Contrasting Performance of Faster Binary Signaling With QAM 1419
G. J. Foschini

Adaptive Transversal Equalization of Multipath Propagation for 1447
16-QAM, 90-Mb/s Digital Radio

Enhancement of ADPCM Speech by Adaptive Postfiltering 1465
V. Ramamoorthy and N. S. Jayant
On Using the Itakura-Saito Measures for Speech Coder 1477
Performance Evaluation
B.-H. Juang
A Packet/Circuit Switch 1499
Z. L. Budrikis and A. N. Netravali
An Approximate Analysis of Sojourn Times in the M/G/1 1521
Queue With Round-Robin Service Discipline
P. J. Fleming
Analysis of a TDMA Network With Voice and Data Traffic 1537
M. L. Honig
PAPERS BY AT&T BELL LABORATORIES AUTHORS 1565

In this paper we determine the performance of Faster Binary Signaling
(FBS), an alternative method to Quadrature Amplitude Modulation (QAM)
for achieving a high bit rate over an ideal, bandlimited, noisy channel. With
this method, signaling is faster than the Nyquist rate. Consequently, there are
fewer points in the signal constellation, resulting in a greater separation of
the points when the average transmitter power is the same as for QAM. Thus,
at the expense of introducing Intersymbol Interference (ISI), there is an
apparent improvement in noise immunity. The ISI can be mitigated with
maximum likelihood sequence detection. We explore the advisability of trading
freedom from ISI for added noise immunity for the extreme case where the
system with faster signaling uses a four-point constellation. The question of
the efficacy of FBS has been difficult to approach, but FBS has loomed as a
possibly strong competitor among alternatives to QAM. We show here how to
analyze FBS, and we give examples involving FBS operating at up to five
times the QAM rate. In the examples, FBS is revealed to be, at best, of
marginal value even if one allows for implementation capabilities far beyond
those of forthcoming processors.

For data communication over channels such as voiceband analog
telephone circuits, satellite links, or terrestrial digital radio hops, there
is a search for practical techniques that are more efficient (in bits per
cycle) than QAM. This search is intensifying because of the growing

demand for data communication services over bandlimited channels
and because of the continuing drop in the cost of high-speed processing
required by advanced communication methods.

The bandlimited channel with additive white Gaussian noise (power
MN), where the average transmitter power 7(Y > _/) is con-
strained, serves as a proving ground for theoretical explorations of the
relative efficacy of proposed techniques. Specific methods have been
discussed as candidates for improving efficiency. Three candidates are
Higher-Dimensional Constellations (HDCs),’ Ungerbock’s Trellis
Coding (UTC),? and Faster Binary Signaling (FBS).? Both HDC and
UTC have lent themselves to analysis and their significant value over
the QAM method has been established. Moreover, we are beginning
to understand the relative value of UTC over HDC.* On the other
hand, the effectiveness of FBS has hitherto remained a mystery. In
Ref. 5* some theoretical results on FBS for some special pulses are
presented but the relative effectiveness issue is not settled.

To understand the FBS method, consider the elementary QAM
method, 4-PSK (phase-shift keying), in a situation comfortably meet-
ing a stringent probability of error (P.) constraint. If the bit rate
requirement increases it can be met without expanding bandwidth by
increasing the number of points in the QAM constellation. FBS is a
natural alternative means of transmitting at higher bit rates. In FBS,
one fixes the constellation at four points and increases the symbol
rate as much as necessary. Maximum Likelihood Sequence Detection
(MLSD) is employed to overcome the consequent Intersymbol Inter-
ference (ISI) in the best way possible (see Refs. 6 through 8 for
treatments of MLSD). The minimum separation between distinct
points in the planar FBS constellation is greater than for the QAM
constellation. One might say FBS trades freedom from ISI for added
noise immunity and then MLSD is used to mitigate the ISI.

With the efficacy of FBS unknown, it looms as a possibly competi-
tive technique. Here we help the process of evaluating the field of
candidate methods for moving beyond the capabilities of QAM by
showing how to analyze FBS. We show by examples that FBS is, at
best, of marginal value relative to QAM, even if one allows for
implementation capabilities far beyond those of forthcoming proces-
sors. Specifically, we allow for complexity of up to 10° states. Given
the strides in processor technology in the last few decades and the
imminent hardware advances, a number like 10° is chosen to avoid
outdating of this paper for a long time. Such prudence is needed.
Indeed, the analysis that follows leaves open the possibility that FBS

* The terminology “faster binary signaling” is not used in [A] and [M]. In [M] the
term “faster than Nyquist signaling” is used.

would offer substantial improvement over QAM if complexity were
not a consideration. Moreover, one must also consider that fast detec-
tors could go far beyond conventional MLSD in processing efficiency.?

The data transmission medium is represented here in the simplest
idealized form as a lossless characteristic with additive white Gaussian
noise. The channelization is shown in Fig. 1. On each channel the
transmitted signal is subject to an average power constraint and it is
assumed that, for all the systems that we discuss, the average trans-
mitted power is much greater than the noise power. [The ramification
of this high signal-to-noise ratio (s/n) assumption is discussed in
Section III.] In the analysis that follows, we assume that the channels
are isolated from each other in that it is not permitted to mitigate
Adjacent Channel Interference (ACI) through some elaborate scheme
requiring coordination among channels. It is required that R bits per
second be transmitted over the channel. Whatever the form of modu-
lation used, soft maximum likelihood sequence detection is employed
at the receiver.’°

It is because of the current prominence of QAM in applications that
the work here is presented with QAM as the benchmark method. Since
a flat channel transfer characteristic is assumed, it is trivial to relate
results to the equivalent baseband channel representing a QAM rail.
We use M = m? to denote the number of points in the QAM constel-
lation. So, constellations with 16, 64, 256, and 1024 points correspond
to FBS operating at 2, 3, 4, and 5 times the conventional rate. We
elected to work with square QAM constellations even though certain
departures from such constellations yield superior performance.’ The
reason for our choice is that we want to analyze FBS in isolation and
the aforementioned departures can be viewed as the first step in using
the HDC method. Finally, we employ the harmless expediency of
dealing with M as if it is a continuous variable in our calculations and
in some of our graphs. When M is a positive integer that is not a

perfect square, one can find a two-dimensional constellation that
realizes the situation covered by the analysis.

Throughout we will assume that the standard QAM method is
meeting a P, requirement (107° or less). When we compare alternatives
to the QAM method, we will associate primed variables with parame-
ters of the non-QAM system where needed to avoid ambiguity.

The generic system structure depicted in Fig. 2 is interpreted here
for the special case of FBS. For FBS the binary data are blocked into
successive 2-bit words. A pair of independent, synchronous, delta
function streams are formed using the two bits to randomly sign the
pair of delta functions that are input to the pair of baseband filters.
The baseband filters nominally cut off at W cycles/second. On each
rail the pulse rate is r = R/2. For convenience we use JT’ = 1/r to
denote the time interval between impulses. The filter outputs combine
to form the in-phase and quadrature rails of a passband signal. Thus,
it may seem that what we have is 4 PSK; but FBS is unusual in that
its symbol rate, 1/T’, is higher than the conventional 1/T = 2 W.

The higher rate does not increase bandwidth but it does cause ISI,
which will be combatted with MLSD. The ISI is assumed to involve a

Fig. 2—Generic system structure. For QAM the transmit and receive filters are low-
pass filters. For the FBS case they are a matched pair with finite memory in the sampled
data domain. The nominal cutoff frequency of the low-pass filters is W= 4/2.

memory of », i.e., the bandlimited baseband filters are such that the
impulse response sampled at times {n/T’}?_-.. have the form h = 0,
ho, hi, --- h,, 0. Thus MLSD requires 2’*! states (2” per rail).*

We would like to, if we could, make the following idealizations
concerning H(f), the Fourier transform of the impulse response of the
baseband filter.

A. Each member of a bank of FBS passband systems is spectrally
disjoint [in baseband, H(f) vanishes outside (~W, W)].

W’ [W’ = 1/(2T’) > W).
The first of the above is needed for spectral efficiency. Statement B
is needed to be consistent with the assumption that MLSD involves a
memory of vy. Mathematically it would seem impossible to meet A and
B. After all, if H(f) is a trigonometric polynomial vanishing on
[W, W’], it vanishes everywhere. There is no real difficulty here. We
will adhere to B with v the degree of H(f). While A will not strictly
hold, one can get as close to ideal as desired as long as v is large enough
to meet out-of-band energy constraints. In MLSD a matched filter is
used to initiate the detection process. The matched filter receiver
serves to select the desired band [-W, W]. We have a dual view of
what the frequency band is. From the point of view of where the signal
power is concentrated, [—W, W] is the band. From the point of view
of MLSD, we are dealing with a sampled temporal response, which
can only correspond to a transform that is a polynomial on [—W’,
W’]. So long as the degree of the polynomial is large enough the two
views of bandwidth can be reconciled.

