider
the intriguing case of FSK, which can be made extremely tolerant of
phase noise by using very wide transmitter frequency deviation. At
moderate bit rates (S500 Mb/s), where direct-deviation laser FSK
transmitters operate fairly well, this modulation technique appears to
be most attractive.

Il. REVIEW OF DIRECT DETECTION AND FUNDAMENTAL LIMITS
2.1 Direct detection

Before commencing our principal investigation of coherent tech-
niques, we briefly review some results related to direct detection of
light signals.

Direct detection of light pulses implies a photodetector that converts
light energy to electrical signals. The detection mechanism is based
upon photon counting, which is subject to statistical fluctuations.
- More specifically, the photon counting process is a time-varying Pois-
son process whose intensity function A(t) is directly proportional to
the information-bearing data wave.

In the case of binary transmission, the choice between a one or a
zero is translated into the presence or absence of a burst of optical
energy. As an illustration, consider the passage of a single pulse
through an ideal transmission model depicted in Fig. 1. In the case of
a one being transmitted, a square electrical signal turns on the laser
or LED and energy is sent into the fiber. In the photodetector, light
will be detected due to the electromagnetic energy present. Exactly
when in time the photons register on the detector is random. The
actual electrical current at the output of this device caused by a photon
is a wideband pulse g-w(t) (which is very narrow compared with the
signal duration T'), where g (gain) is an integer-valued random variable
or g = 1, depending on whether an (APD) or a pin diode is used. In
practical systems where. amplification of weak signals is required,
APD’s are invariably used.

ELECTRICAL
PULSE
ELECTRICAL SIGNAL >) 9x W(t-t,)
| k
(t,) ARE RANDOM ARRIVAL TIMES, POISSON DISTRIBUTED
WITH A MEAN ARRIVAL RATE A(t) PHOTONS/s :
(9): AVALANCHE GAIN OF PHOTODETECTOR
E(n) = AT = EXPECTED NUMBER OF PHOTONS/BIT
-AT
Pe = > - > e~2P p = OPTICAL ENERGY

Assuming that superposition holds for optical fiber transmission,
the single-pulse description may be extended to an entire data wave.
If one transmits a sequence of on or off pulses, then the received signal,
defined as the electrical output of the photodetector on which proc-
essing is performed, is written as

where the time points {t,} form a Poisson process having intensity
function A(t), with

and h(t) is a square pulse, {a,} = 0 or 1 are the data levels, {g,} is
avalanche gain, 7' = signaling interval, and w(t) = output pulse of the
photodetector.

In this simple model, to detect the j th bit, one integrates the output
of the photodetector over the jth T-second interval and compares the
random variable with a threshold. If the output is greater than the
threshold, a one is declared; if it is less, a zero is declared.

In the ideal situation, when a pin diode is used, g, = 1 and when the
threshold is set at zero, the average output of the integrator will yield
fé A(t)dt = \T when a one is sent and zero output when a zero is sent.
Since the number of counts n with intensity AT' is Poisson distributed

1
Pe = 5 eM. | (4)
The average optical energy, photons per bit, is just P = 1/2 (AT) +
1/2(0) and so (4) is written

This is a fundamental limit on the bit error rate and is commonly
referred to as the “quantum limit.”

Equation 5 implies that in order to obtain an error rate of 107°,
about 10 photons/bit are required. This of course is the error rate
achieved in the absence of coding. It has been shown recently® that by
employing coding the number of required photons per bit is on the
order of 2 to 3 provided the information rate is less than a character-
istic rate called channel capacity.

When an avalanche detector is used to gain optical amplification,
the average value of {g,} may be large but the fluctuations are also
large causing amplitude jitter. The penalties incurred by avalanche
detectors have been extensively studied.? Depending on the type of
avalanche detectors used, the loss can be anywhere from 10 to 20 dBs
from the quantum limit (see, for example, Ref. 10). Thus one of the
chief motivations for turning to coherent techniques is to minimize
this tremendous loss in detector sensitivity.

We begin the discussion of coherent techniques and their possible
merits by assuming that the electromagnetic wave at the output of a
laser can be represented as

where A? is proportional to the optical power. Now suppose that this
wave is phase modulated so that a one results in A cos wot and a zero
results in —A cos wot. An ideal homodyne detector adds to the received
wave a local carrier wave of amplitude equal to exactly A. So, the sum
is

When the sum is detected by a photodetector (pin diode) and the
output integrated for JT’ seconds, one obtains for the average number |
of counts dT, either 4A ?T or 0. The average transmitted optical energy |
in this case is P = A’T and so the probability of a bit error is

This result indicates a 3-dB improvement over the quantum limit, and
it is often referred to as the “super quantum” limit.’! Reviewing briefly,
to achieve the super quantum limit, the local laser had to know exactly
the frequency, phase, and the magnitude of the transmitter laser—a
rather ambitious requirement. This detector is depicted in Fig. 2 with
alternative No. 1 used as the input to the photodetector.

Now suppose that we relax the requirements on the local laser and
permit its intensity to be any value B, but still requiring knowledge of
the transmitted carrier frequency and phase. Now the combined waves
become | |

Again (8) is detected by a photodetector and consequently the average
number of counts at the output after integration is now (B + A)’T,
where B is the amplitude of the local laser and it is assumed that
B>A.

To estimate the resulting bit error rate in this situation we invoke
a limit theorem. The theorem has to do with the conditions under
which a “shot noise” process—the output current from the photode-
tector—is well approximated by a “white gaussian” noise process. The
main requirement is that the rate of photon arrivals be large. Since B
in (8) can be made as large as one desires, the average number of
photons is proportional to \T' = (B? + A? + 2AB)T. If the common
bias term (B* + A*)T is subtracted from XT, there is left an antipodal
signal pair +2ABT for the net average counts corresponding to recep-
tion of binary ones and zeroes. The variance of the resulting Poisson
process also equals AT and since B > A by hypothesis, the variance
is essentially TB*. Now in the limit of large number of counts due to
the addition of the local laser to the incoming signal, the output
electrical signal can be modeled by

where n(t) is a white Gaussian noise process with double-sided spectral
density equal to B*. Integrating (9) from 0 to T, results in a Gaussian
random variable. The resulting bit error rate is then |

which is asymptotically (large P) the quantum limit. We have thus
demonstrated that an ideal homodyne detector using a pin photodiode
achieves the quantum limit. This is made possible by the availability
of large “local” optical power that provides indirect amplification of
the incoming weak optical signal. While providing amplification, the
procedure also produces additive noise. This mode of detection is
depicted in Fig. 2 with alternative No. 2 for the input to the photode-
tector.

Finally we consider a detection technique where, instead of trans-
lating the incoming optical wave directly to baseband, it might be
advantageous in some cases to make a frequency translation to an
Intermediate Frequency (IF). This procedure is called heterodyne
reception and it is depicted in Fig. 2 with alternative No. 3 for the
input to the photodetector.

To understand the consequences of this approach we proceed as
follows. Let the local laser frequency be denoted by wy, and the
incoming optical frequency by wo such that the IF frequency is w; =
wo — w,. The addition of the two waves now results in

Expressing s(t) in terms of the envelope and phase about w, results
in the representation

The response of the photodiode to the wave (12) is again a shot-noise
process with intensity function, Ao, equal to the envelope squared.

Using the same limit arguments as in the previous section, we first
subtract B? + A? from X(t), which retains the antipodal signal pair

Because B > A the fluctuating noise is white Gaussian with double-
sided spectral density = B*. Denoting the additive noise by n(t), the
equivalent signal-in-noise problem after heterodyning becomes

This is a standard elementary detection problem and deciding whether
a plus or a minus was sent is accomplished by multiplying (17) by
cos w;t, integrating for T seconds, and comparing the result with a
threshold set to zero. The decision statistic is

_ where we have neglected the double frequency term. Since the random
variable {§ n(t) cos(w;t) dt has variance equal to B?T/2, the bit error
rate in this case is asymptotically

The exponent is seen to be a factor of 2 smaller than in (10) and
because of this heterodyne detection is 3 dB inferior to the quantum
limit. The asymptotic performance of the ideal frequency translation
methods just discussed are summarized in Table I. |

With these preliminaries we are now in a position to discuss more
realistic detection systems, where laser phase noise must be taken into
account. Before doing this however, we briefly review the origins of
this noise.

Phase or frequency noise in lasers is a well-known and documented
phenomenon that sets fundamental limitations on the performance of

coherent optical communications.'!*" It has been observed that the
spectral density of this frequency noise has a 1/f to 1/f* characteristic
up to around 1 MHz, and is flat for frequencies above 1 MHz,"
shown in Fig. 3. The flat, or white, component is associated with
quantum fluctuations and is the principal cause for line broadening.
From a communications theory point of view, the relatively low-
frequency components can be easily tracked, and so we shall not dwell
on this part of the noise.’® Our main focus here will be on the white
component.

Laser phase noise is caused by randomly occurring spontaneous
emission events, which are an inevitable aspect of laser operation.
Each event causes a sudden jump (of random magnitude and sign) in
the phase of the electromagnetic field generated by the device. As time
evolves the phase executes a random walk away from the value it
would have had in the absence of spontaneous emission. The mean-
squared phase deviation grows approximately linearly with time, and
since the average time between steps in the random walk becomes
vanishingly small, the random phase 6(t) becomes in the limit a
Wiener process characterized by a zero-mean, white Gaussian fre-
quency noise p(t) with two-sided spectral eee No.’ Thus, the
phase process is represented as

To determine the parameter No (which is a function of both the
laser structure and the operating conditions), one can measure the
spectral density of the frequency fluctuations in the emitted light and
hence determine Np directly. Experiments of this sort have shown that
the representation in (20) using the white-noise approximation for
u(t) is reasonably accurate for 0.1 ns S 7 S 1 us, which is adequate
for our present purposes. Another technique for measuring No makes
use of the fact that phase noise causes an observable broadening of
the laser emission spectrum. In effect, the accumulated phase error
given by (21) limits the duration of temporal coherence of the laser
radiation to an interval of roughly 1/(27)*No; the corresponding line-
width is therefore proportional to the noise density No. The following
discussion makes this relationship more precise.

where the innocuous inclusion of the uniform phase ¢ renders s(t) a
‘stationary process with correlation function,

A simple calculation reveals that the Fourier transform of (23), the
power spectrum, is

S A?
[ G(f)df = x (25)
as it should. A sketch of the baseband spectrum, commonly referred
to as Lorentzian is shown in Fig. 4. The parameter characterizing
G(f), No, can be experimentally determined by measuring the 3-dB
bandwidth of the spectrum around fp. Denoting the total (two-sided)
3-dB bandwidth by B;, it is seen from (24) that
B,

when B; < fo. |
We will see later that seemingly modest amounts of phase noise can
seriously degrade coherent system performance; thus it is imperative

to make lasers with the narrowest possible linewidth. Unfortunately,
the semiconductor injection laser designs likely to be used in the 1.3-
to 1.6-um lightwave band typically have linewidths in the range 5- to »
50-MHz, which is too broad for many potentially important coherent
lightwave applications.’*”° (For comparison, the reader should note
that microwave oscillators, which are widely used in coherent radio
applications, have linewidths on the order of 1 Hz.) To reduce laser
linewidth, experimenters have exploited the fact that the noise density
No is inversely proportional to PoQ’, where Po is the laser output
power and Q is the quality factor of the “cold” laser cavity resonance;
thus high-power, high-Q lasers tend to have narrow linewidths. The
most impressive line-narrowing experiments have been performed
using a mirror or diffraction grating external to the laser chip to
produce a composite cavity of very high Q.”" Under relatively benign
laboratory conditions, linewidths of tens of kilohertz have been ob-
tained: an improvement of three orders of magnitude! Whether this
approach will prove practical under harsh field conditions remains to
be seen. In any case, it appears that phase noise will be an important
consideration for the foreseeable future, so we turn now to developing
a clear understanding of its consequences.

We saw in Section III that homodyne detection of an optical PSK
data wave makes it possible to achieve the quantum limit. However,
to gain the full benefit of this approach the local laser must have
perfect knowledge of the transmitted optical center frequency and
phase. In this section we explore the possibility of deriving these

crucial parameters from either the optical data wave directly or from
_a heterodyned version.”” At microwave frequencies, carrier recovery
techniques are well established, while at optical frequencies, the avail-
able methodologies are still limited. For example, it is very difficult to
directly multiply two optical signals or square an optical wave, which
makes it difficult to wipe off the binary modulation. In attempting to
derive carrier in the optical frequencies, it is, therefore, necessary to
resort to data-aided techniques.”

At the input to the detector, the power of the incoming optical data
signal is split so that a fraction, A7k’, is devoted to the estimation of
the phase process 6(t) by a phase-locked loop, and the remaining
portion of the power, A?(1 — k?), is used for demodulation. The power
division that is determined by the choice of the constant 0 < k <= 1 is
a parameter that must be optimized. In the phase-locked loop we
presume that the local Voltage Controlled Oscillator (VCO) can be

modulated by the estimated data m(t) using decision directed tech-
niques, and consequently the modulation can be locally wiped off and
a modulation-free carrier thus made available in the lower portion of
the figure to perform the homodyne demodulation function. We also
assume that the optical center frequency can be identically matched
by the local laser. In any event, if this is not the case, an additional
phase-locked loop may be needed to track this mismatch.'® Making
these assumptions, it is possible to analyze the performance of this
detector and to obtain tight bounds on the degradation from ideal
homodyne performance.

For the subsequent analysis we refer to Fig. 5. Adding a fraction c
of the VCO output to a fraction of the incoming optical data wave we
obtain

where m(t) at the output of the VCO is the reconstructed data wave
from past decisions, and it is assumed to be devoid of errors. The
squared envelope of V(t), which is the average value of the Poisson
counting process n(¢), at the output of the photodetector is

After de elimination, the “signal” portion of the “shot noise” process,
which in the limit becomes a Gaussian process since B can be made
large is |

and the resulting zero-mean white Gaussian noise process is denoted
by v(t) with spectral density equal to B?c’.

Thus the equivalent signal-plus-noise process, which controls the
frequency of the local laser VCO is

u(t) = 2ABkc sin Y(t) + v(t), (31)
and so, because of feedback, we must satisfy the equation
t
§(t) = o(t) +K i u(t’)dt’, (32)
where ¢(t) is the phase noise process of the local laser, and K is a
proportionality constant to be determined later.
Subtracting from both sides of (32), 0(t), the phase of the incoming

wave, and differentiating, we get the stochastic differential equation

and yo(t) is now a white Gaussian process with double-sided spectral
density,

where the Lorentzian bandwidth of the transmitter laser is Bz, and
the local laser VCO bandwidth is By».

It is well known**” that (33) obeys a Fokker-Plank equation yielding
the steady-state probability density function (mod27) for the phase-
error process y(t),

The probability density (36) is sharply peaked at y = 0 when a is
large and becomes flat, or uniform when a is small. One strives
therefore to design the phase-locked loop so that a is as large as needed
to obtain minimum degradation from ideal (y = 0). For fixed A and k,
(37) reveals that a cannot be made arbitrarily large because of the
finite Lorentzian bandwidth B,. However, there exist a maximum
value of a (when K2 = 27B_) given by,

where P = A?T—the transmitted optical energy in the received signal.
Equation (39) reveals that for fixed optical energy and k, ao can be
made large only by increasing the ratio R/B,.

We now examine the performance of this optical homodyne detector
using the phase-locked loop output as the reference carrier wave.
In the lower portion of Fig. 5 the sum signal W(t) is

W(t) = AV1 — k? m(t)cos(wot + 0(t))
+ Bv1 — c? cos(wot + O(t)). (40)
The squared envelope of W(t)—the response of the photodetector—
is therefore
E*(t) = A*(1 — k*) + B41 — cc’) |
+ 2ABV1 — k? V1 - ¢? m(t)cos y(t). (41)
The resulting shot-noise process at the output of the photodetector

again becomes in the limit a Gaussian process with average value
(after dc elimination) equal to

2ABV1 — k? V1 — c? m(t)cos Y(t), (42)
and zero-mean white Gaussian noise v(t) with a spectrum equal to
B*(1 — c?).
Thus the equivalent signal plus noise prior to integration is
S(t) = 2AV1 — k? m(t)cos W(t) + volt), (43)

where we have divided signal plus noise by BV1 — c’ thus normalizing
the spectrum of vo({t) to unity.

Integrating (43) over a T-second interval results in the decision
statistic |

where yp iS now a zero-mean Gaussian random variable with unit
variance, the random variable

ically since it requires knowledge of the nth-order probability distri-
bution of the phase error process y(t). The most we have, however, is
the first-order distribution, and therefore we must resort to upper
bounds using the only information we have.

In Appendix A, an exponential upper bound on (46) is developed
with the result

2
glaje *, a/p? >1
Pes kee a/p? <1, (47)
where the coefficient, g(a), is defined in Appendix A, eq. (178).
When a/p? = 1, (47) is reduced to a single bound,
as
Pe <= g(aje *. (48)
Recall that
p* = 4P(1 — k’), (49)
and

where vy is the ratio of the signaling rate to the average Lorentzian
laser bandwidth

2R
i= B, (52)
It is seen from (47) that the error exponent assumes two crucially
different forms depending on whether r = 0 or r < 0. For fixed y and
k, the exponent is linear in P when r = 1, while it behaves as the
square root of P for r< 1. Thus the probability of error decays much
more slowly with P when P > constant y [eq. (51)], indicating a
threshold for Pe versus P. For fixed y and P, however, r is a monoton-
ically increasing function of k, 0 < k < 1, and so there exists a value
k = k,, such that r(k,) = 1, given by

This value of’k is not, however, the optimum value that makes the
negative exponent the greatest, or the probability of error the smallest.
The optimum value of k, or the power division, is found by setting the
derivative of the exponent in (47) for r < 1 to zero. From (47) the
negative exponent is

indicating that there exist a k,» < k which increases the exponent.
Setting (55) to zero, we get the formula for the optimum R,

One can now proceed to solve (58) numerically and use this optimum
value to plot the upper bound on the probability of error versus P for
different values of y. A more incisive way to exhibit the behavior of
the error rate, however, is to use the suboptimum value of k, > Rop
given in (53), which renders r = 1, and then define a penalty, which is
the reduction of the exponent relative to ideal performances. Since k,,
< k,, this still provides an upper bound on the error rate. It can be
seen from (58) that when the term 1/(1 — k3,) is neglected, the resulting
equation is identical to (53) and so, to this degree of approximation,
Roy ~ k, (this is a good approximation when k,, turns out to be small,
which should be the case when the penalty is small). Following this -
approach, we solve (53) for x and write the negative exponent (54) as

The penalty incurred due to the finite value of Y may now be defined
as follows. Equating (59) to 2P,, where P, is chosen to achieve a
desired error rate, say 107°, in which case P, ~ 10, one obtains a
function of P,/P versus y, and for each value of y one can then
calculate, —10 log P,/P), which defines the penalty. |

2P, = 4Pxvx7 +1 —- x, . (60)
and when
1
x= — >= V2/P
8Vr /
is substituted into (60), one gets the penalty function
ee
: P +»vy+16xP, ey

Fig. 6—Penalty versus signaling rate divided by linewidth in homodyne PSK detec-
tion at two different ideal error rates for identical laser linewidths.

corresponding to Pe ~ 107° and the other to Pe ~ 107°. It can be seen
that negligible penalty is incurred when y = 3000 for Pe ~ 10~? and 3
dB is given up at y ~ 500. We will see in the next section that a
heterodyne Phase-Locked Loop (PLL) also suffers a 3-dB penalty at
around y ~ 500. This approach is analyzed in the next section.

Heterodyning the optical data wave down to an IF frequency, and
then deriving the carrier from the resulting microwave signal using
standard well-known techniques may make it easier to wipe off the
modulation, and consequently ease the signal processing burden of the
phase-locked loops.”®”” In the subsequent analysis we presume that
the modulation has been eliminated, and as such the phase-locked
loop can operate directly on the IF carrier wave. So, after heterodyning
the optical signal

to an IF frequency f; and wiping off the modulation, we obtain the
microwave signal plus noise

where A? equals optical power, n(t) is again a white Gaussian noise
process with unit double sided spectral density, and @(t) is now the
difference between the transmitting laser’s phase noise and the local
laser’s phase noise. Consequently, the variance of 6(¢) now is

The signal (63) is‘now the input to a conventional PLL depicted in
Fig. 7. The analysis of the PLL is straightforward. Denote the output
of the PLL by

double-frequency terms from its output. Yet, the cut-off frequency is
placed high enough so that the output noise can still be regarded as
white. Thus, because of feedback, the following equation must be
satisfied

where. now the double-sided spectral peneily of v(t) is equal to K?/2.
The phase error therefore is

t
y(t) = 0(t) — Ke { e(s)ds, (69)
0 : F
and when this is differentiated one obtains |
dy _ dé
a a OF 7
= Qa(ui(t) — wo(t)) — Kee(t), (70)

where p; and po are the frequency noises of the transmitter laser and
the laser involved in the local heterodyner, respectively.

When (70) is substituted into (68), one again obtains the well-known
stochastic differential equation governing the evolution of the phase-
error process,

ae = —K, KA sin y(t), (71)
where
u(t) = —22(ui — pe) + v(t) Ke (72)
- is a white Gaussian noise process with double-sided spectrum equal to >
K?K3 |
D= 27Bz + 9 (73)

Letting K,K2 = K, we observe that (71) is identical in structure to
(33), which again results in a Fokker-Plank equation yielding the
steady-state probability density function (mod 27) for the phase error

_ e*cosy _ .
p(y) = Onley’ nsyesr. (74)
The important system parameter is now given by
2AK 2AK
nr a a
—— 27B,

The algebraic form of the PLL parameter a departs from the
conventional form where, by decreasing the loop bandwidth, a can be
made as large as possible. Here there is a minimum band resulting
from the presence of phase noise. The only way that a can be increased
is by increasing the input optical energy, or by decreasing phase noise
relative to the signaling rate. Consequently there exists a maximum
value of a (when K? = 47B,z) piven by

4.4 Performance |
Now consider the modulated heterodyned wave |
S(t) = +2A cos(2Qrfit + O(t)) + n(t), Ost<T, (77)
and the estimated carrier wave from the PLL
cos(2afit + 6(t)). (78)
Multiplying (77) by (78), integrating from 0 to T, and eliminating the

where y(t) is the phase-error process obeying at any instant of time
the probability density (74). Rewriting (79) as before,
S = +ATE + »p, (80)

where p = V2TA and vp now has unit variance.
The probability of error is as before,

This probability is structurally identical to (46), and we therefore use
the same bound from Appendix A.

With the definition of a, in eq. (76) we can compute the threshold
parameter in this case,

We note that the behavior of this bound is slightly different from
the one applicable in the homodyne case. For one, here ideal perform-
ance is 3 dB inferior to the quantum limit, which is accounted for by
the heterodyning operation, as we have already noted. Next, ideal |
performance with negligible degradation is attained when

B, > 8rP,
and for ratios less than this, the degradation is increased gracefully.
As an example, when P = 20, yielding an ideal error rate ~107~°, the
threshold parameter y = 500. For ratios greater than this, no penalty
in optical power (from the ideal P = 20) is incurred. To assess the
penalty at operations at less than this threshold, we use the same
definition as before. The probability of error exponent when r, < 1 is

Fig. 8—Penalty versus signaling rate divided by linewidth in heterodyne PSK detec-
tion at two different ideal error rates for identical laser linewidths.

b= = or, rs 1. (88)
Substituting the definition of r, [eq. (84)] and solving for y, yields
1 — v1 — 5)?
y = 8rP, eae (89)

In Fig. 8 we plot b in decibels versus y for two different values of
P,. We see that around y ~ 400, the asymptotic degradation of 3 dB
is approached. Interestingly, at around this ratio the optical PLL also
degrades by 3 dB, as we have already mentioned.

In concluding this section we remark that our estimates are only
upper bounds, albeit, we feel, tight bounds. The exact evaluation of
the error rate is not feasible because of the nonlinear functionals that
are involved. If one chooses to ignore the time integrals, than the
probability of error can be evaluated numerically as has been done in
Refs. 28 and 29. |

In the next section we will see that noncoherent techniques such as
frequency modulation and differential phase modulation, where knowl-
edge of carrier phase is not essential, yields performance very near to
what can be attained with heterodyne phase-lock technique at reason-
able signaling rates.

V. BINARY FREQUENCY SHIFT KEYING
Here we discuss and analyze the performance of binary Frequency

Shift Keying (FSK) as one of the modulation options.*°*? In this
modulation method information is conveyed by switching the fre-
quency of the laser between two different values. Thus, during a fixed
interval, T = 1/R, where R is the signaling rate, the receiver has to
decide whether

was transmitted. In (90) A? is again proportional to the received optical
power and 6(t) is the phase noise process associated with the laser.

It appears that this modulation method is impervious to the effects
of phase noise, since the shifted frequencies w,; and we, can be suffi-
ciently separated and if enough bandwidth is available, crosstalk due
to the fluctuating phase noise can be minimized. These are the chief
reasons, then, for considering FSK. We point out that this is not what
is commonly referred to as continuous-phase narrow-band FM. The
latter yields the same performance as differential phase modulation
treated in the next section.

The first step in the processing of the FSK optical signal is to
heterodyne (90) to an IF frequency w;. As was already noted, this can
be accomplished by adding to (90) a locally generated optical signal
and then direct detecting the sum by a photodetector. The sum signal
then is

where the IF frequencies are w; = w; — w;, and ¢(t) is the phase noise
associated with the local laser. The output of the photodetector is
again a “shot noise” process |

When the local laser intensity B >> A (66) approaches a white
Gaussian process with average value equal to \(¢) and standard devia-
tion also equal to A(t). Thus the dc part of the average value of (93) is

while the resultant zero-mean Gaussian noise, n(t), has a double-sided
spectral density equal to B”.
With these preliminaries, we now confront a classical detection

problem. Given the IF signal (94), plus white Gaussian noise of unit
spectral density

how does one process V(t) so as to attain the least probability of
error? While the problem is classical the solution is not tractable in
general because of the presence of the phase noise process A(t).

For calibration purposes, let us first review briefly the performance
under the assumption that the phase noise is effectively a constant.
In the case when A(t) is slowly varying with respect to the rate R =
1/T, or when the symbol rate R is much greater than the bandwidth
of the laser signal, the optimum detector has a well-known structure
depicted schematically in Fig. 9. The results in this case will serve as
a benchmark to which the later more general results will be compared.
Also assume that the frequency shifted signals are orthogonal, i.e., the
two frequencies w; and w, are chosen such that

To proceed with the error rate analysis, assume that w; was sent. In
this case, we expect the x output in Fig. 8 to be greater than y, and an
error is made when x — y < 0. Because of symmetry, when wz is sent,
y is expected to be greater than x and a mistake is now made when
y — x = 0. So, the probability of error is just

In order to evaluate this probability, we express the random varia-
bles x and y as indicated by the mathematical operations in Fig. 7. It

Fig. 9—Structure of the optimum FM detector when linewidth approximates zero.

The desired probability is just the probability that the difference in
the lengths of two 2-dimensional Gaussian vectors is less than zero.
As can be verified, the x’s are independent Gaussian random variables
with identical variances, o” = T/2. For these random variables, (97)
can be expressed exactly™ as

Comparing this with the performance of direct detection, we observe
that the optical signal (90), after direct detection and integration for
T seconds, yields an average photon count equal to A?7. In this direct
detection case, the chance of making an error would be the chance
of detecting zero photons in T seconds. From the Poisson distribu-
tion, for the number of photons detected, this probability is just
1/2 exp[—A’T], which is 3 dB worse than the quantum limit. Com-
paring this with (98), however, reveals an additional 3-dB loss due to
heterodyning. So, heterodyne FSK detection is 6 dB inferior to

Returning now to the more realistic situation where phase noise is
present and must be included in the performance analysis, we recall
that the crucial assumption for the previous analysis was that the
symbol rate baud R = 1/T' be much greater than the linewidth of the
laser, so that the phase process A(t) could be regarded as a constant
during the integration period. The inclusion of the phase noise process
immediately raises fundamental problems concerning the detector
structure.

As is well known,”’ the optimal processor in the presence of phase
noise first estimates the phase process, and then uses this estimate to
coherently demodulate or detect the IF signal. This is precisely what
a PLL does, but we have seen that coherent PLL detection becomes

feasible only at very high data rates. This situation leads us to
postulate a detector that, while not optimum, is reasonable and does
not require a phase-locked loop.

The proposed frequency detector structure is shown in Fig. 10. It is
essentially an energy detector. It consists of two ideal bandpass filters
of total bandwidth equal to 2W. The purpose of these filters is to limit
the added white noise bandwidth as much as possible, while at the
same time to retain most of the energy in the information carrying
signal. A precise number for the bandwidth that satisfies these two
seemingly contradictory requirements is hard to derive because,
strictly, the received sine wave with phase noise has infinite band-
width. The front-end bandwidth must remain as a parameter in our
subsequent analysis, and engineering estimates will be attempted later.

Following these bandlimiting filters with square-envelope detectors
and an integrator essentially provides an estimate of the energy in
each frequency band, and this quantity should be independent of phase
noise provided that the front-end bandwidth is sufficiently large. We
now proceed to analyze the performance of this structure.

In the representation of signal plus noise, eq. (95), assume that a
was sent. Regarding w as the center frequency, we represent the signal
and noise in terms of in-phase and quadrature components as

S(t) = 2A cos w,t cos A(t) — 2A sin wt sin A(t)
+ nj(t) cos wt + ne(t) sin ait
= x(t) cos a;t + y(t) sin art, | (99)
where

At the output of the bandpass filters x(t) and y(t) are bandlimited
versions of the input. As we have already stated the signals x(t) and
y(t) remain essentially undistorted at the output, and the only effect
of the bandpass filters is to limit the noise band. Clearly, as the band
W increases, this approximation becomes better.

The output baseband noises with unit input noise intensity now.
have mean-square values |

Squaring the envelope and integrating as indicated in Fig. 8 yields the
quadratic decision statistic q,,

q = t [x*(t) + y*(t)]dé, (101)
where for a given A(t), x(t) and y(t) are independent Gaussian
processes with

The covariance functions of the ideally bandlimited baseband noise
processes n,(t) and n2(t) are |

As is well known,® associated with this covariance kernel are an
infinite set of orthonormal eigenfunctions {y,(t)} and a set of nonneg-
ative eigenvalues {),}. Using these eigenfunctions, we represent the
processes x(t) and y(t) as

In the bottom leg of Fig. 10 we assume that only the “noise” goes
through (by previous hypothesis), and so the resulting quadratic form
g2 is now comprised only of noise. Of course, this is a mild assumption
since w, can be separated from wz as much as one wishes to provide
minimum leakage. Thus,

where v,(t) and v.(t) are quadrature and in-phase bandlimited noises
in the lower leg, and are independent of the noises in the upper leg
since the spectra occupy nonoverlapping frequency bands. For this
reason we denote these noises by »:(t) and v(t) to distinguish them
from n,; and nz in the upper leg. The decision statistic is the difference
of the quadratic forms (105) and (106) and consequently the probabil-
ity of error is

Pe = Pr[q = qi — gz = Ol. | (107)
This can be expressed in terms of the characteristic function of q,
—— C(w) = Ee, (108)
as the integral** |
Pe = mo [ en dw. (109)

~ The ie, « > 0 in the denominator denotes the fact that in the complex
w plane the contour of integration goes above the singularity at w = 0.
Using (105) and (106) it is straightforward to calculate (108)

and Eaiy(-) denotes the expectation with respect to the phase process
A(t). To proceed further, we invoke an excellent approximation®
regarding the behavior of the eigenvalues {\,,} in this application. Since
these are the eigenvalues of the Prolate-Spheroidal wave functions, it
is shown in Ref. 35 that

We note the nth order pole at z = —1 and the essential singularity
at z = 1. When the time-bandwidth product n = 1, the contour can be
closed in the left-hand plane, and the value of the integral is just the
residue of

This result is identical to (98) and a moments reflection will reveal
_ the reason for the consistency. Note that roughly

where R + kB; = 2W—the bandpass bandwidth of the incoming signal.
This band must equal to or be greater than the signaling rate R, plus
kB,—the bandwidth required to pass the sine-wave signal with the
phase noise undistorted (k is a positive integer), and again B, is the
laser linewidth. Clearly, when B,/R <1, no postintegration is required
and consequently n = 1. Therefore in this case we get the previous
result where we regarded the phase noise as a constant—precisely the
case when kB;/R « 1. When this condition is not necessarily satisfied
and son # 1, we can still evaluate the contour integral. In this general
case, closing the contour in the left-hand plane enclosing the nth order
pole gives for (113) the residue and hence the probability of bit error

As can be seen from (119), the degradation due to excess noise and
postintegration manifests itself only in an algebraic coefficient in the
bit-error-rate expression and not in the exponent.

From the foregoing analysis we present the curves of Fig. 11 that
are plots of Pe versus P in decibels for different values of , which is
twice the ratio of symbol rate to the sum of laser linewidths. The
front-end bandwidths around frequencies f/, and f. in Fig. 9 were
selected to be

of 10 times the laser linewidths is judged to be adequate to pass the
incoming FM signal without appreciable distortion.

When y — ~, Pe — exp{—P/2}, which is the ideal binary FM
performance. As already mentioned, this ideal performance is still 6
dB worse than the quantum limit, since 3 dB is lost from heterodyning
and 3 additional decibels is lost due to the fact that the heterodyned
FM signals are orthogonal rather than antipodal.

Fig. 11 can be used to determine the minimum data rate for efficient
performance. Suppose the lasers have identical linewidths of 10 MHz,
and one wishes the degradation not to exceed 1 dB. What is the
minimum admissible data rate R? From Fig. 11, less than 1-dB
degradation yields y ~ 10, implying R = 100 Mb/s.

What is the degradation if one desires to transmit at 20 Mb/s? For
the same laser linewidths as before, eq. (98) yields y = 2, and from the
figure we see that this value of y yields a degradation of 2 dB.

As is well known, FSK can accommodate more than two frequencies
to convey digital information without appreciably altering the form of
the error rate expression provided again that bandwidth expansion is
not an obstacle. For example, consider using 2” frequences m = 2.
‘Generalizing the structure of Fig. 10 to 2” legs yields a probability of

error that is 2” times the binary error rate but requires only P/2m
photons per bit to deliver the same amount of information as in the
binary case. To cite an example, using four frequencies, m = 2, the
information rate is 2/T’ and so T can be doubled to obtain the same
bits per second as when m = 2. This comes at a moderate increase of
bandwidth and a factor-of-4 increase in the error rate.

Let us consider the previous example where the binary rate R = 100
Mb/s. This can be achieved with ~40 photons per bit resulting in an
error rate ~10~-*. Suppose one had only 20 photons per bit to expand,
how can this data rate be accommodated without increasing the error
‘rate? Suppose we half the signaling rate so that the new T equals
twice the old T. To maintain the same rate in bits per second we must
use four frequencies rather than two. The new signaling rate has now
been halved and so we must recalculate y from eq. (98) corresponding
to R = 50. We find it to be five. From Fig. 11 we estimate that this
value of y results in a ~1-dB loss. The upshot is that FSK with four
frequencies signaling at a data rate of 100 Mb/s can be achieved at a
4-dB increase of optical power over the quantum limit.

In the presence of additive white Gaussian noise, Differential Phase
Shift Keying (DPSK) is known to be very efficient in terms of the
s/n required to achieve an acceptable error rate.*** It is only a fraction
of a decibel less efficient than coherent phase shift keying—the most
efficient known method. Our objective here is to investigate the
performance of this modulation method in optical communications
and to assess the incurred penalty due to phase noise.

We begin our treatment by first considering detection at optical
frequencies and then analyze the heterodyned version. Processing at
optical frequencies may be practically inhibited because of the present
lack of efficient (noise free) amplifiers, and therefore we also analyze
the heterodyned version and compare performance of these two dif-
ferent approaches.

In DPSK, information is conveyed by the phase differences in two
consecutive signaling intervals. Thus when the optical signal in a
particular signaling interval is

So(t) = aoA cos(wot + O(t)), 0O<it<T, (121)
and in the previous interval it was
S_i(t) _ aA cos(wot + A(t)), —-T<t< 0, (122)

where do, a-; are +1 and 6@(t) is again the phase noise process,
information then is conveyed by the product aoa-_1.

Ideally one would measure the time average* of the phase difference
in (121) and (122) during the interval [0, 7] and decide that a_,a) = 1
if it lies between —7/2 and 7/2 and a_,a) = —1 if it lies outside of this
phase range. Consequently, in a practically implemented system a
lower bound on the bit error rate would be

From this lower bound on bit error rate the ratio B,/R is determined,
setting a floor for the minimum admissible rate in terms of B,. For
example, if one desires an error rate ~10~°, one equates

The problem, however, is that in the Gorical processing repertory
the phase differential cannot yet be obtained directly nor can two
optical waves be multiplied directly. Therefore, one must consider a
detection method that provides information about the phase difference
in an indirect manner. Thus consider the following approach: first,
the incoming signal is delayed by 7' seconds, then half of the power of
the delayed signal plus and minus half of the power of the undelayed
versions are passed through separate photodetectors. A schematic
representation of this phase detector is shown in Fig. 12.

* I am indebted to Leonid G. Kazovsky for pointing out the time average of the phase
noise differential provides a better lower bound than just the instantaneous value that
appeared in my original manuscript.

where we set
A(t) = O(t) — (t — T).
The output pulse count from the “sum” photodetector is a doubly
stochastic Poisson process n, with conditional intensity \4 equal to
the squared envelope of (124),

while the “difference” photodetector output is also a Poisson process
ny independent of n, but having intensity

Integrating the outputs of the two photodetectors for T seconds yields
two average random counts |

Because of phase noise, the exact evaluation of the bit error rate is
not mathematically tractable, but an exponential upper bound can be
obtained from the moment generating function of the differential
count, n. The moment generating function of n, conditioned on 6 and
aoa_;, is readily calculated from the Poisson distribution

M,.(s| 6, doa-1) = eA “DHA, (129)
and will be used to upper bound the probability of error.
We begin by writing the probability of error

In the. last two inequalities we used (129) in a Chernoff bound and
made use of the convexity of the exponential function as well as the
fact that A@(t) is stationary.

The tightest upper bound is obtained by selecting an optimum s for
a given pv and u. This value of s can be obtained by setting the derivative
of the exponent to zero. The set of random s’s optimizing the exponent
is then found to be

and since s, has to be positive, we require that v > u, which from (133)
implies that cos A@(t) must be positive. For values of A@(t) such that
cos Aé < 0, the optimum value of s, is seen to be zero. Thus, in order
that the bound (131) be reasonably tight, the average with respect to
A6(t) indicated in (131) must be carried out over two sets of A@é

The first term above can be upper bounded by setting s = 0 and
further upperbounding the probability that cos Aé <= 0, yields for this
term Pr[A@ = 7/2]. So after some calculations and substitutions, (136)
becomes

and again P = AT.
The penalty incurred due to phase noise depends on the behavior of

the expectation in (137). It does not appear feasible to evaluate this
expectation exactly and therefore we must resort to asymptotic anal-
ysis valid for large P. We defer this analysis to after the discussion of
heterodyne DPSK, since the penalty evaluation there involves a sim-
ilar calculation.

Before embarking an analysis of heterodyne DPSK, however, we
remark that even when Aé = 0, the probability of error in optically
processed DPSK is 3 dB inferior to the quantum limit. This is seen
from (17) since when Aé = 0,

= 1 —-A°T __ 1 —P
Pe=<se =a (139)
The reason for this loss is inherent in the demodulation process. As
can be seen, the optical detector turned the optical signal into an “on-
off” signal and the explanation for the inefficiency is that twice as
much average optical power has been transmitted to detect an on-off
signal.

It should also be pointed out that the chief motivation for using
differential phase modulation in the microwave region is that it yields
an error probability close to coherent demodulation without having to
transmit and recover carrier phase.*” We therefore turn our attention
to heterodyne DPSK detection next.

As has already been pointed out, heterodyning an optical signal
results in additive white noise and therefore it is imperative as in the
_FSK case, to bandlimit the heterodyned signal plus noise to a band-
width just sufficient to pass the received signal with the impressed
phase noise undisturbed. In DPSK, a bandlimited signal is multiplied
by a delayed version and the product is post integrated in order to
eliminate residual noise. This is a standard comparison detector’? and
is depicted in Fig. 13.

As has been done before, the heterodyned signal is represented in
the interval 0 <t < Tas, .

where the subscript indicates a delay of T seconds, w; is the IF
frequency, and 6(t) = 6(t) — &(t), (A(t) is the transmitter laser phase
noise and ®(t) is the local laser’s phase noise.) Again, n(t) is a white
gaussian noise process with unit double sided spectral density.

V(t) = 2aA cos 6(t)cos wt + n,(t)cos wt + neo(t)sin wt
— 2aoA sin 6(t)sin w;t
= [2a,A cos 6(t) + n,(t)]cos w;t
— [2aA sin 4(t) + no(t)]sin wrt, | (142)
Vua(t) = 2a_,A cos ba(t)cos w;t + nyg(t)cos wt
+ Nogsin wt — 2a_,A cos 6g(t)sin w;t
= [2a_,A cos ég(t) + nma(t)|cos w;t
+ [2a_,A sin 6g(t) + neog(t)|sin w;t (143)

The baseband noise processes in (142) and (143), mi (t), mig(t), ne(t)
and nog(t) are mutually independent and bandlimited to W hertz.
Since the total noise power at the output of the band-pass filter is 4W,
the identical variances of the baseband noises must also equal to 4W.
Now, multiplying V(t) by V(t), eliminating double frequency com-
ponents and integrating, results in the decision statistic qo,

A detection error is made whenever aja_; = 1 and q S 0 or when
ajoa_; = —1 and q < 0. So, the bit error rate then is just

Pe = Pr{g s Oj. (146)
To facilitate the calculation of (146) let,
x(t) = 2A cos 6(t) + n(t)
Xxa(t) = 2A cos 6a(t) + me(t)
y(t) = 2A sin 6(t) + neo(t)
ya(t) = 2A sin 6a(t) + neoa(t),

where the u,; and v; are again the coefficients in the expansions of y(t)
and u(t) in the eigenfunctions {y,,(t)}.

Structurally (150) is identical to the quadratic forms obtained in the
FSK case and therefore, we can express the bit error rate as

The eigenvalues as before are approximately 1 fork <n =2WT and.
zero beyond and consequently (152) is to a good approximation,

Unfortunately this integral cannot be expressed more explicitly as
in FSK because of the essential singularity at z = —1. When ¢ = 0 (no
phase noise), the essential singularity disappears and (154) is identical
to the previously encountered integral associated with the error rate
in FSK. So in this case we obtain exactly

where P,,-; is again the (n — 1)th order polynomial defined in Appendix
B and P= A°T.
In the important case when 1/T = R > B_;, we obtain from (154),
setting n = I, |
par BoP

This result, again as in the optical case, can be seen to be 3 dB inferior
to the “quantum limit” that is solely attributed to the 3-dB loss in the
heterodyning process.

We now return to the central problem of assessing the penalty
incurred by DPSK due to the presence of phase noise. An exact
evaluation of the penalty is not mathematically tractable in general,
and, as in the previous case, we must resort to upper bounds using the
moment generating function of the quadratic form (153) or (154).

The inequality in (157) is valid because of convexity and the station-
arity of ¢(t). While the form of this moment generating function is
different from (129), similar techniques can be used to bound the
probability of error. We proceed as follows:

Splitting the range of ¢ into two parts, R; such that ¢eR,, cos ¢ > 0
and R, such that ¢eR2, cos ¢ < 0, we write (159) as

By setting the derivative of f(z) to zero reveals that there exists an
optimizing z > 0 namely,

(Bri + Bro
R |
and where B,;, and Bz» are again the linewidths of the two lasers.
From (162) and (137) we see that the degradation from ideal per-

formance, exp(—P), is essentially determined by the average value
over the range ¢ceR, of

We now evaluate (167) asymptotically valid for P — © and fixed C.
Setting the derivative of (166) to zero, we solve the transcendental
equation for the saddle point, 0 < ¢o = 7/2.

Using (170) we can summarize the error rate estimates developed in
the foregoing analysis.

1. Optical DPSK. Substituting (170) into (137) with the definition
of G(@) in (167), the probability of error is

Ignoring unimportant coefficients, the probability of error is seen
to be dominated by the maximum of the two terms in either (171) or
(172). These are seen to be identical expressions barring the coeffi-
cients. It is observed from (171) and (172) that the exponents in the
second term approach the exponents of the first term as P > ~, This
can be verified from (168) since when P — », C — o for fixed o% and
so the solution, ¢9 — 2/2. As a consequence of this limit, our esti-
mate of the error rate versus P has a threshold, or floor, at

and according to our prediction the floor for optical DPSK is R/Br ~
100 while in heterodyne DPSK it is R/B, ~ 200, for identical laser
linewidths. These floor predictions are slightly pessimistic. The lower
bounds in (123) predict a floor of R/B, = 67 for optical DPSK and
R/B,z = 134 for heterotype DPSK. The discrepancy has to do with our
bounding techniques. |

It is now possible to define the exponential degradation or penalty,
from ideal performance (above the respective floors) in either case by

This is seen to be a function of the single parameter C defined in (167)
and @p the solution to (168). |

It is important to observe that (171) and/or both (162) and (137)
exhibit an exponential degradation due to phase noise unlike in the
case of FSK. In both systems widening the front-end bandwidths of
the respective detectors ensures minimal distortion suffered by the
received heterodyned signals. However, in DPSK the static phase
noise differential manifests itself in an exponential degradation, while
in FSK no such degradation occurs.

In Fig. 14 we exhibit the penalty function, (173), as a function of
R/B, for both optical and heterodyne DPSK. We note that the
exponential degradation is infinite below these respective floors. The
arrows shown on the figures at R/B, = 100 and 200 are aimed to
emphasize that according to our estimates, the degradation is infinite
at rates less than these values. In other words, no amount of additional
optical energy can drive the error rate below these respective floors.
Similar curves can be drawn for different Pes and hence different Ps.
A striking feature of these curves is that the 3-dB degradation is

Fig. 14—Penalty in decibels due to phase noise in optical and heterodyne DPSK,
Pe ~ 10°, D = 20.

approached very rapidly at rates greater than 400 times the laser
linewidths (less than 4 dB is given up in either system). Evidently
DPSK is very sensitive to phase noise for R < 300B,.

We saw that on-off keying of an optical wave using direct detection
achieves the quantum limit. Here we wish to analyze the performance
of this modulation method when the on-off optical signal is first
heterodyned to an IF frequency and then direct detected. Since the
microwave version of on-off modulation has the same signal distance
properties as FSK, we expect similar performance. There are however
some differences and for the sake of completeness we include the
following analysis.

Letting the IF frequency be w;, the differential phase noise be 6(t)
and the resultant Gaussian noise with unit spectral density be n(t),
the observed IF microwave signal is then represented as

A reasonable way to process (121) is to first bandlimit it, then
envelope detect the bandlimited version and finally postintegrate the
square-envelope to obtain the decision statistic. This processor is
depicted in Fig. 15 and the output statistic is

where n; and nz are baseband gaussian noise processes with double-
sided spectral densities equal to 2 and are bandlimited to W hertz.
This must be so since the bandpass noise process

In writing (122) we assumed as before that W is sufficiently wide to
pass the in-phase and quadrature signals in the presence of phase
noise undistorted.
Now let
x(t) = 2A cos 6(t) + n,(t)
and
y(t) = 2A sin 6(t) + n.(t), (177)

and make an expansion of x and y in terms of the prolate-spheroidal
orthonormal set of functions

We note that {xn}, {yn}, {min} and {n2,} are mutually independent
conditional Gaussian random variables (conditioned on 6(t)) with the
following parameters:

where {A,} are the eigenvalues associated with the eigenfunctions y(t).
Our method of estimating the error rate will be based on a Chernof

To proceed further with the analysis, we invoke again the excellent
approximation regarding the behavior of the eigenvalues \,. Here it
can be verified that

and when this approximation is used in (180) and (181) we obtain for
the moment generating functions

where 7 is a threshold that will be optimized later. The probabilities
in (185) cannot be evaluated exactly; however, tight exponential upper
bounds can be obtained as follows.

Using (183) in (186) it can be verified that there exists a 6 = 6, which
makes the bound tightest. By differentiation we find

Following the same procedure for tightening the bound in (187) we
find an optimum 9p given by,

Equating the exponent in (189) to the one in (191) we determine the
best threshold 7 given by

which is identical to the FSK performance since here the average
optical energy is half that of FSK. The exponential degradation from
ideal is seen to be

where n = 2WT is the total band required to pass the modulated
signal with phase noise undistorted. A conservative example might
be as follows. P = 80 will yield an ideal error rate of ~107°, n =
1/R(1 + 10B,) (10B; ~ 400 MHz should be sufficient to pass the
signal with phase noise undistorted.) For these parameters a simple
calculation reveals that when R = 0.5 GB/s the penalty is about 0.2
dB. At rates below this the degradation starts to be substantial. Note
however, that this performance is still 3 dB poorer than DPSK and 6
dB poorer than the quantum limit.

I wish to thank many of my colleagues at AT&T Bell Laboratories
for the many fruitful technical interactions during the course of this
research. The contributions of Paul S. Henry have been invaluable.
He acquainted me with this subject matter, provided continuous
guidance, and made significant and substantial contributions to the
evolution of the main results. I would have been honored to have him
as a coauthor but he modestly refused.

I also had valuable discussions with the following: A. S. Acampora,
N. Amitay, B. Glance, D. J. Goodman, L. J. Greenstein, G. J. Foschini,
T. Li, L. A. Linke, B. F. Logan, R. W. Lucky, J. E. Mazo, A. A. M.
Saleh, G. Vannucci, and A. D. Wyner. Thanks are also due to Vincent
W.S. Chan from MIT Lincoln Laboratory for stimulating discussions.

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APPENDIX A
An Estimate of Probability of Error in Optical Homodyne Detection

Here we derive an upper bound for the bit error rate which will be
used in evaluating performance of optical homodyne as well as hetero-
dyne reception. We need to estimate the following probability (eq.
192), | |

where é and i are independent random variables with joint probability
density p(£)p(v%). By definition, (194) is written,

oO x2
Pes { ep(x)dx = Ee* = Eee = e2Ee* (196)
To make progress with (196), we note that since e~ is convex

= —p. (202)
We are thus assured that a \ = Xo exists such that E(Xo) < 0. To find
this optimum value of 4, we examine,

and conclude that when a/p = p, \o = p. On the other hand when a/p
< p, the optimum value of \p is on the boundary \o = a/p and so the
optimum exponent in (201) is |

Evaluation of Residues
I am indebted to B. F. Logan, Jr. of Department 11219 for supplying

material included in this Appendix.
We require the residue of f,(z)/(—z) at z = —1, where

Setting P/2 = \, z =t — 1, the probability of error Pe(P, n) in Section
V (eq. 118) is the coefficient of t” in the Taylor series expansion of
the function

“5a 2 : ped, n)x*. | (208)
Then,
e
Pe(n, A) = panei Dri, n).
The generator function for the generalized Laguere polynomials, de-

Cross-Polarization Cancellation and Equalization
in Digital Transmission Over Dually Polarized
Multipath Fading Channels

A theory for data-aided equalization and cancellation in digital data trans-
mission over dually polarized fading radio channels is presented. The present
theory generalizes and extends previous work by admitting decision feedback
structures with finite-tap transversal filter implementations. Subject to the
assumption that some past and/or future data symbols are correctly detected,
formulas and algorithms for evaluating the least mean-square error for differ-
ent structures are presented. In a sequence of curves we evaluate and compare
the performance of various structures for a particular propagation model and
several fading events. We find improvement in performance for decision
feedback over linear equalization. More importantly, we discovered that in
this application, as in the single-channel transmission case, decision feedback/
canceler structures are much less sensitive to timing phase than linear equal-
izers.

One of the purposes of this article is to call attention to mounting
research results pointing the way toward effective methods for com-
bating the deleterious effects of various impairments arising in digital
data transmission over dually polarized fading radio channels.

Transmission of M-state Quadrature Amplitude-Modulated (QAM)
signals via orthogonally polarized carriers is an effective method for

reusing existing bandwidth with obvious economic advantages. The
main obstacle in the way of realizing these advantages is the unavoid-
able presence of Cross-Polarization Interference (CPI) between the
dually polarized signals that arise due to multipath fading, antenna
misalignments, and imperfect waveguide feeds. The chief purpose of
our current work is to obtain a fundamental understanding and a
solution to this problem.

There is a well established theory of linear and decision psaback
equalization/cancellation to mitigate the effects of intersymbol inter-
ference (ISI) and noise in the transmission of a single digital signal.’
However, consideration of data-aided CPI cancellation in addition to
ISI equalization in the presence of noise has not been treated before.
The work of Amitay and Salz’ establishes a theoretical base for optimal
linear compensation of CPI and ISI in the presence of noise; however,
their work is limited strictly to linear techniques and considers only
ideal infinite-tap transversal structures. |

In this article we generalize previous treatments of this subject in
two major respects. Our first contribution is to cast the problem of
CPI cancellation and ISI equalization in a general theoretical frame-
work that admits data-aided decision feedback techniques. Secondly,
and most importantly, we admit finite-tap transversal structures that
in practice can be implemented adaptively.

The receiver configuration is based on a matrix structure suggested
by the theory of optimal detection and is shown in Figs. 1 and 2. The
optimal structure is comprised of a linear matrix equalizer/canceler
and an ISI and CPI estimator, which is used to subtract: some of the
interference from the received signals. An architecture, previously
proposed by Kavehrad,° is a special case of this generalized structure.

The dually polarized channel is modeled by a particular 4 x 4 real
matrix impulse response or its Fourier transform followed by additive
noise. The 2 X 2 block-diagonal elements of this matrix represent the
copolarized (in line) responses, while the off-diagonal 2 x 2 block
entries represent cross-coupled and cross-polarized interfering re-
sponses. Each matrix channel characterizes a snapshot of a multipath
fading event, which in the presence of noise limits the achievable error
rate of the receiver for a given data rate. We use a propagation model
proposed in Ref. 2.

In comparing the performance of various equalizer/cancelers, the
Mean-Square Error (MSE) is used. The justification for using this
criterion has been amply discussed in the literature.’” But the chief
motivation for its use is due to its mathematical tractability. It turns
out that it also leads to an exponentially tight upper bound on error
rate. In practice, it lends itself to easy estimation and thereby is used
to update transversal filter-tap coefficients recursively.

Section II contains the system model and theoretical developments.
Computational algorithms are provided in Section II], and our numer-
ical results and associated discussions are given in Section IV. Finally,
a summary is presented in the last section.

Consider a dually polarized digital radio communications channel
supporting two independent QAM data signals. This type of commu-
nication channel with an ideal QAM modulator and demodulator is
shown in Fig. 1. The four independent synchronous data signals S,,(¢),
S(t), /= 1, 2, with the generic representation

amplitude modulate two linearly polarized carrier waves in quadrature.
The modulated signal,

Si(t) = Sy (t)cos wot + So, (t)sin wot, (2a)
is transmitted over the vertically polarized channel, while
Sr(t) = Sin(t)cos wot + Son(t)sin wot (2b)

is transmitted over the horizontal channel. The carrier frequency is
wo and the real data symbols

are assumed to be independently drawn from a lattice of points with
odd integer coordinates. The QAM constellations associated with eq.
(2) are, therefore, rectangular. The scalar shaping pulse, g(t), is
selected by the designer to satisfy limitations on transmitted power
and bandwidth.

The individual transmission channels are characterized by bandpass
impulse responses or by their respective Fourier transforms,

The resolution of h,(t) and h;(t) into their respective baseband in-
phase and quadrature components turns out to be convenient in our
application. :

To accommodate coupling between the polarized channels, two pairs
of impulse responses, one associated with the cochannel and the other

associated with the cross-channel, are used to completely characterize
the medium.

At the output, two errr noises are added and the signal plus
noise is then coherently demodulated. The end-to-end system includ-
ing the modulators and the demodulators is shown in Fig. 1. It is
convenient to view this linear system as a four-input port four-output
port network and characterize it by a 4 X 4 matrix impulse response
or its Fourier transform, which is the overall system frequency re-
sponse. 7
It is now easy to verify that the I/O PeNaHONSHIpS can be expressed

In our application, however, it turns out to be more convenient to
work with the real matrix in eq. (6).

We now return to the J/O relationship in eq. (5) and substitute eq.
(1) to obtain in more detail

D(t) =); | a — nT)H(t — r)dr-A, + v(t), (9)
where the real data symbol vector A, is given by
Qivun
Qoun .
An = : 10
Qihn (
Qehn

In an ideal system, eq. (11) would yield D(0) = Ao X constant. This
result is obtained when

1. Ho = constant X J (J is the identity matrix), which implies that
the flat, or nondispersive, CPI vanishes;

Clearly, these requirements cannot be achieved in practice, and the
designer of data communications systems must deal with these im-
pairments and find methods that minimize their effects on system
performance.

A well-known approach? is the use of linear equalization. Our
objective here is to investigate a general cancellation technique in
conjunction with linear equalization, which could potentially yield
better performance than with just the linear equalizer alone. To this
end, we begin our analysis by placing a linear matrix filter in cascade
with the channel prior to sampling, and we choose its characteristics
so as to minimize the total MSE between the actual output sample
and the desired output after canceling some CPI and ISI.