There are several questions before us. Can we make the energy
outside the [-W, W] band so small that the interference between
neighboring systems is negligible, yet the number of states involved in
MLSD is reasonable? If we can accomplish this, does the FBS system
perform better than the comparable QAM system? How much better
does it perform and at what complexity?

We investigate these questions in the context of three kinds of
discrete impulse responses. The first of these is the Nyquist responses
(“brickwall spectra” on [—W, W]) truncated to memory »v. For these
we shall see that the interference from neighboring systems is prohib-
itive for reasonable v. The second set of impulse responses are the
discrete prolate spheroidal wave functions, which, for fixed total energy
and fixed v, have the least interference from neighboring systems. We
demonstrate that their performance is not good. Finally we explore
optimally designed responses and find that for reasonable v, even for
the most favorable cases, the advantage over QAM is very modest.

*Notice that if the constellation were not the product of two one-dimensional
constellations, as we have assumed, the complexity would grow as 4’ instead of 2’”*?.

The probability of bit error, P., is an important performance mea-
sure for data communication systems. For QAM as well as the systems
employing HDC, UTC, or FBS, if MLSD is used the probability of bit
error decays exponentially as the noise spectral density is decreased
(except for an algebraic multiplier). That is, an exponentially tight
bound on P, has the form

where x and & are independent of o”, the noise power spectral density
on a single dimension. For FBS viewed in the MLSD context, the
exponentially tight bound has the form

where * denotes convolution and the minimum is over all doubly
infinite sequences e of the form

Olen 2+ ex0,
where e, belongs to {0, 1, -1} and K can be any nonnegative integer.
Clearly, din < |] h||*. In cases where d2in = || h||? it is common

to say that the matched filter bound is attained. What is meant is that
the exponent of P, is the same as if there were only a single data pulse
to be detected (no ISI). The terminology stems from the fact that,
when there is no ISI, MLSD employs simply a matched filter (along
with a threshold comparator).’”

We take the quantity & as a convenient indicator of performance in
the high s/n realm. (We stress that, for models of specific systems,
more refined computations estimating the actual error probability are
often needed.) The “gain” of one system over another is expressed as

We shall be concerned in this paper with estimating the gain that FBS
exhibits over QAM. Both UTC and HDC exhibit substantial gain over
QAM. For UTC, gains in the range of 3 to 6 dB have been reported
and, for a 3-dB gain, the required complexity is extremely reasonable.”

For the conventional QAM system, ||h||*? denotes the energy per
pulse prior to multiplication by a; belonging to [+1, +3, --- +

(L — 1)], so the modulated pulse has average energy (L? — 1)/3 || h||?.
If there is one symbol every 7’ seconds, the average signal power is
[(L? — 1)/3] [|| h||7/T]. Since the information rate per rail is
(log.L.)/T b/s, FBS must operate at a rate of (logeL)/T pulses/s. Let
h’ be the impulse response for FBS. For the two systems to have
identical signal power we must have

For FBS, accounting for the wider bandwidth, the noise variance per
sample is o” (log.L.)/T. Not necessarily all of || h’ ||? is realized in the
error exponent.

G = 10 logy (2)
where p = log,.L = W’/W = T/T’. One could interpret 10 logio[(4’ —
1)/3p] as a noise immunity gain and 10 logjo(d2in/||h||7) as the
penalty for ISI.

Shortly it will prove useful to allow for replacing the noise power
spectral density on the FBS system by a level greater than that on the
slower system, say (1 + 8)o? with 6 > 0 in place of o”. This will enable
us to compensate for interference from adjacent channels. When, and
if, the matched filter bound is attained the gain is expressed by 10
logio(4° — 1)/[3o(1 + 8)]. Figure 3 depicts this function with 8 as a
parameter. It is evident from the 6 = 0 curve that, depending on p, if
the matched filter bound is attained, the gain can be considerable.

We consider now the interference in the band (—W’, W’) that stems
from those channels (other than the primary channel centered at zero)
whose power spectral density is nonzero in (-W’, W’). The determi-
nation of the additional power due to these interfering channels is
straightforward. When measured for a single rail, at the output of the
matched filter, H*(w), the power is the same as if o” were replaced by

The sum is over all neighboring systems overlapping the (-W’, W’)
band. Because of its genesis, the term that adds to o” in (3) is called
the Adjacent-Channel Interference term or ACI. For H’(f) with nearly
all the energy in the (—W, W) band, | H’(f) |?/fw | H’(g) |?dg has a
mean value of approximately T on (—W, W). Since each channel is
symmetrically disposed relative to its neighbors, the integral in the

1 2 3 4
SPEED FACTOR, p
Fig. 3—Adjusted gain versus speed factor when matched filter bound is attained.
ee

numerator of (3) can be replaced by f W,. It follows that the strength
of the ACI term in (3) is roughly indicated by that energy in a pulse
with transform H’(f) that is out of the band (—W, W). (We use OBE
to denote out-of-band energy.) This approximation becomes more
precise if H’(f) is approximately flat in (-—W, W) (as is the case in
Section IV).

If ACI is not negligible, it is reasonable to modify the gain by
subtracting 10 logio(1 + OBE/o”) or, the more precise but more
complex, 10 log,o(1 + ACI/c”). No matter which gain expression is
most appropriate, if we insist on some degree of spectral isolation it is
unclear how much gain can be attained at specific levels of complexity

(2’*! with v < 26). In the sequel we will find that the answer is not
much gain. For the analysis in Sections IV and V we consider OBE in
the range [0, o”]. As we note in Section VI there is no point in
considering OBE outside this range.

We can see from the formula for minimum distance (1) that, if we
slide a window of size v along an error sequence e, a repeated state is
forced to occur by 3” shifts. It follows that, to attain the minimum,
one need not search over more than 3° events. Since {3° |» =
1, 2, 3, 4, ---} = {27, 19683, 7.63 x 101”, 4.4 x 10°8, ..-}. We see that,
even for v = 3, a brute force search is extremely ambitious and for py =
4 it is completely out of the question. (Reference 13 discusses three
other state symmetries as well as a repeat.)

For future reference, we borrow from Ref. 13 and list four useful
representations and notations for error events in Appendix A.

For each », it is useful to view a set of error events, one of which is
guaranteed to achieve d2,in as a tree. Construct a tree of sequences
with three branches emanating out of each node and with the labeling
illustrated in Fig. 4. The labels along each upward path represent the
beginning string for the nonzero portion of an error event. Once a
string of v consecutive zeros is encountered, the growth out of such a
node is pruned from the tree, since continuing the event with nonzero
elements will correspond to creating labelings for beginnings of events
with a greater || h+*e||? than the all-zero continuation.

To envision a computer search for d2i, for a specific h, one can
think of climbing up the tree and to the left and at each node
computing the accumulated

on the upward path to the node. There is one summand for each node
in the upward path. Climb higher if the record low for a completed

error event (a number < ||/]||”) has not been exceeded. Otherwise,
climb down to the first node that offers an unclimbed branch and then
climb that branch. Whenever a node with y consecutive zeros is
reached, and the old record has not been reached, record the new
candidate event for achieving d2,;, and the new record before climbing
down.

Some additional special search tools prove useful. Specifically, one
can terminate an upward climb whenever any of the four symmetries
A; = +A;(i <j) or A; = +A?(i < j) is detected (see Ref. 3). Two other
search tools are very powerful, one for symmetrical h (see Appendix
B) and one for nonsymmetrical h, which is discussed in Section 6.3.

One of the most elementary results in data communication theory
is Nyquist’s result that, for WT = 1/2, signals of the form
ye. Anh(t — nT) with h(t) = sin t/t are ISI-free. In FBS, we replace
T by T/p with p > 1 and, as we have already mentioned, the signal
bandwidth is invariant to p but ISI arises. A hypothetical FBS system
based on such a pulse incorporates a level of idealization beyond the
standard one associated with the abrupt cutoff. Namely, the system
represents the limits of infinite decoding complexity as well as zero
energy in the bands, W < |/| < W’.

For each system, assuming the pulse energy is normalized to 1 and
W’ is normalized to 1/2, we have
2

The infimum is over all error events with «, belonging to {0, 1, —1}
and K ranging over the positive integers. Expression (4) is considered
in Ref. 5 where it is demonstrated that d2;, > 0 for all p => 1. The
intriguing question of whether, for such systems, a positive “gain” is
available remains open.
Let

denote the FBS pulse transform normalized to unit energy. While
expression (4) allows for infinite complexity, for any implementation,
approximations of H’” must be considered. The optimum least-mean-

square approximations, the Fourier series {HN¥(w)A 6 qne’””}, are
natural responses to use to inquire whether, as v increases, FBS
performs better than QAM. Of course, if » becomes too large the
required detector becomes forbiddingly complex.

The tree search discussed in Section 3.3 was used to determine the
“gain” for the least-mean-square approximations. The symmetry of
the impulse responses allowed the addition of the test of Appendix B
to significantly reduce the running time of the algorithm.

The results are shown in Fig. 5. The “apparent gains” are only
meaningful if the Out-of-Band Energies (OBEs) are sufficiently small.
Indeed, they are not sufficiently small as we now discuss. Figure 6
shows the out-of-band energy for a unit energy response for v = 26.
Also shown are the noise levels for the benchmark QAM system
providing the same information rate at a P, of 10-* and 10°°. Of the
four points p = 2, 3, 4, and 5, only p = 2 shows the out-of-band energy
below the noise level. The margin for P, = 107° is slight (=4 dB) but,
from Fig. 5, we see that for y = 26 the “gain” is negligible (~0.1 dB).
For p = 2, if we reduce v to increase the “gain”, the attempt is
undermined by the increase in out-of-band energy. The out-of-band
energies for vy = 20 and py = 14 are also shown in Fig. 6, for p = 2.

We conclude that, for the least-mean-square approximation of a
Nyquist pulse, FBS signaling under the mild requirement P, = 107°
does not offer any significant gain over QAM. In making the compar-
ison, we have allowed FBS the extraordinary complexity of 27° = 6 x
10’ states per rail (>10® states total). If P. were decreased, FBS would
fare even worse.

Figure 7 illustrates approximate Nyquist spectra for v = 26 and
minimizing error events. We note that for »y = 26 the number of
candidate error events exceed 10%” < 3°,

Allowing for complexity not exceeding 10° states, FBS is not attrac-
tive relative to QAM for the examples considered thus far. As we
mentioned in the last section, for the asymptote of infinite complexity
(vy — o) and stringent out-of-band energy constraint, the limiting
squared distance is expressed in (4). Although we cannot compute the
gain G,, we can find an upper bound using candidate error events. We
used events revealed to be useful in the tree search for v < 26. The list
of trigonometric polynomials below {E,(w)}°-2 was used to bound d3in:

Fig. 5—Apparent gain of FBS over QAM (gains unachievable because of interference from adjacent bands).

They yield Ge < 0.107 dB, G3 < 1.4 dB, G4 < 0.477 dB, and Gs <1.1
dB. This shows that, even allowing for an arbitrarily large number of
states, in the limit of stringent out-of-band energy requirements, the
gains available using a Nyquist pulse are at best very modest. The
E,(w) characteristics are illustrated in Fig. 8.

In Section IV we investigated whether the approximations HY” have
good distance properties for FBS with p = 2, 3, 4, and 5. We found
that, for reasonable complexity and out-of-band energy constraints,

Me . - re ae of | HXY |? and extremal error event (+ and — mean +1 and —1). Vertical dotted line marks transition from in-band to
- - n ‘5

they do not. The H," minimize J, | Hi’ — ¥% qne’”* |?w(w)dw in
the special case when the weight function w(w) = 1. In light of the
results of Section IV, we can reformulate the least-mean-square ap-
proximation using a w(w) that is 0 on (—z/p, z/p) and 1 otherwise.
The weighting reflects the fact that it is essential to keep the out-of-
band energy small but, having seen that the flat transform has no
special distance properties, we have no motivation for keeping the
transfer characteristic flat within (—7/p, z/p).

The extremal responses so obtained are called the Discrete Prolate
Spheroidal Wave Functions (DPSWF).”* Their theory has been devel-
oped by Slepian.’° Wyner has suggested their consideration for use in
data communication systems for reasons other than those we are
considering here.’ Let H?% denote the transform of the discrete
spheroidal wave function of memory v corresponding to an FBS system

with parameter p. Since minimizing the out-of-band energy corre-
sponds to maximizing the in-band energy, the coefficients of H?* are
then components (q,, qi, --: g,) of the eigenvector of the matrix of the
symmetric quadratic form

corresponding to the largest eigenvalue, i,,,. Since we normalize by
constraining HfS to have unit energy, the quantity 1 — ),,, is the out-
of-band energy.

In Fig. 6, 10 logio(1 — X,,,) is plotted against p for various vy (see
dashed curves). Unlike HNY we see that a significant portion of the
loci for H®> are disposed well above curves for the noise levels for
P, = 107° and P, = 10~*. Thus, there are spaces of systems of moderate
complexity with small out-of-band energy, whose distance properties
are of interest. What are the distance properties of H?’S in the range
p = 2, 3, 4, 5? They are not good. Use of the search algorithm of
Section 3.2 demonstrated no gain for any HS whose (», p) coordinate
corresponded to an out-of-band energy below the level of the Gaussian
noise for P, of 10~*. For example, for p = 2 at v = 4, the out-of-band
energy is —26.3 dB, which is below the level of the additive noise.
However, the minimum distance of H¥3 is poor, specifically G = —1.33
dB. For larger v, the out-of-band energy drops precipitously but
distance decreases as well. As p increases in the range 3, 4, and 5, the
situation worsens: G values significantly below 0 dB occur with out-
of-band energy prohibitively above the noise level. As y is increased,
the distance drops markedly.

At this point we have an interesting situation. The least-mean-
square approximations to the Nyquist pulse are shown to have attrac-
tive “apparent gains” relative to QAM but the gains cannot be realized
because the signal spectrum is not adequately confined. On the other
hand, the results for DPSWF’s show that great spectral confinement
is possible, but these pulse shapes do not exhibit any gain over QAM.
The question remains as to how much gain we can achieve under a
spectral confinement constraint for a specific complexity. This is
addressed in Section VI.

optimal h. (Since H can have 2p zeros disposed in inverse conjugate
pairs there can be as many as 2’ possible factors of H that have real
coefficients.) The problem of finding optimal h is essentially a linear
programming (LP) problem. The suggestion of viewing optimum
MLSE system design as an LP problem appears in my paper with R.
R. Anderson.?? It turns out that, in most cases of interest, the number
of constraints corresponding to the various error events is too large
for the LP to be useful by itself. The LP is combined with the tree
search algorithm that serves to eliminate most error events from
consideration. The LP-tree search algorithm solves the design prob-
lem. We proceed now to describe the LP and show how it is integrated
with a tree search algorithm. Then we present the performance results
for optimally designed pulses.

Recall that an LP problem is one of the following type: Given a
vector ¢, find a vector y that maximizes (y, c) subject to a set of linear
constraints of the form (y, a;) = b;, i belonging to Z a finite index set.
The a; and b; are given vectors and scalars, respectively. It is very
useful that < constraints can be converted into = constraints by
changing sign and so equality constraints can be represented by a pair
of = constraints.

In our application, 7 is infinite. Since we shall see that the feasible
y exists in a bounded set we can, in principle, obtain a solution as
close to optimum as desired by solving an LP with sufficiently many
constraints.

Now h=0h.,,--- ,h,, ---, h, 0. We will represent h in a 2v + 2
dimensional space where the first 2v + 1 coordinates are (h_,, --- ,
h,). An additional coordinate augments the projection of h so that we
have (h_,, --- , h,, s). The augmented vector of 2v + 2 components is
denoted y. The additional coordinate, s, is a mathematical convenience
that will facilitate maximization of the minimum squared distance, as
we shall see.

1. As a convenient normalization, we assume that the energy in h
cannot exceed 1, so h, < 1; therefore,

3. H(w) is nonnegative. H(w) is the function that has h for Fourier
coefficients and the operation of Fourier series is a linear one. So the
constraints H(w) = 0, 0 < w < a, can be put in the form (y, a.) = 0
by defining a,, appropriately. There is one constraint for each w on
0 < w < x. In our application we can use a discrete set of the form
{wn = (nr)/N}%-, with N sufficiently large to give adequate accuracy.