Denote the matrix filter impulse response by W(t) and evaluate its
output at t = 0. This yields the column vector for the overall system
response |

To describe our approach, we first discuss the following statistical
problem. Suppose that one observes the vector Do(0), eq. (13), and
wishes to design the best processing strategy that estimates Ay in a
sense of minimizing the probability of error. The precise solution to
this problem remains intractable because of the non-Gaussian nature
of ISI and CPI. While the precise mathematical solution is unknown,
some qualitative aspects of the solution have been discussed.*” It is
easy to argue that the optimal detector structure consists of a matched
filter followed by a least-mean-square estimator of the interference,
which is then subtracted from the matched filter output. After sub-
tracting the estimate of the interference, the problem reduces to
detecting a known signal in additive Gaussian noise, which has a well-
known solution. The difficulty with this formulation, while physically
appealing, is that the least-mean-square estimator of interference is
just as difficult and intractable to evaluate as the detection problem
originally posed. One redeeming feature of this approach, however, is
that if one does not insist on least-mean-square estimation of inter-

ference, a reasonable detector structure can be determined. We argue
that constructing reasonable estimates of CPI and ISI, which are not
necessarily optimum, subtracting them from the incoming signal, and
then constructing an optimum detector essentially satisfies the spirit
of the suggested optimal procedure.

We now formulate our approach more precisely. To start, assume
that over a finite set of sampling instants, S, vector data symbols, A,,
n € S, are available at the receiver and before we make a final decision

is subtracted from D,(0). Actually, this is feasible since prior to n = 0,
symbols have been decoded all along and what is presumed in our
proposal is that we use the already-decoded symbols to improve on
the current estimate of Ap. Since practical systems are not realizable
relative to a large delay, there is a problem in using symbols that have
not yet occurred. This can be overcome by introducing a delay, making ~
tentative decisions, and then returning to modify the Ao decision.

How realistic is this assumption? The answer depends on the system
error rate prior to cancellation. For example, when the error rate is
10°* and the cancellation window size is small relative to 10‘, the
probability that almost all of the symbols in this window have been
correctly detected is fairly large. Thus, after cancellation, the error
rate may be much improved. On the other hand, if the error rate prior
to cancellation is high, no improvement after cancellation can be
expected since the estimation of the interference is not reliable.
Evidently, decision-directed cancellation as proposed here is a boot-
strapping technique. It is very successful over a certain range of error
rates and fails when the error rate is high. Unfortunately, these
qualitative statements are extremely difficult to make precise, and it
is necessary to rely on simulation results.® The assumption that A,, is
known in the canceler window will clearly result in optimistic perform-
ance predictions, and whether the predicted benefits can be realized
must be ascertained experimentally.

We now proceed to include this “genie” in our mathematical anal-
ysis. As already stated, the performance criterion we use throughout
this work is the least MSE normalized to the transmitted symbols
variance, denoted o%. This is a mathematically tractable criterion to
work with, and by minimizing MSE, one also minimizes an exponen-
tially tight upper bound on the error rate. Its use is also practically
motivated because it lends itself to easy estimation, and it can be used
to update transversal filter-tap coefficients in practical adaptive sys-
tems.

Returning to the mathematical problem at hand, we define the error
vector « as the difference between Do(0) minus the canceler output
vector, and the desired vector data symbol, Ao,

¢ = Updo + X Undn— X CaAn + vo — Ao, (16)
a. nés |
where C,, represents canceler-tap values. Total MSE can be expressed
as | |

where “tr” stands for trace of a matrix, E{-} denotes mathematical
expectation with respect to all random variables, and * represents
complex conjugate transpose.

The set of canceler matrices, C,, n © S, can immediately be deter-
mined. If they are not identically set to U,, they can only increase the
value of MSE. Consequently, we set C,, = Un, n € S, and the residual
MSE results in a functional of the matrix impulse response, wid),
and the size of the cancellation window.

The minimization of MSE with respect to the matrix W(t) is
accomplished by the use of the calculus of variations. After substituting
for U,,, defined in eq. (14), we get _

where the matrices, nj, i,j = 1, --- , 4 have the entry “I” in the (/;)th
position and zero everywhere else. By computing the trace of eq. (21),
we obtain

The structure of Wo(r) is practically interesting. It consists of a
matched filter followed by a matrix-tapped delay line where the matrix
taps are zero for n € S. In other words, the linear transversal filter or
equalizer specified in eq. (23) operates over a range of matrix-tap
coefficients where the canceler is not operative. This is to avoid
interaction between collocated taps and possible unstability problems.
The structure is shown schematically in Fig. 2. In practice, this
structure can be approximated and implemented by a finite transversal
filter whose taps can be adaptively updated.

After post-multiplying eq. (23) by W'(—r), integrating, and then
comparing the result with eq. (19), we get an explicit formula for the
optimum MSE,

where Up is obtained by solving a set of infinite linear equations
obtained by post-multiplying eq. (23) by H(z — kT) and then inte-
grating. Thus,

Fig. 2—Cross-polarization interference and intersymbol interference canceler block
diagram.

To evaluate the merits of our. system, we must have a solution for
U,. The task of solving eq. (25) is rather complicated. It is made
difficult by the fact that the matrix equations are not specified over
the finite set, S. While the number of unknowns is infinite, the values
at the gap window are not specified. A way around this dilemma was
found in the scalar case,* and with care applied to matrix manipula-
tions, it is possible to adopt the same techniques here.

We proceed by first separating eq. (25) into two equations, one for
k = 0 and the other for k ¥ 0. Thus,

where do; is the Kronecker delta function. The solution of eq. (28) is
facilitated by introducing a set of matrix variables {V,,}”.. and a set of
unknown matrices [An}=s : Pe these matrices, we write eq. (28) as

For these aati infinite sets of matrix sauation: to identically coin-
| cide with eq. (28), the following constraints must hold: |

If these can be satisfied, the solution to eq. (81) will be identical to
the solution of eq. (28) with V, = U,, n € J, and this is the sole
purpose for introducing new variables. Evidently, eq. (31) is easy to
- solve since it is in a form of a convolutional equation. To this end
define the inverse matrix sequences, {M‘}®.., as

Now, insert this into eq. (31) to obtain explicitly the desired saliicion,
V, = UI — Up) Y (R,— A)ME?, all n. (34)

From this we can obtain a finite set of equations in the unknown |
matrices A,, since V, = 0 forn Ed,

_ By substituting the definition of M, from eq. (30) into the left-hand
side of eq. (85) and making use of eq. (33), we obtain the desired
equations for the unknown constraint matrices Az, k € J, |

where the last equality derives from the fact that V, = 0,n € J; V, =
U,,n €dJ; and R, = M,, n E Jd. Finally, by substituting eq. (27) into
eq. (37), we can write

Substituting this into eq. (24) provides an explicit expression for MSEp
in terms of Ao only,

Our effort in the following will be centered on determining Ao as a
function of the cancellation window size, or the size of set J. |
2.3 The matched filter bound

When the canceler window is doubly infinite in extent, one obtains
the very best possible result. In other words, the genie has eliminated
all ISI and CPI. In this special case, N, = —o and No = ©, and eq.
(36) is now easy to solve since it reads ,

Ting — PMGSY = Yo ARMED, | (41)
k=—0
By evaluating the Fourier series of both sides of eq. (41), we obtain
IT = 6’? M~(6) + A(O)M (6), (42) -

Qa —T
Since M(6) = o7J + R(6) and M“ (6) is in fact the inverse, M-'(6),
we determine from eq. (42) that A(@) = R(@). Consequently, the zeroth

coefficient of A(0) is Ro = 1/(27) f=, R(6)d0, and when this is substi-
tuted into eq. (40) we get the desired matched filter bound,

This will serve as a lower bound to attainable performance to which
we will compare all other results.

In this application it is assumed that all the causal terms, which
depend only on past decisions, are canceled in addition to a finite
number of noncausal terms. This implies that Nz = © and JN, is finite.
When N, = 0, the canceler becomes a decision feedback equalizer’
since causal interference can be canceled by a feedback circuit. Here,
‘we will determine MSE for the more general case when JN, is not
necessarily zero.

To treat this case it is more convenient to solve for Up directly from
eq. (28) rather than through eq. (36). Thus, we rewrite eq. (28) as

-N,
 UrMn-n = (I - Uo)Rm, ms —N,, (47)
k=—00 .
which is recognized to be a matrix Wiener-Hopf equation, and its
solution depends on being able to factor positive definite Hermitian
matrices.® | |

The validity of this expression and the existence of Mj; and M;, were
first proved by Wiener and Akutowicz.?

The procedure for solving these is to first solve for Y(6) from eq.
(49) in terms of M (0), an easy task in terms of the Fourier transforms
of {M7} and {y,}. Having obtained Y(@), one proceeds to solve eq. (50)
for U(@) in terms of M*(6). Note that eq. (48) implies

M(6) = M (6)M*(6),
and since M(6) is Hermitian, M(#) = M‘(6), implying [M*(8)]' =
M~(6), [M-(6)]* = M7*(6), and the factorization problem is reduced to
finding a matrix M*(6) such that

where the entries in M*(6), [M*(6)]i; are such that [M*(@)], has a
Fourier series with only positive frequency coefficients. We shall later
discuss algorithms for determining M*(0) from M(#)—a rather com-
plicated task.’°

We now proceed to determine the sequence y,,. Multiply both sides
of eq. (50) by M=,, and sum m from —~ to —N,. This gives the for-
mula |

-N,
» VmMim = 2 U,M_x. (51)
Now, recall that M, = R, + o76;ol, and, therefore, eq. (49) can be put
into the form

1 0 —1
= oftr (2 Ma M)") (58a)
C n=-N,
Notice that when N, = 0, that is, no anticausal cancellation,
M2|"
MSE) = o2tr ; | , 7 (58b)
oO 7 3

where My = Mo = (M@)". This is the formula for decision feedback
equalization derived by Falconer and Foschini for QAM transmission
over a single channel, which they cast in a matrix formulation.”
Evidently, the form of the answer generalizes to arbitrary dimensions.

The theoretical results we have derived so far apply to an ideal
canceler of any window size and an infinite-tap linear equalizer whose
matrix taps vanish inside the cancellation window. To assess the
penalties incurred by a finite-tap linear equalizer outside the cancel-
lation window, we derive least MSE formulas applicable to this case.
With these formulas we will be in a position to evaluate the merits of
equalization/cancellation using only a finite number of matrix taps
and to gain insight as to how best to deploy the total number of
available taps. Also inherent in the theory derived so far is the
independence of MSE on sampling phase. This is so since the trans-
versal equalizer/canceler is preceded by a matched filter whose struc-
ture presumes knowledge of sampling phase. Here, we shall relax this
condition and derive the MSE for a front-end filter matched to the
transmitter filter only rather than to the overall channel response and,
thereby, bring out the dependence of MSE on timing phase.

neF
where the two sets F and S are disjoint and F now is a finite set,
[F:n& F,n=—-N, — M,, ---,-N, — 1,0, No+1, ---,No+ Mo}.
In eq. (59), g(t), as before, is a scaler pulse shape, while {Q,},er is a

4 xX 4 matrix sequence. The objective now is to select the Q,’s that
minimize the total MSE, eq. (19),

Setting the derivatives of eq. (61) with respect to the elements of
the matrices {Q,}ner to zero, we get a set of linear matrix equations
for the unknowns, {Q,}ner, namely,

The solution of eq. (63) is straightforward and is discussed in a later
section.

For now, label the solution of eq. (63) by Q?—the optimal Qs.
Premultiply by Q°, sum over n € F, and substitute the result into eq.
(63). This yields the desired formula for the least-mean-square error,

The next section will present computation algorithms for numeri-
cally evaluating the formulas developed here.

An examination of Section III demonstrates that the theoretical
analysis of M-QAM signal transmission over dually polarized channels
in the presence of multipath fading is a numerically intensive activity.
In this section we provide an overview of the major computational
issues related to our investigation.

When the linear equalizer in Fig. 2 has a finite-tap window size, the
optimum receiver structure comprises a matched filter followed by a
matrix transversal filter and a matrix canceler. The most general case
under this assumption is when the matrix canceler has a finite number
of causal and anticausal taps and the solution of eq. (36) for Ao provides
a means of calculating minimum mean-square error by use of eq. (40).
To solve for Ao, block matrices M‘~”’s defined in eq. (33) have to be
determined first. One way to determine the M‘-?”s is to solve eq. (33)
by a Levinson-type algorithm’? where the entries are block matrices.
Thus, matrix convolution eq. (33) is then represented as

equation can be solved for M‘”s, with the M;,’s given in eq. (30).
Having the M‘-””s and expressing eq. (36) in the form

When the matrix canceler has knowledge of infinite past data
symbols, it becomes a decision feedback equalizer. In addition, it may
also employ a finite number of anticausal taps to operate on the future
symbols, in which case it becomes a finite window canceler. This can
be accomplished by a finite delay. As shown in eq. (47), to determine
MSE», a matrix Wiener-Hopf equation has to be solved. This involves
determination of anticausal factors of the M(@) matrix as explained
in Section 2.5. |

There are at least two computational algorithms available for solving
a matrix Wiener-Hopf equation. One method as introduced in Ref. 13
converts the matrix that has to be factored directly into a nonlinear
difference equation of a Ricatti type, which converges to a stable
solution. Another method, which we adopt in our present work, is a
Bauer-type factorization of positive definite polynomial matrices.“
This algorithm is suited to sampled data applications and takes
advantage of the periodic and positive nature of the channel covariance
matrix, M(@), as in this work. It performs the factorization in the
following steps. Suppose one desires to factor the n X n matrix M(@)
as follows:

approach 0, as |m| becomes infinitely large. One now follows the
steps:
Step 1: Form the following variable-size Toeplitz matrix

of respective size (m + 1)n X (m +1)n,m=0,---,thatis Hermitian —
andreal. |
Step 2: For every m = 0, perform the Cholesky’s factorization”

where Lis the transpose of L,,, and L» is a square, real, and lower
triangular matrix with positive diagonal scalar entries,

Le Le
Aa os ee 0 |
Im =) 5 3 ae - (72)
Le” Lb” er ee ee ee
All blocks in eq. (72) are real n X n, and all L®”,r=0, ---, m, are

The fetdnsation of eq. (71) and the evaluation of M7_, in eq. (73) are
performed numerically. To find an upper bound on m for stopping the
calculations, a convergence point must be established. This can be
- done by checking the trace of M;_, in each iteration to determine
whether it has reached a level of constancy and, if so, for what value
of m. This completes the factorization. Indeed, in Ref. 14 it is shown
- that a constant trace as a function of m corresponds to a minimum of
a quadratic functional:

where T',, was defined in eq. (70) and x,’s represent the elements of X.
Hence, there is a theoretical base for establishing the convergence
point. |

Finally; we consider the case where the matrix linear equalizer
operates on a finite set of taps that do not overlap with those of the
finite-tap matrix canceler. This is a case of great. practical interest.
Here, the receive filter is assumed to have a square-root-Nyquist
transfer function’® matching the transmit filter. Since it no longer
matches the overall channel and transmitter characteristics, MSEo is
a function of timing phase. Therefore, an optimum timing reference
has to be established before the optimum nonstationary covariance
matrix can be determined. This is accomplished here by minimizing
the mean-square eye closure (MS-EC), which is a measure of the
amount of received level perturbation caused by CPI and ISI.'° In our
present work it is assumed that the demodulator removes the channel
phase at the optimum sampling time reference.!® Once an optimal set
of samples is found, the covariance matrix, Grm, of eq. (62) is formed
as

In terms of the H,,’s defined in eq. (12) the covariance matrix can be
expressed as

Hence, by adding a’ to the diagonal elements of Gin, the matrix Ram
is formed, as expressed in eq. (64). The Q,’s, that is, the coefficients
of the finite window equalizer, can be computed as follows:

In this section, the minimum mean-square error (MSEp) is evaluated
for the various techniques covered in the previous sections. We will
first discuss a channel model, and then we will exhibit and discuss the
behavior of MSEp as a function of the number of equalizer/canceler
taps (M,, Mo, Ni, Ne) (see Fig. 2). |

The cross-polarization fading propagation model employed is the
one that is proposed in Ref. 2 and is briefly reviewed here. The
frequency characteristics of the propagation model are presented by
the complex matrix | |

where the functional form of C,;(w) and C2o(w) is that of a single (in
line) fading channel model documented by Rummler”’ with the generic
representation

where a and p are real variables representing flat and dispersive fading
levels, @ is related to the fade notch offset, and 7 is the delay between

direct and reflected paths assumed to be 6.3 ns in this study. Also in
the model,

which is in the same form as C;(w), except for an additional variable
Aw that allows noncollocated fade notches to occur on the two polar-
ization signal transfer characteristics. From Ref. 2, cross-polarized
paths are assumed to behave as

where K,, Kz, Ks, and Ks are constants that incorporate the nonideal
properties of antennas and waveguide feeds at both ends of the
channel, typically taking on values varying from one hop to another
in the —35 to —20 dB range. The last term in eqs. (83) and (84)
represents a nondispersive cross-polarization response contributed by
an independent ray. In the present work, R3, Re, and Aw are assumed
to be zero and the K;’s are assumed to be —20 dB.

Computation of the channel covariance matrix is the initial neces-
sary step behind all the MSE, calculations. In the case of the infinite
window-size equalizer discussed in Sections 3.3 through 3.5, the receive
filter is assumed to be a matched filter; hence, no reference timing
establishment is necessary. The peak of the correlation function serves
as a timing reference. By computing the sampled correlation matrix
of eq. (26), we can proceed with the normalized MSE, calculations as
explained in previous sections. |

By applying the finite window equalizer, as discussed in Section 3.6,
a set of optimum samples of overall impulse response is found by
establishing a timing reference, to, for which the MS-EC of the received
in-line signal is a minimum, and at this reference, the channel phase
is removed.’® This has to be done for the two polarized signals inde-
pendently.

where C(w) is the propagation transfer matrix and P(w) is the diagonal
Nyquist-shaping filter transfer matrix. Now, for instance, if the im-
pulse response of the vertical in-line signal is

and the upper row block matrices of the overall impulse response
matrix have to be multiplied by exp(—j @(t)) < J (J being the unity
matrix) in order to remove the channel phase at t). With this back-
ground, we now present the numerical results in the following subsec-
tion.

To provide a single set of curves for MSEo, independent of the
number of transmit states in M-QAM signal space, we normalize
MSE, as defined in eqs. (40), (43), (46), (58a), (58b), and (65) by
dividing the formulas by o4, that is, the transmitted symbols variance.
In addition, we only compute the normalized MSE» for one of the M-
QAM signals that comprise the dually polarized signals, namely, S,(t).

If one defines the unfaded signal-to-noise ratio (s/n) by I’, it can be
verified that in the case of a matched filter receiver, the normalized
MSE» in the absence of any cross-polarization interference (K,, Ko,
Ky, Ks, R3, Re = 0) is simply

Od
and, consequently, for a large unfaded s/n, it becomes I~’. Hence, eq.
(88) establishes an ultimate performance bound that can only be
achieved in a utopian environment. This reference will be our baseline
in the following evaluations. In a dually polarized system with a finite
amount of nondispersive coupling (Ky, Ko, K4, Ks > 0), the matched
filter bound is degraded somewhat. For K, = Ky = K, = Ks = —20 dB
we found a small amount of degradation in the ideal MSEo, which is
not a function of the dispersive fade depth and only diminishes when
there is no cross-coupling, that is, in a completely orthogonal system.

In all that follows it is assumed that the transmit filter is square-
root Nyquist shaped,’® and the receive filter either matches the overall
transmitter and channel or the transmitter only. A Nyquist roll-off of
45 percent, both a 40- and a 22-MHz channel bandwidth, and an s/n
of 63 dB are used in our numerical evaluations.

Figure 3 depicts the normalized MSE» as a function of the number
of canceler taps, Q, when a 40-dB centered fade over a 22-MHz channel
band is applied to both polarized signals. The linear equalizer in this
case possesses an infinite number of taps. The case of pure linear
equalization (N, = Nz = 0), no cancellation, exhibits the largest MSEo

Fig. 3— Optimum normalized MSE versus number of canceler taps for a 40-dB
centered fade over a 22-MHz channel.

degradation relative to the asymptotic matched filter bound. This is
due to the noise enhancement experienced by the linear equalizer
during deep fades. When both causal and anticausal canceler taps are
present, all the curves rapidly approach the matched filter bound for
a finite constant coupling (K; = —20 dB, i = 1, 2, 4, 5). The curve for
a decision feedback type canceler starts at an ideal decision feedback
equalizer normalized MSE» and approaches the asymptotic value with
two anticausal taps. The finite window size canceler curve starts at
the linear equalizer case (Ni; = Nz = 0) and reaches the matched filter
bound asymptotic value with a total of four causal/anticausal taps.
Finally, when no anticausal taps are employed the curve asymptoti-
cally approaches the ideal decision feedback case with only two causal
taps.

In Fig. 4, we depict results similar to Fig. 3 for the case when the
centered fade notch depth is reduced to 20 dB over a 22-MHz channel.

Fig.4—Optimum normalized MSE versus number of canceler taps for a 20-dB
centered fade over a 22-MHz channel.

As can be observed, the linear equalizer (N; = No = 0) performance is
improved. In both figures the fade notch is located at the band center;
however, since in both cases the receive filter matches the overall
channel and transmitter, an offset fade notch does not have a serious
impact on the results for the same fade notch depth.’®

In Figs. 5 and 6 we depict the achievable MSE» when the linear
equalizer has a finite number of taps. The fade notch in Fig. 5 is
centered, but in Fig. 6 it is offset from the band center. For ease of
presenting the results in our work, fade notch offset from the band
center is expressed in terms of the ratio of the fade notch distance
from the band center to the channel equivalent baseband bandwidth
in percentage. In Fig. 6, the fade notch is offset by 69 percent over a
22-MHz channel, that is, an offset of 7.6 MHz from the band center.
As observed from Fig. 5, a total of nine taps (including the center tap)
are required to achieve the asymptotic matched filter bound when
decision feedback taps are present. It is interesting to note that the
same asymptotic performance can be achieved no matter how the nine
synchronously spaced taps are deployed between the linear equalizer

Fig.5—Optimum normalized MSE versus number of canceler taps for a 40-dB
centered fade over a 22-MHz channel.

and the canceler as long as the canceler operates in a decision feedback
mode. This is because the equalizer and canceler-tap windows comple-
ment one another; therefore, since the taps do not overlap, for the
same number of taps, the performance remains almost the same in
the decision feedback cases. An important configuration is when the
linear equalizer operates only on the main lobe of CPI by means of its
center matrix taps (M,; = Mz = 0). This is a single tap linear equalizer
structure as opposed to the single tap decision feedback CPI canceler
proposed by Kavehrad.? It is clear that as long as the canceler window
is sufficiently wide, a main lobe CPI canceler can achieve the asymp-
totic matched filter bound. The curves again indicate that deep fades
degrade the linear equalizer (N; = No = 0) performance significantly.

Fig. 6—Optimum normalized MSE versus number of canceler taps for a 40-dB, 69-
percent offset fade over a 22-MHz channel. |

Previous studies’? showed that every 3-dB degradation in MSEp
translates into a loss of 1 bit/s/Hz of data rate efficiency. Hence,
linear equalization may not prove’ adequate rate efficiency in deep
fades. ° |

In Fig. 6 we depict curves similar to Fig. 5, except for a 40-dB offset
fade with the notch frequency offset by 69 percent. Improved perform-
ance turns out to be due to the particular notch position as is brought
out in the discussion of Fig. 8.

Figure 7 illustrates a similar set of curves for a 20-dB centered
fading of dually polarized signals over both a 22- and a MHz
channel. As can be observed, the linear equalizer (N; = =)
performance improves because of the decreased fade depth; ic

_ is more, as expected. This is due to the wider channel band over which
the same fade notch depth causes more dispersion. The degradation
amounts to 2.2-dB loss of MSE» comparing to a matched filter bound,
that is, roughly 1 bit/s/Hz loss of data rate efficiency, and the loss can
even be more for offset fades, as will be seen in Fig. 8. Hence, even

Fig. 7—Optimum normalized MSE versus number of canceler taps for a 20-dB
centered fade over a 22-MHz and a 40-MHz channel.

with more typical fades the use of the linear equalizer can be trouble-
some over a 40-MHz channel.

M, = M,=4 |
N, = Nz = 0 (no cancellation).
2. Center tap only linear equalizer/finite window canceler with
M, = M2. =0
N, = No = 4.

nounced. The center tap equalizer with a finite window canceler
exhibits a very small sensitivity to offset fades. The degradation of
MSE») for some offset fades can be explained considering the fact that
these fades cause cross-coupling of the imaginary part of a complex
QAM signal into its real part, and for a particular notch offset
frequency within the band, the coupling reaches its maximum. There-
fore, the MSE» versus fade notch offset curves exhibit this phenome-
non. In dually polarized systems, as in the case of the problem at
hand, this is even more pronounced than in single signal transmission,
because in the 4 X 4 system under offset fading there is coupling of
three interfering data streams into the fourth one. A decision-feed-
back-type canceler structure, by canceling the major contributors to
CPI and ISI and with a lesser noise enhancement, exhibits an improved
performance compared with the linear equalizer. Note that all the
curves in Fig. 8 have been obtained under optimum timing conditions.

To investigate the sensitivity of the two structures to timing phase,
we plot in Fig. 9 the normalized MSE» and superimpose the normalized
MS-EC of the received signal before equalization/cancellation as a
function of sample timing offset from the optimum timing reference.

Fig. 9—Sensitivity to timing phase for a 40-dB fade notch, offset by 34.5 percent,
over a 22-MHz channel.

This is done for a severe fade, namely, a 40-dB fade with a notch
frequency offset by 34.5 percent over a 22-MHz channel. It is clear
that the finite linear equalizer is much more sensitive to timing phase
than the decision feedback type. This was previously shown in a paper
by J. Salz’® for infinite window structures in single-signal transmis-
sion. We demonstrated the concept here for dual-polarization trans-
mission and finite window architectures. Notice that in Fig. 9 the
optimum timing reference is established based on minimizing the MS-
EC of the received signal in presence of fading, before CPI and ISI
cancellation; hence, after cancellation occurs this timing reference
may not be the one that minimizes the canceler output MSE, and

indeed the linear equalizer curve on Fig. 9 indicates this fact. As seen,
the MS-EC curve has a minimum at the optimum timing reference.
The sensitivity of the matrix linear equalizer to timing phase can be
reduced by applying half-a-baud spaced taps, that is, by deploying
fractionally spaced taps.”° Also, a decision feedback timing method
may prove more robust” than the minimum MS-EC timing that has
been adopted in our work. Needless to say, that decision feedback
timing method is more complex in terms of implementation than the
minimum MS-EC timing, which can essentially be implemented at
intermediate frequency. The degradation in MSHp seen in Fig. 9 is
partly attributed to the asymmetric amplitude and delay responses of
the fading channel that in the presence of a nonzero roll-off shaping
filter cause a destructive addition of aliases.’®

Current work generalizes and extends previous results”* in the
following respects. Data-aided decision feedback and canceler struc-
tures, known to be effective in single-channel data transmission, are
_adopted and included in our class of receiver structures. As a practical
_ feature, we admit transversal filter realizations with a finite number

Because of the departure from ideal linear infinite structures con-
sidered previously, we encountered extremely difficult numerical prob-
lems, which we addressed and solved.

The dually polarized digital radio channel is modeled as a four-input
port, four-output port linear. network followed by additive noise. We
‘determine the optimum admissible receiver structures when the trans-
mitted signals are two independent M-state QAM digital data signals.

The mathematically tractable criterion, the MSE, is used through-
out our work. This figure of merit has several redeeming features in
addition to its being mathematically tractable. For one, it can be used
to determine a sharp upper bound on error rate. More importantly, it
is the quantity which is estimated in practice to provide information
for updating tap coefficients in adaptive systems.

The receiver structure that minimizes the MSE consists of a matrix
matched filter in cascade with a transversal filter combined with an
intersymbol interference as well as a cross-polarization interference
canceler. The canceler uses the detected data symbols to estimate the
interference to be canceled. This is a major assumption on which our
results rest. Since data-aided operations presume correct knowledge
of detected data symbols and since wrong decisions will be occasionally
accepted, our proposal is necessarily a boot-strapping approach. Thus,
cancellation is only feasible when tentative decisions are correct most

of the time, and yet the error rate is not sufficiently low to meet
system specifications. Our approach makes possible the reduction of
the final error rate to an acceptable level. In circumstances where the
initial error rate is very high (=1077), our proposal will not work, and
in order to ensure availability of reliable data symbols, one option is
_to dedicate a small fraction of the main data frame to a sequence a
priori known to both transmitter and receiver for proper acquisition
of data.

We use the assumption of the availability of correct data symbols
to derive our main results. These are expressions for minimum attain-
able MSE as a function of various system parameters and numerical
algorithms for evaluating the mathematical formulas—a rather inten-
sive activity because of the large number of matrix equations that has
to be solved. Inclusion of the effects of errors in the feedback/canceler
loops has proved so far to be mathematically intractable.

From our extensive numerical work, which is exhibited in a sequence
of graphs, we draw these major conclusions:

1. For a reasonable copolarized and cross-polarized propagation
model? and a severe centered fade—40-dB notch depth with a second-
ary ray delay of 6.3 ns over an approximately 22-MHz channel band-
width—the performance of transversal filters with a finite number of
taps deployed in a decision feedback/canceler structure is substantially
(6 dB) better than linear equalization, and the difference can be up to
10 dB for offset fades: It can be shown that a 3-dB increase in MSE
translates into about 1 bit/s/Hz decrease in data rate at a fixed error
rate or an order of magnitude increase in outage probability. Hence,
linear equalization may not be adequate in deep fades. Whether this
gain can be realized in practice depends on the degree of error propa-
gation. This is difficult to assess mathematically and must be studied
by computer simulation and/or by experimentation.

2. Decision feedback/canceler structures achieve the ultimate
matched filter bound with only nine matrix taps provided that error
propagation is neglected.

3. Nine linear equalizer taps essentially achieve the performance of
the infinite-tap linear equalizer. This method is, of course, free of error
propagation. |

4. For milder centered fades —20- dB depth with a secondary ray
delay of 6.3 ns—the linear equalizer configuration with nine taps is
only 1-dB inferior to the decision feedback structure over a 22-MHz
channel. However, if the channel bandwidth is increased to 40 MHz,
the performance of the linear equalizer is worse than that of the
decision feedback structure by 2.2 dB, and the difference can be up to
- 8 dB for offset fades.

5. Decision feedback/canceler configurations are less sensitive to
timing phase than linear structures.

Discussions with N. Amitay; G. J. Foschini; L. J. Greenstein; D. J.
Goodman; T. Kailath and Hanoch Lev-Ari of Stanford University; N.
Kazanjian; A. M. Saleh; and D. C. Youla of Polytechnic Institute of
New York were extremely valuable during the course of this work.

1. G. J. Foschini and J. Salz, “Digital Communication Over Fading Radio Channels,”
B.S.T.J., 62, No. 2, Part I (February 1983), pp. 429-59.

2. N. Amitay and J. Salz, “Linear Equalization Theory in Digital Data Transmission
Over Dually Polarized Fading Radio Channels,” AT&T Bell Lab. Tech. J., 63,
No. 10 (December 1984), pp. 2215-59.

M. Kavehrad, “Baseband Cross-Polarization Interference Cancellation for M-Quad-
rature Amplitude- Modulated Signals Over Multipath Fading Radio Channels,”
AT&T Tech. J., 64, No. 8 (October 1985), pp. 1913-26.

M. S. Mueller and J. Salz, “A Unified Theory of Data-Aided Equalization,” B.S.TWJ.,
60, No. 11 (November 1981), pp. 2023-38.

T. Kailath, “A General Likelihood-Ratio Formula for Random Signals in Gaussian
Noise,” IEEE Trans. Inform. Theory, [T-15, No. 3 (May 1969), pp. 350-61.

A. Gersho and T. L. Lim, “Adaptive Cancellation of Intersymbol Interference for
Data Transmission,” B.S.T.J., 60, No. 11 (November 1981), pp. 1997-2020.

J. Salz, “Optimum Mean- Square Decision Feedback Equalization,” B.S.T.J., 52,
No. 8 (October 1973), pp. 1341-73.

D. C. Youla, “On the Factorization of Rational Matrices,” IRE Trans. Inform.
Theory, J T-7 (July 1961), pp. 172-89.

N. Wiener and KE. J. Akutowicz, “A Factorization of Positive Hermitian Matrices,”
J. Math. Mech., 8 (August 1959), pp. 111-20.

11. D. D. Falconer and G. J. Foschini, “Theory of Minimum Mean-Square-Error QAM
Systems Employing Decision Feedback Equalization,” B.S.T.J., 52, No. 10 (De-
cember 1973), pp. 1821-49.

12. N. Levinson, “The Wiener RMS (Root Mean Square) Error Criterion in Filter
Design and Prediction,” J. Math. Phys., 25 (February 1946), pp. 261-78.

13. W. G. Tuel, “Computer Algorithm for Spectral Factorization of Rational Matrices,”
IBM J. Res. Develop., 12 (March 1968), pp. 163-70.

14. D. C. Youla and N. Kazanjian, “Bauer-Type Factorization of Positive Matrices and
the Theory of Matrix Polynomials Orthogonal on the Unit Circle,” IEEE Trans.
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Comp., 26, No. 106 (April 1969), pp. 221-30.

16. R. W. Lucky, J. Salz, and E. J. Weldon, Principles of Data Communication, New
York: McGraw-Hill, 1968.

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B.S.T.J., 58, No. 7 (May-June 1979), pp. 1037-71.

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IEEE Trans. Commun., COM-32, No..4 (April 1984), pp. 429-43.

Cross-Polarization Interference Cancellation and
Nonminimum Phase Fades

This paper reports on a continuation of the work presented in Ref.
1, that is, cancellation of cross-polarization interference by a trans-
. versal structure at baseband when two M-state quadrature amplitude-
modulated (M-QAM) signals are transmitted over the same channel
using orthogonal field polarizations. In this paper we extend the scope
of the previous study, which only dealt with minimum phase fades, to
include evaluation of nonminimum phase fade effects encountered in
transmission over multipath fading channels such as those experienced
in line-of-sight terrestrial radio applications. _

Recent studies** have examined IF and baseband equalizers in the
transmission of a single QAM signal under nonminimum phase fades.
For example, Ref. 4 examines transition distortions when the fade
phase changes from minimum to nonminimum. In particular, Ref. 2
concludes that, in mitigating intersymbol interference, baseband
equalizers outperform IF structures used for this purpose in the

presence of nonminimum phase fades. Since, as pointed out in Ref. 2,
on the average 50 percent of deep selective fades take on a nonmini-
mum phase, we examine the proposed baseband interference canceler
performance’ in nonminimum phase fades. We use the static modeling
of the dual-polarized channels introduced in Ref. 5 to emulate snap-
shots of fadings of the main (reference) and the cross-coupled paths.
This simple model follows the results of recent measurements cited in
Ref. 5. | |

In this study, we first show that some of the nonminimum phase
fade channel models available in the literature that were meant to be
used in single-signal transmission may not be appropriate for appli-
cation in multicarrier transmission systems such as dual-polarized or
space-diversity systems. Then, employing what seems to be the proper
static model for nonminimum phase fades, we evaluate dual-polarized
system performance with and without the proposed canceler.’ It is
demonstrated that, for a proper model, the type of fading has no
impact on the performance of the baseband cross-polarized interfer-
ence canceler, as long as the fadings of the reference (main) copolarized
path signal and the cross-coupled path signal are of the same type,
i.e., both minimum or both nonminimum phase. As in Ref. 1, dual-
polarized system performance signature (M-curve) is used as a measure
in the performance evaluation.

In this section we compare three commonly used static models of
nonminimum phase fades and demonstrate that only one of them
seems to be appropriate for application in dual-polarized systems. The
first model, which has been used in Ref. 4 and some other studies,
states that if for minimum phase fades we employ a two-ray transfer
_ function shown by | |

| H(w) = a{l — 6 exp(—j (w — wo)7)], 0<b<1, (1)
in which b represents the fade notch depth and w is its displacement
from the midband frequency, 7 is the delay between the arrival times

of the two rays, and a is the flat fade level, then for nonminimum
phase fades we have to use

in which 0’ is the fade notch depth. The second model suggests using
H(w) = alb — exp(—] (w — wo)r)], O<sb<1 (3).

Comparing equations (1) through (4), it is obvious that the magnitudes
of the transfer functions are the same at the fade notch frequency in
all three different models and are all equal to

For the magnitude to be the same at the fade notch frequency in eq.
(2), the value of b’ has to be set to (2 — b) so one can compare
minimum and nonminimum phase fades of equal magnitude. This
stems from the fact that the minimum phase fade depth here is defined
as

and for the nonminimum phase fades of eq. (2) to have comparable
magnitudes with minimum phase fades, b’ has to be equal to (2 — b).
Although the fade magnitudes are all the same, the envelope delay
responses of the different transfer functions cited above are not. That
is exactly why one has to be careful in choosing a model to form static
nonminimum phase fades in multicarrier channels. The envelope delay
response as a function of frequency for the minimum phase transfer
characteristic in eq. (1) can be found by using

Tuer(w) =
where the subscript MPF stands for Minimum Phase Fade. In the
following, for simplicity we use midband fades to prove our point, i.e.,
when w in eq. (6) is equal to wo. The conclusions arrived at, however,
are felt to be general. Hence, the envelope delay of a midband minimum
phase fade from eq. (1) is

For the first, second, and third nonminimum phase fade transfer
functions the envelope delay response is

_ respectively, where the subscript NMPF stands for nonminimum
phase fade. Notice that the first and the second models introduce
constant delays of 27 and 7 in the envelope delay response, respec-
tively, and these delays have nothing to do with the nonminimum
phase fade nature. To prove the point, in Fig. 1 we show the overall
impulse response of a single minimum phase faded channel using a
7.5-dB midband fade notch. As seen, the optimum timing reference,
topt, is to the left of the unfaded signal optimum timing reference, i.e.,
the time origin. This is of course due to the negative delay character-
istic. Using a 7.5-dB midband nonminimum phase faded transfer
function of the third model, the overall impulse response of the system
for 45-percent roll-off Nyquist-shaped transmit/receive filters is
shown in Fig. 2. Note that the optimum time reference is now to the -
right of the time origin owing to the positive delay character, and the
impulse response is time reversed about the timing reference when
compared to the corresponding one for the minimum phase fade of
Fig. 1. The first or the second model will translate to the right the

Fig. 1—In-phase received impulse response of a single-carrier system for a minimum
phase fade.

Fig. 2—In-phase received impulse response of a single-carrier system for a nonmini-
mum phase fade.

position of the optimum timing reference of the nonminimum phase
faded impulse response of Fig. 2 by 27 or 7, respectively. Such a
translation is harmless in single-carrier systems because once the
optimum timing reference is established, taking samples of the impulse
response at every baud period results in the same set of samples for
minimum and nonminimum phase fades regardless of the actual |
position of the timing reference. However, in dual-polarized, or gen-
erally in multicarrier systems, the position of the timing reference can
be as important as the sampling points in time. To demonstrate this,
we use the dual-polarized channel model shown in Fig. 1 of Ref. 1, and
illustrate the received in-phase impulse responses formed by the main
copolarized and the cross-coupled paths in Fig. 3. These responses are
composed of the real part of the main-path signal U;1, cross-coupling
of the main-path quadrature phase part U,1, cross-coupled in-phase
part of the cross-coupled path U;1, and cross-coupled quadrature
phase part of the cross-coupled path U,1. Using notations of Refs. 1
and 5, the in-phase received waveforms shown in Fig. 3 correspond to
7.5-dB midband minimum phase fading of the main-polarization path
and a cross-coupled path fading of (—20., 0., 0.),* that is, an interfering
cross-coupled signal with a flat fading level of 20 dB below the main-
polarized signal level. Hence, the main-path impulse response timing

Fig. 3—In-phase received impulse responses of dual-polarized system for a midband
minimum phase fade on the main path and a flat fade on the cross-coupled path.

Fig. 4—In-phase received impulse responses of dual-polarized system for a midband
nonminimum phase fade on the main path and a flat fade on the cross-coupled path.

reference is to the left of the time origin because of its minimum phase
fading, and the cross-polarized path impulse response is an attenuated
Nyquist pulse centered at the origin because of its flat fading, as
expected. The corresponding shapes for an equivalent nonminimum
phase fade of the main path, using the third model, are shown in Fig.
4. As seen, the interferer impulse-response shape does not change;
however, the main-path impulse-response timing reference moves to
the right of the time origin, and the impulse response shape itself is

time reversed. Now in the dual-polarized system, once the receiver
timing recovery circuit locates the optimum time reference by mini-
mum-peak-distortion or minimum-mean-square error criterion, the
main and the interfering impulse responses are sampled using the
same reference timing, as explained in detail in Ref. 5; hence, the
unnecessary time shifts inherent to the first and the second nonmini-
mum phase fade models will result in establishing an incorrect refer-
ence point and a false set of samples of the cross-coupled interfering
impulse responses. For further illustration, the displacement of the
optimum timing reference from the origin as a function of the fade
notch offset for different fade notch depths is shown in Fig. 5. Clearly,
the upper half-plane represents nonminimum phase and the lower half
corresponds to minimum phase fades. We employed the second non-
minimum phase fade transfer function model in this case. As seen, the
two sets of curves are vertically symmetrical except for a constant
time shift of 6.3 ns which happens to be the value of 7 used in this
example. The constant time shift is what the transfer function model

Fig. 5—Constant time shift in optimum timing reference by the second nonminimum
phase fade model.

The system. model and cross- eeoupled interference canceler are the
same as those introduced in Figs. 1 and 6 of Ref. 1, respectively. The
channel model allows for a single Frequenicyiae lective fade notch, both
in the reference copolarized and the cross-coupled paths; therefore, it
requires the two-ray model parameters of both paths. 7

The canceler structure shown in Fig. 6 of Ref. 1 operates on the
baseband received digitized signals. The main-lobe estimators and
decision circuits together make preliminary estimates of the main lobe
of the impulse response of the reference copolarized path. These
_ estimators can be made of two to three tap transversal equalizers, and
decisions made by detection circuits are provided to the cross-polari-
zation canceler transversal filter to form and then eliminate the
interfering main lobe from the opposite polarization received signal
that has been properly delayed. Then, the cross-coupled interference
canceled signals are equalized for mitigating intersymbol interference
and cross-rail interference. The tap coefficients of the main-lobe
estimators can either be derived from the preliminary decision circuit
error signals or, to get a better performance, they can be set by using
the error signals of the final decision circuits, as shown in Fig. 6 of
Ref. 1. The slow channel time variations allow the use of the final

Ill. NUMERICAL CALCULATIONS OF CANCELER PERFORMANCE IN
NONMINIMUM PHASE FADES |

In this section we assume two 16-QAM signals are dual-polarized
and transmitted over the channel model shown in Fig. 1 of Ref. 1
assuming nonminimum phase fading of the main path. The cross-
coupled path is assumed to be flat or minimum phase faded. For
nonminimum phase fades we use the third model described in Section
II. The transmit/receive filters are the Nyquist type of 45-percent roll-
off. The symbol rate is chosen to be 22.5 Mbaud and a signal-to-noise
ratio of 60 dB is assumed. Performance signature (M-curve) for a 107°
symbol error rate is used in the numerical evaluation, and the Gauss
quadrature method’ is used to derive the probability of error, accu- —
rately.

consider the following two examples. In the first example the main
path is nonminimum phase faded with a fade notch depth of 7.5 dB
and a notch offset of 11.2 MHz from the band center. The cross-
coupled path in this case is assumed to be flat faded by 20 dB. Clearly,
the cross-coupled path signal has a perfect Nyquist shape and is
centered at the time origin. The main-path signal timing reference is
to the left or the right of the origin depending on whether the main-
path fade is minimum or nonminimum phase. However, since the
amount by which the timing reference is translated has the same
absolute value, as seen in Figs. 6 and 7, once the timing reference is
found, the main and the cross-coupled signals are sampled using the
same reference; hence, the same set of interferer samples results,
regardless of the type of the fade. Therefore, the resulting end per-
formance should not change. However, this is no longer true when the
cross-coupled path is faded dispersively because the interfering im-
pulse response is no longer symmetrical about the vertical axis going
through the origin. This is shown in Figs. 8 and 9 where, in addition
to a 7.5-dB depth and an 11.2-MHz offset fading of the main path,
the cross-coupled path is faded by a 5-dB midband fade. Hence,.
depending on whether the main-path fading is a minimum or nonmin-
imum phase, different sets of samples of the interfering impulse
response are involved and the end performance may not necessarily
remain the same.

The system performance signatures for two different fading eonde
tions of the cross-coupled path with and without the canceler of Ref.
1 are shown in Figs. 10 and 11. In Fig. 10 it is assumed that the cross-

_ Fig. 6—In-phase received impulse responses of dual-polarized system for an offset
minimum phase fade on the main path and a flat fade on the cross-coupled path.

Fig. 7—In-phase impulse responses of dual-polarized system for an offset nonmini-
mum phase fade on the main path and a flat fade on the cross-coupled path.

Fig. 8—In-phase impulse responses of dual-polarized system for an offset minimum
phase fade on the main path and a midband minimum phase fade on the cross-coupled
path.

coupled path is only flat faded; hence, the nonminimum phase per-
‘formance signatures with and without the canceler are identical to the
minimum phase faded signatures of Fig. 7 in Ref. 1. As seen in Fig.
10, the single-tap canceler’ improves the dual-pol system performance
to nearly that of a single-pol system shown by a dotted M-curve. For
midband fades, as observed in Fig. 10, the timing reference of the
main-path impulse response is offset from the peak of the main lobe

Fig. 9—In-phase impulse responses of dual-polarized system for an offset nonmini-
mum phase fade on the main path and a midband minimum phase fade on the cross-
coupled path.

Fig. 10—Canceler performance in dual-polarized 16-QAM radio for a flat fade on the
cross-coupled path.

Fig. 11—Canceler eetionnanens in dual- polarized 16- QAM radio for a midband dis-
persive fade on the cross-coupled path.

of the interfering impulse response shape; hence, the canceler does
not perform as well for midband fades as it would for offset fades of
the main path because as the main-path fade notch moves toward the
band edge, the timing reference of the overall impulse response moves
toward the origin; hence, the two peaks tend to align. Of course, the
remedy for this situation is to increase the number of canceler trans-
versal filter taps. In Fig. 11, we consider the case where the cross-
coupled path is dispersively faded by a minimum phase midband fade
of 5-dB notch depth. In this case, since by asymmetry two different
sets of samples of the interfering impulse responses are involved, the
resulting signatures under minimum or nonminimum phase fades are
not the same, and the nonminimum phase fade increases the outage ~
slightly for the same cross-coupled path fading conditions. As the fade
notch moves away from the band center, and since the reference
timing of the main-path impulse response moves toward the time
origin for both minimum and nonminimum phase fades, the two sets
of samples taken of the interfering impulse responses become similar;
hence, the minimum and nonminimum phase fade signatures approach
one another and become approximately the same around the band
~ edge..

A comparison of the signatures for minimum and nonminimum
phase fadings of the main path indicates that the baseband canceler

performance is relatively insensitive to the type of the fade as antici-
pated in Ref. 1. |

Notice that the difference in the signatures for minimum and non-
minimum phase fading of the copolarized channel is because we have
assumed a minimum phase fade for the cross-coupled channel in both
cases. Given that the copolarized and the cross-coupled channel fading
_ have both minimum or both nonminimum phase, the pplenanuaee are
indeed identical in both cases.

In this paper performance of a previously proposed canceler’ is
evaluated under nonminimum phase fades, and it is shown that the
canceler performance is practically transparent to the type of the fade.
An investigation of different static models for nonminimum phase
fades is also performed, and it is shown that some of the known fading
models for single signal transmission may not be appropriate for
multicarrier systems such as dual-polarized channels.

The writer is deeply indebted to W. C. Jakes, Jr. for discussions
during this work.

1. M. Kavehrad, “Baseband Cross-Polarization Interference Cancellation for M-Quad-
rature Amplitude-Modulated Signals Over Multipath Fading Radio Channels,”
AT&T Tech. J., 64, No. 8 (November 1985), pp. 1913-26.

2. L. Leclert and P. Vandamme, “Non-Minimum Phase Fadings Effects on Equaliza-
ears Techniques in Digital Radio Systems,” GLOBECOM Conf., November 1983,

3. H. Sari, “Baseband Equalizer Performance in the Presence of Selective Fading,”
GLOBECOM Conf., November 1983, San Diego.

4, G. Niezgoda et al., “Effects of Non-Minimum Phase Fades on the Performance of
a 49 QPRS 90 ‘Mb/s Digital Radio Using Decision Feedback Equalizations,”
GLOBECOM Conf., November 1983, San Diego.

5. M. Kavehrad and C. A. Siller, Jr., “Performance Signatures for Dual-Polarized
Transmission of M-QAM Signals Over Fading Multipath Channels,” AT&T
Tech. J., this issue.

6. W.D. Rummler, ‘ ‘A New Selective Fading Model: Application to Propagation Data,”
B.S.T.J., 58, No. 5 (May-June 1979), pp. 1037-71.

Analysis/Simulation Study of Cross-Polarization
Cancellation in Dual-Polarization Digital Radio

This paper analyzes cross-polarization cancellation in dual-polarization
digital radio links transmitting M-ary quadrature amplitude-modulation
(M-QAM) signals. We consider the use of three options in the receiver: (1) no
cancellation ; (2) ideal (i.e., total) cancellation; and (3) optimal nondispersive
cancellation. For every option, we assume the canceler to be followed, in each
polarization branch, by an ideal minimum mean-square error equalizer. By
postulating a statistical model for the co-polarization and cross-polarization
responses of a digital radio channel, and then simulating thousands of sets of
these responses, we obtain curves that relate outage probability to the number
of modulation levels. We show graphically that the no-canceler case is unthink-
able; that total cancellation permits results close to those for single-polariza-
tion transmission; and that optimal nondispersive cancellation can have a
limited range of application. We also examine the effects of key system
parameters and the various modeling assumptions.

This paper reports on a theoretical study of cross-polarization
cancellation in dual-polarization microwave digital radio receivers. In
another recent theoretical study, Amitay and Salz derived and evalu-
ated the optimal linear receiver response to cross-pol coupling and
multipath fading in combination.’ By contrast, this paper analyzes
suboptimal structures wherein the cross-pol canceler and multipath

equalizer stages are in cascade. We do this because, in the design of
practical adaptive receivers, it may be desirable to deal with cross-pol
interference and co-pol dispersion separately. For example, cross-pol
cancellation may be simpler to perform at IF, while multipath equali-
zation is best done at baseband. More important, modularity in pro- __
viding the cross-pol cancellation and multipath equalization functions .
- may permit more economy and flexibility in designing and using the
radio system.

Our primary aim is to show that, in using cross-pol cancellation
followed by multipath equalization, overall outage performance can be
made nearly as good as that for an equalized single-pol radio link. Our
secondary aim is to quantify the differences in attainable outage
performance for three distinct approaches to cancellation, namely, (1)
no cancellation; (2) ideal (i.e., total) cancellation; and (3) optimal
nondispersive cancellation. Finally, we aim to show how these results
are influenced by key parameters of the system and various assump-
tions regarding the radio propagation.

To obtain these quantitative results, we use a combination of re-
ceiver analysis and Monte Carlo simulation of the co-pol and cross-
pol responses of the propagation channel. The simulations require
specifying a model for the channel responses. We have done so, despite
the sparsity of published work in this area, by using data, tentative
theories and speculations by the author and others. Model uncertain-
ties are dealt with, in part, by means of sensitivity studies.

We consider digital radio links transmitting independent random
data streams on two nominally identical cofrequency channels using
nominally orthogonal polarizations. For reasons of both practicality
and convenience, we assume the polarizations to be vertical (V-pol)
and horizontal (H-pol). © |

The cofrequency channels are in the microwave common carrier
bands at 4, 6, and 11 GHz, corresponding to channel bandwidths,
respectively, of 20, 30, and 40 MHz. The modulation is M-ary Quad-
rature Amplitude Modulation (M-QAM), with M = 4, 16, 64, 256, etc.
The end-to-end spectral shaping (excluding channel dispersion and

TRANSMITTER
Fig. 1—Schematic representation of a dual-pol M-QAM transmitter with root-cosine-

roll-off spectral shaping and a microwave channel with co-pol dispersion and cross-pol
coupling. The H-functions are slowly time varying during periods of multipath activity.

adaptive filtering) is cosine roll-off, with roll-off factor a. This shaping
is divided evenly between transmitter and receiver. In the receiver,
the H-pol and V-pol signals are applied to a cross-pol canceler that
linearly combines the two input branches, followed by fixed filtering
and adaptive equalization in each output branch.

The above description is reflected in Figs. 1 and 2.* Figure 1 depicts
the transmitter and channel, the latter being characterizable by a pair
of co-pol responses [H,,(f) for the V-pol signal and H>.(f) for the H-
pol signal] and a pair of cross-coupling responses [Hj.(f) and Ho,(f)].
Ideally (and in fact under normal conditions), H,,(f) and Ho2(f) are
identical nondispersive responses. During multipath fading, however,
they become dispersive and possibly smaller as well, as recounted in
numerous papers on statistical models.?°
_ The cross-pol. responses, on the other hand, are ideally zero. In fact,
however, they are generally nonzero, even under normal conditions.
During such times, they are typically nondispersive and small com-
pared to the co-pol responses; but, during multipath fading, they tend
to be dispersive and can become relatively large as well,® giving rise to
_ potentially serious impairments in detection.

Figure 2 shows the structure of the receiver to be analyzed. All the
G-functions are adaptive, and we assume they can be controlled in
practice to satisfy the various criteria specified below. For convenience

* Note that f is referred to the center frequency of the radio channel, and that C(f)
represents the raised cosine function (dimensionless).

Fig. 2—Schematic representation of dual-pol receiver processing consisting of cross-
pol cancellation, root-cosine-roll-off filtering, and adaptive equalization. The G-func-
tions are assumed to be adapted to the prevailing channel response functions in Fig. 1.

In principle, the location of the fixed filtering is immaterial, and each of the various
linear stages can be at RF,IF or baseband, as appropriate.

only, all the processing stages are shown as passband (rather than
baseband) circuits. |

Finally, we assume in this study that the RF carriers and the symbol
timings are nominally identical but asynchronous. This assumption
enables the cross-pol interference to be treated as a noise-like process,
which simplifies the analysis in some respects. The results should
differ little from those for synchronized dual-pol transmissions.

To evaluate the canceler/equalizer structure, we need some analyt-
ical relationships. Let Sy(f) be the Fourier transform of a long random
sequence of (complex) data into the V-pol transmit filtering (Fig. 1),
and let Sx(f) be the same for the H-pol branch. Because of cross-
coupling in both the propagation medium and the receiver, each branch
will show a mixture of V-pol and H-pol data streams at its equalizer
output (Fig. 2). Because of symmetry, however, we need only consider
the output of the V-pol branch, and will follow that custom throughout.

The Fourier transform of the equivalent baseband signal at the V-
pol output is :

_'The second term clearly represents the cross-pol interference (hereafter

abbreviated CPI); the bracketed quantity of the first term represents
the changes in the desired signal due to the propagation medium,
canceler, and equalizer. If H,,(f) = Hoo(f) = 1 and Hi2(f) = Half) =
0, Sv(f) reduces to the case of an undistorted single-pol transmission.
We now state the three conditions on G,;(f) and Gj2(f) that will be
analyzed here:
1. No cancellation, 1.e.,

where gop is that complex value for which the mean square CPI at the
canceler output is minimized. There are other possible criteria for
defining opt (e.g., that value that minimizes the mean-square CPI at
the equalizer output), but the simple criterion used here is good enough
for our purposes.

Finally, we assume that the equalizer associated with each of the
above three conditions is an ideal Minimum Mean-Square Error
(MMSE) equalizer. That is, it maximizes the ratio of the signal half-
distance to the total root-mean-square (rms) distortion (thermal noise,
CPI, and intersymbol interference) as sampled at the baseband detec-
tor. Previous studies for single-pol channels have shown that the
performance of an ideal MMSE equalizer can be closely approached
using fractionally spaced tapped-delay-line filters with just a few taps.’

Obtaining an empirical statistical model for the four channel re-
sponse functions—Aj1(f), Hie(f), Hoi(f) and Hoo(f)—is a very difficult
business. The model to be used here is based only partly on measured
data, the other parts being theories on the underlying physics of the
propagation medium, and speculations on what mathematical artifacts
to include so that important issues are not overlooked. The model
draws on the published data, models or ideas of N. Amitay,’ W. D.
Rummler,”® K. T. Wu,° S. Lin,®? M. L. Steinberger? and M. Liniger,?°
as well as on private discussions with these investigators and others.
The author’s own ideas are included in the mixture, and he accepts
sole responsibility for flaws in the speculative model presented next.

The co-pol responses can be represented using the three-path func-
tion introduced by Rummler.’ For most of the year, H1,(f) = Hoo(f) =
_ 1; but, during 7 seconds per hop per year, the two functions become
Hilf) = ai{1 — bye’*'e~*"]; minimum phase response 5)

where 7» is a function of the radio path and band; 7 is a fixed parameter
of 6.3 ns; (ay), G22, b11, b22611, 622) are slowly varying “fitting” parameters
that collectively provide accurate approximations to the true re-
sponses; 0;; < 1 and bg < 1 at all times; and these representations
apply over bandwidths of 40 MHz or less.