5. The 2v + 2 component, seemingly extraneous so far, now comes
into play. Let {E;(w)};.- be the error polynomials. Project them into a
2v + 2 dimensional space using the successive Fourier coefficients with
index of absolute value < pr to get the first 2v + 1 components and use
—1 for the last component. Call the resulting vectors {e;};._¢. It will
not bother us if some E;(w) have nonzero Fourier coefficients with
index exceeding v. Taken together, the constraints {(y, e;) = 0}j-¢
amount to a statement that, for each admissible h, the squared
minimum distance is never larger than a candidate distance.

The optimal h is the one maximizing the minimum distance. So the
constrained y attaining max (y, 1,42) has the optimum design for its
first 2v + 1 coordinates and the optimal exponent for the last coordi-
nate.

In 2v + 2 space, the set of all y meeting constraints is denoted Y.
Y is not empty. For example, it contains 61,41, where 6 is a number
small enough that energy constraints are met. The optimization will
not degenerate as Y is closed and bounded. Y is closed since it is
expressible as the intersection of closed half-spaces. Y is bounded
since each component of y is bounded by the pulse energy constraint.
To see why, note that e = 1,4; shows y2,42 < 1. For the remaining
bounds on the components of y we note H(w) = ¥”, xm4i1+,e7" = 0
and so factorization is possible, H(w) = H(w) H*(w). Fourier coeffi-
cients (x1, x2, --- , Xey+1) are sums of products and so, by the Schwarz
inequality, y,+1 = x,4; bounds all the components of each y vector.

ples of interest, there are too many error events. For example, for v as
small as 4, the estimate in Section 3.1 indicated that there are over
10*° error events about which we should be concerned. The difficulty
of too many constraints may sometimes be handled by solving a
problem with a manageable number of the constraints. If it can be
verified that the optimum meets all constraints (not just the manage-
able ones), then the solution to the simplified problem is the same as
the solution to the difficult problem. We design, via an LP, a response
maximizing the minimum distance over some error events and then
seek to verify, using a tree search, that the minimum distance is not
reduced if one minimizes over all error events.

If the above procedure is unsuccessful, one can repeat it, enlarging
~ to include the minimizing error event revealed by the tree search.
Eventually, the iteration process will converge. Prior to convergence,
LP gives an upper bound while the tree search gives a lower bound to
the d2.in achievable by the optimum design.

The LP provides an h, while the tree search requires an h. The
minimum-phase deconvolution, fh, is suggested, since, among all h
satisfying h’«h = h, h has the greatest Y., h? for each k.!7"8 This
maximal frontal energy concentration expedites the tree search, which
operates first on the leading coordinates of h. (Orders of magnitude
of difference in running time have been observed between minimum-
and maximum-phase deconvolutions in the tree search algorithm.)

In estimating the performance of the optimum system employing
an h of memory », power levels were set as follows: For the FBS
system, as we mentioned in Section 6.2, the pulse energy was bounded
above by one. The noise level was set to meet the required P, in the
benchmark system operating at maximum power. Finally, the out-of-
band energy constraint was set to a fraction of the noise power.

The resulting gain versus v curves are shown in Fig. 9 with v as a
parameter. The OBE is constrained to 07/10 so a penalty of 0.414 =
10 log 11/10 is included in the gain calculation. It is apparent from
the curves that, even at extraordinary complexity (exceeding 10°
states) and a P, requirement of 107°, the resulting gains are very
modest. This conclusion is not sensitive to the exact premises under-
lying the computation. Calculation shows that, for v = 26, if we allow
6 to be larger than 1/10, the gains generally decrease because of the
OBE penalty. There is little to achieve by making OBE smaller than
o”/10 since the design is merely more constrained, and omitting the
OBE penalty cannot add more to the gain than 0.414 dB. When the
gain is positive, the ACI levels are generally within 0.25 dB of the
OBE level so the gains based on ACI are not different in any important

Fig. 9—Gain limit versus memory under spectral confinement constraint for P. =

way from those based on OBE. There is no point in showing curves
for P. < 107° in the benchmark system, as the gains can only decrease
if the design is further constrained.

Figure 9 has enabled us to determine the relative merit of FBS for
reasonable complexity. The possibility remains that, for extremely
large v, FBS could exhibit substantial gains and that these asymptotic
gains could improve as p increases.

Figure 9 was derived using a list of 50 error events obtained by
running the LP-tree search iteration for successive v values. To con-
clude that FBS offers at best a very modest improvement over QAM,
it is only necessary to present upper bounds in Fig. 9 rather than exact
maximum gains. However, in preparing Fig. 9, we established that it
is reasonable to exercise the LP-tree search algorithm to guarantee
the precise optimum gain that can be attained for up to 10° states. It

Fig. 10a—Example of an optimally designed transmitter characteristic and an error
event characteristic. Vertical dotted line delineates the band edge.

is interesting to note that optimum system design can be accomplished
for systems with such an enormous number of states.

Figure 10a illustrates an optimally designed spectrum and a corre-
sponding minimizing error event. Figure 10b illustrates an interesting
constrast between a pulse spectrum and an extremal error event.

For p = 5/4, the gain is only about 0.5 dB but a very interesting
behavior is observed. Namely, with little complexity, the maximum
distance possible is attained in the sense that the matched filter bound
is obtained. From Section III, eq. (2), we can write the following
expression for the gain (neglecting the OBE penalty):

where the function c(p, v) gives the fraction of the matched filter
energy attained. For fixed p, c(p, v) is a nondecreasing function of vp.

Fig. 10b—The in-band transmitter spectrum is optimized for a limited set of error
events, one of which is shown. The extraordinary flexibility afforded by over one million
states allows the optimum spectrum to have some peaks and valleys in opposition to
those of the minimizing error event.

that the answer is yes. For p > 3 consideration of the error transform
|1 — e|? shows that the answer is no, as

in the limit of stringent out-of-band energy constraints. In light of the
limited gains available with optimal FBS, it would be of only academic
interest to pinpoint the largest » value for which the matched filter
bound is attained as complexity is increased. Consequently, we shall
not pursue this question further.

At this point it is natural to question whether it is worthwhile to
generalize and consider Faster Multilevel Signaling (FMS). Motiva-

tion for considering FBS comes from Ref. 5, where the first theoretical
results on FBS were reported; from Ref. 3, where highly significant
benefits of FBS were suggested but not established; and from discus-
sions with J. Salz, who related that the idea of FBS has been around
for many years and that it is important to settle the question of its
‘merit. Since FBS proves to be unattractive, why should one consider
FMS, especially when we know that increasing the number of levels
toward that of the competing QAM system would seem to blur the
distinction? Can one generally discount the competitiveness of FMS?
We discuss why we cannot dismiss FMS and why, despite the findings
on FBS, FMS systems may have some value.

FBS fared poorly. If we look back on our analysis of FBS it is
obvious that it was the OBE constraint that drove the performance
level of FBS. We noticed in Section IV suboptimal pulses exhibiting
substantial gains that could not be realized because of prohibitive
OBE. The stringency of the OBE constraint was necessitated by the
substantial overlap of spectra between neighboring systems. As we
move away from binary toward more levels, in the class of FMS
systems, to compete with a fixed QAM system, the ratio p = W’/W>
1 decreases. The OBE constraint we need to impose is seen to be more
relaxed.

Moreover, as we decrease p, systems are represented for which the
ACI constraint is not of any direct importance. There is the interesting
class of questions pertaining to transmitter filter smoothness consid-
erations. For example, which performs better—a QAM system em-
ploying a square root raised cosine pulse with roll-off a = p — 1, or an
optimized FMS system with band-edge nulls of specified order and
with system memory v? The two systems are required to have the
same power and information rate. The answer, of course, depends on
M, a, v, and the degree of the band-edge null. The band-edge null is
useful for spectral confinement as well as for easing synthesizability.
The imposition of nulls of specific order at specific frequencies lead to
additional linear constraints and is easily handled by the LP-tree
search program.

The simple partial response 0, 1, 1, 0 can be used to illustrate that
there are situations where FMS can be very beneficial relative to
QAM. Among all systems required to have a band-edge null, the system
0, 1, 1, 0 requires the least number of states, m per rail. It is easily
shown that the response attains the matched filter bound independent
of m. Figure 11 shows a gain versus roll-off plot, which speaks for
itself concerning the substantial gains that are available in certain
cases.