Rummler® has published a widely used joint probability density
function (pdf) for the a-, b-, and ¢-parameters of a co-pol function
such as (5). We will use that pdf to characterize the statistics of both
Hy, (f) and H2o(f). Moreover, we assume each function to be minimum
phase over a fraction p of all fade events, with p a parameter lying
between 0 and 1. |

In evaluating the dual-pol receiver, it will be necessary to assume
something about the similarity in time between H,,(f) and Ho2(f).
According to a limited body of data, these functions tend to track
quite closely, i.e., Hio(f) ~ Hi(f) during most fade events. On the
other hand, significant impairments could accrue during those events
where they are dissimilar. To bracket the range of possibilities, we
have obtained simulation results under two distinct assumptions:
Aoo(f) = Hii(f) in every fade event; and Hi,(f) and Hoo(f) are statis-
tically identical but mutually independent over the ensemble of fade
events.

Hi; (f) = {0.5 ky[Hi(f) + Ao2(f)] + ejeVi} eet (6)
Lx]
where k;; is a fixed complex constant related to the residual cross-pol
coupling in the radio antennas and waveguide runs (typically, 20
log| ky | < —25 dB); Yj is a uniformly random phase over the ensemble
of fade events; «; is a Rayleigh variate over that ensemble (typically,
10 log| «; |? < —35 dB); and 6T is a fixed time “misalignment” between
the co-pol and cross-pol paths to the receiver. We further assume that

This model for the cross-pol responses is more or less consistent |
with data and theories reported previously.®* © The delay parameter
dT is new and is included to add richness to the model; we will see if
6T has any important effects on performance as it ranges from 0 to 4
ns. This way of including delay effects may, in fact, be too mild.
Amitay’ has suggested that 67' might be larger than 4 ns, and that the
factor e’*? might more properly be attached to just ¢,e’*%. Such an >
approach would make the cross-pol responses more dispersive, con-
sistent with some reported measurements. For now, however, we will
use the model for cross-pol responses described above.

To begin, we note that an M- L-QAM signal contains two /M-level |
AM signals in phase quadrature, and that the possible data values in
each quadrature rail are +1, +3, --- + (VM — 1). We also note that,
after cross-pol cancellation, fixed filtering, adaptive equalization, and
coherent demodulation (in whatever order), two baseband pulse
streams are sampled every T seconds at each of two data detectors in
the V-pol receiver branch, and similarly for the H-pol receiver branch.
Our analytical goal is to derive a signal-to-distortion ratio that char-
acterizes performance at either baseband detector of either receiver
branch. To the maximum extent possible, we invoke known relation-
ships and any simplifying assumptions that do not compromise the
generality of the results.

One simplifying assumption, noted and explained in Section 2.1, is
that the cross-pol interference can be treated as a noise-like process.
Another simplification accrues by assuming a zero-percent roll-off
factor (a = 0) for the end-to-end spectral shaping, in which case C(f)
is a unit rectangle on the f-interval [—1/2T, 1/27]. This simplifying
pee an is justified by earlier findings that, over the practical range

Finally, for convenience we ere a carrier-to-noise ratio as follows:
let Po be the average received power in an M-QAM signal in the
absence of fading; and let No be the power spectrum density of the

* The effects of the phases of ki2 and ky, and of any dissimilarities between these
two constants, should be small. We thus remove them from the study to simplify
matters.

noise input to the V-pol (or H-pol) receiver, including the contribution
of the receiver noise figure. Then

CNR 4 PoT/No. (7)
This is the unfaded Carrier-to-Noise Ratio (CNR) in the Nyquist
bandwidth, and is typically on the order of 10° (60 dB).
3.2 Signal and distortions at the canceler output —

From previous discussions and Figs. 1 and 2, we can show that the
data pulse at the canceler output for a data value of unity would have
a Fourier transform

Similarly, the thermal noise at that output has a power spectrum
density

where (see Fig. 2) |
B,(f) = [| Gulf) |? + | Gie(f) | 71. (11)
The cross-pol interference has a power spectrum density
X(f) = PoTC(f)B.Af), (12)

The composite noise power spectrum density* can then be written as
N'(f) =N(f) + X(f) |
= Nol B,(f) + CNR C(f)B.(f)I- (14)

Henceforth, we will make use of our assumption that a = 0, ie., that
C(f) is a unit rectangle on [—1/2T, 1/2T)].

* This spectrum density is for the sum of CPI and thermal noise, which can be treated
as noise-like but is clearly not Gaussian except when the CPI vanishes.

baseband detector, the ratio of sampled-signal half-distance to rms
distortion, the latter consisting of thermal noise, CPI, and Intersymbol
Interference (ISI). We need not concern ourselves with how such an
equalizer is realized, as this is a well-developed art; we merely need to
derive and analyze its frequency response.

The derivation is accomplished in two steps. First, we assume a
noise-whitening input filter, 1.e., one having a real transfer function
proportional to 1/VN’(f), and then we invoke the result in Section
5.1 of Ref. 12 for MMSE equalizers with white input noise. The
equalizer response for a = 0 turns out to be

The data pulse following an seus lick with this response has a real,
nonnegative Fourier transform. Assuming optimal carrier and timing
recovery, we can show that the squared half distance between adjacent
signal samples (constellation points) at the detector is therefore

where the integration limits, both here and in subsequent expressions,
are +1/2T. |

The mean-square intersymbol interference at the sample times can
likewise be shown to be

The signal-to-distortion ratio (p) at either baseband detector is now
defined to be

The significance of P is a the Bit Error Rate (BER) can be tightly
upperbounded by!®

By using the various relationships in Sections 3.1 through 3.3, we can
express p in the form

The focus of our investigation is the signal-to-distortion parameter —
T introduced by (21). To “calibrate” this quantity, let Io denote the
value of I’ that must be exceeded if BER is to lie below some threshold
value BERo. We can find a tight upperbound for Ip given BER» and
M by invoking the equality in (20) and combining it with (21). The
result is :

Some values of Io, expressed in decibles, are given in Table I for.
various BER» and M of interest.

Although the above results are applicable to any assumptions we
make about the canceler, two special cases are worthy of note. The

_ first is the case of total cancellation [see (3)]. Under this condition,
X(f) in (12) vanishes entirely and so N’(f) reduces to N(f). Also, as
the mathematics of this situation makes clear, the choices of Gi;(f)
and G,.(f) individually are immaterial, just so their ratio satisfies (3).
‘This is a consequence of the (presumed) fact that the dominant
thermal noise is introduced before the canceler. |

The second special case is that of nondispersive cancellation [see
(4)]. We assume that g.,: is chosen so as to minimize X(f) [see (12)
and (13)]. For a zero-percent roll-off factor, it is easy to show that

‘For convenience in what follows, we now redefine two ypereinelers
of the channel model, namely, |

We can say that the channel is statistically specified once we (1)
assign numerical values to p, 6T, K and E; and (2) declare H,,(f) and
H2.(f) to be either identical or statistically independent. Similarly, we
can say that the radio system is design-specified once we (1) assign
numerical values to a, CNR, and 1/T; and (2) declare the type of
cross-pol cancellation to be used.

A computer program has been written that apeiae for any joint
specification of the channel and system, the yearly probability distri-
bution of I’. To accomplish this, the program combines Monte Carlo
simulation methods with the channel model of Section 2.3 to generate
a large statistical “ensemble” of channel response functions. That is,
each member of the ensemble is a set of functions, {Hii(f), Hio(f),
Half), Hoo(f)}, generated by deriving the various parameter values
(Qy1, 011, Ge2, bee, €12, €21, etc.) in accordance with the statistics of the
model. For each set of H-functions thus generated, the program
computes I’ using the formulas of Section 3.4. After generating and
evaluating the prescribed number of sets (20,000 in our study), the
program computes a cumulative probability function, P(T), for the
population of I'-values thus obtained.

We present results for the following channel/system parameter
values or conditions, where those in boldface are the ones used the
most extensively:

Statistical dependence between co-pol functions: Totally depend-
ent (Hi:(f) = Hee(f)) and totally independent. |

Type of cancellation: No cancellation, total cancellation, and
nondispersive cancellation. |

Symbol rate, 1/T: 15 Mbaud, 22.5 Mbaud, and 30 Mbaud (typical
values for digital radio systems in the 4-, 6-, and 11-GHz radio
bands, respectively).

Throughout our simulations and computations, we have assumed a
0-percent roll-off factor (a = 0) and a 63-dB unfaded carrier-to-noise
ratio (CNR = 2 X 10°). However, we will also discuss how the computed
results would vary with these parameters.

The results of this study are given by Figs. 3 through 6. Each figure
shows curves of P(I) versus I for a number of different channel/
system specifications. These curves can be interpreted as conditional
outage probabilities and can be used to estimate yearly outage seconds
on a radio hop, as follows: Given a threshold bit error rate (BERo)
and the number of modulation levels (M), the minimum acceptable
value of I can be obtained using (27) or Table I. (Assuming BERo =
107*, To is roughly 20 dB for M = 16 and 26 dB for M = 64.) The
resulting P(IT) is the probability of outage on a given hop conditioned
on the occurrence of fading. To estimate yearly outage seconds on the
hop, one would need to estimate, measure, or obtain from available
models the expected number of yearly fading seconds, 7». (A repre-
sentative value is 16,000 seconds.) Multiplying Ty) by P(I'o) would
yield the expected number of outage seconds per hop per year. This
utilitarian aspect of the curves in Figs. 3 through 6 should be kept in
mind as we make some relative comparisons.

Figure 3 shows P(T) for each of three symbol rates and each of
three cancellation options. The “common conditions” listed on the
figure are assumed to be the most representative for an actual dual-
pol channel.

The top curve shows what can be expected if no cancellation
whatsoever is employed. The heavy line used here contains the results
for all symbol rates from 15 Mbaud to 30 Mbaud. It is clear that the
absence of some sort of cancellation is unthinkable for the dual-pol

Fig. 3—Cumulative probability functions for I. Results are shown for each of the
three canceller options considered and for symbol rates of 15, 22.5, and 30 Mbaud. In
addition to the “common conditions” shown, a = 0 and CNR = 2 X 10° (63 dB).

systems of interest, and we dismiss this option from further consid-
eration. |

The results for nondispersive cancellation reveal an order-of-mag-
nitude improvement (for the lower values of I’) and a fairly strong

dependence on symbol rate. The latter is not surprising since larger
symbol rates involve larger bandwidths and, hence, greater dispersion
in the channel H-functions. In microwave common carrier channels
with narrow bandwidths (e.g., 3.6 MHz in the 2-GHz band), nondis-
persive cancellation might thus be quite adequate. This would depend,
of course, on the number of modulation levels and the MOutage proba-
bility requirements.

The narrow band at the bottom, for the case of total cancellation,
contains results for all symbol rates between 15 Mbaud and 30 Mbaud.
This approach provides, at the lower values of I, another order-of-
magnitude improvement beyond nondispersive cancellation. More-
over, the dependence on symbol rate (or bandwidth) is seen to be
small. Later, we will compare these results with those for single-pol
transmission using ideal MMSE equalization.

Figure 4 shows results for both nondispersive and total cancellation
as p and 67’ range over the sets of values we have specified for them.
The “common conditions” assumed here are typical ones; alternative
realistic assumptions would not alter the trends revealed by these
curves. |

The main conclusion we can draw here is that outage performance
for nondispersive cancellation is sensitive to the details of the channel
model, while that for total cancellation is not. This is reflected in the
wideness and narrowness, respectively, of the bands for these two
cases. Since total cancellation is the obvious design choice for a high-
quality system, this is good news. It implies that certain hard-to-
determine fine details of the channel model need not be accurately
specified to obtain reliable performance estimates.

Figure 5 illustrates, for both nondispersive and total cancellation, |
how performance is affected by the statistical dependence between
Ay, (f) and Hoo(f). The performance variation over the range between
total dependence [Hi;(f) = Hoo(f)] and total independence is seen to
be fairly small. Nevertheless, it would be clearly beneficial if the co-
pol responses were more independent than they apparently are. |

A simple explanation can be given for the improvement shown when
Hy,(f) and H2(f) are independent. Let us consider the case of total
cancellation and assume, for simplicity, that all H-functions are essen-
tially flat with frequency. Combining (10) and (11) with (3) and (1)
under these circumstances, we can show that the receiver output

Fig. 4—Cumulative probability functions for I. Results are shown for nondispersive
and total cancellation, with the symbol rate fixed at 22.6 Mbaud. The parameters here
are the minimum phase probability (p) and the delay parameter (67). All else is the
same as in Fig. 3.

signal-to-noise ratio would be proportional to | Hy,;He2 — Hy.H2, |7/
[| Hoo|° + | Hie|?]. The potential benefit of statistical independence
arises when | H;,| is weak; at such times, the numerator can be quite
small if | H22| is similarly weak, via near cancellation of Hy,H2. by

20 25 30 35 40
TIN DECIBELS
Fig. 5—Cumulative probability functions for [. Same conditions as in Fig. 4, except

that p and 6T are fixed at 0.5 and 2 ns, and results are shown for both totally dependent
and totally independent co-pol response functions.

Hy2H>,. If Hi, and He are independent, however, there is a chance
that Hoo will be strong enough at such times to avoid this near-
cancellation. The statistical effect of all this is reflected in the com-
parative results of Fig. 5. |

Figure 6 shows results for both nondispersive and total cancellation
as K and E range over the sets of values we have specified for them.
Note that these particular results are for 1/7 = 22.5 Mbaud and H,(f)
— Hof ).

For nondispersive cancellation, each heavy line contains results for
a particular E and all K below —25 dB. The negligible influence of K
over this range is apparent. However, FE is another matter. As this
quantity decreases in 5-dB steps from —35 dB, the probability curves
shift to the right by nearly 5 dB. These features reflect the fact that
nondispersive cancellation virtually eliminates the effect of the cross-
coupling gain component kHi;(f) in (6) [remember that H.(f) =
H,,(f) in these calculations] but is less effective against the second
component, whose mean-square value in decibels is E. This reveals
yet another way in which precision in the channel model is needed to
estimate nondispersive canceller performance.

For the case of total cancellation, on the other hand, the lower band
in Fig. 6 shows how much that need is attenuated. Even so, this band
widens measurably as F decreases below —45 dB. The point is made
clear by the dotted curve, for K = E = —o, which is equivalent to the
case of single-pol transmission (i.e., no input CPI) and ideal MMSE
equalization.

We now see how closely dual-pol outage performance comes to that
for single-pol operation when total cross-pol cancellation is used. We
can also make a limited comparison between the cascaded approach
(i.e., canceller and equalizer in tandem) and the optimum linear
receiver (i.e., jointly optimizing {Gi,(f), Gie(f), Golf), Goo(f)} against
both CPI and multipath dispersion, thereby eliminating the separate
equalizer). The latter case is treated comprehensively by Amitay and
Salz in Ref. 1. In that paper, the probability functions are plotted
against spectral efficiency, in b/s/Hz, and so the following correspond-
ences apply: the abscissa values of 4 and 6 b/s/Hz in Ref. 1 correspond
closely to I = 20 and 26 dB, respectively, in the present paper. Now
assuming that E = —35 dB and 1/T = 30 Mbaud, Fig. 3 of Ref. 1
shows the outage probability for the optimal linear receiver to be
roughly four times higher than for single-pol transmission when the
spectral efficiency is 4 b/s/Hz, and roughly two times higher when the
spectral efficiency is 6 b/s/Hz. The corresponding results in the
present study for ! = 20 and 26 dB are quite similar and, for higher
abscissa values, the similarities are even greater. While recognizing
that the two studies used somewhat different models and methods of
analysis, and entirely different random numbers in their Monte Carlo

20 25 30 , “80:34 40
TIN DECIBELS
Fig. 6—Cumulative probability functions for T. Same conditions as in Fig. 4, except

that p and 6T are fixed at 0.5 and 2:ns, and K and E are parameters. Note the dotted -
curve “No CPI,” which corresponds to an ideally equalized single-pol channel.

simulations, we perceive a general truth in this comparison: specifi-
cally, the cascaded approach, in which cancellation and equalization
are functionally separate, leads to outage statistics very close to those
for optimal linear reception. | |

All of the results in Figs. 3 through 6 are for a = 0 and CNR = 2 X
10° (63 dB). Nevertheless, we can say something about the influences
of these parameters. From previous studies,!! for example, we know
that a has a very small effect on outage probability over the practical
range 0 < a < 0.5. Also, the curves for total cancellation would
essentially shift X dB to the right (or left) for every X-dB increase (or
decrease) in CNR. This is because the residual distortion in the case
of total cancellation followed by MMSK equalization is almost entirely
thermal noise. In the case of nondispersive cancellation, wherein the
dominant residual distortion is uncancelled CPI, this sensitivity of the
results to CNR would be sharply reduced.

We have used analysis and Monte Carlo simulation to estimate
conditional outage probabilities in dual-pol digital radio as functions
of a detection measure (I) that can be related to the number of
modulation levels and the bit error rate. In performing the simulations,
we have resorted to a statistical model for the dual-pol channel that
lacks a firm empirical basis. This results unavoidably from the current
incomplete status of dual-pol channel measurement and modeling. We
have dealt with this limitation, in part, by treating various uncertain
aspects of the model parametrically.

e Obtaining reliable estimates of outage probability for the case of
nondispersive cancellation requires accurate, detailed descriptions
of the underlying channel model. The outage performance of this
cancellation approach is also quite sensitive to bandwidth (or
symbol rate).

e In the case of total cancellation, by contrast, outage performance
is insensitive to many details of the channel model as well as to
symbol rate. |

e While far superior to no cancellation, nondispersive cancellation
leads to outage probabilities an order-of-magnitude greater than
does total cancellation. Nonetheless, it may find applications
where symbol rates are low (less than 5 Mbaud) and the outage
requirements are liberal.

e The outage statistics for total cancellation followed by ideal equal-
ization are fairly close to those for single-pol transmission,
wherein there is no cross-pol interference to cancel.

e More significantly, total cancellation in cascade with ideal equal-
ization appears to produce outage statistics very close to those for
optimal linear reception, wherein the effects of cross-pol interfer-

ence and multipath dispersion are jointly minimized in the same
receiver stage. Use of the cascade approach, therefore, may permit
such benefits as design simplicity, manufacturing economy, and
operational flexibility with no serious loss in performance.

I am grateful to Lisa J. (Domenico) Case for her help i in executing
the computer simulation/analysis programs.

1. N. Amitay and J. Salz, “Linear Equalization Theory in Digital Data Transmission
Over Dually Polarized Fading Radio Channels,” AT&T Bell Lab. Tech. J., 63,
No. 10, Part 1 (December 1984), pp. 2215-59.

W. D. Rummler, “A New Selective Fading Model: Application to Propagation Data,”
B.S.T.J., 58, No. 7 (May-June 1979), pp. 1037-71.

W. D. Rummler, “More on the Multipath Fading Channel Model,” IEEE Trans.
Commun., COM-29, No. 3 (March 1981), pp. 346-52.

L. J. Greenstein and B. A. Czekaj, “A Polynomial Model for Multipath Fading
Channel Responses,” B.S.TW., 59, No. 7 (September 1980), pp. 1197-225.

J. C. Campbell and R. P. Coutts, ‘ ‘Outage Prediction of Digital Radio Systems,”
Electron. Lett., 18, No. 25/26 (December 1982), pp. 1071-2.

K. T. Wu, “Measured Statistics of Multipath Dispersion of Cross Polarization
Interference,” Paper 46.3, Int. Conf. Commun., May 14-17, 1984, Amsterdam.
N. Amitay and L. J. Greenstein, “Multipath Outage Performance of Digital Radio

Receivers Using Finite-Tap Adaptive Equalizers,” IEEE Trans. Commun., COM-
32, No. 5 (May 1984), pp. 597-608.
8. S. H. Lin, “Impact of Microwave Depolarization During Multipath Fading on Digital
Radio Performance, ” B.S.T.J., 56, No. 5 (May 1977), pp. 645-74.
9. M. L. Steinberger, “Design of a Terrestrial Cross Pol Canceller, ” Paper 2B.6, Int.
Conf. Commun., June 13-17, 1982, Philadelphia, Pa.

10. M. Liniger, “Sweep Measurements of Multipath Effects on. Cross-Polarized RF-
Channels Including Space Diversity,” Paper 45.7, GLOBECOM ’84, November
26-29, 1984, Atlanta, Ga.

11. W. C. Wong and L. J. Greenstein, “Multipath Fading Models and Adoptive Equal-
izers in Microwave Digital Radio,” IKEE Trans. Commun., COM-32, No. 8
(August 1984), pp. 928-34.

12. R. W. Lucky, J. Salz, and E. J. Weldon, Jr., Principles of Data Communication, New
York: McGraw- Hill, 1968.

13. G. J. Foschini and J. Salz, “Digital Communications Over Fading Radio Channels,”
B.S.T.J., 62, No. 2, Part 1 (February 1983), pp. 429-56.

A Laboratory Simulation Facility for Multipath
Fading Microwave Radio Channels

This paper describes a laboratory facility capable of simulating time-varying
radio multipath channel responses in real time under computer control. Four
independent fading channels are available that can be used for single-polari-
zation nondiversity, combined in pairs for single-polarization dual diversity,
or cross-coupled to simulate the two outputs of a dual-polarization nondivers-
ity channel. Each channel contains a controllable variable network capable of
producing a narrowband intermediate frequency response that resembles the
“three-path” function of Rummler. A wide range of models can be accommo-
dated by altering the computer-stored sequences used to control each variable
channel network. The only assumption implicit in the choice of model is that
the channel response can be fitted to the generic function over bandwidths up
to 40 MHz. The channel responses are controlled by either entering fixed
parameters from a keyboard, or by reading time-varying parameters stored in
disk memory. This description includes the architecture, hardware design,
software implementation, and performance of the simulation facility.

The primary impediment to the operation of digital radio on micro-
wave line-of-sight paths is multipath fading. In dual-polarization

(dual-pol) systems, this problem is augmented by cross-polarization
(cross-pol) coupling. Numerous radio measurements and data analyses
over the past several years have served to characterize multipath and
cross-pol responses, and to translate these characterizations into sta-
tistical models.’!’ Many of these models have been eagerly engaged
by radio systems analysts to estimate, using analysis and/or Monte
Carlo simulation, the expected link outage times for different modu-
lations, link parameters and receiver techniques.’***? Over the same
period, a number of outage measurements have been reported for
specific hardware designs, based on either field trials conducted over
radio paths or laboratory “signature” measurements coupled with
assumed radio channel models.??*!

If one stands back from this myriad of activities, important limita-
tions in all of the above approaches become apparent. The analysis/
simulation of statistical models applied to specified designs permits
rapid comparisons among contending radio schemes, but relies on
idealized models of the hardware behavior. Moreover, such study
methods do not readily take account of the time dynamics of the
channel responses. The coupling of statistical models with lab-mea-
sured hardware “signatures” likewise omits channel time dynamics
and their effects on system techniques. This deficiency is absent only
in the case of field measurements, but these are both costly and time-
consuming. More important, system qualities inferred from this ap-
proach are subject to the particular responses that nature provides
during the test interval. Meaningful comparisons between systems,
therefore, require measurements conducted in parallel under identical
conditions of path and time.

The Channel Simulation Facility (CSF) reported here is intended
to fill the gaps between these various study approaches. It simulates
time-varying radio channel responses in the laboratory in real time,
and it plays the channel responses into actual hardware realizations
rather than idealized system models. The channel responses are dic-
tated by computer-generated control signals, and so the twin benefits
of model selectability (software-controllable changes in the history of
the channel response functions) and repeatability (ability to replicate
the channel response history for different systems at different times)
are realized. And, finally, the ability to simulate channels in a labo-
ratory can sharply contract the time needed to cover a “fading year”
and permit considerable reductions in cost.

The heart of the CSF is a group of four identical, variable channel
networks, each of which produces, under computer control, a narrow-

band Intermediate Frequency (IF) response that resembles the “three-
path” function of Rummler.’ That function is commonly expressed in
the literature as

where a, 6b and ¢ are slowly varying random “fade parameters,” and
w(= 2rf) is measured from the selected intermediate frequency. For
later convenience, the following parameters are defined:

Thus each variable network has a response about the IF (either 70
MHz or 1.070 GHz, as selected) given by

where a and c are computer-controlled attenuations and @ is a com-.
puter-controlled phase shift. In terms of (1), the hardware variables c
and @ are ab and ¢ + 7a, respectively. For later convenience, we define

The four variable channel networks are physically paired, i.e., two
such networks with a common input but separate outputs are con-
tained in each of two identical Dual Channel Units (DCUs). The four
networks can be used in three distinct ways: (1) A single network (any
of the four) can be used to produce the output of a single-pol nondi-
versity channel, e.g., see Fig. 1a; (2) two networks within the same
DCU can be used to produce the two outputs of a single-pol dual
diversity channel, e.g., see Fig. 1b; or (3) all four networks can be
combined, using a Cross-Coupling Unit (CCU), to produce the two
outputs of a dual-pol nondiversity channel, e.g., see Fig. 2.

We thus see that a total of three units comprise the CSF, permitting
either of three modes of use at either of two IF's. These and other
features of the CSF are summarized in Table I.

Two important facts about the CSF are important to emphasize.
One is that the responses of the four networks are completely and
separately controllable, and can be driven either via specified hardware
parameters typed into a computer terminal (dial-up responses, which
are fixed until the input parameters are changed); or via software
control, whereby the electronically adjustable hardware parameters
are time varied by reading stored sequences from a disk and applying
Digital-To-Analog (D/A) conversion and low-pass filtering. In the
latter case, the sequences are produced by software routines tailored
to a specified statistical channel model. A wide range of models can

Fig. 1—Simulation modes of (a) a single-pol channel nondiversity and (b) a dual
space diversity.

thus be accommodated by suitably altering the software routines. This
feature is the key to the flexibility and simplicity of use of the CSF.

The second important fact is that designing the variable networks
about the “three-path” function is not restrictive with respect to
permissible channel models. The only assumption implicit in this
approach is that all channel response functions of interest can be well
fitted by this generic function. For the vast majority of radio paths
that have been measured, the evidence supports this assumption for
channel bandwidths up to 40 MHz. This means that if a given user of
the CSF wishes to consider a model with a different fitting function
(e.g., the first-degree polynomial® or the “two-path” function’), soft-
ware can be written to interface this model with the simulator hard-
ware.

Section II describes the architecture of the CSF, showing block
diagrams of the three main units and discussing how they are used to
provide the various possible modes of operation. Section III gives a
more detailed description of the electronic circuits and components,
while Section IV gives details on the construction. Section V discusses
the software associated with the CSF. Specific topics include the
generation of the sequences that drive the variable hardware parame-
ters, and the calibration and measurement of the variable networks.
The existing software for generating the sequences is tailored to a
specific dual-diversity channel model. Section VI presents perform-

Table I—Definitions and some features of the channel
simulation facility

Capabilities Simulates either a single-pol nondiversity channel
(1 input, 1 output); a single-pol diversity channel
(1 input, 2 outputs); or a dual-pol nondiversity chan-
nel (2 inputs, 2 outputs).

Physical configuration Consists of two identical dual channel units (DCUs)
and one cross-connecting unit (CCU), plus connect-
ing cables.

DCU (Multipath fade simula- Contains two parallel networks (common input, sep-
tor arate outputs), each producing a separate, variable
“three-path” frequency response.

CCU (“Cross-pol coupler”) Connects the four outputs of two DCUs so as to
simulate a dual-pol nondiversity channel.
Variability of the network Computer-controlled; either keyboard entry fixed re-
responses sponses or program-generated time-varying re-
sponses.
Input IF 70 MHz
Input power level 0 dBm
Output IF 70 MHz or 1.070 GHz, as selected.
Output power 0 dBm for each variable channel network at 70 MHz.
~15 dBm for each variable channel network at 1.070
GHz.
Bandwidth Each network provides a “three-path” response over

ance results. This includes specific data on bandwidths, power levels
and calibration stability; assessments of how well the computer-gen-
erated sequences satisfy the underlying statistical model; and assess-
ments of how well the hardware response variations match the in-
tended ones, i.e., those dictated by the computer-generated outputs.

Figure 3 shows, in simplified form, the block diagram of a DCU. A
DCU contains two parallel networks with a common IF input, the top
network being labeled Horn Channel and the bottom are being labeled
Dish Channel. These labels are particularly apt when the two networks
simulate the two paths of a dual space diversity link. For convenience,
we will use these designations throughout our discussions to distin-
guish the top and bottom networks. .

An actual DCU contains circuitry not depicted in Fig. 3, including
bandlimiting IF filtering, upconversions from 70 MHz to 1.070 GHz,
and the provision of the variable phase shifts (@) via local oscillators
and mixers. This circuitry will be discussed in Section II.

A control computer is used to control, through D/A conversion and
low-pass filtering, each of the circuit parameter a, c and 0. It is thus

clear that each of the two networks provides a frequency response
identical in form to (4).

Finally, Fig. 3 shows the manner in which the intermediate fre-
quency of the network outputs is selected. When the external con-
necting cables (shown dashed) are absent, the two outputs are at 1.070
GHz. When the cables are connected as shown, each 1.070-GHz output
is applied to a separate downconverter, with a shared 1.0-GHz Local
Oscillator (LO) providing the other input. In this case, the outputs
will be at 70 MHz, the IF of the DCU input.

Figure 1 shows two obvious ways to use a single DCU. In Fig. 1a,
the control computer delivers fade parameters (A, C and @) to just the
Horn Channel network, and the output of just that network is applied
to the follow-on equipment. Alternatively, one could control, and use
the output from, just the Dish Channel. In either case, the simulator
is used in this way to represent a single-pol nondiversity channel.

In Fig. 1b, the control computer delivers fade parameters to both
the Horn and Dish Channel networks, and both outputs are used. In
this case, the simulator can be configured as a dual space diversity

channel. On a typical space diversity link, the primary (upper) and
secondary (lower) receiving antennas on the radio tower would be of
the horn and dish type, respectively, thus leading to the designations
used here. |

Figure 4 shows how the CCU combines four inputs to produce the
simulated V-pol and H-pol outputs of a dual-pol channel: the co-pol
V and cross-pol H signals are combined in a 10-dB directional coupler,
following a manually set attenuation and phase shift of the latter, to
produce the V-pol receiver input. Similar combining produces the
H-pol receiver input. |

The manual attenuation adjustments in the CCU, achieved via.
panel-mounted controls, extend the dynamic range over which the
cross-pol coupling levels can be varied. Each attenuator can be incre-
mented in 1-dB steps from 0 dB to 43 dB.

The manual phase shift adjustments, also achieved via panel-
mounted controls, provide flexibility in how the co-pol and cross-pol
signals from the same original transmission are relatively phased. The
phase adjustment is continuous over a range of nearly 360 degrees.

Each variable channel network of a DCU contains a direct path
with a variable (analog voltage controlled) attenuator, and a parallel
path with a 6.3-ns delay, a variable attenuator and a variable (analog
voltage controlled) phase shifter. The summed outputs of these paths
yield a network response akin to the “three path” function of Rumm-
ler. Each of the parallel paths must be nondispersive, in fact, over a
bandwidth of 40 MHz or more. Further, it should be possible to vary
each attenuator without introducing phase changes and to vary the
phase shifter without introducing attenuation changes.

‘An input frequency of 70 MHz and a channel bandwidth of 40 MHz
were chosen for the CSF. The variable channel networks could be
built at this or any other frequency if ideal components were available.
However, given the limitations of practical components, we chose the
approach of converting the input signal frequency to a higher fre-
quency, where narrowband circuit techniques can be used. This ap-
proach, moreover, simplifies the task of providing the variable phase
shift, as we shall see.

A network frequency of 1.070 GHz was chosen. This choice trades
off the availability of “phase shift free” narrowband attenuators, and

ease of filtering of unwanted mixing products and local oscillator
leakage through the mixer. The network output at 1.070 GHz can be
either downconverted back to 70 MHz, used directly, or easily upcon-
verted to, say, 4, 6, or 11 GHz for radio equipment tests.

A simplified block diagram of a single variable channel network is
shown in Fig. 5. The input signal at 70 MHz (Point 1) is power divided —
into two paths. One component is upconverted in a double-balanced
diode mixer to 1.070 GHz; the other is delayed and then similarly
upconverted to 1.070 GHz by a second double-balanced diode mixer.
The output from the first mixer is controlled in amplitude by a variable
attenuator and provides one input to a signal combiner. The output
from the second mixer (in the delayed path) is controlled in amplitude
by a second attenuator and in relative phase by the local oscillator
phase shifter. This delayed signal sums with the direct input in the
output signal combiner.

The 6.38-ns delay is provided by adding additional coaxial cable in
the 70-MHz signal path between the input power divider and the
delayed path upconverter. The exact length of this cable is determined
empirically by measuring the signal delay difference at the output
signal combiner (points 2 and 3). This is most easily done in the
frequency domain by sweeping the network input signal, setting the

direct path and delayed path attenuations to be about equal (to form
periodic deep fades), and then measuring the frequency spacing be-
tween successive nulls. When the path difference is 6.3 ns, the null
spacing is 158.73 MHz, which is the reciprocal of the delay.

The output of the variable channel network contains both the upper
and lower mixing products at 1.070 GHz and 0.930 GHz. If the 1.070-
GHz signal is to be mixed back to 70 MHz, the contribution from the
0.930-GHz signal will distort the desired channel response and must
be effectively eliminated by filtering.

A detailed block diagram of a DCU is shown in Fig. 6. This diagram
shows the two variable channel networks driven by a common 70-
MHz input signal and the two downconverters for mixing the simulator
outputs back to the input frequency. A crystal-controlled phase-locked
Continuous Wave (CW) source at 1 GHz acts as a common local
oscillator to all upconverters and downconverters, to ensure frequency
coherence between the input signal and both output signals. When
both DCUs are used in the complete CSF, they are virtually identical
except for the provision to drive all four variable channel networks
from a common local oscillator residing in one DCU (master). This
higher power source in the master DCU can drive the second DCU
(slave). The result is four frequency-coherent outputs, assuming the
input signals to the two DCUs are themselves frequency coherent. (As
described in Section 2.2, the two input signals to the DCUs may or
may not be frequency coherent, depending on the experiment.)

Each variable network uses two local oscillator signals for upcon-
version. One is derived through power dividers from the 1-GHz source;
the other passes first through a voltage-controlled phase shifter. The
phase shifter varies the phase of the 1-GHz CW signal from 0 degrees
to over 360 degrees, when the @ control signal varies between 0 and
+10 volts. This phase is imparted to the signal in the delayed path of
the variable network by the mixing process of the upconverter.

The signals are scaled in the direct and delayed paths of each
variable channel network by voltage-controlled attenuators. The at-
tenuations are variable from 0 dB to over 36 dB when the A and C
control signals vary between 0 and +10 volts. The outputs from the
attenuators are added in a reactive signal combiner. The combined
signal is bandpass filtered to eliminate the lower mixing product at
0.930 GHz. The desired output from the filter at 1.070 GHz is amplified
to provide the final fading channel output.

The output filter, a fifth-order Butterworth filter with a 3-dB
bandwidth of 70 MHz, is the most narrowband circuit of each variable
network. This filter adequately attenuates the unwanted mixing prod-

uct while maintaining a flat amplitude response with small time delay
variation (+1.5 ns) over the design bandwidth of 40 MHz.

to prevent reflections from mismatches that produce unwanted ripples

in the channel response. Fixed coaxial attenuators and ferrite isolators

As we have seen, the four attenuators and two phase shifters within
each DCU are controlled by time-varying analog voltages, the latter
being derived from digital sequences supplied by the control computer.
To accomplish this, two D/A converters mounted within the computer
provide six control voltage sequences to each DCU. Each sequence
consists of 12-bit words delivered at a 1-Hz rate. Each sequence is
converted into a smooth analog variation by a fourth-order Butter-
worth state variable low-pass filter with a 3-dB bandwidth of 0.5 Hz.

line (.141 inch) was used to interconnect components to improve phase
stability. Line lengths in both the direct and delay paths were closely
matched in each variable channel network of each DCU, so as to
obtain nearly identical characteristics. The elongated coils in the
center of the figure are the additional line lengths required to produce
the 6.3-ns path delays.

8. The input and output signal ports are available on the panel for
interconnection convenience. The 70-MHz input port and the 70-MHz
downconverter output ports are the three BNC-type coaxial connec-
tors in the lower center of the panel. The 1.070-GHz outputs from the
Horn and Dish variable channel networks are brought out on N-type
coaxial connectors located at the upper left and upper right sides of
the panel. The inputs to the two downconverters associated with the
Horn and Dish channels are located directly below these N-type
connectors.
_ Screw-trimmer adjustments, paired in 12 holes in the upper center
left of the panel, are provided for shifting and scaling the control
voltages to the four variable attenuators and two phase shifters. The
range of these control voltages is approximately preset by adjustments
on the D/A converter boards within the control computer. The front
panel adjustments allow for convenient trimming at the DCU. The
panel meter can be selectively switched to monitor each of these
voltages.

Figures 9 and 10 show a top view of the inside and the front panel
of the cross-coupling unit. This relatively simple unit contains the two

cross-pol phase shifters (large rectangular devices in the center), the
two cross-pol attenuators (cylindrical devices adjacent to the phase
shifters) and two 10-dB directional couplers.

The front view of the complete CSF is shown in Fig. 11. The cable
interconnections shown represent the cross-coupled case described
in Section 2.2. The lower DCU can be arbitrarily called the
V-pol channel and the upper DCU, the H-pol channel. The V-pol
Horn Channel output is connected to the left cross-pol coupler Main
input. The H-pol Horn Channel output is connected to the left cross-
pol coupler X-Pol input, to be attenuated and phase-shifted before
summing to the main V-pol signal. The dish and horn outputs from
the H-pol and V-pol channels, respectively, are similarly combined in
the right cross-pol coupler. The cross-pol coupler outputs are shown
returned to the respective downconverters in the DCUs to provide 70-
MHz outputs. If desired, instead, the 1.070-GHz outputs from the
coupler can be used directly.

V. SOFTWARE DESIGN
This section deals with two distinct aspects of the CSF software.

One is the computer routines that generate random time sequences
for parameters of the specified channel models; the other is the
software that runs on the control computer to effect real-time opera-
tion of the CSF hardware.

The sequence generation software was developed on the UNIX™
operating system with the option to move it to the control computer.
In the initial arrangement, however, the fade parameter sequences
were generated on a UNIX system minicomputer and down loaded to
the control computer. As described earlier, these sequences were
converted into continuous analog signals and played into the simulator
hardware to achieve time-varying channel responses. Special software
was written to test the accuracy of the generated sequences and of the
resulting hardware responses. These issues are dealt with in Section
VI.

We describe here how to generate the random time-varying param-
eters for a fade model. We will first summarize the generation proce-
dure employed by our software to generate random variations with
any given Probability Density Function (PDF) and Autocorrelation
Function (ACF). Then we will describe the techniques used to generate
random variations with PDFs and ACFs specified by Rummler.’ In
this subsection, we show how to generate the random variations A, B,
and @, as defined in (2), for both the Horn and Dish Channels. The
transformation of these variations into A, C, and @ is performed by
the control computer and is described briefly in a later section on
control computer software.

The problem of generating a random variation with arbitrary PDF
and ACF is difficult in general and sometimes impossible.** In our
case, however, we wish to match the statistics of fade models with
well-behaved PDFs and ACFs. Further, while accurate matching of
the PDF is essential, particularly when the simulator is used for outage
measurements, slight variations in the ACF shape are acceptable since
the ACF is either unknown or only sparsely measured for many models.
These considerations ease the variation generation problem consider-
ably. It should be noted that the second-order statistics are not
uniquely determined by the PDF and ACF and that we will accept
whatever second-order statistics are generated when matching the
PDF and ACF.

This method is general in that it generates random variations with
any specified PDF and with an ACF close to the specified ACF. The
method starts by generating a Gaussian random process with an initial

ACF that we normally take as the desired ACF. From the Gaussian
variation, a new variation with the required PDF can be formed by a
nonlinear transformation. It is easier to think of this transformation
in two steps—a Gaussian-to-uniform transformation followed by a
transformation to the required PDF. The first transformation involves
using the Q function, while the second transformation involves using
the inverse of the required distribution function. While this gives
variations with the correct PDF, the nonlinear transformations will
make the output ACF different from the initial ACF.

To make the output ACF match the desired ACF, we use the
following iterative procedure: The initial ACF is applied to the under-
lying Gaussian variation. Based on the output ACF and an educated
guess, we select a new ACF for the underlying Gaussian variation.
This procedure stops when the desired ACF is obtained. In our case,
the iterative procedure converges quickly. We found that the ACFs of
two of our parameters remained essentially unchanged after the non-
linear transformations.

The nonlinear transformation required to convert the Gaussian
variations into variations with the desired PDF is broken up into a
transformation to uniform followed by a transformation to the desired
PDF. The transformation to uniform can be accomplished through
the Q function, i.e.,

where x is the original Gaussian variation and u is the uniform
variation. This function can also be used as a test of the quality of the
Gaussian variation by computing the new probability distribution
function and comparing the result with the uniform probability dis-
tribution function. This test was used to verify the uniform-to-Gaus-
sian random variation conversions.

We chose to use the Rummler dual-channel space diversity fade
model,’ although Rummler’s rationalized model® or any other dual
channel model could have been employed as well. Familiarity with
Rummler models is assumed in the following paragraphs.

Since ACF measurements are not available for this model but are
available for the single-channel model,** we use the single-channel
model ACFs for both channels of the diversity model. These ACFs,
measured by Rummler, are not true ACFs; they were rescaled so that
R(0) = 1. The scaling factor was determined by extrapolating the
measured ACF to the ordinate and finding the intercept. The ACF for
each of the fade parameters closely approximates an exponential. For
this reason, we applied an exponential ACF to each underlying Gaus-
sian variation and adjusted its time constant to form different ACFs.
Matching the time constants at the 1/e points of the rescaled Rummler
ACFs gives time constants of 129, 53, and 32 seconds for the respective
A, B, and ¢ parameters. These ACF's were used in generating the
dniderlying Gaussian variation discussed above.

The easiest parameters to generate are the ¢’s because evs are
independent of each other and of all the other parameters. For each
@, we begin with U(0, 1) variation, u, generated as discussed in the
previous subsection. The ¢ parameter, with a pedestal-type PDF, is
generated from the u variation by the simple rescaling

Ok oe + a)(2u — 1) + 90(1 — a)sgn(2u — 1); otherwise, (7)
where a is the pedestal parameter (5 for the Horn Channel and 8 for
the Dish Channel) and ¢ is in degrees.

The B parameters are generated next because, in both the Horn and
Dish Channels, the A parameter’s mean value is dependent on the B
parameter. The general technique for converting a U(0, 1) random
variation to a particular distribution can be accomplished by passing
each sample through the inverse of the distribution function. For the
B parameters, finding the inverse function algebraically is rather
complicated, so this was done numerically.

Finally, the A parameters are generated. For this case, the final
distribution is Gaussian so no transformations are need to obtain the
correct PDF. However, the mean of each A is dependent on the
associated value of B. Further, Ajom is correlated with Agis,. To
generate the A’s, two independent N(0, 1) random variations (x, and
X2) with identical ACFs are generated. Next, they are linearly combined
to produce the desired correlation and, finally, the variances are scaled
and appropriate mean values are added. In short,

Adish = POdishX1 + Gaish V1 — p?X2 + Maish(Baisn), _ (9)
where the Greek constants and the u(B) functions are defined in Ref.
vi

The control computer software falls into two categories, namely
software to produce the fades and software to calibrate and test the
hardware. The purpose of the fade software is (1) to translate fade
model variations. (e.g., A, B, and ¢) into the A, C, and 6 variations
supported by the hardware; (2) to perform calibration table lookup
and quantization to the closest binary output sample value; and (3) to
play the binary variations through the analog hardware. The remaining
software generates the calibration tables, tests the overall accuracy of
the fade control and calibration software, as well as the analog hard-
ware, and allows manual control over the fade simulator. These
operations are described in more detail in the following subsections.

The analog hardware implements the transfer function given by (4),
(see Fig. 5), where c = ab and 0 = ¢ + x. The voltage-controlled
attenuators have control signals proportional to 10 log (power atten-
uation). These final log conversions are handled by a lookup table
(described shortly), but the software-generated parameters A and B
are in decibel form and must first be converted to linear form before
computing c. The phase offset of + is required in the hardware to
ensure continuity into the D/A low-pass filter.

To ensure accuracy, calibration lookup tables are used to map and
quantize the analog floating point values generated from the fade
model software into one of the integer output values of the D/A
converter. Software automatically finds the closest table entry to the
floating point value and outputs the corresponding integer value.
Separate software outputs these integer variations from RAM memory
or disk to the D/A converters at a 1-Hz rate.

The calibration lookup tables discussed above are automatically
generated under software control via hardware measurements taken
with a network analyzer connected to a computer. The a and c
parameter calibrations are similar as shown in Figs. 12a and b and
require the disconnection and termination of the delayed path (direct
path for the c parameter). The control computer then measures the
end-to-end attenuation of the connected path for each of the 4096
values and generates a table. These values are smoothed by averaging
within a window centered around each value. This is required to obtain
a monotonic table, especially for large attenuations where measure-
ments become noisy.

To simplify the software, the 6 calibration was made interactive.
With both paths connected as in Fig. 12c, a spectral dip or notch is

(C) CALIBRATION OF 0
Fig. 12—Block diagrams of parameter calibrations: (a) calibration of a, (b) calibration
of c, and (c) calibration of 6.

formed by manually adjusting the relative values of a and c. This
notch is placed at the upper and lower band edges as well as the center
of the channel and the binary control values are stored. This infor-
mation is then used to generate the lookup table via interpolation and
extrapolation. The extrapolation is needed since notch position mea-
surements can only be made in the bandlimited channel.

Additional software allows for manual control over each of the
parameters, A, C, and @, in real time. This provides for quick checking
of the hardware and the lookup tables. This software also provides a
useful mode of operation for the simulator, allowing selected responses
to be easily “dialed” into the hardware.

Finally, there is software for playing a selected response and com-
paring it with the actual response measured by the network analyzer
to obtain an error measure. More details about this are contained in
the performance section that follows.

We will evaluate the performance of the CSF in several ways here.
First, we illustrate some “three-path” responses produced by the
variable channel networks over the 40-MHz bandwidth of interest.
Next, we demonstrate that the joint statistics of the computer-gener-
ated sequences of fade variables (A, B and ¢) conform to the model
from which they are derived. Finally, we demonstrate that the fre-
quency responses of the hardware, over a computer-generated popu-
lation of A, C, 6 values, are indeed the responses these fade variables
are intended to produce. With these demonstrations, we affirm that
the CSF can produce laboratory fading environments similar to those
on actual digital radio links.

The static and dynamic transmission performance of each DCU was
measured using a Hewlett-Packard Model 8505A Network Analyzer.
Here we discuss frequency response measurements on one or both
variable channel networks of a DCU, and show sample results.

In a typical measurement, fixed control voltages are set by keyboard
inputs on the control computer, and applied to the attenuators and
phase shifters of the variable channel networks. The values of the
control voltages produce a particular transmission function in each
network. The Network Analyzer provides a frequency-swept input
signal, in the vicinity of 70 MHz, to the DCU. The two 70-MHz
outputs from the DCU are returned to the analyzer, where the fre-
quency response magnitudes of the two networks are simultaneously
observed. Alternatively, only one network output from the DCU is
returned to the analyzer, along with a reference sample of the input

signal, to permit measurement of both the transmission magnitude
and phase (or group delay) of that network.

In our initial experiment, we set the control voltage of the delay
path attenuator so as to give maximum attenuation (>36 dB). This
enabled us to measure the transmission performance of the direct path
alone, with the output filter and downconverter included. Figure 13
shows the corresponding frequency response for one channel of a
DCU. The upper trace is the magnitude and the lower trace is the
group delay. Over a 40-MHz bandwidth centered at 70 MHz, the
magnitude is flat within +0.12 dB and the group delay is flat within
+1.5 ns. This response is primarily determined by the output Butter-
worth filter described in Section 3.3. |

In another experiment, we set the control voltage to the delay path
attenuator so that the signal level at the variable channel network
combiner was 0.92 dB weaker than that of the direct path. A minimum-
phase fade was thus created, with a maximum fade depth 20 dB below
the direct path gain. The upper trace in Fig. 14 shows the magnitude
of the resulting response. The delay path phase shifter was set to place
the maximum fade in the center of the channel passband. The lower
trace shows the channel group delay, wherein the time delay distortion
can be clearly seen: Frequency components at the edges of the channel
are delayed by more than 60 ns relative to components in the center
of the channel.

When the signal levels in the direct and delay paths are interchanged
so that the delayed signal to the network combiner is 0.92 dB stronger,

a nonminimum-phase fade occurs, with the same maximum fade depth
of 20 dB. The upper trace in Fig. 15 shows the transmission magnitude
of the resulting channel response, with the maximum fade placed once
more in the center of the channel passband. The time delay distortion
is shown in the lower trace. Under these conditions, signal components
at the center of the channel are delayed relative to components at the
edges of the channel.

In general, by appropriately setting the relative signal levels in the
direct and delay paths of a variable channel network, the depth of fade
in the channel response can be selected. This fade can then be
positioned in frequency by appropriately setting the phase shift in the
delay path. Figure 16a shows the transmission magnitude of the

Fig. 16—Transmission magnitude response as a fade is moved in frequency across
channel.

channel response as a 10-dB fade is moved in frequency across the
channel. The fade depth remains constant within better than +0.25
dB over the 40-MHz channel bandwidth. If the delay path signal level
is increased in magnitude to produce a 20-dB fade depth, the variation
in fade depth across the channel increases to +1.0 dB, as shown in
Fig. 16b. These variations are due to minor amplitude response im-
balances between the direct and delay paths. The +1.0-dB variation
for a 20-dB fade depth, for example, is caused by imbalances of about
+0.1 dB.

The DCU was designed to be a general-purpose laboratory instru-
ment. A nominal input signal level of 0 dBm was chosen for the 70-
MHz DCU input. This level is normally available from the modulator

portion of typical radio modems. The unfaded 70-MHz output from
each network of the DCU is designed to have a level of 0 dBm when
there is 3-dB attenuation in the direct path. This 3 dB can be removed,
under software control, to provide some up-fade capability. The un-
faded outputs at the alternative frequency of 1.070 GHz are designed
for a level of —15 dBm.

Finally, test ports are provided to permit frequency response mea-
surements of each variable network prior to the 1.070-GHz bandpass
filter. The signal level at each of these ports is nominally —52 dBm.

The distributions were obtained from a single variation containing
50,000 samples, which corresponds to about 1666 independent samples
for the ACF used. Similar results were obtained for the other fade
parameters but are not shown for brevity.

For the pedestal-type density functions for ¢, accuracy was checked
directly by measuring the slopes of the final distribution functions.
This yielded the desired values of 5 and 8 for the density function
pedestal ratios for the Horn and Dish Channels. The A parameter
distributions were checked using the same Gaussian-to-uniform trans-
formation as in the generation procedures for B and ¢. Again, a linear
plot should result and was obtained.

The ACF for Baish are shown in Fig. 19. The time separation among
ACFs for the underlying Gaussian, uniform, and final variations is
about 2 seconds for this parameter. Results for Bor, and the ¢
parameters are similar but show less variation. The underlying Gaus-
sians variations were adjusted so that their ACF's agree with the fade
model ACFs at the 1/e points.

The ACF for the Gaussian variations of Aporn and Agisn must be the
same so that the resulting distributions both follow the required ACF.
Therefore, the ACF time constants for the Horn and Dish Channels
are set equal and adjusted together. It was found that changes in ACF
resulted when the A parameters were correlated and then each was
biased by a mean related to B, as specified by the model. This is shown
in Fig. 20.

and ¢ parameters, respectively. The agreements are quite good.
Thus, we have carried out our parameter generation method for the

Fig. 21—Comparison of desired Rummler ACF with measured ACF for Apom and
Adish:

. Fig. 22—Comparison of desired Rummler ACF with measured ACF for Bhom and
dish:

accuracy. For reasonable PDFs, our iterative procedure yields a ran-
dom variation whose ACF is close to the desired ACF when the latter
is applied to the underlining Gaussian variation. The B and ¢ param-
eters showed little variation or no variation after the nonlinear func-

Fig. 23—Comparison of desired Rummler ACF with measured ACF for ¢porm and ddish-

tions were applied. The A parameter (which does not use the iterative
generation method) showed a modest variation due to the addition of
the mean which is related to the B parameter. Finally, the PDFs of
the generated parameters have been shown to agree with the PDFs
specified by the model.

In this subsection, we assess the accuracy with which the simulation
hardware produces the responses called for by the software. We begin
by discussing appropriate error measures and end with experimental
results.

Consider any one of the variable channel networks, i.e., the Horn
Channel or Dish Channel of a given DCU. The input control variables
(A, C, and @) to this network at any instant are intended to produce a
short-term frequency response [see (2), (4), and (5)] given by

As the control variables change slowly, H(f) is meant to change
slowly, too, in accordance with this equation. |

Now suppose that, for given A, C, and 6, we measure the actual
response of the network, Ho(/), and compute some index of its depar-

ture from the intended response, (10), over some bandwidth. For
example, we could compute

where the overbar denotes a frequency integration from, say, -20 MHz
to +20 MHz. Next, suppose that « were computed at fixed intervals in
time as A, C, and @ moved slowly through their computer-generated
variations. We could then obtain a population of e-values and compute
a cumulative distribution, P(e). If the joint variations of A, C and @
for this experiment were representative of a particular statistical
fading model, we could say that P(e) is the error-measure distribution,
over the fade response ensemble of that model, of the variable network.
Finally, a set of such distributions for all four variable networks could
be used to characterize—for that particular model—the accuracy of
the simulator hardware. |

First, we chose, for the sake of simplicity, to avoid measuring the
complex response Ho(/); instead, we measured its squared magnitude
(amplitude response) only. This function can be usefully compared
with the squared magnitude of H(f), (10), by means of the following
error measure:

and the frequency averaging is over the specified bandwidth. Given
the uncomplicated nature of the frequency-selective networks, we
assert that little is lost by omitting phase response data from the error
measures; amplitude response accuracy alone is a reliable indicator of
how well Ho(f) matches H(/).

It can be shown that the above error measure can be used to
approximate quite closely the root mean square decibel error between

* We show the control variables A, C and @ here to make their pertinence explicit.
Henceforth, we shall suppress them.

G(f) and Go( f ), as averaged over the specitiad bandwidth. Specifically,
this quantity is approximated by (4.34 ve) dB for all ¢ < 0.2.

Our second change was to separate the error measure into a scale
error and a shape error.* This approach acknowledges the fact that a
small scale difference between G(f) and G)(f) (i.e, a small level
difference in their decibel variations) would be relatively unimportant
given the vagaries of transmitter power, free space path loss, receiver
noise figure, and other link power quantities. We estimate that pure
level differences lying within +1 dB would be quite acceptable in view
of these factors.

Accordingly, we modify ¢ in (12) so as to permit a scaling of Go(f)
by an “optimal” factor, ropt, to be defined shortly. The value of « with
this scaling incorporated is then a measure solely of the shape differ-
ence between the intended and measured responses. The mathematics
follows.

The minimizing r, which we define to be optimal and call ropt, is easily
found to be

(Gol f)/G(f))
opt > oS SS 15
ot GA/GAY. =
and the resulting « is
[(Go( f)/G(f))I°
— 7 16
(Ga fV/GAY? ve
For a given A, C and 6, the quantity
k= 10 logio(Topt) (17)

is the scale error of the variables network, in decibels, for these control
variables; and the quantity

estimates the root-mean square decibel shape error (or just shape
error) of the variable network for these control variables. A conserv-
ative accuracy requirement is that these quantities lie within +1 dB
and below 0.5 dB, Bi over all likely combinations of the
control variables.

* Ifthe power responses G(f) and Go(f) are replaced by their decibel equivalents, this
separation is akin to isolating the “dc” and “ac” error variations with respect to f.

The error measures defined above were obtained for each of the four
variable channel networks of the CSF. The results can be regarded as
more than a check on the hardware alone. They also reflect on the
accuracy of the software translations into electrical signals, the lookup
routines and tables, and the software that generates those tables.

The hardware and software test configuration is shown in Fig. 24.
The control computer first calculates the magnitude of the desired
responses G(/), from the down loaded files containing A, B, and ¢
parameters. These files are also used to generate the binary A, C, and
6 values for the D/A converters by transformations and lookup tables
as previously discussed. The D/A converters are loaded with the binary
values and after a delay to allow the lowpass filters on the D/A control
lines to settle, the response of the channel, Go(/), is sampled at 40
frequencies spaced at 1-MHz intervals over the specified bandwidth.
These frequency samples are compared to the calculated samples, and
error measures are computed and saved on a disk. Specifically, for
each time sample of G(f) and Go(/), we computed ¢ and r.,; using (16)
and (15). In doing so, we used numerical-sum approximations to the
continuous-frequency integrations called for in these equations. Given
the smooth frequency responses produced by the variable channel
networks, we found the 1-MHz spacings between measured values of
G and Gp to be quite adequate.

The final step in the procedure was to obtain, for Sich network, a
cumulative distribution from the 7200 values of e, and another for Fropt.
Results for r.», are given in Fig. 25 for the Horn and Dish Channels

Fig. 25—Cumulative distributions of scale error, R, for each channel of a dual channel
unit. |

Fig. 26—Cumulative distributions of shape error, o, for each channel of a dual channel
unit.

of one of the DCUs; the corresponding results for « [converted to o via
(18)] are given in Fig. 26. Very similar results were obtained for the
other DCU. There are small but discernible differences between the
distributions for the Horn and Dish Channels. Mostly, these are due

to the slightly different multipath fading histories generated for these
two channels by the software.

The important findings from Figs. 25 and 26 are that the scale error
(R) lies well within +1 dB over all fades and that the root-mean-
square decibel error (or shape error, a) lies below 0.5 dB for over 98
percent of all fades. This is true for all four networks of the CSF and
is the result we wanted. The accuracy of the hardware is thus con-
firmed.

The authors want to thank M. F. Wazowicz for his help in hardware
assembly and E. C. Cox, P. Knoetgen and V. R. Dillard for their help
in the development of control software. We also want to especially
thank L. J. Greenstein and Y. S. Yeh for their generous advice and
support in both design and documentation of the simulation facility.

. W. D. Rummler, “A New Selective Fading Model: Application to Propagation Data,”
B.S.T.J., 58, No. 5 (May-June 1979), pp. 1037-71.

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Commun., COM-29, No. 3 (March 1981), pp. 346-52.

. M. Liniger, “Sweep Measurements of the Transferfunction of an RF-Channel and
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Radio Channel,” B.S.T.J., 61, No. 9, Part 1 (November 1982), pp. 2185-219.
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paper E27.
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Interference,” Intl. Conf. Commun., May 14-7, 1984, Amsterdam, paper 46.3.
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GLOBECOM ’84, Atlanta, Ga., November 26-29, 1984, paper 45.1.

11. M. Liniger, “Sweep Measurements of Multipath Effects on Cross-Polarized RF-
Channels Including Space Diversity,” GLOBECOM ’84, Atlanta, Ga., November
26-9, 2984, paper 45.

12. L. J. Greenstein and V. K. Prabhu, “Analysis of Multipath Outage With Applications

13. W. C. Jakes, Jr., “An Approximate Method to Estimate an Upper Bound on the
Effect of Multipath Delay Distortion on Digital Transmission,” IEEE Trans.
Commun., COM-27, No. 1 (January 1979), pp. 76-81.

14. R. P. Coutts and J. C. Campbell, “Mean Square Error Analysis of QAM Digital
Radio Systems Subject to Frequency Selective Fading,” A.T.R., 16, No. 1, 1982,
pp. 23-38.

15. L. J. Greenstein and B. A. Czekaj-Augun, “Performance Comparisons Among Digital
Radio Techniques Subjected to Multipath Fading,” IEEE Trans. Commun., COM-
30, No. 5 (May 1982), p pp. 1184-97.

16. G. J. Foschini and J. Salz, “Digital Communications Over Fading Radio Channels,”
B.S.T.J., 62, No. 2, Part 1 (February 1983), pp. 429-56.

17. D. P. Taylor and M. Shafi, “Fade Margin and Outage Computation of 4¢-QPRS
Radio Employing Decision Feedback Equalization,” Intl. Conf. Commun., Boston,
Mass., June 19-22, 1983, paper F2.1.

18. O. Andrisano, “The Combined Effects of Noise and Multipath Propagation in
Multilevel PSK Radio Links,” IEEE Trans. Commun., COM-32, No. 4 (April
1984), pp. 411-8.

19. N. Amitay and L. J. Greenstein, “Multipath Outage Performance of Digital Radio
Receivers Using Finite-Tap Adaptive Equalizers,” IEEE Trans. Commun., COM-
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20. W: C. Wong and L. J. Greenstein, “Multipath Fading Models and Adaptive Equal-
izers in Microwave Digital Radio,” IEEE Trans. Commun., COM-32, No. 8
(August 1984), pp. 928-34.

21. N. Amitay and J. Salz, “Linear Equalization Theory in Digital Data Transmission
Over Dually Polarized Fading Radio Channels,” AT&T Bell Lab. Tech. J., 63,
No. 10, Part 1 (December 1984), pp. 2215-59.

22. L. J. Greenstein and Y. S. Yeh, “A Simulation Study of Space Diversity and
Adaptive Equalization in Microwave Digital Radio,” AT&T Tech. J., 64, No. 4
(April 1985), pp. 885-905.

23. M. Emshwiller, “Characterization of the Performance of PSK Digital Radio Trans-
mission in the Presence of Multipath Fading,” ICC78, June 1978, Paper 47.3.

24. C. W. Lundgren and W. D. Rummler, “Digital Radio Outage Due to Selective
Fading-Observation vs. Prediction from Laboratory Simulation” B.S.T.J., 58, No.

25. Y. Y. Wang, “Simulation and Measured Performance of a Space Diversity Combiner
for 6 GHz Digital Radio,” IEEE Trans. Commun., COM-27, No. 12 (December
1979), pp. 1896-1907.

26. S. T. Matsuura, “Estimated Performance of a QPR Digital Microwave Radio in the
Presence of Frequency Selective Fading,” Intl. Conf. Commun., Philadelphia,
Penn., June 13-7, 1982, paper 7B2.

27. J.C. Campbell and R. P. Coutts, “Outage Prediction of Digital Radio Systems,”
Electron. Lett., 18, No. 25/26, December 1982, pp. 1071-2.

28. A. L. Martin, R. ’P. Coutts and J. C. Campbell, “Results of a 16 QAM 140 Mbit/s
Digital Radio Field Experiment,” Intl. Conf. Commun., Boston, Mass., June 19-
22, 1983, paper F2.2.

29. C. P. Bates and M. A Skinner, “Impact of Technology on High Ganccigy Digital
Radio Systems,” Intl. Conf. Commun., Boston, MA, June 19-22, 1983, paper F2.3.

30. S. Barber, “Cofrequency Cross-Polarized Operation of a 91 Mb/s Digital Radio,”
IEEE Trans. Commun., Vol. COM-32, No. 1, January 1984, pp. 87-91.

31. M. H. Meyers, “Multipath Fading Outage Estimates Incorporating Path and Equip-
ment fe ” GLOBECOM 84, Atlanta, Ga., November 26-9, 1984,
paper

32. M. M. Sondhi, “Random Processes With Specified Spectral Density and First Order
Probability Density,” B.S.T.J., 62, No. 3 (March 1983), pp. 679-702.

' 33. G. Vannucci and M. C. Teich, “Computer Simulation of Superposed Coherent and
Cohaotic Radiation,” Applied Optics, 19, No. 4 (February 15, 1980), pp. 548-53.

34. W. D. Rummler, “Advances in Multipath Channel Modeling,” Intl. Conf. Commun.,
Seattle, Wash., June 8-12, 1980, 52.3.1.

Harold H. Hoffman, New York University; AT&T Bell Laboratories. Mr.
Hoffman, a member of the Radio Systems Research Department, has worked

on microwave radio relay systems, cordless telephone, satellite orientation,
mobile radio, millimeter wave propagation, and a 7-meter radio astronomy
receiver and antenna. He holds patents for IF amplifier design and signal
processing. He is presently concerned with digital radio fading simulators and
mobile radio propagation studies.

R. S. Roman, AT&T Bell Laboratories, 1970—. Mr. Roman, a member of
the Radio Systems Research Department, has been concerned with digital —
radio system studies. He is currently a student at Brookdale Community
College.

A. J. Rustako, Jr., B.S.E.E. 1965, M.S.E.E. 1969, New Jersey Institute of
Technology; AT&T Bell Laboratories, 1957—. Mr. Rustako, a member of the
Radio Systems Research Department, has been engaged in radio system and
propagation studies. He contributed to early work in UHF and microwave
mobile radio communications, in particular, propagation measurements of the
multipath medium, and the use of diversity techniques. He has also made ©
contributions in the field of satellite communications. These include earth-
space propagation measurements of rain attenuation and depolarization at 12
GHz, and the use of phased array techniques for rapid scanning spot beam
satellite antennas. He is presently concerned with high-speed digital radio
transmission techniques.

Clark B. Woodworth, B.S.E.E., 1977, Monmouth College; M.S.E.E., 1980,
Rutgers University; AT&T Bell Laboratories, 1977—. Mr. Woodworth worked
in the Satellite Systems Research Department and the Radio Systems Re-
search Department. Work in the latter department included digital radio
studies and multipath channel simulation. He is currently working in the
Network Systems Research Department. Member, Eta Paupe Nu, Sigma Pi
Sigma, IEEE.

Single-Frame Vowel Recognition Using Vector
Quantization With Several Distance Measures

One of the most fundamental concepts used in the standard pattern recog-
nition model for speech recognition is that of distance between pairs of frames
of speech. Several distance measures have been proposed and studied in the
context of an overall speech recognizer. The purpose of this investigation was
to isolate the effects of different distance measures in a recognizer from the
other types of processing typically used in recognition. The way in which this
isolation was achieved was to use a recognizer based on single-frame distance
scores, using a vector quantization approach to give the single-frame reference
patterns required by the recognizer. The vocabulary for recognition was the
set of continuant vowels extracted from carrier words. A speaker-dependent
vowel recognition experiment was carried out using seven talkers (four male,
three female) and five distance measures. Results indicated that there were
differences in performance for the different distance measures when the
number of code-book patterns per vowel was one or two; however, when the
number of code-book patterns was four or more, these differences in perform-
ance became insignificant.

In the past several years, interest has focused on defining and
studying distance measures for speech recognition that reflect mean-
ingful differences between pairs of speech spectra.’~’ Although several
different distance measures have been proposed,'~“ and they have been
studied in a variety of recognition systems,’ as yet there is little

consistency in the reported performance of different recognizers using
different distance measures. For example, although Shikano and
Sugiyama’ found consistent recognition performance improvements
using the weighted likelihood ratio distance measure (as opposed to
an unweighted likelihood ratio distance measure) for a Japanese
speech recognition system, Nocerino et al.’ were not able to match
these results in English alpha-digit recognition experiments. Similarly,
although Davis and Mermelstein® achieved the best performance
among several distance measures with a mel-based cepstral distance
measure, this result has not been confirmed in other recognition tests.

There are several possible explanations for the discrepancies in
results obtained in the various investigations of distance measure
performance cited above. One explanation is that the basic feature
measurements of each of the recognizers were different in all cases,
for example, filter bank analysis versus Linear Predictive Coding
(LPC), different recording conditions and bandwidths, etc. These
differences in recognizer front ends could account for the differences
in performance, but if this were the case then the robustness of the
distance measure would become a major issue. A second explanation
is the difference in vocabulary, talkers, and transmission conditions
(e.g., telephone line versus microphone input). Again these differences
could be important, but they should not be factors for a robust distance
measure. A third explanation is that the experimental results did not
just reflect differences in distance measures but were affected by the
interaction between the components of the recognizer and the distance
measure. Thus, for example, improved performance for a distance
measure might be overshadowed by the power of dynamic time warp-
ing, which could compensate for a distance measure with poorer
performance.

It is the purpose of this paper to investigate he last possibility
discussed above—namely to isolate the effects of different distance
measures from all the other temporal alignment processing used in
recognition. The way in which we accomplish this goal is to design a
recognizer that makes its decisions based on single frames of speech.
In this manner any real differences in distance measures wall manifest
themselves as differences in recognition scores. |

The implication of using single-frame distance scores for recognition
is that the only vocabulary that can be considered is the set of
continuant (steady) vowel sounds. We have considered ten such vowel
sounds and they are listed in Table I, along with carrier words.in
which the vowels occur. One side benefit of the experiments to be
reported here is that a range of performance scores for single-frame
recognition of vowel sounds will be established and can be used to
assess future recognition algorithms in much the same way as digit

and alphabet scores have become standardized for isolated word rec-
ognition.®

Based on the above discussion a series of speaker-dependent recog-
nition tests was performed in the following manner. Each of seven
talkers (four male, three female) spoke the carrier words in Table I
ten times each, in two separate recording sessions, over a dialed-up
telephone line. Each talker also created, in a separate recording -
session, a single robust pattern for each of the ten carrier words. For
diagnostic purposes, an isolated word recognition test was performed
on the 100 isolated word tokens for each talker. All words which were
misrecognized were manually checked to make sure that no recording
errors (by either the talker or the automatic recording system) were
made. ,

The way in which the vowel frames, of each of the ten recordings of
each carrier word, were selected was as follows. The energy contour of
the word was measured, and the vowel portion was defined as the set
of frames whose log energies were contiguous to and within 6 dB of
the global energy peak of the word. The first five replications of each
carrier word were used as a training set, and a series of LPC Vector
Quantization (VQ) code books were designed from the vowel frames
for each vowel and for each talker. The second five replications were
used as an independent test set for recognition purposes.

Five distance measures were used in the evaluations, namely the
likelihood ratio;’” the weighted likelihood ratio;* the cepstral distance;°
a weighted cepstral distance;? and a bandpass liftered, weighted cep-
stral distance.”

The overall results of the single-frame recognition tests show that
for speaker-trained code books with moderate size—that is, either four
or eight vectors per vowel—there were no significant differences in
performance for the five distance measures. For code books with one
or two vectors per vowel, the two weighted cepstral distances per-

formed best; the likelihood ratio was third; the (unweighted) cepstral
distance was fourth; the weighted likelihood ratio was last.

The outline of this paper is as follows. In Section II we discuss the
speech analysis system, show how we extracted the vowel frames from
each carrier word, review the process of creating VQ code books, and
present the five distance measures used in our experiments. In Section
III we summarize the experimental conditions and present the results
of the word recognition tests, the code-book design, and the single-
frame recognition tests. In Section IV we discuss the results and give
general conclusions.

A block diagram of the single-frame, VQ-based recognition system
is given in Fig. 1. For each vowel frame, an LPC analysis is performed
to give either an LPC vector or an LPC derived cepstral vector. We
denote the resulting vector as a. This vector is then passed to a series
of ten vector quantizers (VQ’s), one for each of the ten vowels, and
the minimum VQ distance, ¢’, from the VQ for the ith vowel is
computed as |

where we assume that the ith vowel code book consists of the set of
M vectors bi,, 1 < m <= M. The local distance measure of eq. (1) can
be any of five measures, namely the likelihood ratio; the weighted
likelihood ratio; the (unweighted) cepstral distance; the weighted
cepstral distance; and the bandpass liftered, weighted cepstral dis-
tance. The recognized vowel, i*, is chosen as the one whose VQ distance
é is minimum, that is,
i* = argmin [e’]. (2)
1<1=10
In the following sections we briefly review the LPC analysis condi-
tions, the method of extraction of vowel frames from carrier words,
the procedure for VQ code-book formation, and the five distance
measures used in this study.

The speech signal, s(n), was recorded off a dialed-up, local, telephone
line. We used a sampling rate of 6.67 kHz. The speech signal is
digitized and then preemphasized using a first-order digital network
with transfer function H(z) = 1 — 0.9527". The signal is then blocked
into frames of size N = 300 samples (45 ms), with consecutive frames
spaced by L samples (15 ms). A Hamming window is applied to each
speech frame and an eighth-order (p = 8) autocorrelation analysis is
performed. The zeroth-order autocorrelation is the energy for the
frame, and it is used as the basis for word detection™’ and energy
normalization. An eighth-order LPC analysis is done on each frame,
using the autocorrelation method of LPC,” to give the LPC vector for
that frame. If a cepstral representation is required, a simple transfor-
mation of the LPC vector is performed.”

The way in which the vowel frames were extracted from the isolated
word tokens is illustrated in Fig. 2. Basically we used the log energy
contour of the word to find the vowel region—which was arbitrarily
defined as the set of frames—in the vicinity of the maximum energy
vowel frame, such that the log energy of each frame was within Epi
(dB) of the vowel maximum energy, Emax. After some preliminary
experimentation, a value of Ep = 6 dB was used. Thus, for a typical
carrier word, as illustrated in Fig. 2a, we first located the frame of
maximum energy, t,, and then, by searching in the local region around
t,, found the beginning, tg, and ending, tz, frames of the vowel.
Although this procedure worked well, in general, there were some
specific cases in which it failed. One such example is illustrated in Fig.
2b, in which the carrier word had a stop release at the end (e.g., boot)
whose energy exceeded the maximum vowel energy. The simple strat-

Fig. 2—Illustrations of how vowel frames were extracted from the carrier word.

egy of finding the frame with the maximum energy across the word
would fail in this case. Hence a check was made to ensure that all
strong local maxima of the energy contour were found, and that the
correct vowel maximum was located.

The code-book training set for each vowel (and for each talker)
consisted of all the “vowel frames” that occurred in five occurrences
of each carrier word. In fact, there were between 35 and 90 training
vectors for each vowel. From these training vectors a series of VQ code
books were designed with 1, 2, 4, and 8 vectors per vowel, using a
standard VQ code-book design algorithm.’*’* The distance measure
used in the code-book design was the same one used in the single-
frame recognizer—that is, each of the five distance metrics was used.
The centroid of the vectors in each cluster was chosen to represent
the whole cluster. In our VQ design algorithm the centroid was chosen
to minimize the average distortion of the whole cluster.’®

The five distance measures used in the recognizer included the
following: |

. Bandpass liftered, weighted cepstral distance—dgpcrp(a, b).
‘he form for computation of the likelihood ratio is

(6)
oO
where a is the residual error of the LPC analysis of the frame with
autocorrelation R,. (1).

where R,(i) and R,,(t) refer to the signal autocorrelations of the
frames corresponding to vectors a and b, and c,(t) and c,(z) are the
corresponding LPC-derived cepstral vectors. It should be noted that
we use q > p terms, in the summation of eq. (7), to approximate the
infinite summation of the true weighted likelihood ratio distance. In
particular we have used q = 2p (16), where the “extended” autocorre-
lations and cepstra were derived from the so-called “maximum en-
tropy” extension of the first (p + 1) terms.”
The form for the (unweighted) cepstral distance is

ee we have again used the cepstrum extended to q = 2p terms. The
form for the weighted cepstral distance’ is

and o? is the sample variance of the ith cepstral coefficient, where the
averaging is over the individual vowel sounds, that is,

10
Y [o?].-n,
of = ay | (11)
ym
v=1
with [o?], being the variance of c(i) over the n, frames in the training
set for vowel v. Typically the weighting function w; increases mono-
tonically with the index 1.
Finally, the form for the bandpass liftered, weighted cepstral
distance”’ is

where g was set to 12, and w} had the form of a bandpass lifter, that
is, a raised sinewave of the form

A series of recognition tests was run in which each of seven talkers
(four male, three female) first created robust training tokens of each
carrier word!® and then, in separate recording sessions, spoke each
carrier word ten times each. The first five such recordings were used
as a training set for the VQ code books; the second five recordings
were used as an independent test set. The robust tokens were used in
an isolated word recognition test to check the validity of the recorded
carrier words. The results of the isolated word recognition test are
given in Table II. It can be seen that for three of the talkers (1, 4, and

_ Table II—Word recognition errors for carrier words for each talker
(100 recognition trials per talker)

6) there were no word errors; for talkers 2 and 7 there were 2 word
errors (out of 100 trials each); for talkers 3 and 5 there were 10 and 8
word errors. The overall isolated word recognition accuracy for the
seven talkers is 96.9 percent. The word “bit,” which accounted for 8
of the 22 recognition errors, was confused with the word “bet” in all
such cases.

The results given in Table II indicate that there is a lot of variability
in the recognition performance on the isolated words across both
talkers and vocabulary words.

The results of the single-frame vowel recognition tests are given in
Table III and are shown plotted in Fig. 3. The data in Table III are
the average vowel error rates in percent averaged over the ten vowels
and the seven talkers as a function of VQ code-book size and distance
measure for both the training and testing sets, that is, there were
about 4000 recognitions per set. Figure 3 shows these same data in
graphical form. Several observations can be made from these results,
including the following:

1. There are significant degradations in performance, for all dis-
tance measures and for all code-book sizes, between the training and
testing sets of data. Thus for the VQ code-book size of one we see
degradations of 3 to 4 percent, whereas for the VQ code-book size of
eight we see degradations of from 9 to 10 percent in averaged vowel
error rates.

2. The effects of different distance measures can be seen primarily
for code-book sizes of one and two vectors per vowel, in which case
the two weighted cepstral distances consistently outperformed the
other three metrics, and the weighted likelihood ratio consistently
- performed the worst of the five measures. For code-book sizes of four
and eight vectors per vowel, there were no significant performance
differences among the five distance measures.

Table II|—Average word error rate (%) as a function of VQ code-
book size and distance measure for both the training and testing sets
Results on Training Set Results on Testing Set
; VQ Code-Book Size VQ Code-Book Size
TUSCAN se ae
Measure 1 2 4 8 1 2 4
dir 176 122 7.0 3.4 216 169 141 °&129
dwir 188 13.6 12 3.8 224 189 142 125
dcrp 18.6 11.8 (e' 3.0 21.6 17.4 13.7 13.4
dwcrp 16.5 11.0 6.7 3.8 20.0 16.5 14.2 13.3
dsecer 16.7 105 6.4 3.2 194 155 134 124

Fig. 3—Average vowel error rate (%) versus code-book size for each of the four
distance measures and for the testing and training sets of data.

rate of about 20 to 22 percent for a single code-book vector per vowel,
and the error rate dropped to about 13 percent for eight code-book
vectors per vowel. Thus we conclude that vowel recognition (among
the ten vowels in Table I) cannot be performed reliably using any of
the distance measures we have considered, in the framework of a
single-frame VQ code book-based recognizer.

The results presented in Section III can be interpreted as follows.
In the case where we have a good representation of the patterns to be
recognized, the effects of different distance measures on. recognition
performance are small. Such was the case when we used four or eight
code-book vectors to characterize each vowel in the vocabulary. How-
ever, when the representation of the patterns to be recognized becomes
more coarse, then the effects of different distance measures start to
become important. In these cases a better characterization of speech
sound differences, as obtained from a good distance measure, should
give better recognition scores. Such was the case when we used one or
two code-book vectors to characterize each vowel.

results presented in Table III. We see a big difference in average vowel
error rates between comparable test conditions (distance measure, VQ
code-book size) for the training and testing sets, especially when we
have four or eight vectors per vowel code book. Thus, in a sense, the
effects of different distance measures are small when the code-book
vectors begin to characterize well the seemingly insignificant details
of the training set, and are larger when the code-book vectors char-
acterize mainly the gross spectral behavior of the vowels. For real-
world recognition systems it is most probably the latter case that is
the more important one in that the reference patterns would be
expected to characterize the gross behavior of spectral variations with
time. In general there is not enough training data to reach the point
where we have characterized the fine spectral variations of words
reliably.

The conclusion we reach from the above discussion is that the
results for small code-book sizes, in which there were significant effects
of different distance measures, are probably more representative of
real recognition systems than the results for large code-book sizes. In
these cases—as is evidenced by recent investigations by Tokhura,’
Juang et al.,’° and Nocerino et al.,’/—the weighted cepstral distances
and the likelihood ratio would be expected to give better recognition
performance than the unweighted cepstral distance or the weighted
likelihood ratio measures.

We have presented results on speaker-dependent, single-frame, VQ-
based, vowel recognition for five different distance measures and for
four different size VQ code books. Our results indicate that for small
code-book sizes (one or two vectors per vowel) there is improved
recognition performance using a weighted cepstral distance rather
than the likelihood ratio, the unweighted cepstral distance, or the
weighted likelihood ratio measures. For larger code-book sizes (four or
eight vectors per vowel) the performance differences among the five
distance measures decrease. For practical recognizers, the weighted

cepstral distances appear to have advantages for application to
_ gpeaker-independent systems and for large vocabulary recognizers.
These advantages include increased efficiency of representation, re-
duced complexity of computation, and improved performance.

1. F. Itakura and S. Saito, “Analysis-Synthesis Telephony Based on the Maximum
Likelihood Method,” Proc. Int. Cong. Acoustics, Tokyo, Japan, Paper C5-6, 1968.

2. F. Itakura, “Minimum Prediction Residual Principle Applied to Speech Recogni-
tion,” IEEE Trans. Acoust., Speech, Signal Process., ASSP-23, No. 1, Sear
1975), pp. 67-72.

. K. Shikano and M. Sugiyama, “Evaluation of LPC Spectral Matching Measures for
Spoken Word Recognition,” Trans. IECE, J65-D, No. 5 (May 1982), pp. 535-41.

. D.H. Klatt, “Prediction of Perceived Phonetic Distance From Critical Band Spectra:
A First Step,” Proc. ICASSP ’82, (May 1982), pp. 1278-81.

5. A. H. Gray Jr. and J. D. Markel, “Distance Measures for Speech Processing,” IEEE
Hina Acoust., Speech, Signal Process., ASSP-24, No. 5, (October 1976), pp.
380-91.

6. S. Davis and P. Mermelstein, “Comparison of Parametric Representations for
Monosyllabic Word Recognition in Continuously Spoken Sentences,” IEEE
Trans. Acoust., Speech, Signal Process., ASSP-28, No. 4, (August 1980), pp. 357-
66.

. N. Nocerino et al., “Comparative Study of Several Distortion Measures for Speech
Recognition,” Speech Commun., 4, No. 4 (November 1985).

8. G. R. Doddington and T. B. Schalk, “Speech Recognition-Turning Theory to
Practice,” IEEE Spectrum, 18 (September 1981), pp. 26-32.

. Y. Tokhura, “Speaker Independent Recognition of Isolated Digits Using a Weighted
a a Distance,” J. Acoust. Soc. Am., Suppl. 1, 77, Paper £13 (Spring 1985),
p. S11.

11. L. F. Lamel et al., “An Improved Endpoint Detector for Isolated Word Recognition,”
IEEE Trans. Acoust., Speech, Signal Process., ASSP-29, No. 4 (August 1981),
pp. 777-85. .

13. Y. Linde, A. Buzo, and R. M. Gray, “An Algorithm for Vector Quantization,” IEEE
Trans. Comm., COM-28, No. 1 (January 1980), pp. 84-95.

14. B. H. Juang, D. Wang, and A. H. Gray, Jr., “Distortion Performance of Vector
Quantization for LPC Voice Coding,” IEEE Trans. Acoust., Speech, Signal
Process., ASSP-30, No. 2 (April 1982), pp. 294-303.

Lawrence R. Rabiner, S.B. and S.M., 1964, Ph.D., 1967 (Electrical Engi-
neering), The Massachusetts Institute of Technology; AT&T Bell Laborato-
ries, 1962—. From 1962 through 1964 Mr. Rabiner participated in the coop-
erative plan in electrical engineering at AT&T Bell Laboratories, in Whippany
and Murray Hill, New Jersey. He worked on digital circuitry, military com-
munications problems, and problems in binaural hearing. Presently, Mr.
Rabiner is engaged in research on speech communications and digital signal
processing techniques. He is coauthor of Theory and Application of Digital
Signal Processing (Prentice-Hall, 1975), Digital Processing of Speech Signals
(Prentice Hall, 1978), and Multirate Digital Signal Processing (Prentice-Hall,
1983), Member, National Academy of Engineering, Eta Kappa Nu, Sigma Xi,
Tau Beta Pi, Fellow, Acoustical Society of America, IEEE.

Frank K. Soong, B.S., 1973, National Taiwan University, M.S., 1977,
University of Rhode Island, Ph.D., 1983, Stanford University, all in Electrical
Engineering; AT&T Bell Laboratories, 1982—. From 1972 to 1975 Mr. Soong
served as a teacher at the Chinese Naval Engineering School at Tsoying,
Taiwan. In 1982 he joined the technical staff at AT&T Bell Laboratories,
where he engaged in research in speech, coding, and speaker recognition.
Member, IEEE.

Traffic Capabilities of Two Rearrangeably
Nonblocking Photonic Switching Modules

The architectures of two small switching networks are compared as potential
implementations of a 4 X 4 photonic switching module. Such a module would
be made by interconnecting several 2 X 2 photonic directional couplers on a
single LiNbO3; substrate. While both networks are rearrangeably nonblocking,
we investigate whether one network requires significantly more rearrange-
ments than the other. The analysis includes transient, Monte Carlo simulation,
and Markov steady-state techniques. We conclude that the traffic capabilities
of the two structures are not significantly different, and that selection of an
architecture can be based on other criteria, like loss, crosstalk, or ease of
manufacture.

The percentage of the world’s voice and data communications that
is carried by optical fibers increases daily. The importance of research
in photonic switching increases with it. A promising element for the
implementation of photonic switching systems is the photonic direc-
tional coupler, made from titanium-diffused lithium niobate. The
current state of this technology allows a level of integration of tens of
devices on a single substrate. We investigate two competing architec-
tures for a 4 X 4 switching module, fabricated in this technology, that
could be useful in building photonic systems.

of switching networks and describe the two candidate photonic switch-
ing modules. In Sections IV and V, we present the results of transient
analysis and Monte Carlo simulation, respectively, as applied to the
two switching modules. In Sections VI through VIII, we present steady-
state Markov analysis as applied to a generalized module and to each
candidate switching module, respectively. In Section [X, we enumerate
the network configurations in each of the candidate switching modules.

The traffic-handling performance of a switching network depends
on two entities: |

e The topology of the internal elementary switching stages.
e The rule by which paths through the network are selected.

One topological classification applied to switching networks is the
hierarchy of “blockingness.” Networks are classified by their ability
to establish connections, especially sequences of connections with
intermediate disconnections. An excellent tutorial and summary of
the state-of-the-art in switching network topologies is found in Ref. 1.

A switching network is nonblocking if any desired connection be-
tween two idle ports can be completed immediately without interfer-
ence from connections that may be already established in the network.
If this property is independent of any switching rule used to select
paths through the network, then the topology is nonblocking in the
strict sense. If the property is true, provided the paths assigned to the
established connections were selected under some switching rule, then
the topology is nonblocking in the wide sense.

A switching network is rearrangeably nonblocking if any desired
connection between two idle ports, which might be temporarily blocked
by connections already established in the network, can be completed
possibly after some established connections are moved to different
paths. Such topologies are parallel nonblocking in that any set of I/O
pairs can be connected without blocking if the network is initially idle
and the set is known in advance.

A switching network is blocking if there exist network configurations
of established connections from which some connection between two
idle ports can not be completed, even with rearrangement of the
established connections. Both switching modules under consideration
in this paper are rearrangeably nonblocking. ©

A switching network generally allows more than one path by which
ports on either end of the network may be connected. The switching
rule is the means of selecting one path. For certain special cases, such
as the strict-sense nonblocking networks, the performance is inde-
pendent of the switching rule. In general, however, the switching rule
is an important factor in the performance of a switching network.

The optimal switching rule follows: Route a new connection through
the network in a way that least affects the routing of any future
connections. |

.. route a call through the most heavily loaded part of the network that will still
take the call.

Such switching rules are called packing rules because of their analogy
to similar rules used in the problem of packing spheres into boxes. For
one module under consideration here, the four switching rules above
appear equivalent, and the rule is called prudent in the paper. For the
other module under consideration here, a case will be demonstrated
that represents a counter example to the general adoption of switching
rules that recommend close packing as the primary consideration.

Three measures of the quality of switching networks are transmis-
sion, topology, and traffic cepanuy:

Two measures of transmission quality are insertion loss and cross-
talk. Either measure may be in terms of average or worse-case value.
Under a uniform wiring scheme, insertion loss and crosstalk worsen

as a network becomes deeper. In electrical implementations, crosstalk
usually worsens as interchip, interboard, and especially interframe
wiring increase. This should be much less noticeable in photonic
systems. Loss and crosstalk measures will not necessarily recommend
the same switching networks.

Topological complexity is not so much a performance issue as it is
a manufacturing and economic issue. Quantitative measures that
correlate with manufacturing cost are not known, nor is the impor-
tance of this issue relative to transmission or traffic measures. It is an
important open question.

Traffic capability is a well-known analytical measure applied to
blocking networks.? The probability of blocking is that weighted
proportion of (new connection, network configurations) in which the
given new connection cannot be completed through the network in
the given configuration.

A corresponding measure for rearrangeably nonblocking networks
is developed here. The probability of requiring rearrangement P,; is
that weighted proportion of (new connection, network configurations)
in which the given new connection can be completed through the
network in the given configuration, but only after some established
connections in the network are first rearranged.

A dichotomy appears. If the malevolence is the CPU real time used
in effecting the rearrangement, then the demerit is that any rearrange-
ment is required, regardless of the count of switches or connections
involved, because they are probably all rearranged in parallel. If the
malevolence is the count of established connections disturbed by the
rearrangement, then the demerit may not be binary and one choice of
rearrangement may cost less than another. We will show that the
count of established connections disturbed by a rearrangement is ©
different for the two modules under consideration, so the dichotomy
is relevant. We will derive expressions for how many established
connections must be rearranged, on the average, for each of the
competing architectures.

In transient analysis, a sequence of connections and disconnections
is applied to an idle module. Over a set of such sequences, P,, is the
proportion of those sequences in the set in which a rearrangement was
required to complete a connection. The inadequacy of this analysis is

that it does not produce a single number or expression, because many
sets of sequences must be investigated. But the advantages over the
classical analyses are that the results are not dependent on potentially
unrealistic assumptions about traffic statistics, and that this analysis
entails a level of detail (and corresponding tedium) not required in the
other analyses. This detail uncovered an optimal switching rule for
one module that probably would have been overlooked had the analysis
been confined to traditional steady-state techniques.

In Monte Carlo simulation, a sequence of randomly generated events
is applied to an initially idle module. The inadequacies of this analysis
are that repeated simulations with identical event statistics yield
different results and that no closed-form solution is obtained, giving
P,, as a function of those statistics. The benefit of this analysis is that
both transient and steady-state behavior may be observed, and in fine
detail. If properly recorded, the events leading up to anomalies may
be studied.

In steady-state Markov analysis, the module is assumed to be in
some random network configuration and then a new random connec-
tion or disconnection occurs. P,, is a weighted sum over all network
configurations, and all possible new connections from those network
configurations, of those cases in which the module requires rearrange-
ment to complete the given connection from the given configuration.
The weighting is the steady-state probability distribution over the set
of network configurations or states of equivalence classes. Both the
steady-state probability distribution and the interstate transition
probabilities are dependent on the traffic load. This analysis yields a
closed-form result, but it is only valid in the steady state and it is
dependent on potentially unrealistic statistical assumptions.

We review the technology of photonic switching and present the
topologies of two proposed implementations of a 4 X 4 switching
module.

A photonic directional coupler is a two-input two-output device
whose transmission state depends on the magnitude of an applied
external voltage.? (See Fig. 1.) With nominal 0-volt applied, the
transmission is such that the signals cross over from input to output
(called the crossed state) and with a positive voltage applied, the
transmission is straight through from input to output (called the bar
state). This device is a photonic realization of the generic 2 X 2 beta
switching element,* where the control signal is electronic and the
switched data is photonic. |

UPPER UPPER UPPER
INPUT MIDDLE OUTPUT
SWITCH —- SWITCH
LOWER oa), LOWER
INPUT MIDDLE OUTPUT
SWITCH SWITCH SWITCH

Switching speeds in the tens of picoseconds have been reported,°
but speeds in the nanoseconds are more common.® The attainable
switching speed is highly dependent on device packaging, on the
quality of the electronic driver circuit that switches the applied voltage
to the device, and on the magnitude of this required voltage. A 1:16
multiplexer/demultiplexer has been built as an integrated circuit mod-
ule.’ It is a simple tree topology of photonic directional couplers with
one unused port at the input to each photonic directional coupler.
Another integrated circuit is a 4 X 4 switching module built as a square
array.”” These circuits show the level of integration available today in
this technology. Because each photonic directional coupler has length
in the order of a centimeter, the expectation of increasing the level of
integration by several orders of magnitude is not high, unless some
major breakthrough changes the photonic coupling length by orders
of magnitude.

Two waveguides, made by diffusing titanium in a lithium-niobate
substrate, can be made to intersect or cross over. Crosstalk increases
as the angle of intersection decreases, and becomes unpredictable at
very small angles.’° Thus a configuration, like that of Fig. 2, is feasible
if the crossovers are truly at large angles, as the figure suggests.
However, the photonic directional couplers have lengths in the order
of a centimeter and widths measured in microns, so Fig. 2 is not drawn
to true scale. Furthermore, the loss in a waveguide increases rapidly
as its radius of curvature decreases. So, a configuration like that of

Fig. 2 would require a physically large chip to allow geometries with
high crossover angles and high radii of curvature.

A major technological issue with today’s photonic directional cou-
plers is that they have significant insertion loss. While most of this
loss is due to coupling between fiber and chip and would be alleviated
with integrated circuits, the device loss is still high enough that
multidevice systems, like large switching networks, will require signal
amplification. Photonic gain can be simulated, expensively and with
imposing a bit-rate constraint, by a circuit with a photodetector, an
electronic amplifier, and a laser. However, direct photonic amplifica-
tion is believed to be coming available soon.’’ The devices described
in the reference are fabricated from gallium arsenide, while the pho-
tonic directional couplers are fabricated from lithium niobate. Thus,
integration on the same chip is currently impossible. However, when
such devices become practical, they could be integrated onto the same
chip carrier as the chip containing the photonic directional couplers.
If this form of integration is truly practical in the future, it allows
small radius bends and alleviates many of the problems described in
the preceding paragraph.

We call one candidate 4 X 4 module the 222 Module because it is a
three-stage module, where each stage has two switches. (See Fig. 2.)
This topology is in the class of networks called Clos networks, and this
~ exact configuration is a frequently used example in the literature. The
following terminology is used to identify symbolic inputs and outputs
in the 222 Module:

e A and W represent an arbitrary input and output, respectively, each
shown arbitrarily as an upper port on an upper switch.

e B represents the other port on the same input switch as A and X
represents the other port on the same output switch as W.

e C represents either port on the input switch that A and B do not
share, and Y represents either port on the output switch that W and
X do not share, each shown arbitrarily as an upper port on a lower
switch.

e D represents the other port on the same input switch as C, and Z
represents the other port on the same output switch as Y.

Because of the symmetry of the 222 Module, there is no topological
relationship between A and W.

We call the other candidate 4 X 4 module the 2121 Module because
the topology has four stages, where the input and middle stages have
two switches and the central and output stages have one switch. (See

LOWER LOWER
SWITCH SWITCH
Fig. 3—Configuration and terminology of the 2121 Module.

terms middle and central when applied to switching stages. The follow-
ing terminology is used to identify symbolic inputs and outputs in the

e A represents an arbitrary input and B represents the other port on
the same input switch as A, both shown arbitrarily on the upper
input switch with A above B.

e W represents the only output that A or B can reach through two
stages (necessarily on the same horizontal level as A and B).

e C represents either port on the input switch that A and B do not
share, and D represents the other port on the same input switch as
C, both shown arbitrarily on the lower input switch with C above
D.

e Z represents the only output that C or D can reach through two
stages (necessarily on the same horizontal level as C and D).

e X represents either port on the only switch with two outputs on it,
and Y represents the other port on the output switch with X, shown
arbitrarily with X above Y.

Because the 2121 Module is not symmetric, there is a topological
relationship between the inputs and the outputs. Accordingly, W or Z
is on the upper or lower middle switches depending on whether A and
B or C and D are on the upper or lower input switches, respectively.

An interesting and nonintuitive switching rule was discovered for
the 2121 Module. Consider the following sequence of events applied
to an idle 2121 Module. Let A to W be the first connection, and let
the path that avoids the central switch be selected (intuitively, the
best choice), setting the states of A’s input switch appropriately, and
W’s middle switch to the bar state.

If the second connection is B to X, representing two of the nine
second connections, the intuitively preferred path is not available
because the A to W connection has already set the switches, and is
using that junctor that B can reach that skips the central switch. Two

other paths are available and the choice is extremely interesting. One
path shares the state of A’s input switch and W’s middle switch, and
assigns the central switch to the bar state and the output switch,
appropriately. This path requires the assignment of two switches in
addition to the ones already assigned, and it is intuitively appealing.
After A to W is disconnected, A to Y, A to Z, and C to W, representing
four of the nine third connections, require rearrangement, and the
other five do not.

The other path for B to X still shares the state of A’s input switch,
but it assigns the central switch to the crossed state, the unused middle
switch to the bar state, and the output switch appropriately. This path
requires the assignment of three switches in addition to the ones
already assigned, and is, therefore, intuitively less appealing than the
previous path. However, after A to W is disconnected, only A to Z,
requiring some of the junctors and switch connections used by B to
X, requires a rearrangement of the network configuration.

I have two general interpretations that cover the essence of this
switching rule: (1) If you must use the central switch, the crossed state
is preferred, and (2) Minimize the count of switches that are shared
with any other path. It is not clear yet whether either or both of these
statements is applicable to a generalization of the 2121 Module to an
n Xn architecture. Even in the 2121 Module, I was afraid that the
overall use of this switching rule, while improving the performance
with some sequences of events, would degrade the performance with
other sequences. In all the sequences investigated and all the simula-
tions that were run, no such case arose.

We give an example of a transient sequence of events applied to the
- 2121 Module in which the final connection in the sequence requires
the disturbance of two established connections. We then argue that
the worst-case number could not be greater than two.

The example begins, with an idle module, by connecting A to X.
The optimal path avoids the central switch by assigning W’s middle
switch to the crossed state and A’s input switch and the output switch,
appropriately. Let the second connection be from B to Y. Since two
of the paths are blocked, the remaining path must be used, by assigning
the central switch to the crossed state and Z’s middle switch to the
bar state. Now disconnect the first connection, the one from A to X,
freeing the state of W’s middle switch.

Let a new second connection be from C to X. Since the output
switch and Z’s middle switch are in the wrong states to use two of the
_ paths, the remaining path must be used, by assigning W’s middle
switch to the bar state, and using the idle link through the central

switch. The states of all six switches are set, and two of four new
connections will require rearrangement.

Consider a third connection from A to Z. Its only satin requires the
reconfiguration of the input switch shared by A and B and Z’s middle
switch, and also requires the link through the central switch now used
by B to Y. So the B-to-Y connection must be moved to its other
(optimal) path. But it can’t be moved directly because W’s middle
switch and the output switch are in the wrong state. So, the C-to-X
connection must also be moved to its other (optimal) path.

Having established, by example, that the worst-case number is at
least two, the question remains whether it could be higher. Since there
can only be four established connections, the only other number to
consider is three. However, in any network configuration with three
established connections, the unused fourth connection is always avail-
able. While this fact may not be immediately obvious, it can be proven
exhaustively for the 2121 Module. A more elegant proof, applicable to
the general n X n module, is desirable.

The generalized Clos Network has the topology of Fig. 2, but with
n inputs and m junctors on each of r rectangular input switches, with
n outputs and m junctors on each of r rectangular output switches,
and with m square middle switches each connecting to r junctors
on each side. It is known’ that such a switch is nonblocking if m =
2n — 1 and is rearrangeably nonblocking if m = n. The latter is
satisfied in the 222 Module, in which n = m =r= 2.

The 2121 Module is also rearrangeably nonblocking. This result can
be proven easily by exhaustion for the 2121 Module and is obvious by
inspection of the state diagram in Section VIII. A general prow fora
generalized n X n network is being developed.” |

In the 222 Module each I/O pair has exactly two connection paths.
That is, there are exactly two paths through the module from any
input to any output. The 2121 Module does not have this symmetry.
The I/O pairs at opposite corners of the module (A to Z, B to Z, C to
W, and D to W) have only one path through the module, requiring the
central and the appropriate middle switches in the crossed state. The
I/O pairs straight across the module (A to W, B to W, C to Z, and D
to Z) have two paths through the module: one avoiding the central
switch and the other using the central switch in the bar state. Any
connection to an output on the only output switch (A to X, B to X, A
to Y, B to Y, C to X, D to X, C to Y, and D to Y), has three paths

through the module: one avoiding the central switch, the second using
the central switch in the bar state, and the third using the central
switch in the crossed state.

In the 222 Module each path through the module passes through
exactly three photonic directional couplers. The 2121 Module does not
have this symmetry, either. In the 2121 Module some paths pass
through only two photonic directional couplers, some through three,
and some through four. This variation will make deterministic gain
difficult. If crossovers in the 222 Module require photonic directional
couplers, or have similar loss and crosstalk as photonic directional
couplers, then this module would not be symmetric in this sense either.

If crossovers in the 222 Module are insignificant, its inputs and
outputs are uniform. That is, all inputs and outputs have the same
properties of the following: count of paths per I/O pair, count of
switches per path, and equal access to any port on the other side of
the module. The last two properties do not hold true if crossovers are
significant in the 222 Module, but none hold true in the 2121 Module.

The interconnection topology of the 222 Module has two crossovers:
one between the input and central stages and one between the central
and output stages. The interconnection topology of the 2121 Module
has no crossovers.

An implementation is illustrated in Fig. 4 in which the two cross-
overs in the 222 Module are eliminated by off-chip fibers. Its practi-
cality depends on the difficulty of off-chip fibering, the impracticality
of crossovers on the photonic substrate, and the magnitude of the
advantage of the 222 Module over the 2121 Module (if any).

3.6.4 Transmission 7
It is difficult to predict the difference in the crosstalk characteristics

of the two modules. The crossovers in the 222 Module suggest that it
may be worse than the 2121 Module, but the extra stage in the latter
suggests otherwise.

Insertion loss is easier to predict. The worse-case count of switches
in a network path is four in the 2121 Module and is three in the 222
Module. Thus, the 2121 Module would appear to have a poorer
insertion loss characteristic. However, if crossovers must be imple-
mented by photonic directional couplers, or if they have equivalent
loss, then the worse-case count of (equivalent) switches in a network
path is five in the 222 Module and it would have worse insertion loss
than the 2121 Module.

The switching rule used in the 222 Module is the classical packing
rule discussed in Section 2.2. The switching rule used in the 2121
Module is the unusual rule discussed in Section 3.4. Two implemen-
tations of the switching rule are discussed in Section IX and the
network configurations for each module are enumerated.

If some network configuration of the 222 Module must be rearranged
before a new connection can be completed, only one established
connection need ever be disturbed.” It was shown in Section 3.5 that
there exist network configurations in the 2121 Module in which two
established connections may need to be disturbed. Consequently, we
not only compute P,, for each module in the sections below, but we
also compute, N,,, how many established connections must be dis-
turbed, on the average.

By contrast, the 2121 Module has been counter-intuitive. While its
transient behavior is smooth, it degrades with simple connect/discon-
nect sequences. The switching rule and count of established connec-
tions disturbed has been startling to those familiar with such things.
The plethora of network configurations is partitioned, with great
difficulty, into 50 equivalence classes, making Markov analysis tedious.

IV. TRANSIENT ANALYSIS
In transient analysis, we assume the module is idle and study its

behavior under different sets of connect/disconnect sequences. The
network quality, the average probability of requiring rearrangement
for a specific set of sequences, is the proportion of those sequences in
the set that require rearrangement. Four sets are applied to each of
the two modules, giving eight results. Computer programs applied all
sequences in the four sets to each module and tabulated whether
rearrangement was required. The program’s output was studied and
generalized into theorems whose formal proofs are exhaustive and
tedious and not presented in this paper.’® The proofs are available to
the interested reader.

A template is a set of fixed-length sequences of events applied to the
switching module, where an event is a connection or a disconnection
of an input-output pair. The nomenclature for templates is x, ... Xn,
where x; represents the ith event in a sequence; x; has value C or D,
depending on whether this ith event is a connection or disconnection,
respectively; and n is the length of a sequence.

The following rules govern the determination of significant tem-
plates:
oe Since the module is assumed to be idle before any sequence is

e Similarly, no sequence, nor initial subsequence, would have more Ds
than Cs. |

© Since a module can only support four connections, no sequence, nor
interior subsequence, would have four more Cs than Ds.

o No significant sequences would have, nor would contain an initial
subsequence that has, as many Cs as Ds (e.g., no significant sequence
would begin with CD) because it would have the effect of restarting
from an idle module.

e Since we expect no problems Hononne disconnects in these modules,
no significant sequence would end with a D.

oe In no significant sequence would a D apply to an immediately
preceding C because it would have the effect that the C never
occurred. —

haustively. Fortunately, enough templates with lengths 4 and 5 are
significant. Combining these length and content constraints, the sig-
nificant templates are CCCC, CCDC, CCDCC, and CCCDC.

For sequences in a CCCC template, beginning with all switches idle,
the assumed order of events is

4, The last idle input connects to the last idle output.
With 4 x 4 = 16 cases of first connection, 3 X 3 = 9 cases of second
connection, 2 X 2 = 4 cases of third connection, and 1 X 1 = 1 case of
final connection, the CCCC template contains 16 x 9 X 4 xX 1 = 576
sequences.

For sequences in a CCDC or CCDCC template, beginning with all
switches idle, the assumed order of events is
. One of four inputs connects to one of four outputs.
. One of three idle inputs connects to one of three idle outputs.
. The first input-output pair is disconnected.
. One of three idle inputs connects to one of three idle outputs.
. In a CCDCC sequence only, one of two idle inputs connects to
one of two idle outputs.
With 4 X 4 = 16 cases of first connection, 3 X 3 = 9 cases of second
connection, 1 case of disconnection, 3 X 3 = 9 cases of third connection,
and, in the CCDCC sequences only, 2 X 2 = 4 cases of fourth
connection, the CCDC template contains 16 x 9 x 1 X 9 = 1296
sequences and the CCDCC template contains 16 X9xX1xX9x4=
5184 sequences.

For sequences in a CCCDC template, beginning with all switches
idle, the assumed order of events is

5. One of two idle inputs connects to one ot two idle outputs.
With 4 X 4 = 16 cases of first connection, 3 X 3 = 9 cases of second
connection, 2 X 2 = 4 cases of third connection, 2 cases of disconnec-
tion (either the first or second connection), and 2 X 2 = 4 cases of
final connection. The CCCDC template contains 16 x 9 xX 4 X 2 X
4 = 4608 sequences.

In either module, with a prudent switching rule, no CCCC sequences
require rearrangement. Since all combinations of idle I/O pairs can be
simultaneously interconnected in both modules, they are both at least
rearrangeably nonblocking.

The 222 Module outperforms the 2121 Module under CCDC and
CCDCC sequences. In the 222 Module, with a prudent switching rule,
no CCDC sequences, nor CCDCC sequences, require rearrangement.
However, in the 2121 Module with its unusual switching rule, 2 percent
of the CCDC sequences require rearrangement and 6 percent of the
CCDCC sequences require rearrangement.

Conversely, the 2121 Module outperforms the 222 Module under
CCCDC sequences. In the 222 Module, with a prudent switching rule,
8 percent of all CCCDC sequences require rearrangement. However,
in the 2121 Module, with its unusual switching rule, only 6 percent of
all CCCDC sequences require rearrangement. Since neither module is
generally nonblocking but both are at least rearrangeably nonblocking,
they are identically rearrangeably nonblocking.

Summarizing transient analysis, Fig. 5 illustrates module perform-
ance versus template complexity for the two modules. The scale of the
X axis has no mathematical nor physical meaning. The 222 Module
outperforms the 2121 Module under all tested templates up to and
including CCDCC sequences, but is outperformed by the 2121 Module
under CCCDC sequences. I expected the 222 Module to be consistently
better, so I was surprised by this turn of events. A qualitative expla-
nation of this unusual behavior is elusive, but I propose a conjecture.

The 2121 Module is distorted from its optimal connectivity by simple
sequences of events, because of its asymmetric junctor pattern and
nonuniform switch-count in alternate paths. These distortions usually

involve connections, like A to Z, that go diagonally across the module
and have only one path. The severity of these distortions is softened
by the presence of I/O pairs that have three paths, and by the
opportunity to be clever in the choice of switching rule. The 222
Module, however, is not distorted from optimal connectivity until the
event sequences are more complex. When it finally becomes distorted,
it is more severely distorted than the 2121 Module.

In a simulation, a sequence of randomly generated events is applied
to an initially idle module. In this section, the results of simulation
program are plotted in a scatter diagram and interpreted.

Two distinct programs were written—one for each module. The
programs are, however, similar at a high level of description. Program
variables record switch states, paths assigned to connections, inter-
connected parties, and other parameters and outputs.

The program begins by initializing a random number generator and
reading module connectivity information from files. The program then
enters a loop on traffic intensity: varying from 1.1 to 3.8 in increments
of 0.3. Since the double exponential model of traffic is simulated, the
average holding time and the average quiet time of each input are
computed from the traffic intensity.

Rather than simulate the Poisson environment, by stepping through
small intervals of time and simulating events, the program skips
through time from one event to the next. Corresponding to each input
i, is the time to the next event tne[i], associated with that input. These
values are initialized to an exponentially distributed random number
whose mean is the quiet time. The number of rearrangements is
initialized to zero and an interior loop, on the number of events in the
simulation, is entered. |

Time is advanced to.the minimum value of the four tne[i] and all |
tne[i] are reduced by that value. The event that timed out is simulated.
If it was a holding time that expired, the connection associated with
the relevant input is disconnected and tne[i] for that input is set to a
random quiet time. If it was a quiet time that expired, a connection is
made with the relevant input.

The connection is established by first locating a random idle input. ©
A subroutine that implements the switching rule determines the best
available path through the module. If there is no available path, the
count of rearrangements is incremented, and established connections
are disconnected and reconnected by their best paths. The connection

associated with the event is established and tne[i] for that input is set
to a random holding time.

The loop on events terminates, and after printing out results as a
function of traffic intensity, the loop on traffic intensity terminates.

Simulations were run with sequences of 1000 and 5000 events, for
the 222 and 2121 Modules, under varying traffic load. These numbers
of events should give statistically significant results, and should be
large enough that the transient effects from starting idle would be
damped out. The independent parameter is the mean of traffic inten-
sity, the product of global rate-of-origination by per-call holding time,
how many simultaneous connections exist in a module at any time.
The dependent variable is the percentage of new connections requiring
a rearrangement. Traffic intensity is varied from slightly over 1.0 to
slightly under 4.0, and the result of each simulation is plotted as a
scatter diagram in Fig. 6. Simulation results for the 222 Module are
shown with X and for the 2121 Module with +. Results of simulations
with 5000 events are circled and with 1000 events are not.

At low and moderate traffic intensities, it is noted that P,, gets
worse as the traffic intensity increases and that P,, is uniformly, but
slightly, worse in the 2121 Module than in 222 Module. In neither case
is P,, particularly bad, even at their maxima. That there is a maximum
is somewhat surprising. A monotonically increasing P,, might have
been expected.

At high traffic load, there will be times when a 4 X 4 module is ©
completely connected. The next event would have to be the discon-
nection of one I/O pair, and the event after that would be likely to be
the reconnection of the same pair. It would be likely to be a connection
because the traffic load is high, and it would have to be the same pair
because they are the only idle input and output, the only connection
that could be made. Such a connection would never require rearrange-
ment because it was just disconnected. This behavior is consistent,
whether the module is isolated as a simple 4 X 4 network or part of a
large network.

A familiar result in classical statistics comes from applying Che-
byshev’s inequality to Bernoulli trials. This Bernoulli law of large
numbers is used, for example in determining sample sizes of public
opinion polls:

where f, is the observed frequency after n trials, p is the given or
assumed probability, and e is an arbitrary tolerance. For simulations
with n = 1000 events and p = 0.08, the inequality states that the
variation in the outcome should be within +3 percent for 97 percent
of the simulations. The tolerance is even less with smaller values of p.
A casual glance at Fig. 6 shows that the variation is much greater than
this. A conjectured explanation is that law of large numbers is derived
from the assumption that the Bernoulli trials are independent. Since
P,, in the ith event of a network simulation is highly dependent on
previous events, this fundamental assumption is invalidated.

Three analyses were performed. The first, presented in the remain-
der of this section, is of a generalized nature and pertains to both
modules. It establishes a general model to be used as a check for the
models of the modules under investigation. The second analysis, in
Section VII, is based on a Markov model of the operation of the 222
Module. The third analysis, in Section VIII, is based on a Markov
model of the operation of the 2121 Module.

In this oversimplified model, any network configurations in which
the same count of connections are established are deemed to be
equivalent. There are five equivalence classes of network configura-
tions, corresponding to zero to four established connections, inclusive.
Hence, there are five states in the corresponding Markov process,
shown in Fig. 7. The model is general enough to cover both modules.

The stochastic behavior of the model is based on the classical traffic
assumptions: Poisson-distributed service arrivals and exponentially

distributed holding times. Since the Poisson and exponential distri-
butions are mutual inverses, this model is equivalent to having Pois-
son-distributed arrivals of connections and disconnections or expo-
nentially distributed off-hook and :on-hook times (as used in the
simulation program). Hither way of looking at it, there are two random
variables and there are two underlying parameters: the busyness and
the true time scale. If results are considered per unit time or limited
to steady-state behavior, then true time scale is irrelevant, and one
random variable may be arbitrary and the other specifies busyness per
unit time. We will use the double-Poisson process, for this and the
two later models, and arbitrarily set the per-line disconnect rate to 1.

The state transitions are computed from the individual connect and
disconnect rates of the four inputs (or outputs, equivalently), which
are assumed to be statistically identical. Let \ be the rate at which
each individual input requests a connection through the module, and
let 1.0 be the rate at which each individual established connection is
disconnected. In other words, ) is the ratio of the connect rate to the
disconnect rate of each individual input (or output). The global rate
of new connections from state i is (4 — 1)A, because there are 4 — 1 idle
inputs that could make such a request. The global rate of disconnec-
tions from state 1 is i, because there are i established connections, any
of which could request disconnection.

The analysis is a special case of a queue with dependence on the
state of the system,'* but can also be viewed as a queue with finite
customer population and infinite servers.’