Fig. 11—Gain versus rolloff characteristics for partial response 2~”? (0, 1, 1, 0),
where the approximate number of constellation ee are shown for competing QAM
system, and the number of levels equals the number of states per rail.

A class of examples where ACI is not of direct concern occurs with
the voiceband channel, which has severe band-edge attenuation. The
channel shapes are irregular mounds and there is no obvious spectral
support to assume. For a specific information rate, what is the best
baud to use if the transmitter spectrum has a null of given order and
zero rolloff? This is paraphrasal of the FMS issue coupled with a
simpler question of where to center the signal spectrum. (The trans-
mitter design must also account for the effect of nonlinearities and
the fact that the exact modulus of the channel transfer characteristic
is not known at the transmitter.)

Aside from the new information on FBS, a major finding of this
report is that, for the class of MLSD systems considered, optimum
designs can be accomplished involving numbers of states correspond-
ing to the capabilities of forthcoming MLSD implementation technol-
ogy (and far beyond). We have concentrated here on binary systems
and a very special channel. However, the algorithm extends to apply
to designing optimum m-ary systems of prescribed complexity oper-
ating over arbitrary linear dispersive channels. The astronomical
number of error events is not an obstacle.

The extended algorithm, now being programmed in the course of
joint work with G. Vannucci, will provide a basic tool for probing the
fundamental relationship between attainable rates and system com-
plexity for very general systems. Suppose one wants to achieve a
certain information rate, under spectral confinement requirements
and with a specific level of complexity. By exercising the LP-tree
search algorithm for a sufficient number of p values, one can locate
Popt, the optimum p 2 1, and the associated optimum gain over a
corresponding QAM system with cosine rolloff spectral shaping.

It is a pleasure to acknowledge numerous valuable, stimulating
discussions with G. Vannucci. These discussions involved both the
theoretical and software aspects of this work. L. J. Domenico’s assist-
ance with the programming is greatly appreciated.

1. A. Gersho and V. B. Lawrence, “Multidimensional Signal Design for Digital Trans-
sein hts Bandlimited Channels,” Proc. IEEE ICC, 1, Amsterdam, May 1984,
pp. 377-80.

2. G. Ungerboeck, “Channel Coding with Multilevel/Phase Signals,” IEEE Trans.
Inform. Theory, /T-23, No. 1 (January 1982), pp. 55-67.

3. A. S. Acampora, “Analysis of Maximum Likelihood Sequence Estimation Perform-
ance for Quadrature Amplitude Modulation,” B.S.T.J., 60, No. 6 (July-August
1981), pp. 865-85.

. G. D. Forney, Jr., “Lower Bounds on Error Probability in the Presence of Large
Intersymbol Interference,” IEEE Trans. Commun., COM-20, No. 1 (February
1972), pp. 76-7.

7. G. D. Forney, Jr., “Maximum-Likelihood Sequence Estimation of Digital Sequences
in the Presence of Intersymbol Interference,” IEEE Trans. Inform. Theory, /T-
18 (May 1972), pp. 363-78.

8. G. J. Foschini, “Performance Bound for Maximum Likelihood Reception of Digital
Data,” IEEE Trans. Inform. Theory, [T-21 (January 1975), pp. 47-50.

9. G. J. Foschini, “A Reduced State Variant of MLSD Attaining Optimum Perform-
ance for High SNR,” IEEE Trans. Inform. Theory, /T-23, No. 5 (September
1977), pp. 605-9.

10. A. J. Viterbi and J. K. Omura, Principles of Digital Communication and Coding,
New York: McGraw Hill, 1979.

11. G.J. Foschini, R. D. Gitlin, and S. B. Weinstein, “Optimization of Two Dimensional
Signal Constellations in the Presence of Gaussian Noise,” IEEE Trans. Commun.,
COM-22, No. 1 (January 1974), pp. 28-38.

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

13. R. R. Anderson and G. J. Foschini, “The Minimum Distance for MLSE Digital
Data Systems of Limited Complexity,” IEEE Trans. Inform. Theory, IT-21, No.
5 (September 1975), pp. 544-51.

15. D. Slepian, “Prolate Spheroidal Wave Functions, Fourier Analysis, and Uncer-
oe ae The Discrete Case,” B.S.T.J., 57, No. 5 (May-June 1978), pp. 1371-
430.

16. A. D. Wyner, “Signal Design for PAM Data Transmission to Minimize Excess
Bandwidth,” B.S.T.J., 57, No. 9 (November 1978), pp. 3277-307.

18. H. P. McKean and H. Dym, Fourier Series and Integrals, New York: Academic
Press, 1972, pp. 175-6.

Ai, As, As, --+, Ax++1, Where the states A; are defined as the
successive v-tuples of the sequence representation, where the all-zero
y-tuples are omitted, except for the v-tuple abutting ex.
(0, 0, ee 0, €0), (0, 0, sry €0, é1);

A.3 Augmented state representation
Aj, Ag, Ag, --- Ak+41, where the augmented states Aj are defined
as the successive (v + 1)-tuples of the sequence representation

where h® 4 (h,, h,-1, --- , ho) and the inner product is defined in the
usual way. The b superscript is read “backward” and the b operation
is also applied to error events in the memorandum.

The error sequence maps to the nonnegative cosine polynomial
E(w) = |e + ae + --- + exeR |?

We shall refer to each e as an error sequence, error event, or error
pattern. Let H(w) 4 | ho + hie + --- + h,e’””’|? and define (E, H)
1/(27) f2, E(w)H(w) dw. Then by Parseval’s theorem, (E, H)
||e+h||?, so we have

We present some observations other than the four symmetry con-
ditions that are useful for efficient calculation of d?,;, and a minimizing
error event e for a symmetric transmitter impulse response.

Let {S;}2. denote the resulting sequence of scalar products in hee.
Let K be the smallest integer satisfying sjsx ~ 0 with s, = 0 fork ¢
{1, 2, --- K}. If, in the course of searching the tree, an error event
breaking the current record, A, is encountered, then, for it to be a
minimizing event, we must have

Let [K/2] denote the largest integer less than or equal to K/2. For a
record breaking error event or that error event in reverse (or both) we
must have

Since || h*e||? = || h*(+e°) ||? error events for which it is established
that the inequality (5) is reversed need not be explored further in the
tree search for d2,;,.

To expedite the process of seeking d?,;, among error sequences for
which (5) holds, we discuss the calculation of the height at which the
exploration of the growth of nodes terminates. Let L be the first
integer for which the accumulated sum of s? + s§ + +--+ Si4,; > A/2
so that s? + so + --- si < A/2. Clearly, L = [K/2] so2L+1>K.
Once L is detected there is no need to explore any events involving
2L + 2 scalar products. To put it another way, if L’ = L + 1 is the
first index for which A/2 is exceeded, then s2z, = 0 and 2L’ > K.

It is not always necessary to search to height 2L’ to terminate
growth exploration. To see this note that from the height, 7’, of
occurrence of the last nonzero element in the event under exploration
we have that K > ” + v. So once L’ is determined exploration of
growth is terminated if 7 + vy > 2L’.

Gerard J. Foschini, B.S.E.E., 1961, Newark College of Engineering, Newark,
NJ; M.E.E., 1963, New York University, New York; Ph.D., 1967 (Mathemat-
ics), Steven Institute of Technology, Hoboken, NJ; AT&T Bell Laboratories,
1961—. At AT&T Bell Laboratories Mr. Foschini initially worked on real-
time program design. For many years he worked in the area of communication
theory. In the spring of 1979 he taught at Princeton University. Mr. Foschini
has supervised planning the architecture of data communications networks.
Currently, he is involved with digital radio research. Member, Sigma Xi,
Mathematical Association of America, IEEE, New York Academy of Sciences.

Adaptive Transversal Equalization of Muliipatn
Propagation for 16-QAM, 90-Mb/s Digital Radio

Adaptive transversal equalization is an effective and relatively new coun-
termeasure for dispersive multipath propagation in terrestrial digital radio
networks. In this paper we describe the design and performance of a five-tap
baseband analog equalizer developed for a family of 16-QAM, 90-Mb/s radio
systems. Laboratory measurements and field evaluation during a five-month
fading season in Palmetto, Georgia, indicate that the use of this adaptive
transversal equalizer can significantly reduce the need for costly space-diver-
sity equipment.