The steady-state probabilities are calculated from a conservation
law. In the steady state, the mean exit rate from state 1 is the sum of
the rates on all exit arcs from state 1 times the steady-state probability
of being in state 1. The mean entry rate into state 1 is the sum of the
products of a rate on each entry arc times the steady-state probability
of being in the state from which the arc comes. Conservation of calls
dictates that the mean exit rate must equal the mean entry rate in the
steady-state for each state. The resulting simultaneous equations have
the general solution:"*

where n > 0, A; is the global connect rate on the arc from state i to
state i + 1, uw; is the global disconnect rate on the arc from state i to
state i — 1, and g; is the steady-state probability of being in state 1.
Setting A; = (4 — 1) and yp; = 1 gives

Note that each g; is expressed as a function of go. The probability go
that there are no connections is found by setting the sum of all steady-
state probabilities to 1. This gives the neat result

Traffic intensity is a random variable giving the count of established
connections at any time. In the classical traffic model, having Poisson
arrivals with rate r and exponential holding time with mean h, traffic
intensity is known to be Poisson distributed with mean, 7 = rh.

One way. of looking at the model of Fig. 7 is that the system has
four sources, each generating one calling cycle per unit time. The
calling cycle consists of an exponentially distributed off-hook interval
with mean \/(1 +A) and an exponentially distributed on-hook interval
with mean 1/(1 + A). Setting the arrival rate to 4 and the holding time
to the mean off-hook interval, the Poisson-distributed traffic intensity
has mean 7 = 4 X A/(1 + A). This intuitive argument is verified by
computing the mean count of established connections in the steady
state

Consider the two extremes. If \ = 0, the steady-state probability
mass function is

and the mean traffic intensity 7 = 0. If \ —~ ©, the limit of the steady-.
state state probability mass function is

and the limit of the mean traffic intensity 7 — 4. As a better example,
let X = 1, which means that the individual arrival rate equals the
departure rate, or that each input is on-hook for the same average
time as off-hook. The steady-state probability mass function is

showing a trend toward state 2, with decreasing probability in either
direction away from state 2. The symmetry about state 2 suggests an
average of two connections in the steady state and setting \ = 1 in the
equation for traffic intensity gives 7 = 2.

As another example, let \ = 2.0, meaning that each input is off-
hook twice the time that it is on-hook, that is, two-thirds of the off-
hook/on-hook cycle. The steady-state probability mass function is

showing a trend slightly under state 3 and a lack of the symmetry
observed in the case where \ = 1. If \ = 0.5, the steady-state state
probabilities are reversed from the case where A = 2. The mean traffic
intensity in these cases is 8/3 and 4/3, respectively, representing the
center of mass for each distribution.

The state model of the 222 Module is illustrated in Fig. 8.1° Each
bubble in the figure represents a state and contains the state’s name
and a representative network configuration from the state’s equiva-
lence class. State I represents the idle network configuration and is
equivalent to state 0 in the generalized model. State J represents the
16 network configurations with one connection established. These 16
network configurations are all equivalent, for purposes of determining
P,,, and this state is equivalent to state 1 in the generalized model.

States S through V represent four equivalence classes of network
configurations in which two connections are established. This set of
states is equivalent to state 2 in the generalized model. In state S, the
two established connections terminate on the-same input switch and
on the same output switch. In states T and U, the two established
connections terminate on opposite input switches and on opposite
output switches. In state T, both middle switches are used and in state

U, one middle switch is shared by both established connections and
the other is idle. State T is the only malevolent state in the model and
the only transition in which rearrangement is required is the one from
state T to state X. In state V, either the two established connections
terminate on the same input switch and on opposite output switches,
or the two established connections terminate on opposite input
switches and on the same output switch. These conditions are equiv-
alent for purposes of computing P,,.

States W and X represent two equivalence classes of network
configurations in which three connections are established. These two
states are equivalent to state 3 in the generalized model. In state W,
two of the established connections terminate on the same input switch
and on the same output switch, and the third established connection
terminates on the other input switch and the other output switch. In
state X, the two established connections that terminate on the same
input switch terminate on opposite output switches, and the two
established connections that terminate on the same output switch
terminate on opposite input switches. These are the only distinctions
that need be made among all network configurations with three
connections established.

States Y and Z represent two equivalence classes of network config-
urations in which four connections are established. These two states
are equivalent to state 4 in the generalized model. In state Y, two
established connections that terminate on the same input switch
terminate on the same output switch. In state Z, two established
connections that terminate on the same input switch terminate on
opposite output switches. These are the only distinctions that need be
made among all network configurations with four connections estab-
lished.

Corresponding to the transition from state I to J is a connection
through the module in which two paths are possible. The choice of
path is arbitrary and in no way affects the results. Corresponding to
the transitions from state J to S or V is a second connection that has
only one available path. Corresponding to the transition from state J
to U, is a connection in which the selected path shares a middle switch
with the established connection. Selecting the other path would cor-
respond to an imprudent transition to state T.’° Corresponding to
transitions from states T, U, and V to states W and X are third
connections that have only one available path, even with the rear-
rangement required before the connection corresponding to the tran-
sition from state T to X. Corresponding to the transition from state
S to W, is a third connection in which two paths are possible. The

choice is arbitrary, not affecting the results, but the choice determines
that established connection, which if later disconnected, would corre-
spond to the undesired transition to state T. Corresponding to the
transitions from states W or X to states Y or Z, respectively, are
fourth connections in which only one path is available. The transition
from state J to U represents the only connection where the switching
rule is relevant.

The rates on the arcs connecting the states in Fig. 8 are similar to
those in the generalized model, but several need further explanation.
e The sum of the connect rates on transitions from state J to states S

transition from state 1 to 2 in the generalized model. With one
established connection, there are nine possible second connections:
one that terminates on the same input and output switches as the
established connection, four that terminate on the opposite input
and output switches as the established connection, and four that
terminate on one same switch and one opposite switch as the
established connection. Thus the rates on the transitions to states

e The sum of the connect rates on transitions from each of states S
through V must be 2), corresponding to the connect rate on the
transition from state 2 to 3 in the generalized model. In states S and
V, all third connections lead to the same next state, state W or X,
respectively, and so each single transition is labeled with 2A. In
states T and U, however, half the third connections terminate on
the same input and output switches as an existing connection, and
half terminate on the same input switch as one connection and the
same output switch as the other connection. Thus, states T and U
each transit to both states W and X, and the rates on those
transitions are all X. All third connections associated with transi-
tions from state T to state X require rearrangement, and these are
the only connections requiring rearrangement in the entire model.’®
Hither of the two established connections may be rearranged so that
one middle switch is shared by both connections and one is idle—
an intermediate network configuration that conforms to state U.

e The sum of the disconnect rates on transitions from each of states
W and X must be 3, corresponding to the disconnect rate on the

_ transition from state 3 to 2 in the generalized model. In any network
configuration belonging to state X, two of the three single discon-
nections results in a configuration belonging to state V, so that arc
is labeled with a rate of 2. The other single disconnection results in

rate of 1. In any network configuration belonging to state W, a
transition to a configuration belonging to state V is impossible, but
each of the three single disconnections results in a configuration
belonging to state S, T, or U, respectively. Thus, each of these
transitions is labeled with a rate of 1. The transition from state W
to T is the only entry into that malevolent state, and it is unavoidable
under any prudent switching rule. |

The rates on the state transitions and the notion of X in this devel-
opment are slightly different from that of the reference.’® In that
paper, \ represented the rate of origination of connections between
specific inlet and outlet pairs. In this development, traffic is assumed
to originate at an inlet, and terminate on any random idle outlet.

7.3 Steady-state probabilities
The rate equations for each of the ten states are

As is customary with such processes, only n — 1 of the n equations are
independent. Setting the sum of the n probabilities to 1 provides the
nth independent equation. The solution, after several hours of manual
algebra is

As a check, it is verified that p; above equals gp from the generalized
model, py = £1, Ps + Pr + Pu + Pv = &, Pw + Px = 8s, and py + pz =
g4. Of particular interest, of course, is the steady-state probability of
the malevolent state, pr.

P,, is the proportion of those new connections that require that
an (one) established connection be rearranged before a new connec-
tion may be completed. The numerator is the sum over all states of
(the average count of new connections requiring rearrangement from
state i) X (the steady-state probability of state i). For this network,
it is simply 1 X pr. The denominator is the weighted average count of
possible new connections, similar to the calculation of 7 in the previous
section

points for the 222 Module, taken from the scatter diagram of Fig. 6.
Setting the derivative of the expression above to zero yields an extre-
mum at \ = 7 = 3, and the value at that maximum is 1.76 percent.

- Related to this figure is Ny, the average count of established con-
nections that must be rearranged when a new connection is completed.
Since exactly one established connection is rearranged when the 222
Module requires rearrangement,

Since the Markov model for the 2121 Module has 50 states, a natural
nomenclature is a mapping to a familiar set with 50 elements, also
called “states.” The mapping is shown in Fig. 10, where each U. S.
state represents a set of equivalent network configurations from the
2121 Module. A representative configuration is shown with each state
in Fig. 10. The states are grouped according to the number of estab-
lished connections, or their relationship to states in the generalized
model. FL represents the idle Markov state and the six New England
states represent six Markov states in which all four inputs connect to
all four outputs. States in three intermediate east-to-west bands across
the U. S. A. represent Markov states with one, two, and three estab-
lished connections, respectively. To avoid clutter in Fig. 10, the tran-
sitions among the states are shown in later figures.

The upward transitions in the model for the 2121 Module, in which
no rearrangements are required, are shown in Fig. 11. Consider the
three upward transitions from FL. The transition from FL to GA
represents establishing, in an originally idle module, an A-to-Z (or C-
to-W) connection diagonally across the module. This connection has
only one path through the module, and that path requires the central
switch in the crossed state. By contrast, A-to-W (or C-to-Z) connec-
tions have two paths through the module, represented by NM and AZ.
A direct upward transition from FL to AZ, and none from FL to NM,
demonstrates the preference for the path that avoids the central
switch. Similarly, the direct upward transition from FL to AL, and
lack of same to LA or TX, demonstrates the choice of the path for an
A-to-X connection that avoids the central switch over the other two
paths. Similar logic governs the other upward transitions on the graph.
For simplicity, the weights on the arcs are not shown, but they are
similar to those of the 222 Module. |

The upward transitions, in which one or two established connections
require rearrangement, are shown in Fig. 12. The three states with
one established connection from which rearrangements -may be re-
quired—LA, TX, and NM—have no direct upward transitions from
FL, while the only three states that can be reached directly from FL—
GA, AL, and AZ—have no upward transitions requiring rearrange-
ment. Two of the states with two established connections from which
any rearrangements may be required—TN and KS—have no upward
transitions from any state with one established connection. The other
five states with two established connections from which any rearrange-
ments may be required—VA, IL, AR, IA, and NE—have upward
transitions from states with one established connection, but only from
LA, TX, and NM, the states that cannot be reached directly from FL.
The transitions from IA to MD, KS to SD, and KS to ND are
particularly interesting because both established connections require
rearrangement before the new connection, corresponding to the tran-
sition, can be completed. Since the states from which rearrangement
is required are reached only through complex sequences of connection
and disconnection, we expect P,, to be small.

The downward transitions in the model for the 2121 Module are
shown in Fig. 13. The rates on all the arcs connecting the states, as in
the two previous figures, are not shown.

The rate equations for each of the 50 states are given in Appendix
A. A closed-form solution to this set of equations has not been found,
but approximations were derived by a simple, and tedious, iteration.

The states are initially distributed equilikely within their equiva-
lence class such that the sum of the state probabilities within each
class equals the corresponding state probability in the generalized
model. Thus,

Then, temporarily deleting (1 + A)* from every denominator, these
expressions were substituted into the right side of the rate equations
in Appendix A. The resulting intermediate expressions for each q, are
polynomials in X divided by the (a + bd) term on the left side of the
corresponding rate equation. The long division was effected and the
quotient was truncated to a simple polynomial in ), a new expression
for each q,. Most of these new expressions had coefficients between
double and half their analogs in the original expressions.

These new expressions were then substituted into the rate equations,
and a similar simplification and approximation was effected. By the
third iteration, the expressions were surprisingly close to those of the
second iteration, and rapid convergence was observed. These approx-
imations to the steady-state probabilities, without the long division
and quotient truncation in the final iteration, are given in Appendix
B. Of particular interest, again, are the steady-state probabilities of
the ten malevolent states.

P,, is the proportion of those new connections that require rear-—
rangement of an (at least one) established connection before the new
connection may be completed. The numerator is the sum over all
states of (the average count of new connections requiring rearrange-
ment from state i) X (the steady-state probability of state 1). For the
2121 Module, the numerator is

. = (2.1d* + 4.8d° + 1.0X? — 1.1A)/(1 + 38dA)(1 + A),
using the approximate expressions from Appendix B. The denomi-
nator, 4/(1 + A), is the weighted average count of possible new
connections, as in the calculation from the previous section. The ratio
is then

The curve for this expression is plotted in Fig. 14 through the data
points for the 2121 Module, taken from the scatter diagram of Fig. 6.
Setting the derivative of the expression above to zero yields an extre-
mum at 7 = 2.6, and the value at that maximum is 3.1 percent.

Related to this figure is, N,, the average count of established
connections that require rearrangement when a new connection is
completed. Since this count is two on three of the state transitions,
this figure is not equal to P,,:

This expression is also plotted in Fig. 14. We observe that the Markov
analysis of such an innocent-looking network is surprisingly complex.

Two algorithms for a switching rule are by direct calculation or by
table look-up. The direct calculation of the switching rule for either
module would be time-consuming, with the switching rule for the 2121
Module significantly more complicated than that for the 222 Module.
The count of connection combinations in either module is also large,
with little difference between the two modules, so a table look-up
implementation of the switching rules would require a large ROM.

Two distinct realizations of such an algorithm are by a microcon-
troller per module or by a common controller governing all modules

in a network. Direct calculation in a common controller of a large
network suggests a real-time bottleneck and table look-up in a per-
module algorithm suggests considerable replication of a large ROM.
Therefore, the logical choices are distributed control by direct calcu-
lation in a per-module microcontroller, or centralized control by table
look-up. The count of network configurations in a table look-up
algorithm is discussed in this section.

The Markov diagram of the 222 Module is repeated in Fig. 15, except
that an additional number is placed in each bubble, the count of unique
network configurations that are represented by the Markov state.
State I represents the only idle network configuration. Considering
configurations with a single established connection, since any of four
inputs can be connected to any of four outputs, and each connection
has two paths through the network, state J represents 4 x 4 x 2 = 32
network configurations.

States S through V represent all configurations with two established
connections. A configuration in state S derives from a configuration
in state J by connecting the only other input on the same input switch
as the established connection to the only other output on the same
output switch as the established connection by the only available path.
Thus state S also represents 32 configurations. A configuration in
state T derives from a configuration in state J by connecting either
input on the opposite input switch as the established connection to
either output on the opposite output switch as the established connec-
tion by the path using the unused middle switch. Thus state T
represents 32 X 2 X 2 = 128 configurations. A configuration in state
U derives from a configuration in state J by connecting either input
on the opposite input switch as the established connection to either
output on the opposite output switch as the established connection by
the path sharing the used middle switch. Thus state U represents
32 X 2 X 2 = 128 configurations. A configuration in state V derives
from a configuration in state J by connecting the only other input on
the same input switch as the established connection to either output
on the opposite output switch as the established connection, or either
input on the opposite input switch as the established connection to
the only other output on the same output switch as the established
connection by the only available path. Thus state V represents 32 x
(2 + 2) = 128 configurations.

States W and X represent all configurations with three established
connections. A configuration in state W derives from a configuration
in state S by connecting either input on the unused input switch to
either output on the unused output switch by either available path, or

it derives from a configuration in state T or U by interconnecting
either of two specific I/O pairs by the only available path. The specific
pair share both an input switch and an output switch with either
established connection. Thus state W represents 32 X 2X 2X 2=128
xX 2 = 256 configurations. A configuration in state X derives from a
configuration in state U by connecting either of two specific I/O pairs,

those that share an input switch with one established connection and
an output switch with the other one, by the only available path. Each
also derives from configurations in state V by connecting either idle
input to either idle output by the only available path, but each derives
from two different configurations in state V. Thus state X represents
128 X 2 = 128 X 2 X 2/2 = 256 configurations.

States Y and Z represent all configurations with four established
connections. Each configuration in state Y derives from a configura-
tion in state W by connecting the only idle I/O pair by the only
available path. Each configuration in state Z derives from a configu-
ration in state X by connecting the only idle I/O pair by the only
available path. Thus state Y represents 256 configurations and state
Z represents 256 configurations.

The Markov diagram of the 2121 Module, showing downward tran-
sitions, is repeated in Fig. 16, except that a number replaces the state
name, the count of unique network configurations that are represented
by the corresponding Markov state.

State FL represents the only idle network configuration. States GA
through AZ represent all configurations with one established connec-
tion. In the four configurations in GA, any of the four inputs connects
to the output on a middle switch diagonally across the module by the
only path. In the configurations in AL, LA, and TX, any of the four
inputs connects to either output on the output switch. Each state
represents 4 X 2 = 8 configurations and the states are distinguished
by which of three paths is used for the connection. In the configura-
tions in NM and AZ, any of the four inputs connects to the output on
a middle switch directly across the module. Each state represents four
configurations and the states are distinguished by which of two paths
is used for the connection. Summing it up, states GA through AZ
represent 3 X 4+ 3 X 8 = 36 configurations.

States NC through HI represent all configurations with two estab-
lished connections. In one subset of these 21 states, the inputs, one
from each input switch, connect to the two outputs on the middle
switches. Each state in this subset—SC, MS, or HI—represents four
configurations, and the states are distinguished by whether the con-
nections are diagonally across the module or directly across by either .
of two similar paths. In OK, either output on a middle switch connects
to either input directly across the module by the best path, and the

other input on that same input switch connects diagonally across the
module to the other output on a middle switch. In CA, either output
on a middle switch connects to either input directly across the module
by the best path and either input on the other input switch connects
directly across the module to the other output on a middle switch by
the worst path. OK represents 2 x 2 = 4 configurations and CA
represents 2 X 2 X 2 = 8 configurations.

In another subset of these 21 states, any of the four inputs connects
to either output on the output switch and the other input on the same
switch connects to any other output by the best remaining path. Each
state in this subset—NC, IN, MO, CO, or NV—represents 4 x 2 = 8
configurations, and the states are distinguished by the path of the first
connection and/or the output of the second connection. In another
subset of these 21 states, either input on the upper input switch
connects to either output on the output switch, and either input on
the lower input switch connects to the other output on the output
switch, and both connections use similar paths. Each state in this
subset—KY, TN, or [A—represents 2 x 2 X 2 = 8 configurations, and
the states are distinguished by the three possible paths. In the final
subset of these 21 states, any of the four inputs connects to either
output on the output switch and either input on the other input switch
connects to any other output. Excluding the cases, covered in the
previous subset, where the paths are symmetric, each state in this
subset—VA, WV, IL, AR, NE, KS, WY, and UT—represents 4 x 2 x
2 = 16 configurations, and the states are distinguished by the path of
the first connection and/or the output and/or the path of the second
connection. Summing it up, states NC through HI represent 4 x 4 +
9X 8+8 xX 16 = 216 configurations.

States DE through AK represent all configurations with three
established connections. In all 16 states, any of four inputs connects
to either output on the output switch, and either input on the other
input switch connects to some other output. Each state represents
4 x 2 X 2 = 16 configurations, and the states are distinguished by the
path of the first connection and/or the path and/or output of the
second connection and/or the input and/or output and/or path of the
third connection. Summing then, states DE through AK represent
16 X 16 = 256 configurations.

States RI through VT represent all configurations with four estab-
lished connections. In the subset containing RI, CT, ME, and VT,
either input on the upper input switch connects to either output on
the output switch, either input on the lower input switch connects to
the other output on the output switch, and both connections use
similar paths. The remaining two inputs connect to the remaining two
outputs, by the best remaining similar paths. Each state represents

2x 2 X 2 = 8 configurations, and the states are distinguished by the
path of the first two connections and/or the parties interconnected by
the last two connections. In MA, both inputs on one input switch
connect to the outputs on the middle switches by the best paths, and
both inputs on the other input switch connect to the outputs on the
output switch. In NH, both inputs on one input switch connect, one
to the output straight across on a middle switch and the other to one
of the outputs on the output switch, using paths like those in VT.
Both inputs on the other input switch connect to similar outputs, but
using paths like those in CT. Both MA and NH represent 2 x 4 xX
2 = 16 configurations. Summing them up, states RI through VT
represent 4 X 8 + 2 X 16 = 64 configurations.
Summing them up, we count a total of

distinct network configurations in the 2121 Module. The module
supports more configurations, but no others are reached using the
prudent rule. We see that the 222 Module has more than 2.5 times as
many reachable configurations as the 2121 Module and, hence, a table-
look-up implementation of a control algorithm would be more complex
for the 222 Module than for the 2121 Module.

Two architectures for a 4 X 4 photonic switching network were
compared by their traffic-handling capacity. Both networks are rear-
rangeably nonblocking. The percentage of sequences requiring rear-
rangement was found to be tolerable for both modules. Thus, both
modules are judged acceptable, insofar as rearrangeably nonblocking
modules are acceptable, and practically indistinguishable in their
traffic performance. The selection of one module over the other may
proceed based on criteria other than traffic capacity, like loss, cross-
talk, or cost of manufacture.

Acknowledgments are extended to Vic Benes, Nick Maxemchuk,
and Krishnan Padmanabhan for their time, encouragement, and useful
discussion and to Eric Grosse for his help with SMP.""

1. G. Broomell and J. R. Heath, “Classification Categories and Historical Development
of Circuit Switching Topologies,” Comput. Surv., 15, No. 2 (June 1983), pp.
95-133.

2. V. E. Benes, Mathematical Theory of Connecting Networks, New York: Academic
Press, 1965.

3. R. C. Alferness, R. V. Schmidt, and E. H. Turner, “Characteristics of Ti-Diffused
Lithium Niobate Optical Directional Couplers,” Appl. Opt., 18, No. 23 (December

tor for Time-Division Multiplexing and Data Encoding,” J. Lightwave Technol.,

. L. McCaughan, “Design and Performance Limitation of Integrated Electro-Optic
Cross Points,” Proc. IEEE Globecom, Alanta, November 1984, pp. 878-9.

. J. E. Watson, “Polarization-Independent 1 xX 16 Optical Switch Using Ti:LiNbOs
Waveguides,” Conf. Optical Fiber Commun., San Diego, February 1985, p. 110.

. H. S. Hinton, “A Nonblocking Optical Interconnection Network Using Directional
Couplers,” Proc. of IEEE Globecom, Atlanta, November 1984, pp. 885-9.

11. S. Kobayashi and T. Kimura, “Semiconductor Optical Amplifiers,” IEEE Spectrum,
21, No. 5 (May 1984), pp. 26-33.

14. M. Schwartz, Computer-Communication Network Design and Analysis, Englewood
Cliffs, N.J.: Prentice-Hall, 1977.

16. V. E. Benes, “Programming and Control Problems Arising from Optimal Routing
in Telephone Networks,” B.S.T. J., 45, No. 9 (November 1966), pp. 1373-438.

17. SMP, A Symbolic Manipulation Program, Primer and Summary, Inference Corp.,

Richard A. Thompson, B. S. (Electrical Engineering), 1964, Lafayette
College; M. S. (Electrical Engineering), 1966, Columbia University; Ph.D.
(Computer Science), 1971, University of Connecticut; AT&T Bell Laborato-
ries, 1963-1968, 1977—. From 1963 to 1968 Mr. Thompson was involved in
switching systems development at AT&T Bell Laboratories. He was a member
of the Electrical Engineering Department at Virginia Polytechnic Institute
from 1971 to 1977, achieving the rank of Associate Professor. Since 1977 Mr.
Thompson has been a member of the Digital Systems Research Department
at AT&T Bell Laboratories, with a brief stint in ABI/ATTCP in 1983. His
research interests are probabilistic formal languages, fault tolerance and
cellular automata, terminals and the human-machine interface, communica-
tions switching systems, and, most recently, photonic switching. Mr. Thomp-
son is an active participant in IEEE Computer and Communications Societies.
Senior Member, IEEE.

In the present paper we use transform methods (characteristic function
techniques) and contour integrals to derive a closed-form expression for the
performance union bound of a general discrete-time system. We show that
previously published results may be derived as particular cases of the general
formulation developed in this paper. It is well known that the maximum-
likelihood Viterbi algorithm may be employed not only for decoding of con-
volutional codes but also for optimal detection in other situations. Examples
include bandwidth-efficient demodulation, optimal accommodation for inter-
symbol interference and cross-channel coupling, text recognition, simultane-
ous carrier phase recovery and data demodulation, digital magnetic recording,
nonlinear estimation and smoothing. The union bound is a useful measure of
the performance of the Viterbi algorithm. Past closed-form expressions for
the union bound have usually involved considerable approximation.

A Maximum-Likelihood Receiver (MLR) is optimal when the input
signal is distorted not only by noise but also by some deterministic
factors. The MLR compares the received signal with all possible
signals distorted by the same deterministic factors but not by the
noise. The latter comparison signals must be available at the receiver.
Possible deterministic impairments include intersymbol interference,
cross-channel coupling, modem-implementation errors, channel mis-
equalization, signal distortion by the channel nonlinearities, etc.
Guided by some metric, the MLR searches for the comparison signal

Copyright © 1985 AT&T. Photo reproduction for noncommercial use is permitted with-
out payment of royalty provided that each reproduction is done without alteration and .
that the Journal reference and copyright notice are included on the first page. The title
and abstract, but no other portions, of this paper may be copied or distributed royalty
free by computer-based and other information-service systems without further permis-
sion. Permission to reproduce or republish any other portion of this paper must be
obtained from the Editor.

that is closest to the signal actually received and asserts that this
signal, as it existed before being subject to the deterministic impair-
ment, was the transmitted message.

The MLR performance depends on how well we can model the
deterministic distortions of the signal and also on distances between
signals. The signal separation may be increased by coding.

Technical problems arise in MLR implementation. Strictly speak-
ing, we need to store the whole transmission history and generate all
possible comparison sequences. However, if the system may be mod-
eled by a Markov process, then the Viterbi algorithm may be used to
realize a recursive MLR.’

In designing the MLR it is very important to be able to accurately
evaluate the receiver performance. Because of a very large number of
‘possible comparison signals, it is difficult to find the exact formula for
an MLR performance characteristic. The performance characteristic
upper bound (so-called union bound) is more easily found. Viterbi
introduced transfer function techniques to evaluate the union bound
for some performance characteristics of binary convolutional codes.’
These methods have been extended to obtain performance bounds of
the general finite-state system.” However, the original union bound
was loosened to simplify a series summation. Using the transform
methods developed in this paper, the original union bound is expressed
in closed form.

where u, is a source symbol, x; is the transmitted channel symbol, and
s, is the corresponding system (transmitter) state. Symbols x, are
transmitted over a noisy memoryless channel that outputs symbols

where n, are independent identically distributed variables. The re-
ceiver outputs symbols w, = (S;, U,) which minimize the sum

where m(4¥z, wz) is the so-called branch metric and may be treated as
a cost function for making a decision that the transmitter superstate

was w, if y, was received. This metric is usually a measure of the
signal degradation due to noise. For example,

The optimal solution may be found using the Viterbi algorithm. A
sequence Wo, W1, «++ , W;-1, Wj, Which terminates at time J, is called a
survivor if it minimizes the metric accumulated to this time:

J 7 j
DY Myr, We) = Mini...) L M(Yes Wa),
k=0 k=0
where w; = W;. It is obvious that the sequence that minimizes the total
sum M(y, w) must begin with one of the survivors. The survivors may

be determined recursively via the Viterbi algorithm: Wo, Wi, --- , W,;
W +1 is a survivor if and only if Wo, Wi, «++ , Wj-1, Ww; iS a Survivor and

If all the survivors have a common part, then this part also belongs
to the sequence that minimizes the total metric M(y,w) and the
corresponding symbols are output by the Viterbi receiver.

Depending on the application, we may want to evaluate the Viterbi
receiver performance using some distortion measure d = E{d(wz, wz)},
where d(w;, W,) is the distortion characteristic of the symbol w; which
the receiver identifies as w,. For example, if we wish to find a symbol
error probability, then the distortion characteristic d(w,, W,) is equal
to zero if there are no errors in the symbol (u, = &) and is equal
to one otherwise. If we wish to evaluate a bit error probability, then
d(w,, W,) = m,/b, where m, is the number of bit errors in the symbol
u, (the Hamming distance between u, and u,), and 6b is the total
number of bits in the symbol.

In order to find the distortion measure union bound, consider an
error event of length L (see Ref. 1) that is a pair of the correct path

oy = {w,} and an incorrect path o, = {w,} such that s, # §, for k = 1,
2,---,L—1 and s = §, otherwise.
Then the distortion measure is upper bounded by the union bound:?

where the average is taken over all source sequences, P;(c,/c,) is the
conditional probability that the incorrect path o, has a larger metric
than the correct path oz, and 0.5 the prOnen ha of equality of metrics
if a tie is resolved randomly:

Consider first a discrete memoryless channel. Define a generating
function of a variable Am(yp, Wz, We):

D(z; we, We) = X Prfy,/weper Powe), (4)
Yk
The generating function of the sum of variables Am(y;, wz, Wz) is
equal to the product of the generating functions (4), and the probability
P,(¢,/o,) is equal to the sum of the coefficients of the positive power
terms of the product and one half of the zero power term. Using
contour integrals, we may express the sum as

where p > 1. The total distortion along the incorrect path o, may also
be expressed using generating functions:

CE {u:Rg N |v| > 1}, Re is the region of convergence of (8). The
right-hand side of the inequality (7) is a new expression of the union
bound of the average distortion d.

The generating function G(z, v) may be found from the system state
transition graph with branch weights

or the corresponding matrix equation.’° The system symmetry sim-
plifies the construction of the generating function.

In the case of a continuous channel we may also use a transform
(characteristic function) technique to obtain a union bound similar to
(7). For example, if h(x, n) = x +n, where n is a zero-mean Gaussian
variable with one-sided spectral density No (AWGN) and Euclidean
metric is used by a Viterbi algorithm, then (3) takes on the form

Equation (12) is similar to (5), and therefore we may derive a new
union bound for the analog case that is similar to (7):

where “+” denotes mod2 addition (exclusive-or).
The second equation of the system (1) 5,4; = g(w,) may also be
expressed as the system

However, usually it is represented by the system state graph whose
nodes correspond to the system states, and the symbols along the
transition lines indicate the encoder input symbols (see Fig. 2).

The next step is to define the algorithm branch metric. Suppose
that we received a sequence yo, V1, --: , Yn. Chen the MLR will output
the sequence i, t, --- , &y, which maximizes the likelihood proba-
bility |

where z is the number of bit errors in the sequence xo, %1, --: , Xn OF,
in other words, the Hamming distance (HD) between the sequences:
z2=HD{(yo, --+ , yn); (x0, «++ » Xn) }. Since 0 < (p/q) < 1, maximization
of the likelihood is equivalent to minimization of the HD. Therefore
we may define the algorithm metric as m(w, w) = HD{y; x}. This
metric depends only on the signal difference.

The channel model may be expressed by eq. (2), which takes the
form yz = x, + nN», where nz = (nz, nz) has the following distribution:

We wish to evaluate the decoder output error probability. Therefore
the distortion characteristic is

This distortion characteristic depends only on the signal difference.

We assume that the all-zero sequence is transmitted. The generating
function (4) now takes on the form |

where w = (0, 0, 0), u(w) = HD{(0, 0); x} is the Hamming weight of x
= f(w). The distortion measure d(w, w) = t; therefore 24”) = z"
and

If we use these values as branch weights on the system state graph,
where the all-zero state is split into an initial and final state’ as shown
in Fig. 3, we obtain

as the transition from the initial state to the final state.
The union bound for the bit error probability is found from (7):

In this example we find the union bound on the bit error probability
for the same convolutional code but with soft-decision decoding.
Suppose that symbols x, are BPSK modulated. The receive demodu-
lator is followed by a soft-decision sampler with infinite precision.

is the Euclidean distance (11) between the signals and 6? = E,/Np is
the signal-to-noise ratio. |

It is clear that the maximum likelihood corresponds to the minimum
of the Euclidean distance and vice versa. Therefore we may define the
MLR metric as m(wW, w) = ||x — x|]* = (x’ — x”)? + (x” — x’)*. This
metric depends only on the signal difference.

Using the same arguments as in the previous example, we may
assume that the all-zero sequence was transmitted. According to (10)
and (13)

where a = 1 — oe", (B= vin 2), and g = Qe" (1 — e~”), we may
obtain better approximation. For n = 2
Pp < 0.5a77[A erfc(8V5) +B erfc(BV6) + C erfe(BV7)],

Using contour integrals we have derived a closed-form expression
of the union bound on Viterbi algorithm performance. The transform
methods developed in this paper may be used for some other applica-
tions. For example, using eq. (12), a sum

may be expressed as
_ 2B aa ren a
du,
(B? + v2).
where F(z) = )) a,z” is the sequence generating function.
: .

The union bound on the performance characteristic of hard- and
soft-decision codes was found as an illustration.

The author wishes to thank D. Leed and the referees for their
valuable suggestions and comments.

1. A. J. Viterbi, “Convolutional Codes and Their Performance in Communications
Systems,” {EEE Trans. Commun. Tech., COM-19 (October 1971), pp. 751-72.

2. J. K. Omura, “Performance Bounds for Viterbi Algorithms,” ICC’ 81 Conf. Record,
Denver, June 1981, pp. 2.2.1-.5.

3. G. D. Forney, Jr., “Maximum-Likelihood Sequence Estimation of Digital Sequences
in the Presence of Intersymbol Interference,” IEEE Trans. Inform. Theory., IT-
18 (May 1972), pp. 363-78.

4. F. Oberhettinger and L. Badii, Tables of Laplace Transform, New York: Springer-
Verlag, 1973, p. 16, eq. 2.34.

5. K. A. Post, “Explicit Evaluation of Viterbi’s Union Bounds on Convolutional Code
Performance for the Binary Symmetric Channel,” IEEE Trans. Inform. Theory.,
IT-23 (May 1977), pp. 403-4.

William Turin, M.S. (Mathematics), 1958, Odessa State University; Ph.D.,
(Mathematics), 1966, Institute for Problems of Mechanics, Academy of Sci-
ences, USSR; AT&T Bell Laboratories, 1981—. Mr. Turin has worked at New
York University and was an Associate Professor at the Moscow Institute of
Electrical Engineering and Telecommunications. His research activities in-
clude modeling satellite communication channels, simulation, design and
analysis of error-correcting codes, and analysis of Markov processes. He is the
author of three books. Senior member, IEER.

We describe a new File Transfer Protocol (FTP) that provides a simple and
efficient way of transferring files between heterogeneous systems, such as the
UNIX™ operating system, Duplex Multi-Environment Real-Time (DMERT),
and IBM/MVS. This FTP has been adopted as an AT&T standard. In this
protocol, global functions requiring close coordination are separated from local
functions. Functions that require global coordination are mandatory parts of
the protocol and must be implemented uniformly. Local functions, such as file
management and user interface, can be adjusted to local needs and might even
be optional. This results in a flexible protocol that can be implemented at
various levels of complexity. Thus, FTP implementations can range from very
simple ones that provide basic file transfer service to highly complex ones that
provide extensive security checking and allow a variety of file management
services.

File transfers typically represent a large percentage of the traffic
volume on data networks. In one internal AT&T network, for example,
over 50 percent of the data volume is file transfer traffic. Thus, it is
important that file transfers be implemented through an efficient,
flexible, and reliable protocol.

the functionality of all file transfer protocols previously used within
AT&T. This protocol, called the BX.25 File Transfer Protocol (or
simply the FTP), provides a simple and efficient way to transfer files
between heterogeneous systems such as the UNIX operating system,
Duplex Multi-Environment Real-Time (DMERT),’ IBM/MVS, and
UNIVAC/EXEC. This FTP has been adopted as an AT&T standard.
Several groups in the company have implemented or are implementing
this protocol.’

Our approach in designing the FTP was to separate global functions
requiring close coordination (such as parameter negotiation) from
functions that could be done locally at each individual node. Functions
that require global coordination are mandatory parts of the protocol
and must be implemented uniformly. Local functions, such as file
management and user interface, can be adjusted to local needs and
might even be optional. This results in a flexible protocol that can be
implemented at various levels of complexity. One can view the FTP
as essentially an intelligent bulk data transfer facility.

The FTP is a three-party protocol in which a user at a remote node
(the initiator) can transfer files between a source and a destination
node. The FTP offers three types of services: Copy, Cancel, and Status.
The Copy service allows the transfer of a set of files between the
source and destination nodes. The Cancel and Status services, respec-
tively, allow the user to cancel a transfer and request information
about a transfer. All services can be done in the background, requiring
minimal user supervision.

Our approach in specifying the FTP is consistent with the one
currently recommended by the protocol community. This approach
recommends that one first describe the services offered by the protocol
to the upper layer, as well as the services required by the protocol
from the lower layer. Then, it defines the peer-level protocol that fills
the gap between the upper and lower layers. Typical protocol specifi-
cations, such as levels 2 and 3 of X.25,* cover only Peer: -level message
exchange rules and formats.

The application program resident at each node that performs file
transfers will be called a File Transfer System (FTS). Functions
performed by the FTS cover the application and presentation layer
functions as defined in the Open System Interconnection (OSI)
model.°

As shown in Fig. 1, an FTS has several interfaces: an interface with
the user, a data transfer facility interface, and interfaces with the file
system and with the operating system. In addition, an FTS commu-
nicates with the remote FTSs using a peer-level protocol. The user is
a person or a computer program that wants to use the file transfer
service. The data transfer facility is the lower level of protocol, such

as the BX.25 session layer. The file system is the local storage system
for files, and the operating system manages the local computing
resources.

In Section II, we describe our overall design approach in greater
detail. We then describe services offered by the FTP in Section III,
and identify the services required from the data transport layer in
Section IV. The peer-level protocol is described in Section V. Section
VI describes the negotiation procedure during which the source FTS
and the destination FT'S agree on the options for file transfers. Some
examples are given in Section VII. We also describe a formal specifi-
cation of the FTP using the selection/resolution model in Section
VIIL.®’ This specification gives a more complete and precise descrip-
tion of the protocol than an English language specification alone can
provide. In Section [X, we compare the FTP with others reported in
the literature. Finally, we make some concluding comments in Section
X.

Il. DESIGN APPROACH
In our design, we have clearly separated functions requiring global

coordination from those that are local to a node. Every implementation
would be required to implement the global coordination functions that
would form the core of the protocol. Implementors would be free to
implement the local functions as needed. Implementations would thus
range from very simple ones that provide basic file transfer services
to highly complex ones that provide extensive security checking, allow
a variety of file management capabilities, and provide a sophisticated
user interface. Furthermore, the flexibility in implementing local func-
tions results in a file transfer protocol that works in a heterogeneous
environment, is adaptable to diverse local needs, and can evolve as
requirements change. ,

The core of the protocol implements the coordination functions to
transfer a byte stream efficiently. Translation between the file formats
at the source and the destination is done outside the core of the
protocol by local functions called preprocessors and postprocessors.

For example, if the initial requirements are file transfers between a __
UNIX system and an IBM/MVS system, we need only write pre- and
postprocessors for translation between their file formats. As additional
systems, such as a Honeywell computer, are added, we will then add
new pre- and postprocessors.

The FTP does not have a networkwide standard for file naming. A
user must supply all the naming information required for accessing a
remote file. This includes the information about the device name on
which the file is stored, as well as the path name of the file. This

The FTP uses checkpointing to transfer a byte stream efficiently.
During a file transfer, checkpoints are set along the way. After recovery
from a transmission failure, the file transfer is resumed from a check-
‘point up to which the destination has received the file data correctly.
The file transfer does not have to be restarted from the beginning.
This feature is useful when transferring very large files such as those
stored on several magnetic tapes.

Our security policy makes it difficult for a user to break into the
security mechanism of a remote file system. To access a remote file,
the user must provide access information required at the remote file
system. The user without the proper access information is not allowed
to access a remote file. In the following sections, we give further details
of the protocol.

To invoke a service, the user must issue a Sominand to ie local
FTS. The syntax of the command is a local decision. However, the
information contained in the command is required for global coordi-
nation. Thus, in any implementation of the FTP, each command must
contain the information given below.

For the Copy service, the command must contain the following
information:

(File descriptor at the source) (Preprocessors)
{Destination address)
(File descriptor at the destination) (Postprocessors)
(Priority)

where
(User id) identifies the user initiating the file transfer,
(Source address) identifies the computer on which the files to be
transferred reside,
(File descriptor at the source) is the list of file names to be
transferred,
(Pre-processors) is a list of programs executed on the files before
transfer,
(Destination address) identifies the computer to which the files
are to be transferred, _
(File descriptor at the destination) is the list of file names at the
destination. The number of files in this list must be the same as
that in the list (File descriptor at the source),
(Post-processors) is a list of programs executed at the destination
on the files transferred,
(Priority) specifies the priority level assigned to the copy com-
mand. This field specifies how the FTS should treat the present
job compared to the other file transfers waiting to be done at the
local FTS.

The initiator FTS checks the syntactic correctness of the copy
command. The number of file names in the list (File descriptor at the
source) must be the same as that in (File descriptor at the destina-
tion). If there is an error in the copy command, then the FTS returns

an error message to the user. Otherwise, the FTS returns a job number
to the user. The job number identifies a copy command throughout
the network and the command’s lifetime. The format for a job number
ensures that it is unique throughout the network. The job number has
the following format:

where
(Initiator address) specifies the address of the initiator,
(Unique number assigned by the initiator FTS) is a number
assigned to the command; it uniquely identifies the command at
the initiator.
The local FTS must notify the user upon successful or unsuccessful
completion of the file transfer.

A user at the initiator FTS can inquire about the status of a file
transfer request using the following command:
STATUS (Job number).

e Whether preprocessing, file transfer, and postprocessing have
started, and, if they have, whether they have been completed
successfully or unsuccessfully;

in progress. For such a service, a user issues the following command:
CANCEL (Job number). |

The connection-oriented service must transfer messages of variable
length from a transmitter FTS* to a receiver FTS in error-free form
and in the same sequence as originally delivered to the transmitter
FTS. If it is unable to deliver messages in sequence because of a
network failure, it must inform the transmitter FTS. Once a connec-
tion is set up between two FTSs, it can be used to transport the files
from several file transfer requests. The following service primitives,
or their equivalent, must be provided to the FTS:

1. Connect (Destination Address, FTS). Using the Connect primi-
tive, the FTS in the local computer establishes a connection with an
FTS resident in the computer with the address Destination Address.
If the connection is successfully established, the data transport entity
returns to the FTS Connection id, a unique identifier. The FTS uses
Connection id later to reference the connection just established. If the
data transport entity is unable to establish the connection, it notifies
the local FTS about the failure.

2. Send (Connection id, message). This primitive is used to send a
block of data called message on the connection Connection id.

3. Receive (FTS, message, Connection id). The local FTS uses the
service primitive Receive to receive a block of data message from the
connection Connection id.

4, Disconnect (Connection id). The FTS uses the Disconnect service
primitive to break the connection Connection id. The data transport
facility delivers any messages in transit: before the breakup.

5. Abort (Connection id). This primitive is used to abnormally break
up the connection Connection id. The data transport facility does not
ensure delivery of the messages in transit before the breakup.

A Connectionless service is useful for the transport of single mes-
sages between FTSs. It ensures that such message exchanges are done
with minimal network resources. Establishment and termination of a
connection, such as a BX.25 session, require a significant amount of

* A transmitter FTS is a program that establishes connection with a receiver FTS to
transfer a sequence of messages. A transmitter FTS can be either the initiator or the
source. Similarly, the receiver FTS can be either the source or the destination.

network resources, which is not justified for short message exchanges.
A Connectionless service must provide the following service primitives:

1. Send (Destination Address, message). This primitive is used to
send a block of data called message to the FTS in the computer with
the address Destination Address.

2. Receive (message). The local FTS uses this primitive to receive a
block of data called message from the data transport entity.

Since the peer-level protocol gives the rules for interaction between
any pair of FTSs, it is the core of the global coordination functions.
The rules and the formats specified in this section must be uniformly
implemented in all FTSs. |

All fields in the peer-level messages have been encoded using the
International Organization for Standardization (ISO) communication
heading format standard.**® Figure 2 shows a general message format.
The first field, the message type field, identifies the message type; it
is one byte long. Encodings for various message types are given in
Table I. The first field is followed by one or more parameter fields,
each of which carries a parameter, such as a file descriptor. The first
byte uniquely identifies the parameter type, such as file descriptor.
Codes for various types of parameters are given in Ref. 2. Each
parameter field can be of variable length, with a maximum of 255
bytes. The list of parameters is followed by an end of heading field,
H’80’.* The last field contains data for a file data message. |

*H stands for hexadecimal. H ’xx’ is used to represent a two-digit hexadecimal
number.

Message
Message Message Type
Type Name Code
Command Authenticated COPY H’24’
command, CAC
Negotiation H’31’
package, CNEG
Checkpoint ‘H’32’
command, CCM
Inquiry about H’33’
checkpoint command, CIC
Positive confirmation H’34’
message, CPC
Negative confirmation H’35’
message, CNC
Inquiry command, H’36°
CIO
Cancel command, CCC H’37’
Interrupt command, CIN H’38’
Restart command, CRS H’39’
Responses Acknowledgment, RAC H’41’
Reply negotiation H’42’
package, RRN |
Checkpoint response, H’43’
RCH
Status Response, RSR H’44’
Cancel Response, RCR H’48’
Notification message, RNM H’46’
Data File Header H’01’
Messages Message, DHM | .
File Data H’03’

The rules for message exchanges for the Copy and Status services
are given in Sections 5.2.1 and 5.2.2, respectively. The rules for the
Cancel service are similar to those for the Status service and are not
given here.

e Transfer of an authenticated copy command from the initiator
FTS to the source FTS (if the initiator FTS is not the same as
the source FT'S),

access privileges of the user initiating the file transfer and the syntactic
correctness of the copy command. If the user does not have the

necessary access privileges and the syntax of the copy command is
incorrect, the initiator FTS returns an error message to the user and
the file transfer is abandoned. Otherwise, the initiator ships the copy
command to the source only if the initiator is different from the source
in a peer-level message called the Authenticated Copy Command
(CAC). After sending the CAC, the initiator FTS waits for an acknow]-
edgment from the source FTS. If it does not receive an acknowledg-
ment within a time-out period FT1, the initiator will again send the
message CAC to the initiator. The initiator makes at most Al attempts
at sending the authenticated copy command.

After receiving an authenticated copy command, the source FTS
sends an acknowledgment to the initiator FTS and then enters the
negotiation phase. During the negotiation phase, the source and des-
tination FTSs negotiate about options for the file transfer, such as
selection of pre- and postprocessors, the values of the intercheckpoint
interval and the window size for checkpointing. Intercheckpoint inter-
val and window size are defined later in this section. Further details
about the negotiation phase are given in Section VI.

If the negotiation phase is successful, the source and destination
FTSs enter the file transfer phase. Each file is transferred completely
before the transfer of the next file is initiated. The source FTS sends
the following sequence of peer-level messages to the destination FTS
for each file transmitted:

e File header message: the file header message carries the name of ©
the file in which the file to be transferred will be stored, the file’s
attributes, such as code set type and file length,

e One or more checkpoint commands (a checkpoint command is a
peer-level message which marks a checkpoint during a file trans-
fer).

The checkpoint commands are assigned sequence numbers. Since
the sequence numbers cannot be infinitely large, a cyclically reusable
sequence numbering scheme is used. We take the sequence range to

be 0 to (2°? — 1), where 2* is the size of the sequence space. The
number of outstanding checkpoint commands is limited to w, the
window size for checkpointing. The checkpoint command and the
checkpoint response carry the sequence number of the checkpoint.
The source’s reception of a checkpoint response with the sequence
number x means that the destination FT'S has received everything
correctly up to the checkpoint assigned the sequence number x.

End of File (EOF) is marked by sending a checkpoint command.
The source FTS must receive the checkpoint response to the check-
point command indicating EOF before it initiates the transfer of the
next file. The checkpoint command indicating EOF for the last ele-
ment in the file transfer request must also indicate whether additional
‘file data will be transferred during the current session.

Now, we give the rules for the Status service. A user can issue a
status command at the initiator FTS to inquire about the status of a
file transfer identified by a job number. The initiator FTS checks if it
has sent out the authenticated copy command corresponding to the
job number given in the status command. If it has not, the initiator
FTS informs the user that preprocessing, file transfer, and post-
processing have not started. Otherwise, the initiator FTS sends an
inquiry command to the source FTS. The source FTS checks to see if
it has completed preprocessing and file transfer. If not, it sends a
status message to the initiator FTS, stating the steps it has already
taken and the step presently in progress. If the source F TS has already
completed preprocessing and file transfer, then it sends an inquiry
command to the destination FTS. On receiving the inquiry command,
the destination FTS sends a status message indicating whether post-
processing has been completed successfully or unsuccessfully. The
source FTS combines this information from the destination FTS with
the status information it has about preprocessing and file transfer.
The source FT'S sends a status message to the initiator FTS. The
status message contains the following information:

A file transfer using the FTP has available a choice of several
options. For example, checkpointing may or may not be used during

file transfer; if checkpointing is used during a file transfer, then several
parameters, such as the intercheckpoint interval and the window size,
must be agreed upon between the source and the destination. These
parameters will determine the buffer requirements during the file
transfer at the source and the destination. Another parameter to be
chosen is the list of postprocessors to be used at the destination.

Before the file transfer phase starts, the source FTS and the desti-
nation FTS must negotiate regarding the parameters to be used during
the file transfer and come to an agreement. For this negotiation, a
three-way message exchange between the source FTS and the desti-
nation FTS is used, as shown in Fig. 3. _ :

In the first step, the source FTS sends a negotiation package to the
' destination and then waits for a reply negotiation package. The
negotiation package contains the following information about the
parameters to be used during the file transfer: | |

e Block size. This field specifies the size of the file data transported
in one file data message.

e Intercheckpoint interval. This field specifies in units of bytes the
amount of file data transmitted. between two checkpoints. This
field is required only if the checkpointing option is chosen.

e Window size for checkpointing. This field contains the maximum
number of outstanding checkpoint intervals at the source FTS. If
this field is blank, then the default value is 1.

In the second step, the destination FTS examines the options for
file transfer after receiving a negotiation package from the source
FTS. The destination FT'S decides whether it can accept them. If the
destination FTS cannot accept some or all options, it selects alterna-
tives that it can accept. It puts acceptances, rejections, and alternatives
in a response negotiation package and sends them to the source FTS.
A reply negotiation package contains the following information:

In the third step, on receiving a reply negotiation package, the
source FTS replies with a positive confirmation message if the desti-
nation FTS accepted all options or if the source FTS can accept the
alternatives given in the reply negotiation package. Otherwise, the
source FTS will send a negative confirmation message to the desti-
nation FTS.

If the source FTS does not receive a reply negotiation package from
the destination FTS within a time-our period FT2 after sending a
negotiation package, it sends the negotiation package again. If the
source FTS does not succeed within A2 attempts, it informs the
initiator FTS about the failure and aborts the file transfer.

One parameter that the source and destination FTSs might negoti-
ate on is the time when the file transfer should be started. The FTP
described here does not negotiate on the starting time of the file
transfer, but it can be easily extended to do so. If the underlying
network only carries file transfer traffic, then.such negotiation can be
very useful.? Otherwise, we feel that it should not be used for the
following reason. Such negotiation requires that the FTSs should be
allowed to schedule the future allocation of connections, but this is
not consistent with the OSI reference model approach.” Several appli-
cation layer protocols must share the same network resources. Sched-
uling of the connection allocation, if any, must be done by the lower
layers, not by an application layer protocol, such as the FTP.

For networks that carry file transfers exclusively, negotiation about
the starting time can be very useful. File transfers typically require a
heavy commitment from the network. A connection used for a file
transfer typically utilizes nearly 100 percent of its maximum transfer
rate as defined by its throughput class. On the other hand, a connection
carrying interactive data utilizes only 1 to 2 percent of its maximum
transfer rate. As a result, the number of outgoing and incoming
connections used for file transfers is very small. Scheduling of connec-
tions and file transfers to optimize a cost function, such as network
utilization, or minimizing delay can be very useful.

We give two examples to illustrate the use of the FTP: one for the
Copy service and one for the Status service.
- In the first example (Fig. 4), a user issues a copy command to the
file transfer system in Computer-C to copy a file FSOU from Com-
puter-A to the file FDEST in Computer-B. The file transfer system
in Computer-C (henceforth, referred to as FTS-C) checks the com-

@ AUTHENTICATED COPY FROM FTS-C TO FTS-A
@ NOTIFICATION MESSAGE FROM FTS-A TO FTS-C

mand for its syntactic correctness and checks the access privileges of
the user to find out whether the user is allowed to initiate this file
transfer.

If the copy command has a syntax error, then FTS-C returns an
error message to the user. If the user has the proper access privileges,
then FTS-C returns a job number to the user. The job number is a
unique identifier assigned to the copy command. It uniquely identifies
the command throughout the network. The format for the job number
is given earlier in Section III.

FTS-C sends an authenticated copy command to the FTS in Com-
puter-A (henceforth, referred to as FTS-A). Then, it waits for an
acknowledgment from FTS-A. If FTS-C does not receive an acknowl-
edgment within the time-out period FT1, then it retransmits the

FTS-A sends a file header message to FTS-B. A file header message
contains the information such as the file length and the file name.
Then, FTS-A processes the file data using the preprocessors and
divides it into blocks. Each block is packed into a separate file data
message. Then, FTS-A sends these file data messages to FTS-B. It
also sends the checkpoint commands at every intercheckpoint interval.
If the session being used for the file transfer breaks down, then FTS-
A can resume file transfer from an intermediate checkpoint up to
which FTS-B has received the data correctly. FT'S-B unpacks the file
data in the messages received. Then, it processes the file data received
using the postprocessors agreed upon during the negotiation ees
FTS-B.-then stores the file data in the file FDEST.

After the successful file transfer, FTS-B sends a notification mes-
sage to FTS-A informing it about the successful completion. FTS-A,
in turn, sends a notification message to FTS-C informing it about the
successful completion. Finally, FTS-C informs the user about the
successful completion of the file transfer.

e Similar information about file transfer and postprocessing. The
status can further include additional information, such as how far
the file transfer has progressed.

FTS-C checks the status command for its syntactic correctness. If
the command has a syntax error, then FTS-C returns an error message
to the user. Then, FTS-C checks the access privileges of the user to
determine whether the user can inquire about the file transfer. If the
user has the necessary privileges, then FTS-C checks the local records
to find out whether the file transfer has been completed. Other-
wise, FTS-C sends a status command to the destination FTS, FTS-B.
FTS-B checks the local records to locate the status of the file transfer.
It checks whether it was ever involved in that file transfer. If it was
involved in the file transfer, FTS-B checks whether the file transfer

has been completed or if it is still in progress. FTS-B sends the status
information in a status response message to FTS-A. FTS-A also
collects the local information about the file transfer and combines it
with the information received from the destination FTS, FTS-B. Then,
it sends the status information to FTS-C. Finally, FTS-C delivers the
status information to the user.

description of the protocol. In this section, we first present an informal
discussion of the s/r model; the reader is referred to the references for
more details. We then discuss how the FTP was formally specified
using this approach.

The selection/resolution model is a method in which a complex
system such as a protocol can be represented as a set of coordinating
finite state machines. Each component finite state machine, called a
process, is described by an appropriately labeled directed graph. For
instance, suppose we wished to describe the coordination of two
processes A and B that can be executing segments of code in either a
noncritical section or a critical section. We wish the coordination to.
be such that both processes are not simultaneously in their critical
sections. Figure 6 has two labeled graphs representing the processes.

In the labeled graphs of Fig. 6, the vertices represent states of the
process, and the edges represent possible single-step transitions. Thus,
process A (also B) can be in the states NCS (Noncritical Section),
TRY (trying to enter the critical section), and CS (Critical Section).
A state encapsulates the relevant past of the process needed to deter-
mine future behavior and is known only to that process. The processes
coordinate by exchanging information about their future intentions.
~ That is, they communicate their selections (intentions) to all the other
processes. For example, in state NCS, process A can only choose the
selection ok, while in state TRY it can nondeterministically choose
between head and tail. Note that a process can make only one selection
at any time. Selections of a process are given in curly braces next to
the state.

Each edge has a label next to it. The edge labels are Boolean
conditions based on the selections of all the processes, and determine
the possible transitions of a process. In conditions, + is the same as a
logical or and * is the same as a logical and. For instance, the condition
for process A to go from CS to NCS is simply (A:nok), that is, it really
doesn’t depend on B, and is in fact always true since the only selection
of A in state CS is nok. However, the condition for A to enter the CS
state from the state TRY is [(B:ok) + (A:head)*(B:head) +
(A:tail)*(B:tail)]. In simple words, the condition says that A can go to
the CS state if B selects ok, or if A and B both select head, or if A and
B both select tail. The transition of process A from state TRY to its
next state clearly depends on what B is selecting. Both processes
essentially toss coins to determine who wins if they are both trying to
enter the critical section. The transition of a process to one of the
possible next states (there may be more than one transition enabled)
is called resolution.

In the above example, developed in more detail in Ref. 11, both ©
processes initially start in the state NCS at time-step 0. Then, in each
unit of time, the processes first make a selection based on the state
they are in, and then they resolve (transition to a new state) based on
the selection of all the processes. It can be seen that the processes will
never both be in the states CS at any time step.

The s/r model gives precise algebraic descriptions of processes that
can be combined to give a precise description of the entire system.
Furthermore, many computational aspects of this model can be auto-
mated. In fact, the formal specification can be used to test the
correctness of the FTP by conducting a reachability analysis using the
procedure given in Refs. 6 and 7.