The impairment of terrestrial digital microwave reliability due to
multipath propagation is widely recognized.' Unlike FM radio systems,
where system outage is predominantly determined by the thermal
noise aspect of fading, digital radio is also affected by the dispersive
character of multipath fading. This dispersion, engendered by signifi-
cant amplitude and delay distortion across the channel bandwidth,
causes considerable Intersymbol Interference (ISI) that degrades dig-
ital radio reliability well beyond that expected from the flat fade
margin alone.” Multipath-induced distortion thus is considered the
predominant cause of digital radio outage for frequencies under 12
GHz.

tipath fading. These include frequency diversity,® space diversity,* and
adaptive Intermediate Frequency (IF) equalization. Examples of the
latter include slope equalizers® and notch or resonance equalizers.®
However, frequency-selective fading corrupts both the amplitude and
phase of a transmitted signal. While IF equalizers can be designed to
condition a channel properly for minimum phase fading, they double
the delay distortion during periods of nonminimum phase fading.
(Minimum phase and nonminimum phase fading is clearly defined by
Giger and Barnett’ for a two-path statistical model of multipath
propagation.) This effect naturally impacts the outage of those digital
radio systems that rely solely on amplitude correction.’

Although adaptive transversal equalizers are a relatively new coun-
termeasure to multipath fading in digital radio systems,°® their prior
application in mitigating the effects of linear distortion in other, lower-
speed, digital communication networks is firmly established. Current
practice is to use transversal equalizers in conjunction with IF equal-
izers. In a recent study, Foschini and Salz® considered the application
of equalization techniques to digital data transmission over radio
channels subject to frequency-selective fading. Their theoretical study
of ideal transversal equalizers with an infinite number of taps clearly
established the utility of linear equalization during multipath propa-
gation. These equalizers are especially noteworthy in that they are
capable of providing amplitude and delay equalization for minimum
and nonminimum phase fades.

The baseband synchronous transversal equalizer briefly described
here was designed for a family of 16-QAM (Quadrature Amplitude
Modulation), 90-Mb/s radio systems. Designated DR 6-30 and DR 11-
40, these digital systems provide 3-bit/Hz operation in the 6- and 11-
GHz common carrier bands, respectively.’° In this paper, we focus on
the design and performance features of a high-speed (approximately
22.5-MHz) synchronous transversal equalizer. A theoretical develop-
ment of equalization principles is specifically omitted since those
points are amply discussed in the technical literature (for example,
see Chapter 6 of Ref. 11).

Figure 1 functionally depicts the DR 6/DR 11 digital radio system.
Two independent, 45-Mb/s random data streams are differentially
encoded to form two rails, each with four-level amplitude states, and
then modulated in quadrature to form a 16-QAM, 90-Mb/s IF signal
at 70 MHz. The Radio-Frequency (RF) transmitter modulates the IF
signal up to 6 or 11 GHz for transmission over a line-of-sight terrestrial
path to the digital receiver. At the receiver the signal is down-converted

to IF, where it is processed by an Automatic Gain Control (AGC)
amplifier and adaptive slope equalizer. The baseband receiver de-
modulates the IF signal into in-phase (I) and quadrature (Q) rails,
where the baseband data states are detected and estimates of the
original transmitted data are made. An error detector provides for
system performance monitoring.

baseband equalization, the transmitted symbol states are estimated at
the decision point, and the decoded binary signals are used in subse-
quent digital processing.

To remove in-rail and cross-rail intersymbol distortion, two adaptive
transversal filters (each with five complex-valued tap weights) are
configured for baseband equalization of QAM signals. The selection
of five taps is based on theoretical studies of equalizer performance as
a function of equalizer length. For example, Amitay and Greenstein’?
have investigated the multipath outage performance of digital radio
receivers using finite-length adaptive equalizers. Using Rummler’s
statistical description of multipath channels,’® equalizer performance
for a broad ensemble of fading scenarios was simulated. Their study
indicated that five synchronous taps considerably reduce ISI relative
to performance attained with three taps and that equalizers with seven
or more taps, while obviously further reducing ISI, exhibit a rapidly
diminishing relative reduction in linear distortion. (Independently,
Murase et al.,/4 and Takenaka et al.!° have also selected five-tap filters
for their transversal equalizer designs.)

The equalizer tapped-delay lines are fabricated using lumped-delay
elements isolated with buffer amplifiers. The buffer amplifiers are
Hybrid Integrated Circuits (HICs) and provide high isolation between
the delay line and coefficient-weighting taps. Tap weighting is accom-
plished with variable gain amplifiers. These, too, are hybrid integrated
circuits fabricated in single in-line packages, thereby permitting high-
density electronics on each circuit board. Summing amplifiers (also
HICs) then add the individual tap-weighted signals to form the filter
output.

The coefficient control portion of the equalizer uses zero-forcing
adaptation and is implemented with high-speed Emitter-Coupled
Logic (ECL). The control circuit accepts error polarity and estimated-
symbol polarity from the in-phase and quadrature decision circuits.
Appropriately delayed versions of these bits are then correlated during
each symbol period using exclusive OR gates. The time-averaged
values of these correlations determine the weight of each tap in the
two transversal filters. Time averaging is achieved using operational
amplifier filters optimized for the trade-off between coefficient noise
and dynamic multipath tracking ability.

The entire equalizer consists of three 1-inch plug-in circuit packs in
a format compatible with the DR 6/DR 11 terminal or regenerative
equipment. A photograph of these circuit packs appears in Fig. 3. Two
of these packs are identical analog transversal equalizers, one for in-
phase and cross-rail equalization of the I rail, the other for similar

Fig. 3—Adaptive transversal equalizer consisting of two transversal filter circuit
packs and one zero-forcing control circuit pack.

equalization of the Q rail. Equalizer coefficient control is generated in
the third circuit package.

Il. EQUALIZER PERFORMANCE
3.1 Theoretical performance
3.1.1 Reduction of peak distortion

As we noted above, five-tap synchronous transversal equalizers are
theoretically capable of substantially reducing intersymbol interfer-

ence caused by frequency-selective fading. For zero-forcing coefficient
adaptation, the peak distortion, D,, of the corrupted digital signal is
minimized."’ (As used here, D, is equivalent to Peak Eye Closure
(PEC) for binary transmission.) Representative theoretical perform-
ance is illustrated in Fig. 4. In Fig. 4 we consider one digital rail (I or
Q) and show the variation of peak distortion—with and without
equalization—of a digital signal for a 20-dB fade notch depth as a
function of notch position in a +18-MHz channel. (Ideally, distortion
in the other rail would be identical.) The ordinate on the right side of
this figure provides the corresponding peak eye closure for a four-level
signal, given by PEC = D,(L — 1), where L is the number of discrete
transmitted symbol states on each rail. This illustrative fade grossly
closes the digital eye with D, > 1 over at least a portion of the channel

FREQUENCY FROM CARRIER IN MEGAHERTZ
Fig. 4—Theoretical reduction in peak distortion, D,, with a five-tap QAM transversal

equalizer arrangement. Peak eye closure is noted for four-level transmission, that is,
one rail of a 16-QAM system.

bandwidth. This latter condition highlights an analytic limitation of
zero-forcing equalization: if D, > 1, the coefficient set may be subop-
timal.!’ In spite of this, other analysis (to be discussed shortly) and
our own measured data show that adaptive transversal equalizers do,
in fact, notably reduce intersymbol interference in just such an envi-
ronment. Moreover, zero-forcing is known to assure a global minimum
if D, < 1, affords comparative ease of circuit realization, and minimizes
Bit Error Rate (BER) in the high signal-to-noise ratio that typifies
quiescent digital radio operation. The other dominant adaptation
approach for automatic equalizers, Least-Mean-Square (LMS) algo-
rithmic control, is more difficult to realize in high-speed circuits and
has a proclivity for unsatisfactory local minima when used in the
decision-directed mode.*®

of equalization capability since they can be directly related to outage
predictions for digital radio systems. The signatures are 10°? BER
contours: at each point on the contour, the fade notch depth corre-
sponding to a 10-° BER (defined as a digital radio outage) is specified
as a function of notch position for a fixed-delay statistical model of
multipath propagation. Figure 5 presents theoretical signatures com-
puted by M. H. Meyers’? for no equalization, slope equalization, and
transversal equalization. Figures 4 and 5 confirm that five-tap trans-
versal equalizers theoretically provide a significant reduction in linear
distortion. Indeed, even the use of zero-forcing control for fades with
D, > 1 yields a degree of equalization that mitigates digital radio
outage. The data of Fig. 5 indicate that a fade notch depth as shallow
as 7 dB can cause an outage in the absence of countermeasures. When
the radio receiver is equipped with a transversal equalizer, outages are
not experienced until the notch depth reaches approximately 23 dB,
which can occur for an unequalized D, > 1, as shown in Fig. 4. Also
observe from Fig. 5 that slope equalizer performance is influenced by
the minimum or nonminimum phase character of the fade, as we
mentioned earlier. This is not a limitation of transversal equalization.