The formal specification of the FTP consists of descriptions of 45
processes divided for convenience into 9 clusters (see Fig. 7). A cluster

is a convenient grouping of a set of processes. We have grouped
processes into clusters based on their functions. Thus, there are
clusters that correspond to the basic service functions such as Copy
service, Status service, and Cancel service. There are also clusters that
correspond to interface functions such as interfacing with the user,
the data transfer facility, and the file system. Finally, there is a main
controller cluster that acts as an overall coordinator, and there are
clusters for the timers and the variables.

Each process is defined in a language that essentially provides the
same information as the labeled graph corresponding to that process.
For example, we can define a process that does a syntax check on the
user command. We model this as the process syntc shown in Fig. 8.
As discussed in Section III, the exact choice of the command syntax
is a local decision. Notice that we model only the portion of the process
behavior that describes global coordination with other processes. The
process syntc makes the selection ok, only if the command contains
all the required information and is syntactically correct. Otherwise, it
selects nok. ?

In this figure we have not indicated the conditions for the self-
_loops.* We shall assume that the self-loop condition is like an other-
wise; that is, it is the negation of the or of the conditions on the other |
edges. The process syntc is defined using the specification language in
Fig. 9. Notice the close correspondence with the graphical specifica-
tion. We use $ as shorthand to signify the current state. In this

IDLE {idle, nok, ok}
> OK :((user:copy) + (user:status) +
(user:cancel)) * (Syntc:ok))
=~ NOK :((user:copy) + (user:status) +
(user:cancel))*(syntc:nok))
>s = ¢
OK {ok}
> IDLE :(usit:idle)
>Ss soe.
NOK {nok}
> IDLE :(usit:idle)
>$ 33

example, the empty condition on the self-loop is equivalent to other-
wise.

Process syntc is one of the processes in the User Interface Cluster.
This cluster consists of four processes: usit, syntc, secur, and jobnum.
Each process’s description is on the average about as complex as that
of syntc. The process usit gets into one of the three states—COPY,
CANCKL, and STATUS—if the user command is syntactically correct
and if the user has the permission to execute the command. The
process secur is similar to syntc; it checks the security privileges of the
user. It makes the selection (secur:ok) if the user can execute the
command. Otherwise it makes the selection (secur:nok). The process
jobnum generates a job number for each copy command. The job
number uniquely identifies a copy command throughout the network.

The full formal specification (about 50 pages long) consists of similar
descriptions of all the clusters. The descriptions of the processes, of
course, vary in complexity. The formal specification can be used as a
precise guideline by FTP implementors. In fact, the high-level design

of the FTP implementation may be readily derived from the formal
description. A development group at AT&T Communications used the
formal specification in their design process. They found it to be
extremely useful in developing a high-level C-like language design of
FTP.

For implementation, each user command can be viewed as invoking
an independent copy of the 45 processes. If an FTS must handle
several file transfer commands at the same time, then an independent
copy of the processes is invoked for each command. Since our FTP is
a three-party protocol, a file transfer at an FT'S can be initiated from
a remote FTS. For a file transfer initiated from a remote FTS, an
independent copy of the processes is invoked at the local FTS. The
FTS can thus be implemented as a reentrant program that can be
used independently by each command.

As discussed previously, our approach here is to separate the file
transfer issues from local service functions. This allows great flexibility
in local matters and avoids proscribing rigid formats for the user
interface, for file management and naming, and for presentation
functions such as code sets. Our FTP differs in this regard from other
file transfer protocols that generally are rather inflexible and less
adaptable. For example, none of the other file transfer protocols has
a general mechanism such as pre- and post-processors that allow
additional capabilities to be incorporated in a standard way. In this
section, we compare the FTP with four file-transfer protocols: Arpanet
FTP,” Network Independent (NI) FTP,’*”* Autodin II FTP,’® and
the International Organization for Standardization (ISO) File Service
Protocol.!’

The Arpanet FTP requires the user (initiator) to establish separate
command and data connections with both the source and the desti-
nation of the file transfer. The user must keep track of these connec-
tions. The user is also involved at too detailed a level with the file
transfer operation; it is not possible to simply copy a set of files from
the source to the destination with a single macro command. It is
expected that the user remain on-line at each phase of the transfer,
and the user must be aware of details such as the socket number of
the data connection. The protocol does provide for both copy and
append services, as well as status notification. File management serv-
ices such as creating and deleting files are also available. Security
measures are fixed and encompass only user name, password, and
account number. There is no provision for adding other local security
options.

The NI FTP is a two-party protocol (no provision is made for third
party initiation of a file transfer) that divides the transfer operation
into three phases. The protocol handles only transfers of single se-
quential files; multiple files must be sent as separate file transfers.
The NI FTP protocol provides for a high degree of flexibility in the
actual data transfer operation. This is accomplished through the
negotiation of a large number of possible alternatives. The possible
alternatives are defined in any particular implementation, and cannot
be readily modified. There is a very limited file management capability
in NI FTP. The protocol assumes only minimal support from the
lower layers, and specifies presentation services. It allows for modular
implementation, but does not have provisions for easily adapting to
new requirements. The protocol provides for only foreground file
transfers.

The Autodin II FTP provides high- level service primitives and
allows background file transfers. However, it goes too deeply into
defining the structure of a file at the network level; this results in the
negotiation of a large number of parameters for the actual data
transfer, rather than allowing this to be handled through local proc-
essing. Furthermore, each of the three parties involved in the file
transfer must keep an elaborate list of parameters that define all the
options that have been agreed upon. There is an extensive security
mechanism, but it is not modifiable. Similarly, the user interface is
very specific, and cannot be adjusted to local preferences. Autodin II
FTP is definitely comprehensive but all the detail is defined at the
network level. This results in a protocol that requires a a complex
implementation at each. node. |

A committee of the ISO is working on the File Access, File Transfer,
and File Management (FTAM) services and on a corresponding pro-
tocol. The FTAM services allow a user to access and transfer a remote
file. They also allow a user to manipulate a file in a remote file system.
An example of a service is a command to open a file in a remote file
system. These are micro services. In contrast, our FTP provides macro
services, such as the copy service. Each service of our FTP can be
provided by a long sequence of the ISO service calls. We have carried
out a study to define these sequences of the ISO service calls.'®

The ISO file service uses the concept of the Virtual File Store. The
Virtual File Store is a common model for describing file names and
their attributes. Different file systems have a wide range of styles for
describing the storage of data and the means by which it can be
accessed. The Virtual File Store allows the differences in style and
specification to be absorbed in a local mapping function. The Virtual
File Store attempts to encompass all possible variations of file defi-

nition, resulting in an excessively detailed description of a file that
cannot be easily modified in the future. —

In the FTP presented here, we separated global functions requiring
close coordination (such as parameter negotiation) from functions that
could be done locally at each individual node. In an FTP implemen-
tation, the global functions must be implemented. Local functions can
be implemented depending on user needs. As a result, a minimal
implementation of the protocol is fairly simple. Furthermore, it re-
quires only incremental efforts to extend the minimal implementation.

We feel that the FTP is flexible, adaptable, and powerful enough to
meet most file transfer requirements. Furthermore, it has been speci-
fied precisely enough so as to make implementation a relatively
straightforward task.

1. M. E. Grzelakowski, J. H. Campbell, and M. R. Dubman, “DMERT Operating
System,” B.S.T.J. 62, No. 1, Part 2 (January 1983), pp. 303-22.

Operations Systems Network Protocol Specification: BX.25 Issue 3 Addendum-A,
AT&T Bell Laboratories, Holmdel, NJ, August 1983.

Operations Systems Network Protocol Specification: BX.25 Issue 3, AT&T Bell
Laboratories, Holmdel, NJ, June 1982.

. A. S. Tanenbaum, Computer Networks, Englewood Cliffs, N.J.: Prentice-Hall, 1981,
Chapter 1, pages 10-21.

. S. Aggarwal, R. P. Kurshan, and K. Sabnani, “A Calculus for Protocol Specification
and Validation.” In Protocol Specification, "Testing, and Verification, III, edited by
H. Rudin and C. H. West, Amsterdam: North-Holland, 1983.

10. S. Aggarwal, R. P. Kurshan, and harma, “A Language for the Specification
and Analysis of Protocols,” In aise Specification, Testing, and Verification,
IIT, edited by H. Rudin and C. H. West, Amsterdam: North-Holland, 1983.

11. S. Aggarwal and C. Courcoubetis, “Distributed Implementation of a Model of
Communication and Computation,” Proc. 18th Hawaii Int. Conf. on System
Sciences, January 1985, pp 206-218.

12. N. J. Neigus, “File Transfer Protocol for the ARPA Network.” In ARPANET
Protocol Handbook, edited by E. Feinler and J. Postel, Defense Communications
Agency, 1978.

13. C. J. Bennet and D. N. Frost, “Network Independent File Transfer,” Tech. Report,
University College, London.

14. R. W. S. Hale, “File Transfer Protocols—Comparison and Critique,” NPL Report
DNACS 48/81, Teddington, United Kingdom, 1981.

15. High Level Protocols Group, “A Network Independent File Transfer Protocol,”
HLP/CP(78), NPL, Teddington, United Kingdom, 1977. |

16: H. C. Forsdick, “Autodin II File Transfer Protocol,” Bolt Beranek and Newman
Inc., Report No. 4246, Boston, 1980.

17. Second Draft Proposal on File Transfer, Access, and Management, ISO/TC97/SC21
No. 8571, 1985. (Available from ANSI)

Sudhir Aggarwal, B.S. (Mathematics), 1969, Stanford University; M.S.
(Mathematics), 1971, and Ph.D. (Computer and Communication Sciences),
1975, University of Michigan; Assistant Professor of Computer Science, Uni-
versity of Oregon, 1975-1976; Research Scientist, Lawrence Livermore Labo-
ratory, 1976-1977; Assistant and Associate Professor of Mathematics, Univer-
sity of California, Riverside, 1977-1982; AT&T Bell Laboratories, 1982—. Mr.
Aggarwal is a member of the Mathematical Sciences Research Center. His
research interests are computer communication protocols, local area networks,
and modelling and simulation.

B. Gopinath, M.Sc. (Mathematics), 1964, University of Bombay; Ph.D.
(E.E.), 1968, Stanford University; Research Associate, Stanford University,
1967-1968; Alexander von Humboldt Research Fellow, University of Gottin-
gen, 1971-1972; Gordon McKay Professor, University of California, Berkeley,
1980-1981; AT&T Bell Laboratories, 1968-1983; Bell Communications Re-
search, 1983—. Mr. Gopinath is Division Manager for Systems Principles
Research, and is engaged in research in communications and computer science.

Krishan Sabnani, B. Tech. (E.E.), Indian Institute of Technology, New
Delhi; Ph.D. (E.E.), Columbia University, New York, 1982; fellowship, School
of Engineering and Applied Sciences, Columbia University, 1977; research
assistant, Columbia University, 1978 to 1979; he was employed by RCA,
Princeton, 1979-1981, AT&T Bell Laboratories, 1981—. Mr Sabnani’s current
interests are computer communication protocols and fault-tolerant computing.
Member, Sigma Xi, Eta Kappa Nu, and Epsilon Pi Upsilon.

Automated protocol validation tools are by necessity often based on some
form of symbolic execution. The complexity of the analysis problem however
imposes restrictions on the scope of these tools. The paper studies the nature
of these restrictions and explicitly addresses the problem of finding errors in
data communication protocols of which the size precludes analysis by tradi-
tional means. The protocol tracing method described here allows one to locate
design errors in protocols relatively quickly by probing a partial state space.
This scatter searching method was implemented in a portable program called
Trace. Specifications for the tracer are written in a higher-level language and
are compiled into a minimized finite state machine model, which is then used
to perform either partial or exhaustive symbolic executions. The user of the
tracer can control the scope of each search. The tracer can be used as a fast
debugging tool but also, depending on the complexity of the protocol being
analyzed, as a slower and rather naive correctness prover. The specifications
define the control flow of the protocol and may formalize correctness criteria
in assertion primitives.

Protocol validation by symbolic execution is inherently a time- and
space-consuming task. For lack of better methods, though, many
automated protocol validation tools do use symbolic execution algo-
rithms,!~° and even methods based on validation algebras such as CCS°
or PVA’® still implicitly formalize symbolic executions.’ Unfortu-

* AT&T Bell Laboratories.
T Cf. the expansion theorem in CCS and the shuffle operator in PVA.

Copyright © 1985 AT&T. Photo reproduction for noncommercial use is permitted with-
out payment of royalty provided that each reproduction is done without alteration and
that the Journal reference and copyright notice are included on the first page. The title
and abstract, but no other portions, of this paper may be copied or distributed royalty
. free by computer-based and other information-service systems without further permis-
sion. Permission to reproduce or republish any other. portion of this paper must be
obtained from the Editor.

nately, the assumption that a computer will always be able to take
over when the complexity of a complete analysis surpasses our ability
to perform algebraic expansions by hand is decidedly wrong.?”°

A protocol of a realistic size can generate a state space of in the
order of 10° system states and up. As little as adding one single
message type, one protocol variable, or one slot to the message queues
can further expand the number of reachable system states by orders
of magnitude. For a protocol of this size a symbolic execution algorithm
can at best analyze in the order of 10 to 100 system states per second
of CPU time if the state space is built in core."! To analyze 10° states
exhaustively would then take at least 115 days of computation. Fur-
thermore, assuming that each state can be encoded in no more than
10 to 100 bytes, storing a state space of this size would still require a
machine with several gigabytes of main memory.

So, if this appears to be infeasible, what is the best that can be
done? In the design phase one would like to have tools that can trace
the most glaring bugs in a protocol in no more than a few seconds of
real time. The completeness of an analysis is not really at issue here;
speed is. To find more subtle design errors of a completed protocol one
may be willing to spend more time, but not much more than perhaps
10 hours or in the order of 10° seconds of CPU time. For symbolic
execution algorithms, this requirement sets an upper limit to the
number of states that can be searched at roughly 10’ states. At 10 to
100 bytes per state, however, we cannot expect to do anything useful
with a state space of more than in the order of 10° states. Therefore,
it is preferable that the tracer be able to perform complete analyses in
small state spaces holding just a fraction of the total number of states.
In the remainder of this paper we concentrate on these two issues: the
effectiveness of partial searches and the possibility of performing
complete searches in partial state spaces.

In the following discussion we assume that the protocol submitted
to a tracer is likely to contain errors and that a designer is interested
in seeing any nonempty subset of these. A protocol tracer may, for
instance, scan the state space in an effort to quickly discover typical
violations of user-specified correctness requirements. It is important
to note that the objective of such a partial analysis, or scatter search
as we Shall call it, is to establish the presence rather than the absence
of errors.

What we are aiming for is a protocol tracing method that allows us
to spend a small fraction of the time required by an exhaustive analysis
to find a substantial portion of all design errors. The emphasis is on
speed, not on completeness. If a protocol contains an error, an ex-
haustive search would meticulously report every possible circumstance

under which the error could make the protocol fail. For our purposes,
tracing a single variant of the error in a partial search will suffice.

Section II explains how a general symbolic execution algorithm
based on depth-first search can be organized. It discusses a variant of
symbolic execution called scatter searching and compares its perform-
ance with exhaustive searching. Section III discusses in more detail
heuristics that can be used to perform a partial search, and Section
IV shows how depth-first searches can be organized in incomplete
state spaces. Section V shows how protocol specific correctness criteria
can be verified with a standard symbolic execution algorithm. Section
VI gives a small example of the use of correctness assertions in tracing
bugs in a protocol. A larger example is presented in the Appendix.
Section VII summarizes the main results.

In this section we discuss some experiments with a program called
Trace, which performs a simple depth-first search in a partial state
space generated by a set of interacting finite state machines, where
the state space is maintained as a tree of system states. To determine
the effectiveness of partial searches, the performance of exhaustive
searches and scatter searches was compared, using a search depth
restriction as a parameter. But, first let us consider the working of the
tracer in a little more detail. |

With the exhaustive tracing method a state space tree is searched
starting from the initial system state, exploring every possible execu-
tion path until an end state, a previous state, or an error state is
reached, or until the search depth limit is encountered. A return to a
previously analyzed state terminates the search under two conditions:

1. If the previously analyzed state is in the execution path that
leads from the root of the state space tree to the current state, or

2. If the previously analyzed state was encountered elsewhere in the
state space tree either at the same depth or closer to the root of the
tree than the current state.

In the first case the tracer has discovered an execution loop in the
protocol. The loop could be checked further on liveness, but to save
time the program Trace simply checks that its repetition does not
violate the user-specified correctness criteria and continues. In the
second case the subtree that would be explored by continuing the
search down to the search depth restriction would be contained in the
subtree of the previously analyzed state, and cannot lead to new results.
The tracer can therefore ignore the subtree and continue exploring
new leaves in the tree. )

described in, for instance, Ref. 1, the design of the experimental tracer
_ go far is fairly standard. The exhaustive trace method, however, can
be considered to be a special case of the scatter search. In a scatter
search not every possible execution sequence is explored. The tracer
makes an estimate of the likelihood that exploring a new sequence can
lead to the discovery of a new error, and will search only those
sequences that optimize its chances of finding the largest set of unique
errors in the smallest amount of time. The tracer’s estimate will be
based on a heuristic that should be general enough to work on any
type of protocol. One straightforward way to do this, for instance, is
to restrict the amount of nondeterminism that will be taken into
account by the tracer. These and other techniques will be discussed in
more detail in Section ITI. |

To test the performance of a partial search, we want to compare its
coverage or “scope” with that of an exhaustive search. The test
protocol chosen for these comparisons was large enough to show the
necessity of partial searches and also to give some room for experi-
menting with different flavors of partial searches. However, the size
of the test case precluded, by the nature of the problem, the compila-
tion of a definitive list of “all” errors for reference. As a measure of
the scope of the scatter search we will therefore take the number of
errors traced and compare it with the number of errors traced by an
exhaustive search method.

protocol, generating a state space in the order of 10° system states."
The protocol was analyzed several times, both for the exhaustive
search and the scatter search, with a search depth restriction that was
incremented in steps of 10 levels for each new analysis run.

An exhaustive search for this protocol became infeasible beyond a
depth of 80 levels, that is, numbers of states down from the root of
the state space tree. The tree scanned by the scatter search method
had a maximum depth of 189 steps. Setting the search depth restriction
beyond 189, therefore, no longer affects the scope of the analysis. To
illustrate this, the curve for the scatter search was continued in Fig. 1
up to a depth of 230. The longest scatter search required less than 4
minutes of CPU time to complete. The run time of the exhaustive
search tends to be exponential in the search depth. Using Fig. 1, it
can be estimated that searching the state space tree down to the same
depth (189 steps) with the exhaustive search would take some 3000
years of CPU time. |

Fortunately, the test protocol analyzed contained a generous number
of design errors. No attempt was made to classify them. In Fig. 2a the
number of deadlocks reported by the tracer is shown as a function of
the search depth, and in Fig. 2b the number of deadlocks versus the
time it took to find them is plotted on logarithmic scales.

No deadlocks are found at search depths 10, 20, and 30. The first
error is reported with the scatter search for a search depth of 40 steps,
requiring 4 seconds of CPU time. For the same search depth restriction
the exhaustive search reports the first 3 errors in 6 minutes. By
repeating the analyses for intermediate levels between 30 and 40, we
found that the first error is reported both in the scatter and the
exhaustive search mode at level 35, requiring 4 seconds for the former
and 3 minutes for the latter search. The two intermediate tests were
included in the results shown in Fig. 2.

Very probably, no protocol designer would be interested in tracing
this protocol beyond the first 100 error sequences generated. For the
given test case this would mean that with an exhaustive search the

first 70 steps in state space can be searched in roughly 3 hours of CPU
time. Alternatively, the first 100 steps can be traced with a scatter
search in only 30 seconds of CPU time. |

Note that the time required to find the first error, the minimum
search depth required to trace it, and the relation between search
depth and the number of errors reported are favorable for the scatter
search method. |

The protocol used for these tests requires roughly 40 bytes in the
state space per system state. A total of 332,527 system states is
generated in the longest exhaustive search analysis performed. As a
result, for every new state generated, in the exhaustive search a data
base of up to 15 megabytes must be probed for a state match. Even
with the best hashing methods, this is bound to slow down the analysis
noticeably. In the scatter search the largest number of states seen is
172,402 at a depth of 189 in the tree, corresponding to a database of 8
megabytes. The scatter search therefore should slow down less rapidly.
This effect is illustrated in Fig. 3. The time efficiency is expressed in
the average number of states analyzed per second for each analysis
run.

The steep left-hand side of the curves can be attributed to the
overhéad involved in the setup of a state space, which is felt more if
the number of states explored is small. With the current tracer, the
optimum speed for both search methods is reached when the state
space contains roughly 1000 states.

It is relatively straightforward to give preference to the shortest
complete execution sequences and to defer analysis for longer se-
quences. We have already used this method in the preparation of the
figures above by bounding the depth of the tree explored during a
search. In this section we consider some other partial search heuristics.

A method for reducing the run time of an analysis effectively is to
restrict the amount of nondeterminism in the protocol model. In an
exhaustive search, each node in the state space tree is root to one
subtree for each executable option in each finite state machine in the
protocol. Not all interleavings of these actions are necessarily relevant.
Consider two executable actions: one action a local to machine M1,
for example, an assignment to a local variable,.and the other an
external action b in machine M2, for example, a send or a receive.
There are two possible orders in which these two actions could be
executed, corresponding to the two sequences:

Each of the two sequences leads into a new state that forms the root
of an entire subtree in the state space. The question is of whether or
not the two subtrees are equivalent with respect to the errors to be
traced. Note that the execution of internal action a will not change
_ the environment for the remote machine M 2, so neither the executa-
bility nor the result of b can be any different when a is executed first
or last. Similarly, the execution of a is independent of the environment
affected by M2 and also its executability and result is independent of
whether b preceded it or not. In this case, then, it suffices to search
one of the two possible interleavings and to ignore the other. Unfor-
tunately, there are not many cases where a complete subtree can be
ignored without restricting the scope of an analysis. In some cases,
though, we can predict in what way the scope will be affected. It would
be unwise to restrict the nondeterminism that is local to a finite state
machine, as shown in the following Argos fragment:”

Argos is a CSP-like’® guarded command language”™ defined on buffered
message channels. A detailed discussion of the language itself can be
found in Ref. 12. In the above example a and B are channel names
(bounded buffers declared elsewhere), one and two are message names,

and P and Q are procedure names. If message one ‘is the first message
in A and message two is the first message in B, both input statements
are executable and the process executing the above fragment can make
a nondeterministic choice between the two alternatives, and then
proceed with the execution of either P( ) or Q( ). The protocol tracer
cannot foresee which of the two alternatives may produce an error
without executing them. Note that ignoring one of the two alternatives
in an analysis implies ignoring a potentially important code fragment,
that is, either P( ) or Q( ), without having reason to assume that this
code would be error free. In this case then both alternatives will have
to be explored. The situation is different for the nondeterminism that
results from concurrency, as illustrated by the following Argos frag-
ment: ,

It defines two processes P 1 and P2. Assuming that both initial actions
are executable, it must be decided in what order they will be executed
by the tracer. This time it may, but it will not always make a difference
in what order these two I/O statements are executed. In an exhaustive
search both orders are always analyzed. Ignoring one of the two
possible orders, however, can halve the amount of work to be done for
this node in return for the chance that it will cause the tracer to miss
error sequences. No code fragments are ignored here, only a potentially —
erroneous timing of executions. Fortunately, not all orderings have
the same probability of leading into error states. For instance, if we
are primarily interested in finding deadlock states, that is, states in
which all message channels are empty and not all processes have
reached their end states, we may choose to explore the sequence
starting with a receive action and ignore the other. In practice this
heuristic performs remarkably well, as illustrated by the results dis-
cussed earlier. |

If at some node in the state space tree there are N concurrent
processes, all executable, the tracer can decide to ignore any M <= N
of the processes to reduce the search. In the tests reported in Figs. 1
to 3 we set M = 1 for the scatter search and M = N for the exhaustive
search. In the case where M = 1 the search heuristic is implemented
as a priority scheme that determines which process should be executed
next. Highest priority is given to internal actions. At the next level we
place receive actions, since these tend to bring the system closer to a
deadlock state with empty channels. A lower priority is given to send
actions, and a lower priority still to channel time-outs. Time-outs are

given lowest priority in the partial searches since they tend to create
many spurious error reports. In partial search mode the correct work-
ing of the time-out mechanism is assumed, that is, a time-out is only
considered to be enabled when there is no other option to continue
the protocol. Though this definitely reduces the scope of an analysis,
it does allow us to trace for another class of errors first and defer the
costly tracing of timing errors.

The capacity of a communication channel for holding messages can
also have an important effect on the size of a state space. In the
specification language Argos the channels are modeled by finite
queues. A channel then can be in only a finite number of states

where N is the number of slots in the channel (i.e., the queue size),
and S is the size of the channel sort, that is, the set of all messages
that can be recognized by the channel. Reducing the number of slots
N by 1 can reduce the size of the state space and speed up the analysis
by a factor of up to

In the scatter searches of Figs. 1 to 3 the queue sizes were restricted
to two slots. To study the effect of a variation of the queue size the
tests were repeated for a small range of sizes. Figure 4 shows the effect

of a variation in the number of slots between one and four for both
the exhaustive and the scatter search.

In the Introduction we mentioned that the tracer should be able to
perform searches in even incomplete state spaces since the size of a
complete state space generally precludes its storage or even its usage
during the search. In this section we show how this can be accom-
plished.

First it should be noted that in a depth-first search, at each execution
step only those states that lead from initial state to the current state
are indispensable in the state space. The presence of these states is
necessary for the detection of system execution loops. Not every
system state, though, can be found at the start of such an execution
loop, and therefore it is not necessary to remember each state along a
single execution path. The only states that must be remembered are >
those in which at least one of the interacting finite state machines is
at the start of a local execution loop. Figure 5a shows a small but
consistent reduction in the numbers of states if we restrict the state
space to such “loop states.”

Since the analysis is performed on finite state machines we can try
to minimize the machines in an effort to reduce time or space com-
plexity without affecting the scope of an analysis. The machines
cannot be reduced under the standard notion of language equivalence,
since that will change the behavior or the protocol. A stronger notion
of state equivalence,’ similar to that defined in CCS® can be used.

Figure 5b compares the analyses of minimized state machines and
nonminimized state machines. The protocol tested defines 4 processes,
34 message types, 6 message channels, and 3 local variables. The state
machines generated for the processes contain 69, 47, 7, and 5 states,
respectively. The strongly equivalent minimized machines contain 35,

31, 7, and 5 states. As it turns out, the number of states generated in
the state space is roughly the same in both cases. The connectivity of
the state space tree, however, is different, causing the same states to
be visited more frequently for the nonminimized machines, resulting
in a small increase in run times.

The effort to minimize the amount of work to be done in the search
algorithm is concentrated on minimizing the theoretical maximum
number of states in the product space of the individual finite state
machines. We can do this by reducing the number of states per state
machine (e.g., by masking a variable or a message queue) or, less
straightforwardly, by reducing the number of state machines as such.
The last thing we would like to do is, of course, to extend the number
of state machines that we begin an analysis with.

Somewhat paradoxically, this approach seems to conflict with the
more conventional structured approach to program design that tells
us to identify functions and to separate these in a relatively large
number of logical entities. For protocol design this approach was most
recently suggested in Ref. 15, which describes a method where each
logical entity is formalized in a small finite state machine that interacts
with the others. Dividing a single automaton of 16 states into 2 state
machines of 8 states each, however, quadruples the number of states
in the product space. Similarly, dividing it into 4 even simpler state
machines of 4 states each expands the product state space to 16 times
its original size. In general, increasing the number of state machines
leads to an exponential growth of the product state space and is
counterproductive in analyses.

We noted above that, unlike the more commonly used breadth-first
search (see Refs. 5, 9, and 12), the state space in a depth-first search
need only contain the states in a single execution path from the root
to the current state. Storing other states can avoid double work, but
does not affect the scope of the analysis. This property of the depth-
first search method gives us greater flexibility in controlling the state
space size during analysis runs. If more states can be stored, though,
the search will be more time efficient. Figure 6 shows the effect of the
size of the state space on the time and space requirements of a search,
for a state space cache of 150,000 states that is reduced in steps of
1,000 to a cache of 50,000 states. “Double work” is measured here as
the total number of states created or recreated while searching. Note
that the number of states stored in the cache could roughly be halfed
without noticeable effect on the runtime or the total number of states
created. With a partial state space cache it has to be decided which
state will be deleted from a full cache when a new state must be

created. A simple blind round robin selection of states was found to
outperform a series of other, more subtle, schemes.*»”” It is the strategy
used in the test of Fig. 6.

By default a protocol tracer can check a protocol for the observance
of general correctness requirements such as absence of deadlock and
completeness. The validation language Argos allows for the specifica-
tion of assertions to check on the observance of other correctness
requirements. Assertions are defined as a restricted class of processes.
They specify global system behavior in terms of external actions: For
example, the specification

is a requirement on the order in which messages of the type mesg are
sent to the two channels large and smail. The assertion is that in
each execution sequence a message on channel large must precede a
message on channel sma11, and that these two actions will be executed
repeatedly (they are enclosed in a do loop) in precisely this order.
The main restriction to assertion specifications is that they can only
refer to external actions, that is, sends and receives, and not to
variables. Assignments and Boolean conditions are only allowed in
process definitions, not in assertions. The control flow constructs are
the same as those for process specifications: concatenations, selections,
iterations, jumps, procedure calls, and macros. In other words, the
assertions specify global constraints on the execution of the system as
a whole in terms of message exchanges only. The scope of the assertion,
that is the set of external actions that is traced to verify or to violate
an assertion, is implicitly defined by the set of external actions it

refers to. If an external action occurs at least once in an assertion
body, all its occurrences in an execution of the protocol are required
to comply with it. Compliance with the assertion then means that the
execution of these actions should match the context specified in the
assertion.

Since we define assertions as restricted processes, the assertion
primitives can be compiled into a restricted class of state machines |
and minimized with the same algorithm that is used for the compila-
tion of the protocol processes. The protocol tracer uses the assertion
state machines to monitor the external actions on which they are
defined. Alternatively, though our tracer does not exploit this possi-
bility, it may be possible to develop a heuristic that allows the tracer
to select those executions in a partial search that have the best chance
of violating the correctness requirements expressed in the assertions.

If an action is within the scope of an assertion, the state of the
corresponding state machine will be updated as a side effect of the
execution of that action, as if the assertion machine itself generated
it. Since the assertion primitives cannot access variables or channels,
the “state” of an assertion machine is uniquely defined by its control-
flow state. The “execution” of an assertion machine then costs very
little in the tracing algorithm. When the protocol system reaches an
end state, compliance with the assertion can be established by verifying
that the assertion machine can reach an end state, too. If this is not
true, the assertion is violated and the current execution sequence can
be listed as a counter example. Similarly, if the assertion machine .
cannot be executed for an action that is within its scope, the assertion
has been violated and a counter example can be produced. With little
overhead or added complexity, the finite state machine model can thus
be exploited to combine the depth-first search with assertion checking
capabilities.

The assertion states that the two messages a and b will be appended
precisely once to queue c when the three processes are executed, and
that they can be sent in that order only. Process a starts by sending
message a to queue c and then waits for a response a to arrive in
queue A. Similarly, process b first sends b to queue c and then waits
for a message c. The third process waits for a message to arrive in
queue C, which is assumed to be either an a or a b, anything else would
be an error. Process c then responds by sending a c message to queue
c and waits for a second message to arrive: a b if the first received
message was an a, or an a if the first message was a b. It will complete
by sending a c into queue B.

The protocol is compiled into four finite state machines of three
states each for a and b, three states for the assert primitive, and
seven states for process c. The protocol tracer then takes over and
completes an exhaustive search in 1.35 seconds, reporting the obvious
assertion violation for the execution sequence that starts with C!b.
The violation is reported by the tracer in the following format:

Each column corresponds to a queue and each line to a time step.
The first event is the sending of message b to queue C, which already
violates the assertion. Then message c is sent to A by process c,
message a is sent to c by process a, and finally a c message is sent to
queue B by process c.

Changing the assertion to a more reasonable statement such as
assert {A!c:B!c} will avoid‘the problem. The exhaustive search

for this assertion completes in 1.32 seconds. Omitting the assertion
completely will trigger a default search for deadlocks and incomplete-
ness (e.g., unspecified receptions), which completes in 1.18 seconds.
Note that it is relatively straightforward to formalize liveness criteria
in assert statements. In this case, as for many protocols generating up
to 10° system states, exhaustive analyses are quite feasible. The real
problems of partial searching only occur for the larger protocols
comparable in size to the experimental protocol used for the tests
reported earlier in this paper.

The main assumption we make in this paper is that in a design
phase a protocol is typically known to contain errors and there is a
need for a protocol tracing tool that can quickly find a representative
subset of these errors. The user of such a protocol tracer is not so
much interested in completeness but is very much interested in speed.
With these assumptions important reductions in the time and space
requirements of a tracer become feasible.

The protocol tracer described here consumes only a small fraction
of the time and space required by an exhaustive analysis algorithm to
find a relatively large fraction of the errors present. The run time of a
state space search is reduced by several orders of magnitude by
restricting the number of interleavings, by using search depth and
queue size restrictions, and by using compile time minimizations (Figs.
1, 2, 4, and 5(b)). A more general method of reducing run time would
be the definition of equivalence classes, or state space foldings, as
described in Ref. 8. The experimental protocol debugger Trace does
allow for the definition of such foldings, but too little experience with
this technique has yet been gained to report any results.

The number of states stored in a state space can be reduced by
carefully selecting the states that may be revisited (Fig. 5a). More
importantly, though, the depth-first search technique used allows one
to perform searches with an incomplete state space cache. For the
protocol tested, the cache could be reduced to less than 50 percent of
the state space size (Fig. 6). |

The experiments with assert primitives and the state space cache
were inspired by discussions with Bob Kurshan and Sudhir Aggerwal.
I am also grateful to Doug McIlroy, Lee McMahon, Rob Pike, Ed
Sitar, and Ken Thompson for discussions, support, and inspiration
during the development of the protocol tracer.

1. T. P. Blumer and R. L. Tenney, “A Formal Specification Technique and Implemen-
tation Method for Protocols,” Comput. Networks, 6, No. 3 (1982), pp. 201-19.

. D. Brand and W. H. Joyner Jr., “Verification of Protocols Using Symbolic Execu-
tion,” Comput. Networks, 2 (1978), pp. 351-60.

. d. Hajek, ‘ ‘Automatically Verified Data Transfer Protocols,” Proc. 4th ICCC, Kyoto,
September 1978, pp. 749-56.

C. West, “Applications and Limitations of Automated Protocol Validation,” Proc.
2nd IFIP WG 6.1 Int. Workshop on Protocol Specification, Testing, and Verifi-
cation, USC/ISI, Idyllwild, Calif., May 1982, pp. 361-73.

P. Zafiropulo et al., “Toward Analyzing and Synthesizing Protocols,” IEEE Trans.
Commun., COM-28, No. 4 (1980), pp. 651-61.

R. Milner, “A Calculus for Communicating Systems,” Lecture Notes in Computer
Science, 92 (1980).

G. J. Holzmann, “A Theory for Protocol Validation,” IEEE Trans. Comput., C-31,
No. 8 (August 1982), pp. 730-38.

G. J. Holzmann, “The Pandora System—An Interactive System for the Design of
Data Communication Protocols,” Comput. Networks, 8, No. 2 (1984), pp. 71-81.
D. Brand and P. Zafiropulo, “Synthesis of Protocols for an Unlimited Number of

10. P. R. F. Cunha and T. S. E. Maibaum, “A Synchronization Calculus for Message
Sele tris Programming,” Proc. Int. Conf. on Distributed Systems, IEEE 1981,
pp. 433-45.

11. G. J. Holzmann, “Trace—Performance Measurements,” AT&T Bell Laboratories,
internal report, Jan. 1, 1985.

12. G. J. Holzmann, “Automated Protocol Validation in ‘Argos,’ Assertion Proving and
Scatter Searching,” 1984, available from the author.

14. E. W. Dijkstra, “Guarded Commands, Nondeterminacy and Formal Derivation of

15. R. P. Kurshan, “Proposed Specification of BX.25 Link Layer Protocol,” AT&T
Tech. J. 64, No. 2 (February 1985), pp. 559-96.

16. National Bureau of Standards, Specification of a Transport Protocol for Computer
Communications, 4, “Service S ecifications,” June 1984.

The following specification describes a transport protocol defined
by the National Bureau of Standards.’® The specification is based on
the model given in Ref. 17. Four processes are defined: a local user
process AU connected to a server process A, and a remote user BU
connected to server process B. The control flow constructs and the
I/O statements in Argos are based on CSP, using buffered message
channels instead of rendezvous. Process A, for instance, receives mes-
sages via two channels: one is named ua and is used by the user
process to request services, the other is named ca and is used here to
receive control messages from the remote server. Messages from server
to user are sent through channel ua. The communication between the
two servers is modeled with control messages m1 to m7, as defined in
Ref. 17. The analysis discussed here uses no assert primitives and is
thus a general one for completeness and absence of deadlocks. The
arrow and the semicolon are syntactically equivalent statement sepa-
rators. A double colon flags the start of an option in a repetitive

construct (do --- od) or in an alternative construct (if --- fi). In
this case, the state transition diagram defining the protocol is most
conveniently, though not most elegantly, modeled by assigning a label
to every state and including a goto-jump for every transition. Processes
A and B are symmetrical. Null transitions from the original protocol
were deleted from the model.

do
: ca?m1 — UA! conn_ind — goto rcvd
2: ua?conn_req — cb!m1 — gotocrsent
:: uaPabort — cb!m4
:: ca?m4 > UA! d

2: ca?m2 — UA! conn_conf — goto estab
:: ca?m7 — UA! disconn — goto closed
2: uaP?abort — cb!m4 — goto closed

3:3: ua?conn_resp — cb!m2 — goto estab
2: ca?m7 — UA! disconn — goto closed
:: ua?abort — cb!m4 — goto closed

Aclose:
do
2:3: ca?m3 — UA! close_ind — goto closed
2: ca?m5 — UA! data_ind
ca?m6 — UA! expid_ind
ca?m7 — UA! disconn — goto closed
ua?abort — cb!m4 — goto closed
ca?m4 — UA! d — goto closed

Pclose:
do
:: uaPdata_reg — cb!m5
2: uavexpid_req — cb!m6
2:3: ua?abort — cb!m4 — goto closed
:: ca?m4 — UA! d — goto closed
:: ua?close_req — cb!m3 — goto closed
2: ca?m7 — UA! disconn — goto closed

:: cb?m2 — UB! conn_conf — goto estab
cb?m7 — UB! disconn — goto closed
ub?abort — ca!m4 — goto closed

revd:
if
2:3: ub?conn_resp — ca!m2 — goto estab
:: cb?m7 — UB!disconn — goto closed
:: ub?abort — ca!m4 — goto closed
:: cb?m4 — UB!d — goto closed
fi;

ub?close_req — ca!m3 — goto Aclose
cb?m3 — UB!close_ind — goto Pclose
ub?data_req — ca!m5
cb?m5 — UB! data_ind
ub?expid_regq — ca!m6
cb?m6 — UB! expid_ind
cb?m7 — UB! disconn — goto closed
ub?abort — ca!m4 — goto closed
cb?m4 — UB!d — goto closed
od;
Aclose:
do
:: cb?m3 — UB!close_ind — goto closed
:: cb?m5 — UB! data_ind
cb?m6 — UB! expid_ind
cb?m7 — UB! disconn — goto closed
ub?abort — ca!m4 — goto closed
:: cb?m4 — UB! d — goto closed
od:
Pclose:
do

od
The queue sizes were arbitrarily set to 8 slots per channel. The protocol
tested is defined by the behavior of the two server machines, as visible
to the users. The user behavior is no part of the formal protocol. An
arbitrary set of user processes was defined specifically for the test.
The local user Au will open the connection by sending a conn_req
message to ua and some arbitrary time later it will close it with an
abort message. The remote user BU is considered to be passive,
responding only to close messages and accepting, but ignoring all
others. Default is a keyword for receptions that match any input
from the queue specified.

The protocol as specified above is compiled—in 23 seconds of CPU
time on a VAX-11/750*—into four finite state machines of 27, 27, 10,
and 6 states, respectively. The compiler flags a series of incompleteness
errors, noting for instance that control message m7 can be received
but is never sent. Ignoring those warnings, an exhaustive analysis with
trace takes just under 3 seconds of CPU time and reports 4 error
sequences that reduce to two types of errors. The first one is an |
unspecified reception of the message conn_resp In state closed, for
instance, after the following message exchange:

Each column corresponds to a queue and each line to a time step. The
_ first event recorded is the sending of a message conn_req into queue
ua, followed by an m1 into queue cb, etc. The last message sent is

_ The second problem is an unspecified reception of m2 also in state
closed:

It is now straightforward to study the behavior of the protocol for
different user behaviors, which can reveal, for instance the possibility
of the unspecified reception of a message conn_resp in state Pclose,
or the more obvious deadlock after a simultaneous conn_req message
from both users.

Gerard J. Holzmann, Kand. Ir. (B.S.), 1973, Ir. (M.S.) (Electrical Engi-
neering), 1976, Ph.D. (Technical Sciences), 1979, Delft University of Tech-
nology, The Netherlands; Mr. Holzmann obtained a Fullbright Fellowship in
1979; with the University of Southern California in Los Angeles, 1979-1980;
AT&T Bell Laboratories, 1980-1981, 1983—. Before returning to AT&T Bell
Laboratories, Mr. Holzmann was an Assistant Professor at the Delft Univer-
sity of Technology. In 1981 he was awarded the Prof. Bahler prize of the Royal
Dutch Institute of Engineers (KIVI) for his work on telecommunication
systems. His current research is in distributed systems and computer graphics
in the computing science research center.

We describe a Videocassette Recorder (VCR)-based information system
whereby we can distribute frequently updated large pictorial databases to
individual users and provide a variety of interactive video services. The four
key advantages of this system are: (1) economics, (2) good picture quality, (3)
capability to reach nationwide users, and (4) ability to update the database
frequently (say, daily, preferably in early morning hours when many trans-
mission facilities are unused). An experimental home terminal consisting of a
VCR driven by a personal computer for random-access searches was con-
structed to demonstrate this concept. The pictorial database used in the
demonstration includes real estate listings, vacation guides, autos and Sears-
type merchandise catalogs. We also make comparisons of this system to other
video services and conclude that the present approach has potential advantages
in many applications.

There are presently many systems under development for providing
interactive visual displays.! For example, videotex uses the switched
telephone network to send and receive digital data, which are then
used by a microprocessor terminal to construct color graphics on a TV
screen. Teletext is another technique whereby digital data for the same
purpose are imbedded in the vertical blanking period of a video signal
broadcasted to the end users. The graphics capability of these two

methods is limited in representing those real objects, e.g., clothing and
furniture, for which visual attractiveness is more important than
functional appeal. -

Limitations in the computer-graphic representation of real objects
can be overcome by storing the pictures on an interactive videodisc
(as in the “electronic book” application).? In such a case, hard copies
of the disc have to be distributed by mail or through stores to the end
users. If real-time access to a central database is really required, then
the so-called frame-grabber approach is frequently suggested. In this
latter system, single frames are sent to individual users, time multi-
plexed on a dedicated video channel.? The user terminal must then
store the received frame so that it can be examined by the human
viewer. Digital frame stores for this application are expensive, although
their cost should decline eventually. Nevertheless, a multiuser com-
puter system at a central location must manage the data requests and
send different video frames to different users. Such a system can be
overloaded easily, and practical solutions for giving nationwide service
to thousands of users simultaneously have yet to be worked out.

Here we suggest an alternative arrangement which appears to be
more economical than the videotex or frame-grabber systems. We
propose that the home terminal consist of a home Videocassette
Recorder (VCR) connected to a personal computer, and that a “frame-
search” capability be provided whereby the home computer can specify
which frame of the videocassette is displayed (in still frame) at any
one time. The VCR must of course have good still-frame performance.
The overall system for distributing pictorial databases from a central
station to end users with such home terminals is outlined in Section
II. Then we describe an experimental home terminal for demonstrating
the feasibility of this idea (Section III). Finally, we make comparisons
with other systems (Section IV) and conclude that our present proposal
has potential applications in both the business and consumer markets.

We propose the direct distribution of pictorial information to the
end users’ VCRs via a TV broadcasting channel, as illustrated in Fig.
1. The pictorial database is assembled in one central location, where
individual color pictures (35 mm photographs or slides) are recorded
onto a master videotape (one-inch type) in a frame-by-frame manner,
i.e., a single color picture on a single video frame. A two-hour videotape
can then store up to 216,000 single-frame pictures. In the vertical
blanking period of these video pictures, we insert a frame number for
identification as analogous to the page number in a book. This tech-
nique of numbering the video frames can be implemented easily with
the conventional Vertical Interval Time Code (VITC). In addition,

As suggested in Fig. 1, the database from the central station can be
transmitted through a satellite broadcasting system (C-, K,-band, or
DBS) whereby nationwide users can either receive the information
directly or through a local broadcaster. In the latter case, the local

distributor simply takes the received video signal from the satellite
and retransmits it to the users via his own broadcasting system (e.g.,
cable TV, off-the-air VHF/UHF channels, optical fiber, etc.). Attach-
ing additional information from a local database is optional. Since
many transmission facilities and T'V channels are idle in early morning
hours, this operation can most conveniently be done in the middle of
the night with the VCRs programmed or pretimed for unattended
recording. Once the database is recorded on a videocassette, the users
can assess the information at their home terminals at their leisure.
Direct distribution in this manner avoids the cost of recording and
shipping thousands of tapes (or discs). Moreover, the database is
always up-to-date, depending on how often it is sent (once a day should
be adequate for most applications). No special equipment is required
at the TV station or CATV head-end for sending out the databases.
Indeed with satellite transmission, nationwide distribution is possible
with transmission systems already in place. For example, the Sears
catalog could be sent in 4.5 minutes, assuming 1600 pages and 5 frames
per page. One thousand real-estate listings could be sent in 5.5 minutes,
assuming 10 frames per listing. Custom orders for still-picture or full-
motion information (e.g., instruction manuals) could even be served if
more transmission time were available.

With the database recorded on a videocassette, we can use the
interactive home terminal suggested earlier to browse through the
pictures in a random-access manner. We constructed an experimental
terminal to demonstrate this idea, which is discussed in the next
section. But first, we should point out that the assembly of large
databases at the master station is no easy task. In fact, the production
cost could be an important consideration in systems of this type. The
need for automation in database production is mandatory and indeed
manageable with modern production equipment.

We show in Fig. 3 the block diagram of a VCR-based interactive
terminal suitable for examining stored video data from a videocassette.
The solid-line portion represents what was implemented in our labo-
ratory aS an experiment to demonstrate the proposed concept, while
the dashed-line part stands for an alternate approach using a remote
computer (via telephone hook-up) to control the operation. They are
functionally the same, but offer different kinds of user flexibility. More
is discussed about their differences after we explain the terminal itself.

The terminal consists of three major components: the computer,
which serves as a controller for the entire system; the VCR/computer

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PERSONAL |
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| PHONE
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OPTIONAL i
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interface; and the VCR* (with TV) itself. The video signal from the
VCR is passed through the VCR/computer interface before being
displayed on the TV. Inside the interface (Fig. 4), a field number
decoder is used to examine the video frame identification number
(VITC) recorded in the vertical blanking interval. This information
should be made available to the computer at all times, i.e. when the
tape is in still frame or in motion. But for simplicity, it is sufficient to
provide valid frame numbers-for tape motions from still-frame to play
speed (30 frames/second). For higher tape speeds, the computer can

* Both the VHS and the Betamax formats would work as long as the VCR has good
still-frame performance. We used VHS in our terminal.

use calculated estimates based on the initial known tape location, the
current known tape speed, and the measured elapsed time plus some
precalibrated adjustments. As an adjunct, the status encoder monitors
the current operational status of the VCR (such as stop, fast forward,
play, etc.) and conveys it to the computer. The command decoder, on
the other hand, receives commands from the computer and translates
them into actual instructions to the recorder, i.e., emulating a human
pushing buttons to control the VCR. This was done simply by wiring
up computer-driven electronic switches over the contact points of the
pushbuttons on the remote control unit of the recorder.

Although the field number decoder is designed to detect VITC for
frame identification, it is easy to extend the idea for decoding digital
data recorded in all or part of an active TV field. Thus, part of the
database can be devoted to digital information such as table of con-
tents, programming or search instructions, synthesized audio, etc.
They can be copied directly to the computer memory for use. _

A sample video database was put together for the experimental
demonstration of the terminal. It consists of four sections: (a) real
estate listings; (b) automobiles; (c) a vacation package; and (d) mer-
chandise catalogs. The interactive access to these data is done via a
touch-sensitive screen on which menus are printed to prompt the user.
The choices are all self-explanatory and are available to the user as
touch-sensitive buttons on the personal computer. Instructions have
been kept to a minimum, and the operation is so user friendly that a
user manual is seldom needed. The majority of the material in the
demonstration database is still-frame pictures. However, the full-
motion video segments (with sound too) in the Hawaii Vacation Guide
and also in a merchandise catalog on how to use the riding lawn mower
consume much more recording time than the still-frame pictures. Note
that our use of the personal computer with a touch-sensitive screen
was merely a choice for experimental convenience. Any other user
interface may be substituted to serve the same purpose in an actual
system.

In addition to the user-prompting menu, 16 additional touch-sensi-
tive buttons are always available in the bottom of the computer screen
(see Table I). As an example, if one selects the button for real estate
listings, then a table of choices for different towns appears on the
touch-sensitive screen (Fig. 5). Touching one of these choices would
bring up the next menu for specific price ranges. After that we have
one more menu selection for specific house types, e.g., two-story
colonial, log home, etc. The computer then fetches the text listing of
the house selected from its memory and displays it on the touch-
sensitive screen. Meanwhile, the search routine to locate the picture
for the house is executed to the VCR. Therefore, the user sees the

Brings up the Hawaii Vacation Guide consisting of “Map of
the Hawaii Islands,” “Scenes From Hawaii,” “The Polyne-
sian Culture Center,” and “The Sea Life Park,” all of which
(except the map) are full-motion video with sound.

Brings up the real estate listings: still pictures of the houses
from the VCR and their corresponding text descriptions on
the touch-sensitive screen.

Brings up the merchandise catalogs: still pictures of individual
items plus full-motion-and-sound segments of merchandise
demonstrations.

Returns to the previous menu.
Allows user to place an order or obtain order information.
Brings up function definitions.

Single-frame advance in the forward direction.
Single-frame advance in the reverse direction.

Causes video to move forward with momentary pause at each
still-frame picture for browsing.

Causes video to move backward with momentary pause at
each still-frame picture for browsing.

Pauses for still-frame viewing and brings up text description
of current item.

color picture of the house on the TV and has the text listing simulta-
neously from the computer. The 16 buttons at the bottom of the screen
provide plenty of choices and flexibility for the next step.

The search algorithm implemented in our software is designed to
perform random access on the VCR in the most expedient manner. It

should not be a surprise, however, that the VCR access time is
considerably slow compared to that of a device truly capable of random
access, e.g., a videodisc player. It takes a VCR typically 3 to 5 minutes
to rewind a standard T120 (2/4/6-hour) cassette, and this would be
the worst-case waiting time for getting to a picture from one end of
the tape to another. Such a long access time is probably unsuitable for
many applications. To circumvent this inconvenience, we propose that
the whole database be divided into segments of, say, 1000 pictures
each. Then the search within each segment is quite fast. As an
illustration, if a 10 times play (10X) speed is used for the search, i.e.,
300 frames/second, the maximum access time within such a segment
is only 3.3 seconds. In our experimentation, we found that the speed
search feature of the VCR was the most appropriate function to use
for random access (i.e., fast visual search with 10X, 5X, 2X, etc.)
because the video heads did not have to be disengaged and then
reengaged during the process, and the continuous display of fast
motions on the T'V helped to make the waiting time less objectionable.
In the event that the desired TV frame is very far away from the
current position, the normal fast forward or rewind might have to be
used. The optimization process is based on information previously
obtained by an automatic calibration program which sizes the start
and stop times, the video head engage/disengage intervals, and so on
for each specific VCR. In short, we have developed an automatic
calibration process that can characterize the access performance of a
VCR, and the random-access software is intelligent enough to use this
information for optimal control.

Laying out the database in segments of 1000 frames each requires
intelligent partition of the overall source into groups of “humanly
correlated” materials. For example, we don’t want to put the listings
of all different tires in one segment as in a conventional merchandise
catalog. Instead we want to group all the accessories of a specific car
model together. The situation is analogous to that of a large library
where the books are carefully categorized and authored in such a way
that most of the information desired for a subject is confined to a
book of 1000 pages. The user can browse through each book very
quickly, but probably does not mind spending more time in looking
for another book. The assumption here is that most readers would
spend some time reading a specific book of interest rather than reading
pages of uncorrelated materials from different books in a random
manner. We recognize that such an idea needs much more research
before it can be put into practice. Nevertheless, recent experience
from the videotex experiments seems to suggest evidence supporting
the validity of this concept.

It is clear from the foregoing discussion that software plays a key
role in this system. In our experimental setup, all the software resides
in the personal computer (Fig. 3), and the user has complete control.
The dashed-line portion in the figure indicates the other alternative
of having the controlling software in a remote computer, and the user
interacts with the system via a key pad connected to a modem. This
latter approach has the advantage that more powerful software can be
used at the expense of less user control as well as less privacy. The
basic idea remains, however, that very intelligent software is needed
in managing the enormous database made possible by a simple con-
sumer-type VCR.

Finally, let us point out that the VCR system can be used as a high-
density digital storage unit, which can provide digital high-fidelity
music as well as synthesized voice and text data.

The VCR-based information system has the same applications as
other interactive video services. Some of the generic examples include
real estate listings, vacation/entertainment guides, merchandise cat-
alogs, product demonstrations, and service/instruction manuals.

We summarize in Table II a comparison with videotex and the
videodisc-based system. It should be emphasized before we discuss
these results that there is no single criterion possible for judging the
relative merits or shortcomings of any approach. Instead, most systems
tend to be application oriented. In other words, each individual video
service tends to appeal more for the application it is intended for, and
there is probably no single system that is universally “better” than all

VCR Based Disc Based Videotex
Video quality Good. Good Graphic
Database creation Automatic Automatic Manual photo-
photo-to-tape photo-to-tape to-graphic
Distribution VHF/UHF, By mail or Telephone ©
CATV, DBS, through stores
fiber, etc.
Number of hard copies 1 — 100,000+ 100+
Frequent updates Yes No Yes
Real-time interaction with Limited Limited | Yes
data suppliers
Response time Slow Fast Depends on
number of

The picture performance of VCRs is usually designed to be compat-
ible with the characteristics of other devices they are connected to in
most home use, e.g. resolution is similar to that of a popular consumer
TV (without comb filtering) and signal-to-noise ratio is comparable
to most cable TV systems. In any event, their picture quality in our
subjective viewing was found to be remarkably close to that of cable
TV, while the videotex picture tends to be cartoon-like. The videodisc
is potentially capable of noticeably better quality than the VCR
although this is usually not so in practice.

The database creation process for the VCR and the videodisc is
almost the same. In both systems, the original material (e.g., photos
or slides) is recorded on a one-inch video tape serving as the master,
and the difference between the two cases is that the videodisc requires
further processing in transferring the tape material onto a master disc
before mass duplication. As for videotex, the original material has to
be recreated in the computer (with computer-aided tools) as a graphic
representation of the real object.

The VCR system uses the TV broadcasting for distribution. Video-
tex uses the phone lines to connect customers to a central computer,
while videodiscs have to be distributed via mailing or store sale.

Only one master copy of the database needs to be maintained
nationwide in the VCR system. Videotex service requires a hard copy
at each computer center, and hundreds are thus needed nationwide. —
The videodiscs are, of course, hard copies of the master, and a national
market would require hundreds of thousands of them.

Because the database resides in the user terminal, real-time inter-
action with the data supplier tends to be limited in both the VCR and
videodisc systems. On the other hand, videotex users enjoy continuous
real-time interactive picture transmissions with the central computer.

The VCR access time is considerably slow compared to the videodisc,
as discussed earlier. The videotex response time depends mainly on
the number of simultaneous users and is probably slow (tens of
seconds) for a fair size of simultaneous accesses (say thousands).

The VCR approach could become extremely economical if there was
a mass availability of VCRs and personal computers, both of which
have tremendous appeal in their own right and are gaining popularity
among businesses and consumers. The custom interface necessary to
connect the VCR to the personal computer is so simple that it can
easily be incorporated into either the VCR or the computer. In any
case, its cost should only constitute a small fraction of that for the
total user terminal. Videodisc players capable of true random-access
search are fairly expensive, and their popularity with consumers seems
to be on the decline. Videotex terminals are also quite expensive, but
their cost could decrease dramatically if more large-scale integration
were employed.

We have proposed a system for distributing large pictorial databases
to home videocassette recorders (VCRs). Distribution is done by

broadcasting from a master station where the picture information has
been assembled in a frame-by-frame manner, i.e., one picture per video
frame, resulting in 30 independent pictures in a 1-second video seg-
ment. The broadcasting medium could be a combination of direct
broadcast satellites, cable TV, conventional VHF/UHF TV channels,
or custom fiber systems. The main idea is that this can take place in
the middle of the night when many transmission facilities become
vacant, and the home VCRs can easily be pretimed. for unattended
recording. In fact, distribution or updating is possible as often as
transmission time permits, but once a day is probably adequate for
most applications.