The definition and significance of equipment signatures were pre-
viously mentioned. The laboratory measurement of these signatures
is facilitated through the use of a new computer-controlled multipath
fade simulator that continuously varies notch depth and notch fre-
quency to achieve a 10°? BER. The simulator is inserted in the IF
path of the receiver just before the AGC amplifier (see Fig. 1).

Fig. 5—DR 6 theoretical equipment signatures for 16-QAM digital radio. Performance
for radio without equalization, with an adaptive slope equalizer, and with a five-tap
synchronous transversal equalizer using zero-forcing control.

Signatures were measured using two DR 6 receivers, the first
equipped with an adaptive slope equalizer (the standard arrangement)
and the second equipped with both the adaptive slope equalizer and a
five-tap adaptive transversal equalizer. The 10? BER minimum phase
and nonminimum phase equipment signatures appear in Fig. 6. As the
data reveal, the adaptive slope equalizer performs best when used for
minimum phase fades, with a performance deterioration experienced
for nonminimum phase fades. We commented earlier that IF equalizers
typically double delay distortion during nonminimum phase fading,
and this effect can degrade equipment signature performance. The
same effect naturally occurs when the adaptive slope and synchronous
transversal equalizers are used together. Comparing both sets of
curves, however, we also observe the significant improvement in
equipment signature performance that can be ascribed to the trans-
versal equalizer alone.

Fig. 6—DR 6 measured equipment signatures for 16-QAM digital radio. Performance
for radio with adaptive slope equalization and adaptive slope and transversal equaliza-
tion for 6.3-ns path delay. (Adapted from Ref. 20.)

are compared. The predicted relative outage reduction factor, derived
from theoretical equipment signatures for combined ideal slope and
transversal equalization (see Fig. 5) is 5. This assumes equally probable
minimum and nonminimum phase fading. The predicted relative
outage reduction factor for the measured equipment signatures (see
Fig. 6) is 4.5, again assuming equally probable minimum and non-
minimum phase fading. The relative reduction factors for other ratios
of minimum to nonminimum phase fading range from 4 to 5. The
measurements in Fig. 6 attest to the quality of the transversal equalizer
circuit design itself. Regarding this point, baseband implementation
of the equalizer permits integration of substantial portions of the
circuitry, thus simplifying design and manufacture. The development
of new carrier and timing recovery circuits also helps place laboratory
performance close to the theoretical limit shown in Fig. 5.

An important aspect of multipath propagation is its rapid temporal
variation. To assure optimal equipment performance in the field,
dynamic (time-varying) tests were performed during the development
phase. Dynamic multipath fading is produced in the laboratory by

controlling the continuously variable fade simulator with a microcom-
puter. Realistic time sequences of multipath behavior were pro-
grammed into the simulator. Equalizer performance was monitored
during the simulation of these dynamic fades, thereby permitting
optimization of the equalizer timing-recovery, carrier-recovery, and
coefficient-updating loop parameters.

Several aspects of an equalizer’s response to dynamic multipath
propagation are exercised with the following test sequence (schemat-
ically depicted in Fig. 7): starting with a shallow fade notch depth d,
at a particular notch frequency f,, the notch depth increases at a rate
$; until a notch depth d, is reached. The notch then sweeps across a
band of frequencies from f; to f. at a rate so. At the notch frequency
fe, the notch depth decreases from dz back to d,; at a rate s3. This
fading trajectory retraces itself and is repeated several times for
statistical averaging of the receiver’s error performance. A test se-
quence like this tests the receiver’s ability to track notch depth and
notch frequency dynamics. For trajectory parameters of d; = 6 db, de
= 15 dB, s; = s3 = 9 dB/s, f, = —12 MHz (12 MHz below the IF
frequency), fe = +12 MHz, and s, = 24 MHz/s, the transversal equalizer
consistently operates error free. Those test velocities are also faster
than 90 percent of all observed notch depth and notch position rates
of change reported by Sakagami et al.””

The adaptive transversal equalizer was installed in a DR 6-30 field
test facility at Palmetto, Georgia, on June 4, 1982. This modified

baseband receiver was compared with a standard DR 6 receiver
(equipped with an adaptive slope equalizer) during a multipath season
from June 6 to November 6, 1982. Propagation data collected during
the field evaluation period are shown in Fig. 8.”° The abscissa of this
figure reports fade notch depth; the ordinate indicates time faded
below the respective abscissa value. A considerable amount of fading
exhibits notch depths in excess of 10 dB, which, from Fig. 5, could
correspond to an outage in the absence of suitable countermeasures.
The two baseband receivers shared the same RF and IF front ends.
Field measurements, monitored by AT&T Bell Laboratories personnel

from Merrimack Valley, were grouped into 11 two-week intervals. In
Fig. 9, the number of seconds for which BER > 10° is presented for
both receivers for each of the two-week intervals. Also presented is
the ratio of these two time measurements, representing a composite
improvement factor attributable to the transversal equalizer, alone.
Figure 10 presents similar data for a BER > 10~*. In Fig. 11 we show
the incidence of frame loss with and without the equalizer, as well as
the corresponding reduction in loss of frame.

transversal equalizer provided composite improvement factors of 3.6
for BER > 10~°, 3.7 for BER > 10~+, and 2.9 for frame loss. The 107*
BER improvement factor of 3.7 is only 20 percent below the predicted
improvement factor of 4.5, based on laboratory-measured equipment
signatures.

Because of their ability to adaptively equalize multipath-induced
amplitude and delay distortion for minimum and nonminimum phase

fading, synchronous transversal equalizers promise to play an impor-
tant role as a multipath countermeasure for terrestrial digital micro-
wave networks.

In this paper we summarize the major design and performance
features of a five-tap analog transversal equalizer for the baseband
receivers of two 16-QAM, 90-Mb/s digital radio systems. The equal-
izers heavily rely on HIC technology for their tapped-delay line buffer
amplifiers, tap-weighting coefficients, and summing circuitry. The

zero-forcing adaptation portion of the equalizer is realized with high-
speed ECL logic. The entire equalizer is packaged in three 1-inch plug-
in circuit packs.

During design, the equalizer was tested for its static equipment
signature performance and dynamic tracking capability. The latter
evaluation was facilitated with a special-purpose, computer-controlled
multipath fade simulator. During a 22-week field trial evaluation in
Palmetto, Georgia, the equalizer reduced the overall incidence of DR
6-30 radio outage by more than a factor of 3. System estimates indicate
that this improvement factor could eliminate the need for space-
diversity reception on more than 50 percent of the short-haul digital
radio hops that currently use it. Use of the baseband adaptive trans-
versal equalizer thus can provide considerable cost savings.

Completion of this project required the synergistic efforts of many
individuals. The authors are pleased to acknowledge the following
contributors: J. S. Bitler, for designing many of the analog integrated
circuits; G. L. Frazer, for special insights into carrier and timing
recovery; M. H. Meyers, for theoretical BER and equipment signature
calculations; K. L. Seastrand, for initially proposing transversal equal-
ization in our digital radio systems; M. A. Skinner, for overseeing the
project during its development phase; and R. B. Ward, for contribu-
tions to the carrier recovery design.

12. N. Amitay and L. J. Greenstein, “Multipath Outage Performance of Digital Radio
Receivers Using Finite-Tap Adaptive Equalizers,” NATO/AGARD 33rd Sympo-

sium itd Electromagnetic Wave Propagation Panel, Spatind, Norway, October
4-7, 1983.

13. W. D. Rummler, “A New Selective Fading Model: Application to Propagation Data,”
B.S.T.J., 59, No. 5 (May-June 1979), pp. 1037-71.