With the complete database stored in a videocassette, an end user
can retrieve the data of his particular interest at his leisure with the
aid of a simple home terminal. We constructed such a terminal
comprising a VCR capable of good still-frame performance, a personal
computer serving as a controller for random access, and a custom
interface connecting them together. The VCR/computer interface
translates digital commands from the computer into actual operational
instructions to the VCR, i.e., emulating a human pushing the control
buttons on the recorder. It also feeds back the operational status of
the VCR to the computer. Most important of all, it examines the video
signal and decodes a frame number previously recorded in the vertical
blanking interval during data assembly. This frame number is the
“page number” of the electronic book and is supplied to the computer
so that it knows which video frame or picture is being displayed on
the TV. Software on the computer was implemented to do random-
access search through the database, and the interface to the user is in
the form of a touch-sensitive screen with menu-driven selections. The
capability of this experimental terminal was demonstrated with a
sample database consisting of real estate listings, new car models,
vacation packages, and merchandise catalogs:

The main attraction of our proposal is its potential economics. That
is, it takes advantage of other potentially low-cost and widely available
terminal equipments, namely, the personal computer and the VCR,
both of which have consumer appeal in their own right. Distribution
requires only a single database plus transmission facilities that are
already in place. Thus, a service supplier need only provide a hardware
interface plus software, a networker need only supply a satellite or a
TV station plus a video production unit, and purveyors of information
need only furnish color photos and text.

The authors wish to thank R. F. Weihs for his work on the experi-
mental terminal.

2. R. D. Gordon, “An Intelligent Electronic Book System and Publishing Facility,”
Conference Record of Outlook For Optical and Video Disc Systems and Appli-
ene February 20, 1985, The Institute for Graphic Communications, Miami,
Florida.

3. H. Ando and H. Yamine, “Still-Picture Broadcasting—A New Informational and
Instructional Broadcasting System,” IEEE Trans. Broadcasting, BC-19 (Septem-
ber 1973), pp. 68-76.

The IEEE 802 standard for local area network based on Carrier Sense
Multiple-Access with Collision Detection (CSMA/CD) operates at a peak rate
of 10 Mb/s on a cable of maximum length 2.5 km using baseband signaling.
In many situations, larger channel rates are required over a much larger area.
However, the efficiency of the CSMA/CD access method decreases rapidly if
either the length of the cable is increased for a fixed bit rate or if the bit rate
is increased for a fixed cable length. In this paper, we propose a broadband
network for computer communications containing several CSMA/CD-type
systems, each operating in a different frequency band. In addition, in order to
have a wide area access, while minimizing the loss of performance associated
with large collision delays, terminals in a small given geographical area are
given one of the frequency bands for transmission. Two access protocols are
developed. Using these schemes, it is possible to increase the channel through-
put and the access area and to reduce the collision delay. We present a
simplified analysis to quantify the improvement in performance using our
schemes.

Local Area Networks (LANs) share computing and other resources
among many users and, if properly designed, increase the reliability
by reducing the dependence of a user on one processing unit or a
peripheral. Unlike long-haul networks, where channel utilization has
to be optimized owing to high cost of communication over long

* AT&T Bell Laboratories.
t Visiting from the Department of Electrical and Electronic Engineering, University
of Western Australia, Nedlands, Australia. |

Copyright © 1985 AT&T. Photo reproduction for noncommercial use is permitted with-
out payment of royalty provided that each reproduction is done without alteration and
that the Journal reference and copyright notice are included on the first page. The title
and abstract, but no other portions, of this paper may be copied or distributed royalty
free by computer-based and other information-service systems without further permis-
sion. Permission to reproduce or republish any other portion of this paper must be
obtained from the Editor.

distance, local networks use bandwidth somewhat extravagantly to
reduce the switching costs. Several network topologies, such as rings,
buses, and trees, have been proposed along with access methods such
as carrier sense, token passing, etc.!”

The IEEE 802 standard for local area networks uses CSMA/CD
(Carrier Sense Multiple-Access with Collision Detection) as one of its
access methods.® It uses baseband transmission on coaxial cables
(although other media are possible) at a peak rate of 10 Mb/s. For a
variety of reasons, length of the cable (and therefore length of each
segment of the network) is limited to 2.5 km. Within the limitations
of the above parameters, the CSMA/CD-based access method provides
an efficient means of computer communication for low loads on the
channel. However, if the channel loading is increased, or if the require-
ments dictate either higher bit rates or longer cable lengths—for
example, to serve a metropolitan area—there is considerable loss of
efficiency. Much of this inefficiency comes from the use of the CSMA/
CD protocol. In CSMA/CD, a source transmits a packet when the
channel is sensed as idle, but this injection of the packet can be known
to the other sources only after it has propagated throughout the length
of the cable, during which time another source may attempt to transmit
on the channel. Thus, the number of bits wasted due to collision is
proportional to the propagation delay and the peak bit rate. Also, the
need to detect collisions makes it necessary that each packet have a
duration equal to at least the round-trip delay. With very large nets
and high bit rates, that may represent an unreasonably large minimum
number of bits.

Baseband CSMA/CD has been extended to broadband CSMA/CD
by several CSMA/CD networks, each in a different frequency band
put on the same cable (see, for example, Ref. 4). However, each of
these networks operates almost independently, connected usually by
a signaling channel. Also, the cable length limitation still applies,
making it difficult to use for a metropolitan area. In this paper, we
propose schemes that extend the capabilities of both the baseband and
the broadband CSMA/CD networks by allowing higher bit rates on a
cable, larger cable segments, and at the same time smaller collision
delays. We do this by dividing the available bandwidth of the cable
into several frequency bands and operating a network (or channel) in
each frequency band. Since coaxial cables can easily carry up to 400
MHz, several networks can be accommodated on one cable rather
easily. By modulating the baseband data from devices connected to
the network to a high-frequency band, total channel bit rates of higher
than 50 Mb/s can be obtained easily. However, since the bit rate of
each of the nets is kept low, channel inefficiency due to the use of
CSMA/CD protocol is not increased. To increase the length of the

cable segment, and at the same time limit the collision delay, we divide
the users into communities based on their location and give each user
community a network (i.e., one band of frequencies) to transmit most
of the time. Thus the “effective” end-to-end delay is reduced although
the cable length is increased. The principal characteristics and advan-
tages of our system are the following:

2. Larger cable length but smaller collision delay by dividing the
cable into several parts and operating a network in a given frequency
band for each part to be used by a user community, while retaining
the listening ability on the full cable length.

4, Different grade of service, depending on the complexity of net-
work interface.

5. Restricting most of the high-speed processing to the analog
domain and baseband processing to digital domain. Thus, although
the network may have channel throughput over 50 Mb/s, individual
networks may carry at much lower bit rates.

In this section, we describe one possible implementation of our
system. A block diagram is shown in Fig. 1. Each terminal has a
frequency-agile Radio Frequency (RF) modem that can modulate
binary data for transmission on the cable and demodulate the signal
from the cable to extract the transmitted binary data. Unlike long
distance transmission, since. the intent is not to maximize the data
transmission rate, simple inexpensive modulation schemes can be
chosen with enough separation between the various frequency bands
to keep the filtering simple and to reduce the crosstalk. As an example,
if modems based on Frequency Shift Keying (FSK) with 1/4 bit/Hz
are used, then a cable of bandwidth 300 MHz can support six CSMA/
CD networks of peak rate 10 Mb/s each with a guard band of 10 MHz
to separate each of them.

Our block diagram in Fig. 1 shows bidirectional transmission, that
is, signals injected on the cable at each tap travel in both directions
and the amplifiers are bidirectional. It is necessary for the taps to be
bidirectional so that they can receive signals from either direction.
Although this is a straightforward extension of the baseband CSMA/
CD network, bidirectional amplification and taps may present engi-
neering difficulties, particularly at high frequencies. Alternative de-

In Fig. 1, there are two networks (called Homenets) and therefore
two frequency bands. The transmitter can, by the agility of the modem,
transmit on any of the two frequency bands, and the receiver can
receive and demodulate data from both the frequency bands. Termi-
nals attached to homenet 1 transmit mostly on frequency band f1 and
those attached to homenet 2 transmit mostly on frequency band f2.
If several simultaneous conversations with terminals on different
homenets are required (as in the case of a host computer), then a
terminal may need multiple transmitters and receivers. Details of the
protocols for the access are given in the next section.

Below we give two types of access protocols; the first does not
require synchronization of the different terminals, whereas the second
does. Some desirable characteristics of any protocol should be noted
first. The access delay should be decreased by scheduling the trans-
mission on a net that is either least busy or has the least chance of
collision. The load on the different networks should be distributed
such that a situation does not arise in which many terminals are trying
to transmit on a network and are unable to do so, while the rest of the

networks are carrying a very light load. In carrier sense multiple-
access systems, collisions are a result of the terminals knowing the
transmission by other terminals only after the propagation delay.
Therefore, by requiring all the terminals that are close to each other
to initiate their transmission on a particular network, the collision
delay can be reduced considerably. Thus, in both the protocols, each
terminal is assigned to a particular network. This network is called
the Homenet of the terminal. Homenet assignment is included as a
part of the address of the terminal. Each terminal maintains a list of
the homenet assignments of the other terminals. The homenet assign-
ment, although made primarily by geographical location, may also
take into account the desired connectivity, traffic patterns, etc.

Figure 2 shows a system in which there are two homenets and three
terminals per homenet. The transmission initiated by any of terminals
1, 2, and 3 is mostly on homenet 1, whereas the transmission by
terminals 4, 5, and 6 is on net 2. Since net 1 and net 2 are on two
different bands of frequency, the collisions are now localized. That is,
data from terminal 1 can only collide with data from terminals 2 and
3. Since the distance between the taps on the cable of terminals 1, 2
and 3 is much shorter compared with distance between the taps of
terminals 1 and 6, the probability of collision and, therefore, of data
wasted due to collision is significantly reduced. This increases the
channel utilization and decreases the delay. Of course, the protocols
must and do allow communication between the terminals on different
homenets.

1. Every terminal has at least two receivers and, therefore, is capable
of listening to at least two networks. One of these receivers always
listens to the homenet. The other receiver is free to listen to any net.

PUT S REC2
ON HS
B — CHANNEL BUSY SNM — SCHEDULE TRANSMISSION OF
C — COLLISION DURING TRANSMITTING NEXT MESSAGE
HD — HOME NET OF DESTINATION SNP — SCHEDULE TRANSMISSION OF
HS — HOME NET OF SOURCE NEXT PACKET
LBT — LISTEN BEFORE TRANSMITTING SRB — SCHEDULE RETRY AFTER BUSY
LM — TEST !F TRANSMITTED MESSAGE WAS LAST CHANNEL
LP — TEST IF TRANSMITTED PACKET WAS LAST SRC — SCHEDULE RETRY AFTER COLLISION
LWT — LISTEN WHILE TRANSMITTING TR — TRANSMITTER
REC1 — RECEIVER 1 WIC — WAIT FOR IDLE CHANNEL
REC2 — RECEIVER 2 WNM — WAIT FOR NEXT MESSAGE
S — SOURCE XMIT — TRANSMIT

homenet. The other receiver of that terminal becomes active only after
the first receiver starts listening to a network other than its homenet.

3. Any terminal A desiring to transmit to terminal B goes through
the following sequence.

c. If net, is idle (i.e., absence of carrier), transmit net, carrier for a
period 7, the two-way propagation delay through the total network.
This amounts to a priority preempt on nety,.®*

d. If during the second half of the period T there is collision on
net,, then it implies a preemptive transmission from another terminal,
not on net,. In that case, terminal A backs off and attempts a
transmission on net, with reduced probability at the next time slot of
T. If there is no collision, then terminal A follows its preempt with a
message to terminal B.

e. Terminal B always has one receiver listening to net,; therefore,
it receives information from every collision-free packet on net».

f. If packet communications is to be continued, then terminal A
starts listening on net,, terminal B on net,, and both terminals
transmit on their homenets. Thus, if a message has several packets,
only the first packet may be transmitted on a homenet different from
the homenet of the source; all the subsequent packets are transmitted
on their own homenet with standard CSMA/CD protocol with retrial
period equal to round-trip delay of the homenet.

g. If at step e terminal B is already in communication with some
other terminal on a different network, then it still has a receiver on
net,. If terminal B’s transmitter is on net, (as it normally is, except
when it is trying to set up an initial connection with a terminal on a
network other than net,), even if it is in communication with some
other terminal, it can send an acknowledgment back to terminal A on
net,. If, however, Terminal B’s transmitter is transmitting on a
different channel, there may be delay in sending the acknowledgment.*

h. If after successful connection there is no communication for
a given amount of time and if the receiver on the homenet receives
a message for communication from another source, then the
other receiver of both the home terminals go back to their respective
homenets.

* Alternatively, there could be a signaling channel in a different frequency band
accessed by all the terminals, and the first packet could be transmitted on the signaling
channel. This alternative is attractive for a large network, since it confines collision to
a common homenet and, hence, does not constrain minimum packet length. With the
utilization very low and the network large, the most appropriate protocol on the signaling
channel would be ALOHA.

TIf several simultaneous conversations are required, then a terminal may need
multiple transmitters and receivers.

i. Many different broadcast modes are possible. If the broadcast to
only the terminals in the particular homenet is desired, then data is
transmitted on that network only. However, if broadcast is required
to all the terminals, then the transmitter has to successfully transmit
on each network.

The above protocol is reasonable in that it reduces collisions and
works well when the traffic is quite bursty, with many terminals
trying to transmit messages containing large numbers of small packets
frequently. However, when there are large file transfers, use of the
homenet by a terminal prevents other terminals with the same
homenet from using the channel even though the other networks may
be idle. Thus, a reasonable protocol is needed that will share the
channels more evenly in the presence of large file transfers by one of
the terminals. Protocol 2 attempts to accomplish this at the cost of
slightly larger average delay in establishing a connection. In this
protocol, networks on which a given group of terminals begin trans-
mitting are switched on a periodic basis. The period is of the order of
several packets long (or tens of milliseconds). It thus requires a clock
at every terminal, which may be provided from a central clock on a
different band of frequencies. Details of the protocol are

1. As in protocol 1, terminals in a given geographical area are
grouped together. This grouping is made known to all the terminals
(similar to homenets).

2. A group has a homenet and an assigned transmission network.
The homenet is fixed, whereas the transmission net changes cyclically.
A terminal may, at any time, initiate a transmission only on the
transmission network to which its group is then assigned. Once initi-
ated, the transmission may spill beyond this fixed interval, since the
packet size is not fixed.* As an example, Fig. 4 shows the case of three
groups and three sectors of time.

3. A terminal has at least two receivers. When the terminal is idle,
both of them listen to the homenet. After establishing a connection,
however, one of the receivers switches to the network on which it has
established connection with the other terminal (and, therefore, the
network to which it listens changes cyclically) and the other receiver
remains at the homenet.

*In CSMA/CD the entire packet must be received and CRC checked before the
destination address is verified. Thus, if the entire packet is not received before the
period ends, the receiver may miss it. To overcome this problem, a separate CRC is
provided for the header information and a source terminal starts a transmission
sufficiently before the end of a period such that the destination i is able to receive the
header information before the period ends. ,

Fig. 4—Protocol of Section 3.2. The assignment of group j terminals to net, is made
by rotating the inner circle at a given speed. The terminals in group j may initiate
transmission on net, if the group j pointer is in the sector corresponding to nety,.

4, Terminal A, desiring to transmit to terminal B, goes through the
following sequence:

d. If net, is idle (during the assigned time slot), then transmit to
terminal B on net,.

e. If there is no collision, a packet is assumed to have been received
by its intended receiver. If there is a collision, terminal A ceases
transmission immediately and tries again, using a standard retry
strategy, but the additional constraint that its starting time must be
when terminal A is allowed to transmit on nety.

f. If communication is to be continued, then terminal B switches
one of its receivers to the net on which terminal A will be transmitting
(this will periodically switch), and terminal A will set its receiver on
the net on which terminal B will be transmitting.

g. If at step f terminal B is already in communication, ‘ign its
acknowledgment to terminal A will so indicate. |

h. If after a successful connection there is no communication for a
given amount of time, receivers of both the terminals go back to their

Number of stations continuously queued to transmit a
packet; represents the total offered load.

Number of networks on the frequency-multiplexed cable.
Il, = Probability that a message from a source contains m pack-

The average delay in sending a packet (including transmission time)
when Q stations are continuously queued to transmit a packet, is given
by

This assumes an optimum retry strategy. Since the Offered Load (OL)
is Q packets, in terms of bits it is given by

Here if a message from a source contains m packets, then the first
packet may be transmitted on a different network, but the subsequent
(m —.1) packets will be transmitted on the homenet of the source.
Thus, the total transmission time is divided into two parts: time to
transmit the first packet and time to transmit the remaining packets.
It is assumed for simplicity that the length of each homenet is the
same and it is 1/N times the total cable length. The following analysis
can be easily modified for other configurations.

If the first packet of a source on homenet / is transmitted on homenet
k, then

Time for a packet = < + contention time. (3)
Assume that a source on the jth homenet has probability p;, of wanting
to communicate with a station on the kth homenet. Further, assume
that the number of stations is the same for all nets; distribution, II,,,
of packets is uniform for all messages; and {p,;,} are a constant*
independent of j, k. Then the total traffic offered to the kth homenet
is

_Q |
ak = N’ (4)
of which the offered load from out-of-net is
N-1
qk = —N Qi, (5)
and from within homenet
— Q N-1
qk = N 1 N IT, }. (6)
Therefore the time per out-of-net packet is
tP
Dy = + TI - 1/qi)*-* — 1). (7)

This transmission is on the homenet itself. Therefore, the time for
each packet is simply obtained by
tP T
=— +—[(1 — qi) — 1]. 8
| CN [( qk) ] (8)
This neglects the traffic generated by first packets of terminals from
other homenets. It is assumed that the first packet is a small fraction
of the total message and does not result in any significant traffic.

Since the probability that a message contains k packets is II;, the
average time per packet is given by

* We have made no measurements of traffic on real systems to justify this assumption.
It is made only so that a closed-form expression can be derived for the delay. If other
values of p;, are more realistic, they can be substituted easily in the equations that
follow.

Thus the delay versus the offered load characteristic will be given by
Dz versus OL.

With Protocol 2, all packets are sent on homenet, and the delay for
optimum strategy is given as

When II, is small, that is, messages consist on average of many
packets, then D3 and Dz differ very little from each other.

It is possible to compute the optimum number of nets based on the
above expressions for average delay per packet. This can be done for
the case when Q (and Q/N) is large and the messages contain a large
number of packets, implying that the average delay per packet is
dominated not by the first packet, but by the subsequent packets.
From eq. (1), for single network, the delay is given by

If t = N, then the total capacity C is divided among N nets equally.
However, in most cases, the individual N nets may have a capacity
such that the capacities add up to more than C. Thus let t = N/s. In

this case, each net has a capacity (sC)/N and the total capacity is
given by sC. The delay then becomes

For large (Q/N) (or small N/Q) we can expand Dp» as a function of
N/@Q in Taylor’s series

Thus with standard Ethernet* parameters, from the point of view of
average delay per packet, N should be 1.

This implies that as the length of the cable increases, more networks
are required.

This implies that if each net is operated at 12.5 Mb/s, adding up to
a total capacity of 12.5 x N Mb/s, average delay is minimized when
N= 4.

The average time per packet derived in Section 4.1 was evaluated
for a variety of cases and is plotted in Figs. 5 and 6. In all the cases, a

Fig. 5—Average delay per packet versus length of the cable. Performance comparisons
are made between two networks on a cable and a single net.

Fig. 6—Average delay per packet versus length of cable. Comparison is made between
four networks on a cable and single network.

packet size of 500 bits was used. The curves in Figs. 5 and 6 are for
Q = 200, that is, 200 packets are continuously queued. It is assumed
that a message has 512 packets, that is,

As expected, the delay increases with length, and depending upon the
other parameters of the network, the delay corresponding to multiple
network becomes smaller than that corresponding to single network
if the length is increased beyond a certain value. Figure 7 shows the
variation of the average delay with respect to number of networks.
The capacity of each of the nets is equal and is such that the total
capacity of all the nets adds up to capacity of the single network. We
find that, as expected, for the parameters chosen in Fig. 7, the average
delay does show a minimum around N = 4. This verifies our eppron
mations of the previous section.

Fig. 7—Average delay per packet versus number of networks. Each network has a
eapaaly of C/N, where N is the number of nets and C is the capacity of the single
network.

We have described a broadband local area computer network. It
consists of several local area networks whose data is frequency multi-
plexed on a single cable. The entire cable length is divided into parts;
each part is assigned a network and a frequency band for transmission.
We have also described two protocols that overcome some of the
limitations of the present baseband as well as broadband CSMA/CD
networks. Using our schemes, it is possible to increase the channel
throughput and the length of the cable network, reduce the delay due —
to collisions, and at the same time allow complete connectivity among
all the terminals and devices logged into any network. Approximate
analytical results are also presented to substantiate these claims.

We are grateful to Jerry Foschini and Nick Maxemchuck for many
helpful discussions.

. The IEEE Project 802, Local Area Network Standards, CSMA/CD Access Method
and Physical Layer Specification, IEEE P802-3-82/0.1 10, December 1982.

. N. F. Maxemchuck and A. N. Netravali, “A Multifrequency Multiaccess System for
Local Access,” Proc. ICC 83, Boston.

6. N. F. Maxemchuck, “A Variation on CSMA/CD That Yields Movable TDM Slots
in Integrated Voice/Data Local Networks,” B.S.T.J., 61, No. 7 (September 1982),
pp. 1527-50.

. R. M. Metcalfe and D. R. Boggs, “Ethernet: Distributed Packet Switching for Local
Computer Networks,” Commun. ACM, 19, No. 7 (July 1976), pp. 395-404.

On Binary Differential Detection for Coherent
Lightwave Communication

Motivated by the communication problems caused by phase noise in those
semiconductor lasers that may be used for fiber-optic data transmission, we
consider heterodyned binary Differential Phase-Shift Keying (DPSK) in
conjunction with high-rate (short time chip) redundancy as provided by
repetition or by more complex coding techniques. In surprising contrast to
repetitive coherent phase-shift keying where only a loss of a 2/x (2 db) in
power is incurred in the limit of infinitely many infinitesimal time chips, we
show that DPSK requires, in this limit, an infinite number of photons per bit.
This is true regardless of the coding scheme used with the DPSK modulation.
Next we find the bandwidth expansion that minimizes the number of received
photons per bit required to hold the error rate at 10~° for two situations: first
for a simple repetition code, and then for a repeated (24, 12) Golay code with
maximum likelihood detection. The performance of the latter is assumed to
be representative of other optimally detected codes of the same rate, such as
convolutional codes with Viterbi decoding. Explicit curves relating required
photons per bit to the bandwidth expansion are given for B/R ratios of 0.01
to 10, where B is the laser linewidth and R is the data rate. An example of the
results is that for B/R = 0.1 and a bandwidth expansion of 10, about 23
photons per bit are required for the repeated Golay code to perform as well as
uncoded DPSK without phase noise (which requires 20 photons per bit for Pe
= 10-°). If B/R = 0.01 the bandwidth expansion is reduced to 2, and 12 photons
per bit are required, thus outperforming the phase-stable, but uncoded, situa-
tion. :

Semiconductor lasers that may be used for coherent data transmis-
sion over optical ‘fibers* have severe phase instabilities. Attempts to
transmit a sine wave of frequency f, result in outputs that are modeled
as

where w(t) (measured in radians) is a random process representing
the phase instability. The process w(t) is usually taken to be a Wiener
process and that then implies a Lorentzian [see eq. (4)] line shape for
the power spectrum of (1). Such spectra are indeed observed and 3-db
bandwidths as large as 10 to 20 MHz have been measured.’ These
bandwidths imply that the standard deviation of the change in w(t)
over a ws can be as large as 47. Severe problems would be encountered
with any conventional coherent detection scheme if one is transmitting
data at ten-megabit rates (or lower) rather than gigabit rates. Never-
_ theless, one may wish to do precisely that, and our purpose here is to
mathematically explore one very natural approach, Differentially co-
herent Phase-Shift Keying (DPSK) in combination with code symbols
that have short transmission time. The purpose of using code chips of
short duration is to mitigate the effects of phase wander between
adjacent chips.

We emphasize that the scheme we are about to investigate is not
the only possible one. One could use on-off keying of the optical carrier
(1) with photon counting for detection. Theoretically this outperforms
DPSK by 3 db, even with a stable transmitting carrier assumed for
the latter. However, photon counting is not easy to implement, and
practical avalanche photodiodes can introduce 20 db of loss. Thus
other techniques, which involve heterodyning, are of interest, in hopes
that their implementations can be closer to their own theoretical
ideals. For a general survey of lightwave communications we recom-
mend Ref. 1. In Ref. 2 a large number of modulation schemes for
coherent optics are evaluated with the main purpose being that of
determining the range of B/R values for which coding is not required.
We choose here to examine DPSK in detail, but the general behavior
of its performance with coding is expected to be representative of
modulation formats that do not involve tracking the phase w(t) with
a phase-locked loop. The latter was one of the methods considered in
Ref. 2 and was shown to be feasible only if B/R < 0.003.

Returning to the repetition-DPSK scheme, we note that it might be
expected that in the limit of an infinite number of infinitely rapid

* In optical-fiber work, coherent transmission refers to any modulation format where
an optical oscillator is required at the receiver.
t The carrier wavelength of interest is 1.55 um, or f, is roughly 2 x 10'* Hz.

code chips, the performance would approach something like that of
full interval DPSK with a stable oscillator. One of the surprises that
we have uncovered (and perhaps the major point, of interest of this
work) is that this is far from the truth. We show that in this limit an
infinite number of incident photons per bit are required for fixed error
rate.

Decreasing the number of repetitions while holding the error rate
fixed also will ultimately require the photons per bit to become
unbounded. This occurs when one approaches (from above) the num-
ber of repetitions required to achieve the given error rate, with phase
noise being the sole impairment. Consequently, one expects that there
will be an optimum bandwidth expansion. This is in fact true, and it
is discussed in Section IV, while the similar problem for more sophis-
ticated coding is treated in V.

In Section II we begin by presenting the mathematical details of the
model, while the two limiting cases of large and minimum bandwidth
expansion are investigated for the repetition code in Section III.

and set the (two-sided) spectral density of the white noise n(t) to be
N/2. The variance o2,(t) of w(t) at time t is then :

The power spectrum of (1) can be calculated in terms of these
quantities and is given by®

Often in coherent optics one converts (1) to microwave frequencies
(GHz) where conventional signal processing techniques are available.
This heterodyning is accomplished by mixing (1) with a locally gen-
erated optical wave. The local oscillator also has phase instabilities
that add to those of the received signal, and thus in this paper the
effective bandwidth at-microwave is taken to be double that at optical
frequencies. Furthermore, shot noise fluctuations in photon counts
during the heterodyning causes a white noise background to be added
to the microwave signal. In our model we assume heterodyning to be
done, and thus our received unmodulated carrier is modeled as

Assume, momentarily, that ¢ = 0 and ¢(t) = 0 over a bit interval T,
and we wish to coherently detect the modulation +A. We simply
multiply by cos w,t, integrate the result for T’ seconds, and observe the
sign of the output. The chance of making an error, Pe, is then

In (9), E, = A?T/2 is the energy per bit in the transmitting signal.
When (7) arises, as it does in our case of interest, from heterodyning
of an optical wave, F,/No is not an arbitrary parameter but is (see
Ref. 1) numerically equal to the average number of photons per bit in
the optical wave at the receiver. An E,/Npo corresponding to 18 photons
per bit yields an error rate of 10~° for coherent detection.

The quantity E,/Npo is also numerically. equal to the signal-to-noise
ratio (s/n) if the noise power N is measured in a bandwidth equal to
1/T.

To motivate a later discussion, consider the coherent case further
and instead of integrating the received signal over (0, J’) and making
a decision (what we might unconventionally call soft-decision decod-
ing), we make n = 2m + 1 hard decisions based on time chips of length
T/n, and then use a majority vote to decide the sign of the transmitted
bit. If p, is the chip error rate, the bit error rate, Pe(n), would be

Since we want to keep E, constant, the energy in each chip decreases
as H,/n. For n large, then, we have from (8) and (9)

In (13) we see the ubiquitous 2/7 penalty in s/n for using hard
decisions. Equation (13) was derived for coherent detection. When we
later consider repetition-DPSK to overcome phase noise, we will be
concerned with a corresponding limit for differential detection. Then
_ the fortunate limiting behavior we have just observed will not occur,
because, with DPSK, the chip error rate approaches 1/2 more rapidly
with n than it does in the coherent case exemplified by (11).

To make a tractable model for DPSK, we assume that the received
waveform is

The pulse g(t) is assumed brick-wall Nyquist with g(0) = 1, and energy
T.. The Gaussian noise processes n,(t) and n,(t) are independent, flat
spectrum, and of equal variance o” = N,/T,. As stated earlier, the chip
time, T., for the repetition code is related to the bit time T via T, =

Fig. 1—Geometry of variables for differentially coherent phase-shift keying.

T/(2m + 1), and (2m + 1) repetitions* of the bit are differently
encoded into the chips a, = +1. |
We imagine recovering the differentially encoded chips a, by de-
modulating (14) with cos w,¢ and —sin w,t and sampling the demodu-
lation outputs at the appropriate T, second intervals. At each sampling
instant, then, we obtain a pair of real numbers, which we regard as a
point in the plane. In the absence of phase and additive noise, this
point would simply be (+A, 0), relative to a coordinate system fixed
by the unknown phase @. With noise included, the geometry for two
‘consecutive samples is equivalent to that shown in Fig. 1, drawn for
the case of different consecutive a,. All the components of the two
vectors (A, 0) and (—A, 0) are perturbed by additive independent and
identically distributed zero-mean Gaussian noise of variance o”. Owing
to the phase noise, one of the perturbed vectors is also rotated by an
angle ¢, where ¢ itself is a zero-mean Gaussian variable of variance

where B is the laser linewidth. If the angle y between the resulting
vectors is less than 2/2, we would declare that the two consecutive
chips were the same and, in the present case, an error would be made.
Of course, for y > 2/2 we decide the chips were different.

From (15) we see that as the chip interval decreases, o% approaches
zero and phase noise will make a negligible contribution to the chip
error rate p,. Note also that the additive noise variance o” increases
as T, decreases.

Before we proceed, a few comments about the model are in order.
An equivalent detection procedure is to delay the signal represented

* More generally, our results for the repetition code are unchanged if the bits are
mapped into any two complementary patterns of n chips.

in (14) by a chip time, T,, and sample the product of the delayed and
undelayed signals. The resulting decision statistic is identical to the
angle that we consider, being a representation of it as a quadratic form
equal to the inner product of two noisy vectors. Secondly, a more
realistic description of the signal uses a pulse g(t) that has unit value
over the interval T, and is zero elsewhere. The bandwidth of the signal
then is not precisely defined, but if one estimates the bandwidth of a
flat front-end filter required for noise filtering to be 1/T, (and ignores
any intersymbol interference), then the numerical results are un-
changed. Finally, we note that Salz” integrates the product of the
signal and the delayed signal, rather than simply sampling. This
complicates the analysis considerably. We have not attempted to
investigate the difference in detail over the full range of B/R values,
but we note the following. Salz estimates the B/R value required in
order that DPSK detection suffers only 1-db degradation compared
with B/R = 0 and finds B/R < 0.003 is sufficient. We calculate this
precisely for our model and find B/R < 0.002. This suggests that the
postdetection processing provided by the integrator might not be
significant.

We use (10) later to calculate the bit error rate for repetitive DPSK
(with an appropriate p,). It may be objected that the use of this
formula for the repetitive code is not rigorously justified, since samples
used for detecting consecutive chips have one noise sample in common
and thus the chip detection probabilities are not independent. This
objection is easily overcome by assuming that the chips for two
successive bits are interleaved in the manner abab --- .

In this section we treat two limiting cases of repetitive DPSK. One
case is concerned with a very large number of rapid repetitions. Here
since the chip interval T, becomes small, the phase noise is neglected,
and for fixed E, the chip s/n is small. The other case examines the
minimum number of repetitions required to achieve a fixed error rate
if phase noise were the only impairment. To approach this limit would
require a large E, provided that more than one repetition is required.

We begin with the high repetition rate limit. It is well known that
the error probability for DPSK (see Ref. 4) is

where EF, = E,/n is the energy per chip. For small E, (large n) (16)
yields for the chip error probability, p,,

Note the difference in behavior for the differentially coherent problem
[see (18)] versus the coherent one [see (11)]. In (18) the chip error
rate approaches 1/2 as 1/n, whereas in (11) the behavior is as 1/Vn.
Thus, for DPSK the lower limit of (10) is O(1/Vn) standard deviations
above the mean, and for large n we have that the bit error probability
Pe(n) obeys

In essence, then, we have that if, in Fig. 1, only one of the vectors
is noisy and ¢ is set to zero (coherent phase-shift keying), then Pe(n)
is small in the limit of many repetitions with constant E,, but if both
are noisy (DPSK), Pe(n) limits to 1/2.

For the second limiting case, when the only perturbing influence to
the transmission is the Gaussian phase noise variable ¢, the chip error
rate Is

In (20), p. is explicitly a decreasing function of the number of
repetitions via the phase noise variance o4. Further, if Pe(n) is fixed
in (10), that expression implicitly determines p, as an increasing
function of n. Requiring that both (10) and (20) determine the same
value of p. fixes the number of repetitions required to achieve the bit
error probability Pe. Including additive noise in the calculations can
only increase the required repetition rate for fixed Pe. Setting Pe =
10~°, we have computed the number of repetitions, n, required when
phase noise is the sole impairment. The probability p, is the chip error
probability (20) for n repetitions. Both are displayed in Table I for
several values of B/R. Although our main focus will not be on numer-
ical values of p,, it is worth emphasizing that throughout this paper
values of p, above 0.1, and even approaching 1/2, are possible for the
larger values of B/R.

Table I—Minimum number of
repetitions A required so that bit
error rate does not exceed 107?

The ratio signal-bandwidth/laser-linewidth equals nR/B. From Ta-
ble I we see that this number is sufficiently high so that the implicit
neglect of the wideband filtering on the phase noise that was made
when writing the model [see (14)] seems justified for the parameters
of interest.

In this section we do a general investigation of DPSK detection
with repetitions, including both phase noise and additive noise as
impairments. Our main interest will be to determine the optimum
repetition rate and the corresponding bits per photon required to
achieve a bit error probability of 107°.

There are probably several useful general expressions for the chip
error rate p, when both phase noise and Gaussian noise are present.
We shall work with the one given in (22), namely,

where p is the chip (s/n) [see (17)], o3(n) is the phase noise variance
with n repetitions [see (21)], and J,(p/2) are the modified Bessel

An expression similar to (22) for constant phase error was first
derived by Blachman’ and is given in eq. (62) of Ref. 4. Performing a
simple average when this angle has Gaussian statistics yields (22). An
independent derivation is given in the Appendix.

The successive terms of the sum in (22) decrease in magnitude, and
hence, as for any such alternating series, the first neglected term

bounds the error. We find that 15 terms in double precision (single
precision on a Cray I) is enough to duplicate (16) when o3(n) = 0, and
p = 20 (p, = 107°).

We first use (22) to calculate the deterioration in p, as B/R increases.
Table II shows some results for p = 20, n = 1. We note that B/R =
0.002 yields about a 1-db degradation. |

Next in Figs. 2 through 5 we plot as a function of the number of
repetitions, n, the number of photons per bit, y, which are required at
the receiver to maintain a bit error rate of 10~° for B/R values of 0.01
to 10. This is done using (10) and (22) in the following way. First n is
chosen and the required p, is determined from (10). For fixed B/R and
known n, og in (21) is known, and (22) is then used to compute the
chip s/n p that will achieve that p,. Finally, y = np. These figures
show quantitatively how the optimum value of n (and the correspond-

ing value of y) increase with B/R. From these figures, we derive Table
III, which lists n*, the optimum repetition rate, and y*, the minimum
number of photons per bit required to hold the error rate at 10°°..The
last column in Table III compares ;* with 20, the number of photons
per bit required at this error rate for DPSK with no repetitions and a
stable phase.

We note that the curves are often fairly flat around the minimum,
and hence less bandwidth may be used without significantly increasing
the required number of photons per bit.

The repetition (or complementary) code for transmitting at rate
R = 1/T is but one way to use the discrete time Binary Symmetric

Table IV—Minimum and optimum bandwidth expansion for capacity
and repeated Golay code

Channel (BSC) that we have to work with.* Therefore, to see what is
theoretically possible, we next consider the required number of pho-
tons per bit that would be required to transmit at channel capacity. If
C is capacity in bits per second, we will fix B/C and plot y versus n
where the chip time is JT, = 1/(nC). That is, the binary code using the
chips should have a capacity C = 1/n bits per chip. Here n is the
bandwidth expansion and need not be an integer.

The required chip error probability is found by equating the channel
capacity for the BSC to 1/n, that is,

The chip s/n p required to yield this p, is then computed from (22),
and y = np. |
Curves for B/C = 0.01 to 10 are presented in Figs. 2 through 5, and
summarized in the first half of Table IV. Once again, a feature is the
existence of an optimum chip rate. The necessity of this is easily
argued. The lowest value of n possible is determined by the phase
noise alone, which causes p, to increase as n decreases. Eventually,
the capacity C drops below 1/n, and this fixes the minimum n = fi.
But to approach the pure phase noise situation, yy must increase
indefinitely as n is approached from above. To see why yy must increase
as n is very large, consider plotting C versus n with y fixed. Then, as
we have seen [see (18)], p. = (1/2) — O(1/n) owing to Gaussian (shot)
noise. Setting p, = (1/2) — « and expanding the left side of (24) in

Qe... ;
C = — bits/chip. (25)
In2
* In fact, interleaving (explained at the end of Section II) creates two parallel BSCs.

The total capacity is the sum of the capacity of each and is the same as the capacity
calculated here as if all chip errors were independent.

Since « = O(1/n) and we have O(n) chips per second, the capacity
measured in bits per second vanishes like 1/n. To avoid this, y must
increase.

Finally, we consider specific coding schemes that are more involved
than repetition. We explicitly consider the extended binary (24, 12)
Golay code. Thus for block length 24, 12 information bits are specified,
yielding a rate 1/2 code. This is a linear code and any code word has
759 nearest neighbors at minimum distance dmin = 8. The coding
scheme that we consider is simply to construct low-rate code words by
repeating a given Golay code word J times, make hard decisions on
the chips at the receiver (assuming appropriately interleaved DPSK),
and then use maximum likelihood decoding on the resulting binary
code word of length 24J. Note that the new code has dyin, = 8 and
that the code rate has decreased to 1/(2/).

Assuming that a word error results, on the average, in six bit errors,
we set the word error rate, P,,, to be (1/6) 107”. If p, is the chip error
rate, then the union bound yields

P, = 759 » ( ; ) pia peje (hich ) (26)
where “higher terms” represents the probability of decoding into words
further away than the minimum distance. These terms are neglected
for the low word error probability of interest here.

We have in mind that actual attempts at coding would use repeated
convolutional codes and Viterbi decoding. Our introduction of Golay
codes is simply to make the analysis easier, but overall performance
gains for the same repeated code rate are expected to be the same. In
fact, Chase® finds that a (properly chosen) repeated 16-state convolu-
tional code performs slightly better than a repeated Golay code. The
effectiveness of repeating an appropriately chosen code to obtain a
good low-rate code was, in fact, proposed by Chase,° who was concerned
with codes when p, is large, as is often the case for the present
problem.* Perhaps better rate 1/(2J) convolutional codes exist than
can be generated by repeating the symbols of a given one, but Chase
shows that, at least from a minimum distance point of view, a repeated
rate 1/2 convolutional code (suitably chosen) is close to optimum for
code rates at least as small as 1/128.

Returning to the Golay code, note that J repeats (of any rate 1/2
code, in fact) corresponds to a bandwidth expansion of n = 2d.

* A point emphasized in Ref. 6 is that for p, > 0.25 and asymptotically large block
length, bounded distance algebraic decoders cannot operate (Pe — 1) and maximum
likelihood decoders must be used.

are done in a similar manner, as earlier. The bandwidth expansion
n= 2 is picked, p, is found from (26) with P,, = 1.67 x 107'°, and
then (22) is used to solve for p. The improvements obtained over
simple repetition are displayed in Figs. 2 through 5, and essential
features of the results are given in the second half of Table IV. In
general, we see that the optimum bandwidth expansion is less than
with repetition, and, of course, so is the required number of photons
per bit. An important feature is that for B/R = 0.01 and 0.1, the
required number of photons per bit is less, or comparable to, that
required when no phase noise is present and no coding is used. In
these cases the bandwidth expansion is relatively modest as well.

To reduce the harmful effects of phase noise, coding schemes that
use DPSK detection of code symbols having short duration were
examined. We first considered in detail a simple repetition code and
determined the optimum bandwidth expansion that minimized the
number of received photons per bit required for an error probability
of 10-°. If B/R = 0.01, we found n* = 5 and y* = 37. The corresponding
numbers for a repeated Golay code were n* = 2 and y* = 12.5. By
contrast, 20 photons per bit are required for phase stable but uncoded
DPSK transmission.

The performance of the repeated Golay code is intended to be typical
of that obtained with other moderate coding efforts using maximum
likelihood detection. In particular, it should be comparable to a re-
peated 16-state convolutional code (of the same overall rate) with
Viterbi decoding. It is our understanding that the fastest commercially
available Viterbi decoders operate at about 20 Mb/s (with 64 states).

My interest in, and exposure to, this subset of communication
problems in fiber optics is due to several enjoyable and informative
discussions with Paul Henry and Jack Salz.

2. J. Salz, “Coherent Lightwave Communications,” AT&T Bell Lab. Tech. J., this
issue.

3. J. E. Mazo and J. Salz, “Spectra of Frequency ee With Random Wave-
forms,” Inform. Contr., 9, No. 4 (August 1966), p. 4

4. R. F. Pawula, S. O. Rice, and J. H. Roberts, ‘ faa ney of the Phase Angle
Between Two Vectors Perturbed by Gaussian Noise,” IEEE Trans. Commun.,
COM-30 (August 1982), pp. 1828-41.

5. N. M. Blachman, “The Effect of Phase Error on DPSK Error Probability,” IEEE
Trans. Commun., COM-29 (March 1981), pp. 364-5. |

7. I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products, Fourth
ed. New York: Academic Press, 1965.

We begin by deriving a Fourier series for the angular distribution
g(01) of a vector (A, 0) perturbed in each component by additive, zero-
mean, independent and identically distributed Gaussian noise of var-
iance o”. In what follows,

polar coordinates (r, 6,) and integrating over the r variable after
applying (29), we obtain, after a variable change,

In our problem, we have a second noisy vector that is also perturbed
by rotation through ¢, ¢ being zero-mean Gaussian. The modulo 27
angle 6. of this vector has density h(@2), where

E, being expectation with respect to ¢. In (31), g(@2 — @) is evaluated
by the periodic extension of g(@). We are assuming here that the two
consecutive chips are identical, but the error probability is the same
when two consecutive chips have opposite signs.

Since 6, and 6, are independent, the difference angle y = (02 — 0;)
mod 27 has density p(w) [see eq. [6] of Ref. 4], where

James E. Mazo, B.S. (Physics), 1958, The Massachusetts Institute of Tech-
nology; M.S. (Physics), 1960, and Ph.D. (Physics), 1963, Syracuse University;
Research Associate, Department of Physics, University of Indiana, 1963-1964;
AT&T Bell Laboratories, 1964-1985; AT&T Information Systems, 1985—. At
AT&T Bell Laboratories, Mr. Mazo had been concerned with theoretical
problems in data transmission. He now supervises the Data Theory group at
AT&T Information Systems.

Performance Signatures for Dual-Polarized
Transmission of M-QAM Signals Over Fading
Multipath Channels

Performance signatures for dual-polarized transmission of M-state quad-
rature amplitude-modulated signals over dispersive multipath digital radio
channels are theoretically derived in this work. We report on two major
findings. Firstly, we show that for the assumed propagation model, a cross-
coupled interferer exhibits noiselike behavior and impacts on digital radio
outage time in direct relation to its power level. Secondly, our theoretical
finding is based on a new application of performance signature curves for two
cross-coupled multipath channels. This treatment permits the prediction of
multipath-induced digital radio outage for specified ratios of power in the
copolarized and cross-coupled signals. Theoretical findings are qualitatively
supported by measured performance signatures obtained from a laboratory
simulation of the model. |

The last decade has witnessed a surge of interest in terrestrial digital
radio transmission, with the newer high-capacity systems relying
almost exclusively on single-polarization microwave transmission of
M-state Quadrature Amplitude-Modulation (M-QAM) signals. In
these digital radio systems, the performance degradation associated
with multipath propagation has been of paramount importance and

the subject of considerable prior investigations. For single-polariza-
tion systems, the effects of multipath are well understood;)” suitable
countermeasures—particularly adaptive equalization—have been
studied;?* and these countermeasures are now widely deployed in a
variety of digital radio systems.*®

When compared with single-polarization systems, dual-polarized
operation obviously engenders economic and efficient spectrum-utili-
zation advantages. Unlike single-polarization transmission, however,
an understanding of the effects of multipath propagation on this latter
mode of operation is still in its infancy. For single-polarization oper-
ation, countermeasures to frequency-selective fading mitigate inter-
symbol interference (ISI) in the presence of Gaussian noise; the
transmission of orthogonally polarized signals over the same bandlim-
ited facility is similarly vulnerable to the effects of ISI and noise, but
now Cross-Polarization Interference (CPI), normally suppressed by
the polarization selectivity of the receiving antenna, is also present.
Consequently, an optimal receiver must recover the transmitted signal
in the presence of ISI, CPI, and noise.

Kavehrad’ has previously studied dual-polarized M- QAM transmis-
sion over nondispersive media. He concluded that satisfactory trans-
mission is not feasible without some form of cross-polarization inter-
ference cancellation. Furthermore, the work showed that in an optimal
detection process, the total noise and CPI power must be adaptively
minimized. In this paper we extended the scope of the previous work
by considering dual-polarized transmission of M-QAM signals over
dispersive fading channels like those experienced in line-of-sight ap-
plications. |

The dual-polarized channel is modeled using Rummler-like® multi-
path transfer functions to describe both the copolarized and cross-
coupled paths. The transfer functions emulate snapshots of independ-
ent multipath fading events on the copolarized and cross-coupled
paths, which, in the presence of noise, limit the achievable error rate
at the receiver. Our results are predicted on the major assumption that
‘the two simultaneous fading events are statistically independent.
Performance “signatures,” defined by a locus of fade notch depths and
frequency positions corresponding to a 10™ error rate,’ are used as a
system DeTORMAnce measure since they are conveniently related to
digital radio outage.’°

For the assumed propagation model, we find that the cross- eae
interfering signal exhibits a noiselike behavior. That is, the power loss
associated with a cross-coupled signal subject to flat or dispersive
fading brings about an actual reduction in system outage time. Appar-
ently, the deleterious effect of ISI contributed by dispersive fading of
the cross-polarized interference is more than offset by the power loss

of the same associated with the fading event, consequently improving
net system outage. Furthermore, this finding is supported by labora-
tory measurements in which the fading of two independent 16-QAM
signals is simulated.

The theoretical and experimental finding cited above is based on a
novel application of the aforementioned system performance signature
curves for the propagation model adopted in this work. In particular,
it permits an immediate comparison of multipath-induced dual-polar-
ized digital radio outage for specified ratios of power in the copolarized
signal and cross-coupled interferer.

In the following section we begin by providing a complete model of
the dually polarized communication channel, including the influence
of frequency-selective fading. In Section HI the computational meth-
ods and performance measurers are discussed. Numerical results and
laboratory measurements that support our finding are presented in
Section IV. Our conclusions are summarized in Section V.

The system model for dual-polarized operation is illustrated in Fig.
1. Two independent data sources (one for each of the dual-polarized
channels) are assumed to generate complex-valued symbols at the
baud rate, 1/T,, where T, is a symbol period. We denote these complex

at consecutive instants kT,, k = 1, 2, 3 ---. The index 7 denotes
symbol sets transmitted on the main (i = I) and cross-coupled (i = II)
channels. The real and imaginary parts {5}, 61,} take on discrete levels
+0, +380, ---, +(L — 1)Q with equal probability. Parameter L = JM
is selected in accord with the number of states in the M-ary signal;
the symbols 6 and 6 are independent and identically distributed; and
Q is a measure of signal-point distance from the nearest decision
boundary in signal space.

We assume each of the Pulse-Amplitude-Modulated (PAM) signals
has a raised-cosine (Nyquist) spectral shaping with a roll-off factor I.
The corresponding time-domain impulse shape is therefore!

and the frequency-domain transfer function is designated P(w). On
each of the orthogonally polarized channels, the shaped signals are
modulated by quadrature carrier signals of frequency f,. Furthermore,
the two independent carrier local oscillators for i = I and 1 = II at the
transmitter are out of phase by 6@,,, where this phase difference is a
uniformly. distributed random variable over the range 0 < 6,, < 27.
The complex baseband symbol sequences are assumed misaligned by
Tm, Where Tm is also a uniformly distributed random variable over the
range 0 <7, = T;.

As previously stated, we follow Rummler’s prescription for modeling
multipath fading. The model is applied to both the copolarized (1 = I)
and the cross-coupled (1 = II) signal paths and assumes the presence
of a single inband notch in each of the main and cross-coupled path
transfer functions. This latter assumption of notched fading in the
cross-coupled signal band and notched fading in the main polarized
signal band agrees with recent measurements that indicate the possi-
bility of a shallow fade notch in the interfering cross-coupled path.”

For our analysis we use “static” fading models to emulate “snap-
shots” of multipath fading events.’* The passband transfer function
for the two-ray propagation model can be expressed as

where a represents the flat fade level and all other parameters are
related to dispersive fading as follows: the fade notch depth is —20 log
|1 — p|, where p is the relative amplitude of the secondary ray;* the

relative delay between rays is 7 and wo is the fade notch frequency.
For a fade notch centered in the passband,

In the absence of a cross-coupled interfering signal, the main polar-
ization baseband waveform after demodulation at the receiver, as
shown in Fig. 1, can be expressed as

is the low-pass equivalent of the channel impulse response. The index
I in eqs. (5) and (6) explicitly denotes reference to the main polariza-
- tion channel; n;(t) is filtered Gaussian noise with variance o7,; and ¢y
represents the relative phase difference between the modulator and
demodulator oscillators. In terms of a specific sampling epoch, to,
distortion in the overall channel impulse response is minimized by an
optimum relative carrier phase®

Extending the above discussion to dual-polarized operation, we give
the received baseband signal in the main polarization channel as

where h; and hy represent the low-pass equivalents of the main and
cross-coupled paths impulse responses, respectively. As long as the
main signal [the first term in eq. (8)] is much stronger than the cross-
coupled interferer, the carrier phase tracked by the demodulator of
the main polarization receiver is the relative phase difference between
local oscillators of the main polarization modulator and demodulator,
Ptopt(to). In its expanded form, eq. (8), representing the demodulated —
composite signal at the main polarization receiver, can be expressed
as

Notice that the parameters of the two independent fading events can
be varied independently.

To establish the relationship of eq. (9) to probability of error, and
ultimately derive the performance signature (M curve)’ that describes
outage performance for a 107° symbol error rate, we can focus on the
in-phase or quadrature rail signal on the received main polarization.
Because the signal constellations are symmetrical, the associated
probability of error for each of those two rails is identical and the total
symbol error rate can be determined. From eq. (9) the in-phase
component of the received signal is clearly Re[r;(t)] = r;3(t):

preceding remarks are germane to modeling the channel. In the
following subsection we relate the received in-phase baseband signal,

given by eq. (10), to the associated probability-of-error performance
of the dual-polarized QAM radio system.

Denoting the coefficient of the desired detected in-phase state as
Zo, we have

where fp, the sampling time, is optimized by mimizing the distortion
contributed by the second and third terms in eq. (10). The set of
slicing levels at the in-phase detector of the main polarization receiver
can be set to

{— (L — 2)QZo, Oe gem 20QZo, 0, 20QZo, eeu (L 7 2)QZo},
and, with no loss of generality, we set 2 = 1.
Designating the sum of ISI contributed by dispersion in the main

polarization and cross-coupled interference at the optimum sampling
time to by x(dk, 5k, Bk, Br), eq. (10) reduces to

Considering the automatic gain control operation at the receiver, the
associated probability of error is then

L-1
Pat, =2 A= PA(niz + x) > Zo}. (13)
Also, note that the noise variance o7,, is equal to the total filtered
noise variance, subsequently denoted by o2. Additionally,
PA(x + nit) > Zo} = Ey fPrlnit > (Zo -— x) 1x = x]}, (14)

where E{-} denotes statistical averaging and x is the conditioned value
of random variable x. First, taking the average over the Gaussian
noise, we obtain

1 Zo —- Xx
P,A(x + nis) > Zo} = 1 Bert c = )p. (15)
where the complementary error function, erfc(e), is defined by
a
erfc(e) = —= i) e "dn. 16
(e) can 1 (16)

By symmetry of the constellation, the total symbol error rate can
therefore be expressed as

where F(x) is the cumulative distribution function for the random
variable x.

The integration in eq. (17) can be evaluated using Gauss Quadrature
Rules (GQR).'4 Equation (17) is thus expressed as

In this equation, N is the number of terms in the finite series, and w;
and &; are weights and nodes in the GQR approximation. At this point,
P. in eq. (18) is calculated from the No = 2N + 1 moments of yx.
Because x is functionally dependent upon the independent transmitted
symbol states, we have adopted Prabhu’s algorithm” to determine the
moments. Note that the moments obtained in this manner are condi-
tioned on the values of the two random variables 7,, and 6,,. Hence,
the resulting moments must be averaged over {tm, 6m} before they can
be used in the GQR method.

To derive the probability of symbol error as a function of signal-to-
noise ratio (s/n), we define

where we have normalized Q to one. In the next section we expand
further on the computational aspects of the GQR method.

As previously stated, the relationships above are all dependent upon
the timing phase. This parameter can be selected a posteriori to
minimize P,, thus making probability-of-error computations a formi-
dable, numerically intensive task. In this work we choose a priori
sampling epoch by minimizing the peak distortion of the received
signal prior to equalization. The timing phase is thus dependent on
both the in-phase and quadrature rails of the main polarization M-
QAM signal.

In addition to performance signature, we have also evaluated at
select points along the M curve a corresponding signal-to-interference
ratio (SIR). This ratio is defined as the relative power in the main
polarization to that of the cross-coupled interfering signal. Using
parameters previously defined, this ratio is simply

An examination of Section IJ demonstrates that the theoretical
analysis of M-QAM signal transmission over dual-polarized facilities
in the presence of multipath fading is a computationally exhaustive
activity. In this section we provide a brief overview of numerical issues
related to our investigation. |

Distortion contributed by u;1 and u,n is specifically excluded from eq.
(23) since the complex data sources I and II are not coherent; these
latter signals appear noiselike to the timing recovery circuit in the
main polarization receiver.

From a computational standpoint, the operations described above
are carried out by first computing @yopt(to) and D, (to) for all t in the
range {[—T,, T,], and then selecting the single sampling epoch, topt,
that minimizes D,. With top identified, the symbol-period spaced
samples of u;1, Ug, Ui, and Uz are used for the subsequent probability-
of-error computations. An a priori selection of to,.p, obviously expedites
the computer time necessary to perform the probability-of-error com-
putations. The use of a timing phase that minimizes peak distortion
is one such choice. Another choice could be minimization of mean-
square eye closure. This can be shown to be equivalent to an IF timing
recovery.

For illustrative purposes, we present plots of u;;, Ug1, Uz, aNd Ug
in Fig. 2 for pI = 0.80, Pu = 0.75, qi = 1.0, ay = 0.5, for = 1.63 MHz,
fon = 0 MHz, I = 0.45, tm = 0, Om = 0, and T; = 1/(15 Mbaud). For
this illustration example, top, = —0.47;, and $y opt = 5.13°. In particular,
note that this carrier phase nulls the u,1 response at t = toopt, as it
should. However, even though the cross-coupled interferer corresponds

Fig. 2—Dispersive effects on impulse responses of a dual-polarized digital radio
system.

to a notch-centered fade and 0, = 0, Ug # 0 since the carrier phase
optimization is independent of channel II.

The signal-power-to-interference-power ratio was evaluated from
eq. (21). The presence of | P(w) |? in this expression derives from the
fact that at the receiver, the main and interference signals have P(w)
spectral shaping and thus a |P(w)|* spectral power density. The
a?{1 + p?— 2p;cos[(w — wo;)7i]}, 1 = I, Il, terms correspond to spec-
tral power density associated with the dispersive fade and the undis-
torted signal power. The integral is conveniently evaluated using
Romberg-quadrature methods.

As explained in Section 3.1, the in-phase component of the received
signal on the main polarization has basically four parts. The desired
symbol to be detected is the k = 0 term of u;;(t) in eq. (22a). Each of
the four aforementioned terms consists of a sum of weighted, inde-
pendent, identically distributed random variables that together were
designated as x(6}, Bk, 62, 824) and discussed in Section 3.2. Since the
random variable x is in the form of a sum of sums of independent
random variables, it can be considered a long sum of weighted, inde-
pendent random variables. Thus, to determine a finite number of
moments of x, Prabhu’s” recursive algorithm can be used: If

Based on this recursive formula, a computer program was written to
calculate No moments of the random variable x.

Since all zero-mean random variables involved in the sum are evenly
distributed, the odd moments of the sum become zero. In all our
computations, No = 31 moments were found to be adequate for the
probability-of-error calculations.

Following the computation of conditional moments of x by averaging
over uniformly distributed variables 7,, and 6, the absolute moments
of x were found and subsequently used with the GQR method for
computing the error probabilities. (For a brief summary of GQRs see
Appendix C of Ref. 17.) For additional detail, interested readers are

referred to Ref. 14. Observe that by averaging over ai,(i = I, ID), tn,
and 6,, when computing the No moments wy,, a great deal of compu-
tation time is saved because these calculations are made once for all
s/n values.

In this section we present theoretical and measured performance
signature curves for dual-polarized digital radio. It is well known that
signature curves provide a locus of fade notch depths (in decibels) and
relative fade notch positions (Hz) for a 107° probability of error.
However, unlike their more customary presentation, we must also
include parameters that define the character of the interfering cross-
coupled multipath channel.

As a reference, we have computed the signature of a singularly
polarized 16-QAM system, that is, ay = 0. The data are presented in
Fig. 3 and labeled “1.” Along this contour we also designate the SIR
(in decibels) computed using methods previously described. For the
case of no cross-coupled interferer, the SIR is obviously infinite. For
all other cases, the SIR value at each point on the signature curve is
functionally dependent upon specific fading parameters for the main
and cross-coupled signals. Moreover, we associate with each curve a
triplet representing the dispersive fading status of the cross-coupled
interferer. This triplet is

In addition to curve 1 in Fig. 3, we show three other cases corre-
sponding to (—30, 10, 0), (—30, 5, 0), and (—30, 5, 11) and denoted 2,
3, and 4, respectively. A comparison of curves 2 and 3, with the same
—30 dB flat power levels and 0-MHz notch offsets, reveals that the
fade of curve 2, with a 10-dB inband notch, results in less outage time
than the fade of curve 3, with 5-dB inband notch. Hence, the greater
power loss associated with curve 2 leads to reduced outage, even though
the intersymbol interference for curve 2 exceeds that of curve 3. Now

Fig. 3—Performance signature curves for dual-polarized 16-QAM radio; I = 0.45,
T, = 1/(22.5 Mbaud), s/n = 60 dB. Curves 1 through 4 show relationship of SIR to
relative digital radio outage.

consider curves 3 and 4. This data corresponds to identical flat power
levels and fade notch depths, with the notch position moving from 0
MHz (notch centered) to 11 MHz (near the band edge). The notch-
centered fade apparently causes less outage than the notch offset fade.
When we remember that the unfaded signal spectral energy at 0 MHz
is much more than that near 11 MHz, it should be clear that the
relationship of curves 3 and 4 is again that of diminished net signal
power in the interferer that equates to a reduced outage.

SIR values are listed at certain points along curves 2, 3, and 4. It
will be noted that moving toward higher notch offset values on each
performance signature results in greater SIRs, specifically for the
reason cited above. A fade positioned in a region of normally high
spectral energy will pull out more power than the same fade in a region
of reduced spectral energy. Moreover, at any main polarization notch
frequency position, the SIR increases as the performance signature
drops, because interference signal power, rather than intersymbol

interference, is the dominant factor for outage due to an independent
interferer.

To further verify that interference power plays a major role in
outage performance, Fig. 4 presents two relevant cases. In that figure
we consider (—30, 0, 0) and (—25, 5, 0) dispersive fading of 16-QAM
cross-coupled signals. In both cases the total interference power loss
is approximately the same. The former case corresponds to a 30-dB
flat fade of the interferer, while in the latter case the flat fade is 25
dB with a 5-dB shaped fade positioned in the center of the passband
(a region of high spectral energy). Observe that the outage performance
and SIR are virtually identical along the entire signature curves. We
therefore conclude that whether fading of the cross-coupled signal is
mildly dispersive (i.e., generating intersymbol interference) or flat,
outage performance improvement is accompanied by interference
power reduction provided the flat level of the cross-coupled signal is
considerably below that of the main polarization signal (this is nor-
mally the case because of antenna/waveguide polarization isolation)
and the dispersive fading of the cross-coupled signal is shallow.

To confirm that the aforementioned property holds at lower isola-
tion levels as well, we have repeated the performance signature cal-

T, = 1/(22.5 Mbaud), s/n = 60 dB. The total interference power loss is approximately
the same, as is the relative digital radio outage.

culations for an interfering signal flat fade level of —20 dB. The
results, presented in Fig. 5, confirm those in Figs. 3 and 4.

Up to now 6, and 7, were taken to be random variables. To gain
further insight, we now impose 6,, = 0 and 7,, = 0 conditions on the
interfering signal, thereby illustrating a case for which timing and
carrier phase of the two signals are aligned. We have repeated com-
putations for the case presented in Fig. 5. These computations are
presented in Fig. 6. Observe that the same qualitative properties hold.
In the case of a (—20, 0, 0) interfering signal, the synchronous system
exhibits a lower outage time than the general system for the same
interferer (see Fig. 5).

Data in Figs. 3 through 6 were all computed for a 60-dB s/n, 22.5-
Mbaud symbol rate, [ = 0.45 roll-off, 16-QAM dually polarized radio
system. We have repeated these performance signature computations
for a 66-dB s/n, 15-Mbaud, I = 0.45, 64-QAM dually polarized radio
system. The resulting curves, shown in Fig. 7, are expectedly much
wider than the 16-QAM case. However, the general phenomenon
exhibited by the data of Figs. 3 through 6 remains appropriate to Fig.
7, as well.

T, = 1/(22.5 Mbaud), s/n = 60 dB. These cases correspond to lower isolation levels than
those in Figs. 3 and 4.

Fig. 6—Performance signature curves for dual-polarized 16-QAM radio; IT = 0.45,
T; = 1/(22.5 Mbaud), s/n = 60 dB, 6, = 0, and 7, = 0

Fig. 7—Performance signature curves for dual-polarized 64-QAM radio; I = 0.45,
T, = 1/(15 Mbaud), s/n = 66 dB.

CHANNEL I
Al
DATA | DR-6 IF FADE (+) (+) DR-6
GENERATOR TRANSMITTER SIMULATOR RECEIVER

All
DATA DR-6 IF FADE ERROR-RATE
GENERATOR TRANSMITTER SIMULATOR TEST SET

‘Fig. 8—Experimental facility for the measurement of dual-polarized digital radio
performance signatures.

Using the experimental arrangements functionally illustrated in Fig.
8, we have confirmed the theoretical conclusions discussed in the
previous section. As noted in the figure, two independent data sources
provide inputs to DR-6* transmitters for channels I and II. The flat
power levels of the two outputs are separately adjusted using atten-
uators A; and Ay, and dispersive fading is provided by IF fade simu-
lators. The channel I and II signals are added together with low-level
noise (60-dB s/n) from an IF noise load set. The composite signal,
simulating independent fading of a main polarization and a cross-
coupled interferer, is then applied to a DR-6 receiver and error-rate
test set.

Attenuators A; and Ay were used to adjust the ratio ay/a;. Power
was measured at the output of the IF fade simulators to establish the
SIR and again at the input to the radio receiver to assure the demod-
ulator had the appropriate signal level.

Error-rate measurements were made for a number of different fad-
ing events. The first set of measurements correspond to the triplets
(—0, 0, 0) and (—25, 0, 0), the latter triplet representing a flat,
nondispersive fade of the interferer. The performance signatures are
presented in Fig. 9 and show the same trend as the theoretical data of
Figs. 3 and 7. The quantitative differences are to be expected owing
to equipment imperfections, such as nonideal Nyquist and bandpass
filters, and imperfect carrier and timing recovery circuit operations.
We next examined the influence of a dispersive fade characterized by
(—25, 5, 0). As expected, the performance signature improved, but not
to the point of reaching the (—~, 0, 0) case. This observation agrees
with the data of Figs. 3, 5, and 7.

Fig. 9—Measured performance signatures for dual-polarized 16-QAM radio; I = 0.45,
T, = 1/(22.5 Mbaud), s/n = 60 dB.

The next phase of the experimental study was to increase the flat
fade level from —25 to —20 dB and to introduce a notch-centered 5-
dB fade, thus the triplet (—20, 5, 0). The resulting performance
signature curve is also shown in Fig. 9 and supports the theoretical
data of Fig. 4. In general, conclusions drawn from the experimental
data support our theoretical findings. Note that in all of the experi-
mental tests, the two transmitters were completely independent of one
another; hence, the conditions 0 <7, = T, and 0 <= 6, S 27 actually
held.

In this paper we present computed performance signatures for dual-_
polarized transmission of M-QAM signals over dispersive multipath
digital radio channels. For the assumed model, we show that a cross-
coupled interferer exhibits noiselike behavior, and its power loss,
whether flat or midly dispersive, brings about an improvement (reduc-
tion) in dual-polarized system outage time. The theoretical findings

are supported by measured performance signatures obtained from a
laboratory simulation of the analytic model.

1. A. J. Giger and W. T. Barnett, “Effects of Multipath Propagation on Digital Radio,”
IEEE Trans. Commun., COM-29, No. 2 (December 1979), pp. 1870-5.

2. G. J. Foschini and J. Salz, “Digital Communications Over Fading Radio Channels,”
B.S.T.J., 62, No. 2, Part I (February 1983), pp. 429-59.

3. H. Sari, “A Comparison of Equalization Techniques on 16-QAM Digital Radio
Systems During Selective Fading,” Globecom ’82 Conf. Rec. (November—Decem-
ber 1982), pp. F3.5.1-6.

4, N. Amitay and L. J. Greenstein, “Multipath Outage Performance of Digital Radio

Receivers Using Finite-Tap Adaptive Equalizers,” IEEE Trans. Commun., COM-
32, No. 5 (May 1984), pp. 597-608.

. G. L. Fenderson et al., “Adaptive Equalization of Multipath Propagation for 16-
QAM 90-Mb/s Digital Radio,” AT&T Bell Lab. Tech. J., 63, No. 8 (October
1984), pp. 1447-63.

6. C. A. Siller, Jr., “Multipath Propagation: Its Associated Countermeasures in Digital

7. M. Kavehrad, “Performance of Cross-Polarized M-ary QAM Signals Over Nondis-
persive Fading Channels,” AT&T Bell Lab. Tech. J., 63, No. 3 (March 1984), pp.
499-521.

. W. D. Rummler, “A New Selective Fading Model: Application to Propagation Data,”
B.S.T.J., 58, No. 5 (May-June 1979), pp. 1037-71.

9. M. Emshwiller, “Characterization of the Performance of PSK Digital Radio Trans-
mission in the Presence of Multipath Fading,” ICC ’78 (June 1978), pp. 47.3.1-6.
10. C. W. Lundgren and W. D. Rummler, “Digital Radio Outage Due to Selective
Fading—Observation Versus Prediction From Laboratory Simulation,” B.S.T.J.,
5 (May-June 1979), pp. 1073-100.

11. R. W. Lucky, J. Salz, and E. J. Weldon, Jr., Principles of Data Communications,
New York: McGraw-Hill, 1968.

14. G. H. Golub, and J. H. Welsch, “Calculations of Gauss Quadrature Rules,” Math.
Comp., 26, No. 106 (April 1969), pp. 221-30. |

15. V. K. Prabhu, “Some Considerations of Error Bounds in Digital Systems,” B.S.T.J.,
50 (December 1971), pp. 3127-52.

16. J. J. Kenny, “Digital Radio for 90-Mb/s, 16-QAM Transmission at 6 and 11 GHz,”
Microw. J., 25, No. 8 (August 1982), pp. 71-80. |

17. M. Kavehrad, “Performance of Nondiversity Receivers for Spread Spectrum in
Indoor Wireless Communications,” AT&T Tech. J., 64, No. 6 (July-August 1985),
pp. 1181-210.

AT&T Bell Laboratories, 1969-1978, 1979—. At AT&T Bell Laboratories,
Mr. Siller has analyzed and designed reflector antennas for terrestrial micro-
wave communications; performed exploratory investigations of digitally im-
plemented adaptive transversal equalizers for high-speed digital radio systems;
identified novel approaches to digital FIR filtering and quadrature amplitude
modulation; and explored new algorithms for the stable control of fractionally
spaced equalizers. Mr. Siller has authored nearly 30 papers in the aforemen-
tioned areas and is presently involved in system engineering of future digital
transmission systems. He is the recipient of an AT&T Bell Laboratories
Distinguished Technical Staff Award. Senior Member, IKEE, where he serves
on the Signal Processing and Communication Electronics Technical Commit-
tee, and has helped plan the technical program for several international
conferences; Member, Phi Eta Sigma, Eta Kappa Nu, Tau Beta Pi, Phi Kappa
Phi, Sigma Xi, New York Academy of Sciences.

There have been many developments and researches in the queueing
networks for modeling of computer systems in the last two decades.
Most researchers believe, however, that it is very difficult to concep-
tualize how to derive steady state solutions for general nonproduct
form queueing networks, which do not have a product form solution.’
For instance, a steady state solution for a general queueing network
with first-come-first-served (FCFS) queueing discipline and general
service times is unavailable at present. Several recent empirical papers
have shown that service time distribution can have a significant effect
on performance with particular emphasis on the modeling of computer
systems.” Recently several works describe steady state solutions for
the restricted cases of the nonproduct form queueing networks.*®
Neuts investigated a single server queue with phase type distribution
and also found the solution to the M/G/1 queue.® Carroll et al.

This paper attempts to generalize these studies. It focuses on a
closed network consisting of a nonexponential server and a service
station with two identical nonexponential servers in parallel, which is
a typical model of a computer system; a central processing unit and
two input/output processors. There are a finite number of jobs, and
the queueing discipline is FCFS at each node. An explicit steady state
solution for this network is derived in a quasi matrix geometric form,
where matrices are recursively defined in certain product spaces.
Obviously this form is a generalization of the matrix geometric form.
This result may provide insight for obtaining exact solutions for
general closed networks that do not have a product-form solution.

Section II of this paper introduces notations and definitions for
describing the algebraic description of the general distribution server.
In Section III, both external states and internal states are introduced
to represent the states of the network. In Section IV, the global balance
equations of the network is derived in a matrix form. Section V gives
the steady state solution of the global balance equation. The conclu-
sions are summarized in Section VI.

One of the common approaches for representing general service time
distributions is to use the method of stages. That is, each nonexpo-
nential service time distribution is replaced by a subnetwork of expo-
nential stages with the constraint that the subnetwork can only be
accessed by one job at a time. The principle on which the method of
stages is based is the memoryless property of exponential distribution.
Thus this method is both general and compatible with the definition
of Markov processes. All works to be studied in this paper are based
on Cox’s statement that any server whose service time distribution
function has a rational Laplace transform could be replaced sas
by a subnetwork of exponential distributions.” |

In this paper, we consider a closed network consisting of two service
stations, a nonexponential server (station 0) and a service station
(station 1) containing two identical nonexponential servers (see
Fig. 1).

Let N denote the total number of jobs (there can be only one job
active at any time within each of the servers). When a job completes
service at either server of station 1, it leaves station 1 and joins the
queue of station 0, while another job (if any) in the queue of station 1
takes its place.

Each of the nonexponential servers in Fig. 1 is represented by a
subnetwork of exponential stages, as shown in Fig. 2. There are m;
exponential stages, whose service rates are p;,, ..., i,m, in the server
of station i (t = 0, 1). The server of station i can be characterized by
vectors and matrices. These vectors are denoted by lowercase letters,
whereas matrices are denoted by uppercase and bold letters. This
convention is followed without exception. The notation that will be
used is consistent with that of Refs. 5, 7, and 8.

Definition 1: For each server of station 1, define the following:
p; = the entrance probability vector,

For instance, (p;)z, the kth component of the vector p;, is the
conditional probability that a job, upon entering the server of station
i, will first go to stage k. Similarly, (q;); is the conditional probability
that a job, upon completing service at the stage Fk in the server of
station 1, will leave the server, and (P;);; is the transition probability

from stage k to stage j within the server of station 1. One simple
example is provided to show how to characterize the nonexponential
server.

Example 1: If a server of station 1 is Erlangian-2 as shown in Fig. 3,
then the server is characterized by the following vectors and matrices:

Lemma 1: For i= 0, 1, let I; be an identity matrix of order m; and e; a
vector containing m; ones. The following relationships hold:

Proof: Using the law of total probability, we obtain the following
relationships:

Equations (1) and (2) are obtained from the above relationships. This
proves the lemma.

Next, three important matrices in the paper are introduced to
simplify the balance equations and express the properties of the server.

Note that Q; is an idempotent matrix of rank 1. We can assume
that I; — P;, i = 0, 1, has an inverse since this expression means that
a path exists from each stage out of the server of station 1. Carroll et
al.° and Neuts* have shown that traditional probability notations can
be expressed by vectors and matrices.

Lemma 2: For i = 0, 1, let b;(x) be the probability density function (pdf)
of the server of station 1, E;(x") its nth moment, E;(x) its mean service
time, and B}(s) its Laplace transform.

If there is only one job present (N = 1) in the system, queueing will
never take place. The system is easily solved. If there is more than
one job (N > 1), the state of the network is characterized by a number
of jobs (called external state) and positions of active jobs (called
internal state) at each station. The queueing problem is thus trans-
formed from one involving the remaining service time for a job to one
involving the position of the active job in the subnetwork.

In defining the states of the network, it is important to keep the
internal state separate from the external state in order to make the
balance equation easier to solve. Each external state has a set of
internal states which we call the internal state space. Thus the state
of the network can be represented by a pair, external state and internal
state.

The set of possible external states, which we call the external state
space, is determined as follows:

The cardinality of the external state space is N + 1. Next consider the
internal state space. Since the number of internal states at station 0

is the same as the number of stages in the server, the internal state
space of station 0 is defined as follows:

If there is only one job at station 1, it must be active at one of the
servers. Thus the internal state space for one job at station 1 is defined
as follows:

When there are two or more jobs at station 1, two of them must be
active. In this case, a possible internal state would be a pair of integers
(j, k), where one job is at stage j and the other is at stage k. Since the
two nonexponential servers in station 1 are identical, (Rk, ]) is the
same as (j, k). Thus the internal state space for two or more jobs at
station 1 is defined as follows:

Let d(i) be the dimension (cardinality) of the state space S;.
There are d(0) = mo states in So, d(1) = m, states in S; and d(2) =
m,(m, + 1)/2 such states in So.

Definition 3: For N > 2, the state of the network is represented by a
vector [x1, x2], where x, is a vector representing an external state and
x2 1s a vector representing the internal states of each station. That is

[(N —n, n), (1,j7)] = the state that N — n jobs are at station 0, with
internal state 1 € So, and n jobs at station 1,
with internal state j € So, where 2 <n S
N -1. |

Note that the cardinality of the state spaces is d(0) + d(0)d(1) +
(N — 2)d(0)d(2) + d(2). Next, the steady state probability vectors are
defined similarly.

Definition 4: For.N > 2, the steady state probability vectors, describing
the internal states, are defined as follows:

[bo(N, 0)]; = the steady state probability that the network is in
| the state [(N, 0), (7,)], where 1 € So. :
bo(.N,0)®pi, where ® denotes Kronecker produc
(see Appendix).

[7i(N —1,1)]i; = the steady state probability that the network is
in the state [(N — 1, 1), (1,7)], where i € Sp and
JES,

[r2(N — n, n)]i,; = the steady state probability that the network is
in the state [(N — n,n), (i, j)], where2 <n <
N-1,1€ So, andj € Sz.

[b.(0, N)]; = the steady state probability that the network is in
the state [(0, NV), (,7)], where j € So.
m2(0, N) = po ® b,(0, N).

For k = 1, 2 and n = 0, 1, ---, N, the steady state probability
[a1,(.N — n, n)];,; can be ordered lexicographically to create a vector
ap(N — n,n). The subscript of these probability vectors denotes the
dimension of the objects. For instance, the subscript & is used to denote
the steady state probability vector of order d(0)d(k). This vector is
essentially decomposed into d(0) subvectors, each of order d(k). The
vectors bo( N, 0) and 6.(0, N) are of order d(0) and d(2).

Definition 5: For N > 2, the steady state probabilities, describing the
external states, are defined as follows:

Pr(N, 0) = m(N, 0)(e)*
= the steady state probability that there are all N jobs
at station 0. |
Pr(N —1,1) = m(N-—1, 1)(e™)?
= the steady state probability that there is one job at
station 1.
Pr(N — n,n) = mo(N—n, n)(e™)?
= the steady state probability that there are n (2 <n
< N — 1) jobs at station 1.
Pr(0, N) = 72(0, N)(e)*
= the steady state probability that there are all N jobs
at station 1,

Next, objects of the state space S» are defined similarly. These are
also necessary to connect between S, and So.

M, = the diagonal matrix whose (1, i)th element is the probability
rate of leaving state i = (11, 12) © Se. That is, (Me) i = wii, +
H1,i5
(P.);; = the probability of transition from state (k, 1) to state (k, J),
where k € So, 1 = (h, l2) € Se andj = (ji, Jo) € So.

where 7 is the other member of the 1-pair, ] is the other member of the
j-pair, and M means set intersection.

(Rz);; = the probability that upon a job entering station 1, the state
is changed from (k, 1) to (k,j), where k € So, i € Sy, andj =

(Q2)i; = the probability that upon a job completing at station 1, the
state is changed from (k, 1) to (k, 7), where k © So, 7 © Si,

A simple example is provided to show how to construct the objects of
the state space So.

Example 2: If station 1 has two Erlangian servers (m, = 2) in parallel
as shown in Fig. 4, then the internal state spaces and the dimension
of the spaces are

Note that R» is d(1) X d(2) dimensional rectangular matrix and Q, is
a d(2) X d(1) dimensional rectangular matrix. Thus Q2R» is the
probability matrix that upon a completion at station 1, a job leaves
and another takes its place. This event can happen only if there are
more than two jobs at station 1.

Lemma 3: For the state space S; and So, let I; be an identity matrix of
order d(t) and e; a vector containing d(t) ones. The following relation-
ships hold:

Proof: This lemma follows immediately from lemma 1 and the law of
total probability.

Next, note that we have encountered objects of different state spaces
So, Si, and Sg. They are of different dimensions d(0), d(1), and d(2).
Objects of order d(0)d(1) and d(0)d(2) are said to be on the product
spaces. 'The product space of order d(i)d(j) is represented by Foca.
Before any operation can be performed, all objects have to be replaced
by their images under embedding into product spaces. For instance,
objects of order 1 are scalars. Since scalars constitute subspaces of

spaces So, S;, and S2 and of the product spaces, and there is a 1-1
correspondence between scalars and diagonal matrices all of whose
diagonal elements are equal, scalars are isomorphic to scalar multiples
of matrix I. Similarly, the spaces So, S;, and S. are subspaces of the
product spaces, so that objects from these spaces have a counterpart
in the product spaces while algebraic relationships are preserved.
These counterparts will be denoted by adding both superscripts and a
hat, these counterparts are the image of an embedding (for details, see
Appendix). For instance, Ao is a matrix in the state space So, and A;
is a matrix in the state space S,. Aj’ is a matrix in the product space
F.a, and has characteristics inherited from matrix Ao. Also A{” is a
matrix in the product space F'4,.4, and has characteristics from Aj. If
there is no possibility of confusion, the superscript (0) is omitted. That
is, AS and A® are denoted by A; and Ao.

In order for the network to be in steady state, the probability of
leaving a particular state must be equal to the probability of entering
that state. Thus the total flow into a particular internal state must be
equal to the total flow out of that state. The steady state balance
equations can then be derived for the internal states. All internal
states belonging to one external state are combined in one mathemat-
ical entity, a vector. The balance equations can then be written in a
matrix form, using Kronecker products and generalized embedding of
vectorspaces (see Appendix).

Theorem 1: For the closed network with a nonexponential server and a
station containing two identical nonexponential servers and N > 2, the
global balance equations are:

7m, is a vector in the product space Fa,:4,, whereas the other factors are
objects from space So. In order to write eq. (20) in a matrix form, these
objects must be augmented into objects from product space Faj.a,. By
postmultiplying eq. (20) by p; and using generalized embedding, eq.
(20) can be written in a matrix form where vectors and matrices are
in product space Fa,xa,:

™(N, 0)M® = a(N, 0)MYPPYP) + m(N -— 1, Mig? py. (21)
It can be simplified by substituting (i; — P;)é? for G7 and B; for
NEI 2B:
m(N, 0)BS? = m(N — 1, 1)Bié? p,. (22)
By substituting Q, for 67p,, balance equation (14) is obtained.

By simplifying eq. (23) and using generalized embedding, balance eq.
(15) is obtained.

By simplifying eq. (24) and using generalized embedding, balance eq.
(16) is obtained.

By simplifying eq. (25) and using generalized embedding, balance eq.
(17) is obtained.

By simplifying eq. (26) and using generalized embedding, balance eq.
(18) is obtained. That completes the proof.

Next consider the entire Markov chain of this network. The state
probability vector of the entire Markov chain can be represented by

This satisfies balance eq. (10) and the law of total probability. The
infinitesimal generator Q* of the Markov chain can be written as
follows:

Wallace also termed such processes Quasi Birth and Death (QBD)
processes.'* He has shown that if a QBD process is boundary leading,
the process has a matrix geometric form solution.

The global balance equations were derived by equating the flow in
and out of a certain state. Thus the flow into an external state must
be equal to the flow out of the external state because the external
state is a collection of states.

Lemma 4: (Flow-balance equations between the external states.) For
N > 2, the flow out of a certain external state equals the flow into the
external state. That ts

Proof: The proof will be done in three parts, one part for each of eqs.
(27) through (29).

(3) By induction on n (n = 2, 3, ---, N-1),
Basis step: By postmultiplying balance eq. (16) by (e)”, it follows
that

The solution of the global balance equations (Theorem 1) can be
derived now. A definition is introduced to obtain the steady state
solutions.

Since 7;(N, 0) depends on one unknown vector bo(N, 0), we can
express all state probability vector 7’s in terms of it. With notations
defined above we are ready to derive the steady state solutions.

Proof: The proof will be done in three parts, one part for each of eqs.
(37) through (39). |

Definition 8: For N > 2, define recursively the following matrix:
R,(N — 1) = (BS? + By, — R,U2(N — 2)M2Q.)" (45)

This proves the lemma.
To make the solution explicit, a normalization vector must be
determined. The normalization vector can be derived now.

bo(N, 0)BoQo = c(N)po, (48)
1
where c(N) = ————— a 15 a Scalar.
ey) PofiK(N)(e™)*
Proof: The normalization vector is proportional to po:
bo N, 0)BoQo = bo(N, 0)Boes po = c(N) po, (49)

where c(N) = bo(N, 0)Boeé is a scalar. Thus, the proportionality is
normalization constant c(N). By substituting eq. (49) for eq. (47), it
follows that

An explicit solution is derived for a closed network consisting of a
nonexponential server and a service station with two identical nonex-
ponential servers in parallel (e.g. CPU and I/O devices). There is a

finite number of jobs, and the queueing discipline at each node is
FCFS. By applying an algebraic approach to the method of stages, the
properties of each nonexponential server are represented by vectors
and matrices in a space identified with that server. Product spaces are
introduced to combine the properties of these servers and to allow
their interactions to be described. Both external states and internal
states are introduced to represent the states of the network. That is,
the queueing problem is thus transformed from one involving the time
of service remaining for a job to one involving the position of an active
job in the subnetwork of exponential stages. In the mathematical view,
the problem of integral equations (continuous) is transformed to one
of algebraic equations (discrete) over a finite dimension.

where 7p is a normalization vector chosen to make the steady state
probabilities sum to 1, and each U(n) is a matrix which is recursively
defined by matrices involved. Obviously this form is a generalization
of the matrix geometric form. All of techniques used in this paper are
drawn from linear algebra. This algorithm is easy to implement, even
for a person not familiar with queueing theory, because it involves
only matrix multiplication and inversion (we can use commercial
matrix packages).

The author would like to thank Phil J. Fleming and the referees for
their valuable comments and suggestions.

1. R. R. Muntz, “Queueing Networks: A Critique of the State of the Art and Directions
for the Future,” Computing Survey, 10, No. 3 (September 1978), pp. 353-9.
2. M. Reiser, and H. Kobayashi, “The Effects of Service Time Distribution on System
Performance,” Info. Proc. 74, Amsterdam: North-Holland, 1974, pp. 230-4.
3. C. H. Sauer, and K. H. Chandy, “The Impact of Distributions and Disciplines on
yee Processor Systems,” Commun. ACM, 22, No. 1 (January 1979), pp. 25-
oo;

. M. F. Neuts, Matrix-Geometric Solutions in Stochastic Models—An Algorithmic
Approach, Baltimore: John Hopkins University Press, 1981.

5. J. L. Carroll, A. Van de Liefvoort, and L. Lipsky, “Solution of M/G/1//N-Type
Loops With Extensions to M/G/1 and GI/M/1 Queues,” Op. Res., 30, No. 3
(May-June 1982), pp. 490-513.

.M. F. Neuts, “Explicit Steady-State Solutions to Some Elementary Queueing
Models,” Op. Res., 30, No. 3 (May-June 1982), pp. 480-9.

_ A. Van de Liefvoort, “An Algebraic Approach to the Steady State Solution of G/G/
1/N Type Loops,” Ph.D. Thesis, Univ. of Nebraska, March 1982.

. S. W. Yoo, “An Explicit Steady State Solution of Closed Networks Consisting of
Two General Stations With Parallel Servers,” Ph.D. Dissertation, University of
Kansas, June 1983.

9. D. R. Cox, “A Use of Complex Probabilities i us a Theory of Stochastic Processes,”
Proc. Cambridge Phil. Soc., 51 (1955), pp. 3

10. A. Graham, Kronecker Products and Maines Calculus With Applications, Ellis
Horwood ‘Limited, 1981.

11. V. L. Wallace, “The Solution of Quasi Birth and Death Processes Arising from
Multiple Access Computer Systems,” Ph.D. Thesis, University of Michigan, Ann
Arbor, Michigan, March 1969.

The Kronecker product is the direct product of two disjoint operator
spaces. In particular, if K is an m X n and L anr X s matrix, then the
Kronecker product of K and L, denoted K © L, is the matrix of size
mr X ns which is obtained by multiplying each element (K);; of matrix
K by the full matrix L.

In this paper, there are equations involving matrices of different
dimensions. Before any operation can be performed, all matrices have
to be replaced by their images under embedding into product spaces.
According to the definitions in the previous sections, d(i) denotes a
dimension of state space S;(1 = 0, 1). Let Fu;za, be the additive group
of d(1) X d(j) matrices.

Hen A.1: Define the following mappings between Fa,xa,, Fa,xa,;
- d.xd. and F dod, xdod,» F dpdoxdodo as follows:

where ® stands for the Kronecker product and i = 1, 2,...,¢
These mappings are group homomorphisms and preserve all alge-
braic characteristics of the matrix. The proof is omitted.

Seung W. Yoo, B.S., 1972, Seoul National University, Korea; M.S., 1980,
Ph.D. (Computer Science), 1983, University of Kansas; AT&T Bell Labora-
tories, 1983-1985, AT&T Information Systems, 1985—. Since joining AT&T
Bell Laboratories, Mr. Yoo has been involved in architecture study and
performance analysis of computer systems. He was transferred to AT&T
Information Systems in 1985. Member, ACM.

The transmission performance of digital radio systems is controlled by
spectral distortion caused by multipath fading. To evaluate this performance
for digital systems with high-order modulation schemes, a statistical model
for frequency-selective fading is needed. New propagation data obtained in
Gainesville, Florida, were used to generalize Rummler’s model to include group
delay response. The introduction of the delay response data into the model of
the fading channel enabled the classification of the fades as minimum phase
and nonminimum phase. We found that 24 percent of all fades have significant
delay distortion, and can be characterized as being minimum phase or non-
minimum phase. In the range of practical interest, there are as many minimum
phase as nonminimum phase fades. The results of this work will facilitate a
better understanding of the fading channel, which will be beneficial in the
engineering of radio routes and digital radio design. The results also demon-
strate the need for a description of the geographical occurrence of dispersion,
which will differ from that for multipath fading at a single frequency. This is
based on the observation, presented in this paper, that the relative amount of
dispersive fading is significantly greater in Gainesville, Florida, than in Pal-
metto, Georgia. The availability of a dispersive fading map will facilitate the
accurate engineering of digital radio routes.

Digital radio communication systems are sensitive to selective fad-
ing. To evaluate their performance, a statistical model for frequency-

selective fading is needed. Although the amplitude response of multi-
path fading in a microwave communication channel has been exten-
sively modeled previously,’® statistically based results for the group
delay response were not available until recently,?"' and work on
modeling the delay in a fading channel is just beginning.”

A successful model for multipath fading using amplitude data was
developed by W. D. Rummler.** The delay response in this model was
inferred from the measured amplitude. This led to an ambiguity, since
both minimum phase and nonminimum phase fades are possible. In
this paper, we generalize the use of this model by including group
delay data. The model parameters are defined to characterize both
minimum phase and nonminimum phase behavior, and statistical
distributions of these parameters are compiled. Rummler’s model,
although developed for data obtained in the Palmetto experiment for
a 6-GHz carrier with a 25.3-MHz bandwidth, has proven itself robust
and applicable for a wide range of frequencies. Our new results are
based on amplitude and delay measurements at 6 GHz in a 30-MHz
bandwidth over a 23:3-mile path between Gainesville and Sparr,
Florida.

Inclusion of the delay response into the statistical model of the
dispersive line-of-sight channel will benefit both the engineering of
digital radio routes and digital radio equipment design. The description
of the occurrence of events with minimum and nonminimum phase
response is of immediate practical interest. The need for delay descrip-
tion, such as provided by our model, will increase with the growth of
sophistication of digital radio.

Section III presents the statistical distributions of the modified
model parameters for the data measured in Gainesville. Comparison
with Palmetto data have been made that show that the Gainesville
fading is more dispersive than that at Palmetto.

tape. The amplitude and delay were measured during alternating
frequency scans. Each scan duration was 0.1 second followed by a 0.1-
second retrace. A measurement period was thus 0.4 second. The
amplitude scale was set up for fades between 15 to 65 decibels and the
delay between +50 nanoseconds.* Amplitude fades smaller than 15
decibels were out of scale and recorded at 1 scan/min. Each scan was
sampled at intervals of 2 MHz or 16 samples over the 30-MHz
bandwidth. |

The database used for this study consists of approximately 43,000
scans that represent 17,000 seconds of fading activity. The data were
recorded in Gainesville for 11 months during the period from 1982
through 1984.

A fading event recorded in Gainesville is shown in Fig. 1. As a rule,
fading events were slow enough to be considered static during the
frequency scan. Although on some scans a certain amount of distortion
that could be attributed to the dynamics of the system was noticeable,
it was small enough to be negligible. Many scans were incompletely
recorded because of the limited range of both amplitude and delay
measurement equipment. Therefore, the recorded fading scans were
first screened in order to eliminate fading scans that are not suitable

Ee a a and incomplete amplitude scans. Scans IJ and III rejected. Scan IV
marked.

for modeling. In addition, scans that do not need dispersive modeling,
but have to be included in the total fading statistics, were identified
during the screening process. This screening was done as follows.

A. Only those responses recorded at a rate of 2.5 scans/s were
selected for processing. Responses recorded at different rates were
regarded as unfaded.

B. Data scans were categorized on the basis of amplitude response.
Figure 2 shows several amplitude scans as they appear in the recorded
data. Only complete scans of types I and IV were modeled. Type II
was rejected because part of the scan is out of scale at the high end
(>65 dB). This is a rare event; about 30 such scans were recorded.
Type III was eliminated because the trace is out of scale at the low
end (<15 dB). This is the most common type that was eliminated;
about 20 percent of all scans were type III. Flat fades of type IV were
marked to indicate that dispersive modeling is not required.

C. Data scans were also categorized on the basis of delay response.
Figure 3 shows several delay scans. Scans of types I and II were
included as they are. Flat delay scans of type III that have a dispersive
amplitude response were marked to indicate that delay modeling is
not required. Scans of types IV and V are incomplete and were marked
so that they can be modeled using an alternate technique.

Accordingly, 51,953 scans were selected using criterion A. From
these, 9103 were eliminated because the amplitude was out of scale;
16,892 were marked because the amplitude was flat; 13,580 were
marked because the delay was flat; and 2269 were marked because the

delay was out of scale. Consequently, 12,538 scans, or 24 percent of
the total, had sufficient delay information to be modeled using both
the delay and amplitude measurements. For the rest, only the ampli-
tude measurements were used.

The modeling was done using Rummler’s model for selective fading.?
In addition to wide acceptance and extensive validation this model
has the advantage of separating the flat component of the fade from
the dispersive component. The dispersive component can thus be
statistically characterized by one parameter. The transfer function of
the model is

where we can regard 7 as the relative path delay of the second ray, b
and wor the relative amplitude and phase of that ray, and a as the
amplitude scale factor.

Fig. 4—Amplitude and delay characteristics of the fading model channel response:
a, = 0.1, b; = 0.9, ap = 0.09, and bz = 1.111.

amplitude and delay frequency responses based on eqs. (2) and (3) are
shown in Fig. 4.

Although the model was developed using measured amplitude data
only, we found it adequate for modeling of selective fading using both
measured amplitude and group delay data. The statistical distribution
of the parameters naturally differs from those obtained for amplitude
only, because we can distinguish between minimum and nonminimum
- phase scans.

For convenience, the channel scan is measured in frequency incre-
ments of 2 MHz, resulting in 16 samples over the 30-MHz range

Choosing M = 80, the delay is 7 = 6.25 ns. Thus the in-band frequencies
correspond to n values between 1 and 16, and the model transfer
function is periodic with M = 80, which corresponds to a nee
period 1/7 = 160 MHz (see Fig. 4).

The first modeling step is to choose a, 8 and w so that the sequence
of measured values { Y,,}, n = 1, 16, is closely matched by the sequence
{Y(w,)} using eqs. (4) and (8). Betimates of a, 8 and wo were found by
minimizing the mean square error, Ems, between these sequences. We
define En5 as

where C,, is a weighting coefficient. Since Y,, is derived from data that
was uniformly quantized on a logarithmic scale, we use a weighting
that 1s approximately logarithmic. Hence,

1/2
Eas = 53 x (Y,[dB] — Y(w,)[dB]) i ; (11)
A plot of the distribution of this error for the total population is shown
in Fig. 5. As seen from the plot, nearly 99.9 percent of the errors are
below 2 dB.

The second step in the modeling is to find a and 6 given estimates
for a and £. From these estimates the parameters a and b are obtained
by inverting eq. (5). The set [a, | has two possible solutions [a,, };]
and [de, be]

It is clear from eq. (4) that a = 6 in every scan, since Y(w) must fit a
sequence of nonnegative powers. Thus, both solutions for 6 are real.
It also follows that b,b. = 1.

The two solutions in eqs. (12a) and (12b) satisfy the amplitude
response fit. The solution [a,, 6:] where b; < 1 corresponds to a
minimum phase transfer function, since it produces a zero in the left

half of the complex frequency plane, and the solution [a2, bz], where
b. > 1 corresponds to a nonminimum solution, since it produces a zero
in the right half of that plane.

The decision of whether a given scan corresponds to a minimum
phase or nonminimum phase fade is made here by using the delay
data. For a substantial fraction of the available data, the delay response
was flat. For this population, the estimations were done by using
amplitude data only and assuming the minimum phase solution.

Figure 6 shows a few typical delay scans. T’o determine whether a
scan is minimum phase or nonminimum phase, we have to examine
the shape of the delay response. Scan I shows clearly a nonminimum
phase delay shape, but so does scan II, although it is negative. The
group delay scans usually exhibit a constant delay offset that has to
be subtracted during the fitting process. Scan III is caught in the
transition between minimum phase and nonminimum phase; this
sometimes happens at very deep fades.

The usual time trajectory of a deep fade is as follows: The fade
appears as a minimum phase fade, gradually deepens, crosses to a
nonminimum phase fade, then becomes shallower, and disappears. It
usually does not cross back to the minimum phase shape.

Each delay scan is sandwiched between two amplitude scans that
are separated by 0.2 second. The initial parameters wo;, 61; and b2; used
to fit each delay scan [eq. (3)] were interpolated from the two ampli-
tude scans. Two error estimates were then set up, namely,

where D,,(61, wo) is the minimum phase delay response, and D,,(be, wo)
the nonminimum phase delay response, as derived by fitting the
amplitude data. Further, D,, is the measured group delay at frequency
wn. The group delay scans usually exhibit an unknown constant delay
offset Do (see Fig. 6, scan JI) that has to be estimated during the fitting
process. —

The above errors were minimized using a quasi- Newton optimization
algorithm. First, E,; and HE, were minimized with respect to the delay
offset Do. By selecting the response with the better fit, this step was
sufficient to decide if the shape is minimum phase or nonminimum
phase. The delay responses thus obtained usually fit within a 7-ns
root-mean-square error, for scans having swings smaller than 50 ns.
The delay response fits were further optimized, by adjusting the
parameter b, to yield root-mean-square errors smaller than 4 ns. These
further optimization did not change the values of 6 significantly. The
Eap values [eq. (11)], of the resulting amplitude fits changed by less
than 1 dB.

Delay scans with swings larger than 50 ns were out of scale and
could not be fitted.* For these scans, the choice between minimum
phase and nonminimum phase was made by minimizing the error with
respect to the offset Do only. The initial parameters, interpolated from
the adjacent amplitude scans, were used in the statistics.

One type of poor fit occurred when a delay scan was in a transition
from minimum phase to nonminimum phase. These scans were elim-
inated from the modeling.

The delay data could not be fitted more tightly to the model because,
as seen from Fig. 7, the residuals have strong harmonic components.
The harmonic components indicate long delay echos in the system.
We have concluded from our experiments that this distortion is caused

* The instrumentation was changed recently to record delays in the range of +100
ns.

by multimoding in the antenna and waveguide system of the trans-
mitter and receiver.

The statistical distributions for the parameters of the fading model
transfer function in eq. (1) were characterized previously using ampli-
tude response data from the Palmetto experiment.** We used a similar
approach to generate parameter distributions for the present case.
Essentially, the data scans can be divided into four distinct groups:
Minimum-phase fades, nonminimum phase fades, flat amplitude fades
and flat delay fades. These four groups are characterized best if the
estimated statistical distributions of the model parameters are de-
scribed separately. The flat amplitude and flat delay groups are degen-
erate cases, where only limited modeling was required. A scan with a
flat amplitude response has also a flat delay response. However, a flat
delay scan does not necessarily correspond to flat amplitude response.
Since the delay notch is narrower than the amplitude notch (see Fig.
4), many scans with a sloping amplitude response, corresponding to
an out-of-band notch, have flat delays (see Section 3.5).

The statistical characterizations of the minimum phase fades and
nonminimum phase fades are done first, followed by the characteri-
zations of the flat amplitude and flat delay fades. The statistical
distributions of the model parameters for the whole fading population
are estimated next, based on amplitude data only. This is done in
order to compare the fading characteristics of the Gainesville channel
with those of the Palmetto channel.

For minimum phase fades the distributions are generated for the
parameters

where a, and b, are the minimum phase solutions of eq. (12a), wo is
the notch frequency, B, is the relative notch depth in decibels and A,
is the amplitude scale factor in decibels.

For nonminimum phase fades, the parameter b. is unbounded and
is not easily characterized. Equation (1) is then rewritten as

As shown below, the choice of these parameters leads to distributions
very similar to those of their minimum phase counterparts.

3.3 Distributions for minimum phase and nonminimum phase fades
3.3.1 The distributions of B, and Bz

The cumulative distributions of B,; and Bz are shown in Fig. 8. The
abscissa is B, or By in decibels, and the ordinate shows the time T
that B, or B, is greater than the abscissa. It is clearly seen that, if we
ignore the fades with low dispersion (B < 8 dB) and assume that the
data above 30 dB have too few samples to be reliable, the two curves
can be fitted to within +1.75 dB by one epee distribution. This
distribution is

B
T = 1.37-10' e 482 = 1.87-10* L}*, for 8<B<30dB, (17)
-B .
where L = 10”, and represents the minimum fading voltage relative
to the amplitude scale factor a, (or azb2).

As seen from Fig. 8, there are overall more minimum phase than
nonminimum phase fades (58 percent). However, if we restrict our-
selves to the practically significant range of 8 < B < 30 dB, the
proportion of nonminimum phase fades is 50 percent. Both the mini-
mum and nonminimum phase fades distributions can be expressed,
within the limits of the approximation, in eq. (17).

3.3.2 The distribution of the parameters A, and A2
The cumulative time distributions for A; and Az are shown in Fig.

9. It is clearly seen that the two distributions are almost identical for
larger A (above 14 dB). The means and standard deviations are

A; = 15 GB, o, = 3.0 dB
A, = 15.5 GB, oo = 3.0 GB. (18)
The unnormalized probability density functions (pdf’s) of A; and
A, are shown in Fig. 10. (These pdf’s are scaled in such a way that the
ordinate displays on a logarithmic scale the number of seconds that A
was within +0.5 dB of the abscissa. This same format is used in all

the pdf’s shown later.) The pdf’s show a definite asymmetry about
the mean. This deviation from a Gaussian distribution* could be

partially attributed to the data gathering method used in Gainesville
(see Section 2.2).

The means and standard deviations of A; and A2 as functions of B,
and B, are shown in Fig. 11. These plots show clearly that the
parameters A and B are uncorrelated for the Gainesville data.*

3.3.3 Possible discrepancies in the A and B distributions caused by
instrumentation

1. The amplitude and delay characteristics were sampled every 2
MHz by the recording equipment over the 30-MHz frequency scan.

The 2-MHz resolution may be too low for very deep dispersive fades,
and cause flattening of these curves during the fitting process. This
may cause the distributions of B,; and Bz to appear steeper for very
large B values than they really are. |

2. The Microwave Link Analyzer (MLA) was set up to measure
amplitudes between 15 to 65 dB only. Therefore, some of the shallow
fading events with small A and small B-are not recorded. This could
cause the distribution of B for small values to appear flatter than they
really are, and the distributions of A to appear less symmetric.

Figure 12 shows the density function of the notch frequency, for
both the minimum and nonminimum phase fades, as estimated by the

modeling procedure. This model estimates that approximately 70
percent of the scans have an in-band notch. This is to be expected
because, as seen from Fig. 4, the delay shape is steeper and narrower
than the amplitude notch. Therefore, most of the scans that have an
in-band slope (out-of-band notch) have a flat delay response, and are
therefore not characterized in this group (see Section 2.2).

Fitting the model to amplitude shapes that have in-band slopes is
not unique, since small perturbations in data can produce large vari-
ations of out-of-band notch depth and frequencies. The modeling
algorithm in its present form has a tendency to move the notch for
slope shapes close to the edges of the band. We have tried, with
variable success, to use also the delay data in placing the out-of-band
notches. However, the delay data for shapes with out-of-band notches
are limited and heavily corrupted with multimode distortion that made
the fitting of the model rather difficult.

Amplitude scans that were flat within 1 dB were assumed to be
nondispersive. For such scans, therefore, A was the only parameter

Fig. 12—Unnormalized probability density function of the notch frequency Fo, mea-
sured from midband.

modeled. The cumulative distribution of A for this case is shown in
Fig. 13. As in previous cases, the distribution exhibits a truncated
Gaussian behavior, with

Scans that did not have a flat amplitude but exhibit a small delay
distortion, D(w)peak to peak = 7 ns, were regarded as having a flat delay.
Scans of this type can be put into two categories:

(a) Scans with in-band, or close to in-band, notches and small
amplitude deviations. A 7-ns delay notch corresponds to an amplitude
notch with B = 6.5 dB.

(b) Scans with amplitude slopes that may have a notch so far out
of band that the delay distortion is negligible.

In both of the above cases the delay information is too limited to be
used to decide whether the scan is minimum phase or nonminimum
phase. Since it is relatively unimportant to differentiate, in these

Fig. 13—Unnormalized probability density function of A for flat amplitude fades.

Figure 14 shows the distribution of B. As seen from the plot, 96
percent of the scans have B < 6.5 dB, which puts them in the category
(a) above. Only 4 percent belong to category (b).

3.6 Parameter distributions for the total fade population and comparison
with Palmetto

The distributions of the model parameters for the whole fade pop-
ulation, derived entirely from amplitude response data and comprising

Figure 16 shows the distribution of the parameter B, as measured
in Gainesville. Within reasonable accuracy the distribution can be
modeled by the exponential curve

Fig. 15—Unnormalized probability density function of A for flat delay fades.

The fading data were recorded for 11 months, well distributed over all
seasons during 1.5 years in 1982-1984. The curve in Fig. 16 can
therefore be regarded as an annual average distribution of dispersive
fading in Gainesville. The other curve in Fig. 16 shows the annual
distribution of B for the Palmetto data. This distribution was extrap-
olated from data measured in Palmetto in 1977.* The total fading for
the heaviest month of the year was estimated to be 8000 seconds,
where the total fading was defined as the intercept of the B distribution
with the B = 0 axis. To approximate the average annual fading in
Palmetto, we multiplied the seconds for the heaviest fading month by
a scale factor of 3.5. The average annual fading in Palmetto was thus
estimated to be 28,000 seconds.

Fig. 16—Average annual probability distributions for B for the total populations of
Gainesville and Palmetto.

The Palmetto distribution is steeper than that for Gainesville and
can be modeled over its central region by
B
T=1.6-10' e 98 =2.10' 2?% 4dBs Bs 25dB. (22)
It can be surmised from the shape of the Gainesville distribution that
fading on that path is more dispersive than the Palmetto fading. By
examining, for illustration purposes, the total dispersive fading for
B = 20 dB, we see from Fig. 16 that there is five times more of it in
Gainesville than in Palmetto.

Figure 17 shows the probability density function of parameter A. It
has a truncated Gaussian shape, with mean and standard deviation

Fig. 17—Unnormalized probability density function of A for the total population.

The corresponding distribution in Palmetto was Gaussian, with mean
and standard deviation
Ap = 25 dB; op = 5 GB. (24)
Moreover, the mean Ap was correlated with Bp in the Palmetto
_ experiment. The Gainesville parameters Ag and Bg are independent
(Fig. 18).
The apparent truncation of the A distribution is probably due to
the data gathering method in Gainesville. Specifically, fades below 15
dB were not recorded. This could also be a reason why A and B are

uncorrelated in Gainesville. In Palmetto, A was correlated with B, for
B< 10 dB only.

The probability density function of the notch frequency for the total
fade population is shown in Fig. 19. Most of the scans have an in-
band notch (55 percent). The notch frequency does not exhibit the
bimodal distribution shown in Palmetto.

The fading data obtained in the Gainesville experiment were for
both amplitude and group delay. We generalized the method of
Rummler*“ to include group delay response data, and thus obtain both
minimum phase and nonminimum phase model parameters.

The distributions for minimum phase and nonminimum phase fades
can be regarded as identical, and can be given by a single description
[see eqs. (17) and (18)], in the significant range of interest,8 < B<
30 dB. However, if the shallower fades are included (0 <= B <= 8 dB),
there are, overall, more minimum phase fades (58 percent).

Fig. 19—Unnormalized probability density function of the notch frequency Fo for the
total population, measured from midband.

nel. This is seen from the estimated average annual distributions for
the total fade populations. For the dispersive part of the fading, the B
parameter, the distribution is steeper in Palmetto. By examining Fig.
16 for illustration purposes, we can deduce that there is five times
more fading with B = 20 dB in Gainesville. In addition, the mean of
A is smaller in Gainesville; Ag ~ 16 dB, as compared with Ap ~ 26 dB
in Palmetto.

AT&T long-haul digital-radio performance objectives limit service
failure time to 0.02 percent (two way) annually on a 4000-mile route
due to all causes. One-half of this is allocated to causes associated
with equipment and maintenance. The allocation to dispersive fading,
therefore, is 0.01 percent annually.’* The one-way annual dispersive
fading allocation for the average 25-mile hop is thus 10 seconds.

One measure of the amount of dispersive fading in a radio channel
is the distribution of the parameter B. As seen from Fig. 16, a 10-
second annual failure limit in Gainesville requires a robust digital
radio system designed to operate with notches B < 36 dB. The same

failure limit in Palmetto requires a radio system designed to operate
with notches of B < 26 dB. These comparisons demonstrate the need
for a description of the geographical occurrence of dispersion, which
will differ from that for multipath fading for a single frequency. The
availability of such a dispersive fading map would facilitate the accu-
rate engineering of digital radio routes.

I would like to acknowledge the contributions of G. A. Axeling for
aligning and maintaining the measurement equipment in Gainesville,
B. J. Engelmann for utilizing her computer expertise to produce the
statistical outputs, and G. L. Calhoun for maintaining the database.
For many stimulating discussions on this work and comments on this
manuscript, I would like to thank W. T. Barnett, L. J. Greenstein, V.
K. Prabhu, W. D. Rummler, and A. Vigants.

1. G. M. Babler, “A Study of Frequency Selective Fading for a Microwave Line-of-
Se Narrowband Radio Channel,” B.S.T.J., 51, No. 3 (March 1972), pp. 731-

215. 3 Greenstein, “A Multipath Fading Channel Model for Terrestrial Digital Radio,”
IEEE Trans. Commun., COM-26, No 8 (August 1978), pp. 1247-50.

. W. D. Rummler, “A New Selective Fading Model: Application to Propagation Data,”
B.S.T.WJ., 58, No. 5 (May-June 1979), pp. 1037-71.

. W. D. Rummler, “More on the Multipath Fading Channel Model,” IEEE Trans.
Commun., COM-29, No. 3 (March 1981), pp. 346-52.

L. J. Greenstein and B. A. Czekaj, “A Polynomial Model for Multipath Fading
Channel Responses,” B.S.T.J., 59, No. 7 (September 1980), pp. 1197-225.

. M. H. Meyers, “Multipath Fading Characteristics of Broadband Radio Channels,”
Globecom’84 Conf. Rec., 3 (November 1984), pp. 1460-5.

. W. C. Jakes, Jr., “An Approximate Method to Estimate an Upper Bound on the
Effect of Multipath Delay Distortion on Digital Transmission,” IEEE Trans.
Commun., COM-27, No. 1 (January 1979), pp. 76-81.

L. J. Greenstein and V. K. Prabhu, “Analysis of Multipath Outage With Applications
to 90 Mbit/s PSK Systems at 6 and 11 GHz,” IEEE Trans. Commun., COM-27,
No. 1 (January 1979), pp. 68-75.

9. M. F. Gardina and A. Vigants, “Measured Multipath Dispersion of Amplitude and
Delay at 6 GHz in a 30 MHz Bandwidth,” ICC’84 Conf. Rec., 3 (May 1984), pp.
1433-6.

10. M. Liniger, “Sweep Measurements of Multipath Effects on Cross-Polarized RF-
Channels Including Space Diversity,” Globecom’84 Conf. Rec., 3 (November
1984), pp. 1492-6.

11. L. Martin, “Phase Distortions of Multipath Transfer Functions,” ICC’84 Conf.
Rec., 3 (May 1984), pp. 1437-41,

12. M. Sylvain and J. Lavergnat, “Modelling the Transfer Function of a 55 MHz Wide
Radio Channel Multipath Propagation,” ICC’85 Conf. Rec., June 1985, 3, pp.
1541-6.

Philip Balaban, Diplom Ingenieur (Electrical Engineering), 1950, Technical
University Munich; Ph.D. (Electrical Engineering), 1965, Polytechnic Insti-

tute of Brooklyn; Scientific Department, Israel, 1950-1956; Contraves, Zurich,
Switzerland, 1956-1958; Computer Systems, New York, 1958-1962; Polytech-
nic Institute of Brooklyn, 1962-1967; AT&T Bell Laboratories, 1967—. Prior
to joining AT&T Bell Laboratories, Mr. Balaban was engaged in research and
development in the areas of circuit design, computers, and communications.
At AT&T Bell Laboratories he worked on statistical modeling, simulation and
analysis of transmission systems and circuits. Recently he has been involved
with problems of channel characterization, and system performance of digital —
radio communication systems. In 1965-67 he taught at the Polytechnic Insti-
tute of Brooklyn, and in 1970-71 he was a visiting professor at the Technion
in Israel. Senior Member, IEEE; Member, Sigma Xi.

Comments on “Integration With the 5ESS ™ Switching System,” by S. A. McRoy,
J. H. Miller, J. B. Truesdale, and R. W. Van Slooten*

This letter amplifies the footnote on page 2426 of the article titled,
“Integration With the 5ESS™ Switching System,” in the December
1984, issue of the AT&T Bell Laboratories Technical Journal.

The information given here has been used in the field for engineering
Remote Terminals (RTs) of the SLC® 96 digital subscriber loop carrier
system on the Digital Carrier Line Unit (DCLU) in a 5ESS switching
office. For engineering RT's, the telephone company engineer can
choose one of the three service criteria—1-1/2 percent blocking, 4
percent blocking, and 7 percent blocking. The 1-1/2 percent blocking
criteria is recommended for average busy season busy hour engineer-
ing, the 4 percent blocking Once-A-Month (OAM) for extreme value
engineering, and the 7 percent blocking for high-day engineering. The
DCLU traffic capacities for the two types (Mode I and II) of RTs
corresponding to the three criteria are given in Table I.

1. A two-peripheral-interface-data-bus DCLU has 64 time slots.
2. A four-peripheral-interface-data-bus DCLU has 128 time slots.

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Haque C. A., Temperature Time Effects on Film Growth and Contact Resistance
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Komlos J., Greenberg A. G., An Asymptotically Nonadaptive Algorithm for
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