14. T. Murase et al., “200 Mb/s 16-QAM Digital Radio System With New Countermea-
sure Techniques for Multipath Fading,” ICC ’81 (June 1981), pp. 46.1.1-5.

15. S. Takenaka et al., “A Transversal Fading Equalizer for a 16-QAM Microwave
Digital Radio,” ICC ’81 (June 1981), pp. 46.2.1-5.

16. J. E. Mazo, “Analysis of Decision-Directed Equalizer Convergence,” B.S.T.J., 59,
No. 10 (December 1980), pp. 1857-76.

17. M. Emshwiller, “Characterization of the Performance of PSK Digital Radio Trans-
mission in the Presence of Multipath Fading,” ICC’78 (June 1978), pp. 47.3.2-6.

18. C. W. Lundgren and W. D. Rummler, “Digital Radio Outage Due to Selective
Fading—Observation vs. Prediction From Laboratory Simulation,” B.S.TWJ., 58,
No. 5 (May-June 1979), pp. 1073-100.

20. C. P. Bates and M. A. Skinner, “Impact of Technology on High-Capacity Digital
Radio Systems,” ICC’83 (June 1983), pp. F2.3.1-5.

22. S. Sakagami et al., “Inband Amplitude Dispersion Characteristics During Multipath
Fading on Microwave Links,” Rev. Elec. Commun. Lab., 29 (November-December
1981), pp. 1295-303.

Gerald L. Fenderson, B.S.E.E., 1960, University of Maine; M.S.E.E., 1963,
Northeastern University; AT&T Bell Laboratories, 1960—. Since joining
AT&T Bell Laboratories, Mr. Fenderson has participated in the development
of FM and digital microwave radio systems. His most recent responsibility
has been in the design of digital modems, including adaptive transversal
equalization. Mr. Fenderson received an AT&T Bell Laboratories Distin-
guished Technical Staff Award in December 1982. Member, Tau Beta Pi, Eta
Kappa Nu.

James W. Parker, A.S.M.E.T., 1978, Vermont Technical College; AT&T
Bell Laboratories, 1978—. Mr. Parker has worked on a variety of physical
design projects for digital radio systems and is currently involved in the design

and development of advanced and international radio bays. Member, Tau
Alpha Pi.

Patrick D. Quigley, A.S.E.E.T., 1978, S.U.N.Y. at Alfred; B.S.E.E.T. 1983,
University of Lowell, Lowell, MA; AT&T Bell Laboratories, 1978—. Mr.
Quigley has worked on analog, digital, and firmware development projects for
the AR6 and DR6 radio systems. He is presently a Member of Technical Staff
in the Microwave Radio Systems department, working on the development of
a digital monitoring receiver for advanced digital radio systems.

Scott R. Shepard, B.S. (Electrical Engineering) and B.S. (Physics), 1979,
Kansas State University; M.S. (Electrical Engineering), 1981, The Massachu-
setts Institute of Technology; AT&T Bell Laboratories, 1979—. While at
K.S.U., Mr. Shepard was employed as a research assistant for various problems
in electromagnetic theory, including the design and construction of optical
logic devices and the focusing of charged particles in a linear accelerator. His
master’s research involved the analysis of a nonlinear optical pulse-shaping
device by means of symbolic manipulation software at M.I.T.’s artificial

intelligence laboratory. Mr. Shepard’s major responsibilities at AT&T Bell
Laboratories have been the design of the adaptive transversal equalizer and
the design of the dynamic multipath fade simulator. Currently, in addition to
channel modeling and channel conditioning, he is involved in high-speed,
high-accuracy, analog-to-digital and digital-to-analog signal conversion. Mem-
ber, IEEE, Eta Kappa Nu, Sigma Pi Sigma, Tau Beta Pi, Phi Kappa Phi, and
Sigma Xi.

Curtis A. Siller, Jr., B.S.E.E., 1966, M.S. (Plasma Physics), 1967, Ph.D.
(Electrical Engineering), 1969, The University of Tennessee, at Knoxville;
AT&T Bell Laboratories, 1969-1978, 1979—. Mr. Siller’s earliest experience
at AT&T Bell Laboratories was in the analysis and design of reflector antennas
for terrestrial microwave communications. He subsequently initiated an ex-
ploratory investigation of adaptive transversal equalization for advanced dig-
ital radio systems. His more recent research interests were in digital signal
processing, particularly equalizer control algorithms and new techniques for
digital filtering. Mr. Siller is presently involved in system engineering of future
digital transmission systems. Mr. Siller is the recipient of an AT&T Bell
Laboratories Distinguished Technical Staff Award. Member, Phi Eta Sigma,
Eta Kappa Nu, Tau Beta Pi, Phi Kappa Phi, Sigma Xi, American Association
for the Advancement of Science, and the IEEE, where he is a Senior Member
and serves on the Signal Processing and Communication Electronics Technical
Committee.

Adaptive Differential Pulse Code Modulation (ADPCM) systems can pro-
vide high-quality digitizations of telephone-bandwidth speech at a bit rate of
32 kb/s. At a lower bit rate such as 24 kb/s, the quality of the speech is limited
by an easily perceptible level of quantization noise. This paper proposes an
adaptive postfiltering procedure that can significantly enhance the quality of
lower bit rate ADPCM. The coefficients of the postfilter are easily derivable
from the predictor coefficients in the ADPCM decoder. In a subjective test
involving 18 listeners and two sentence-length test inputs, the enhanced
24-kb/s speech with an optimized postfilter design ranks very close to conven-
tional 32-kb/s speech. A suggested application of the postfiltering procedure
is in packet voice or mobile radio systems where substandard bit rates such as
24 kb/s or 16 kb/s are sometimes necessary. The postfiltering algorithm has
also been successfully tested in non-DPCM situations, such as in the enhance-
ment of speech degraded by additive white Gaussian noise.

Recent algorithms for adaptive prediction’ and adaptive
quantization? have led to the realization of high-quality ADPCM
systems at 32 kb/s. This bit rate is the result of 8-kHz sampling and
quantization using 4 bits/sample. The quality of 24-kb/s speech using
the same prediction algorithm and 3-bits/sample coding is limited by
a clearly perceptible level of quantization noise. This paper proposes
a very simply implemented postfiltering algorithm, which provides a
significant enhancement of 24-kb/s quality. In a subjective test to be

A natural application of the postfiltering procedure would be in
variable bit rate ADPCM systems such as packet networks or mobile
radio where substandard bit rates such as 3 or 2 bits/sample are
occasionally encountered. The postfiltering technique described in this
paper is particularly effective at the bit rate of 3 bits/sample. It is also
effective in non-DPCM situations such as in the enhancement of
speech degraded by additive white Gaussian noise. When the signal-
to-noise ratio (s/n) at the input to the postfilter is too low (as in 2-bit
ADPCM or with white Gaussian noise at a relative noise level exceed-
ing approximately —3 dB), noise suppression can only be achieved at
the expense of severe distortion of the speech signal itself. When the
24-kb/s ADPCM is enhanced, the introduction of speech distortion is
perceptible, but the effect of noise reduction is by far the more
dominant phenomenon.

The philosophy of the postfiltering technique is represented in Fig.
1. Part (a) of the figure shows a signal spectrum with two narrowband
components in the frequency regions W, and Ws, and a flat noise
spectrum that is 15 dB below the first signal component but 5 dB
above the second signal component. An ideal postfilter for this situa-
tion would have a gain of unity (0 dB) in the regions W; and W2 and
a gain of zero (— © dB) in the rest of the frequency range. In real
speech applications, implementation of such all-or-none responses is
impractical except in the special cases where the stopband regions of
the postfilter are merged into a single contiguous frequency region as
in a low-pass or high-pass postfilter.**

A more practical approach, proposed in this paper, is the use of a
postfilter frequency response that peaks in the regions W; and Wo,
but is significantly lower in the rest of the frequency range. Figure 1b
illustrates an extreme example of this approach. Here, the transfer
function of the postfilter is chosen to be identical to the input signal
spectrum in Fig. la. The resulting spectra of postfiltered signal and
postfiltered noise preserve the original signal-to-noise ratios of 15 dB
and —5 dB in the regions W, and W2, respectively. However, the noise
in the rest of the illustrated frequency range is now much lower,
relative to the signal levels, than in part (a) of the figure. Specifically,
the signal-to-background-noise ratios for regions W, and W2 are now
45 dB and 10 dB, in place of 15 dB and —5 dB in the absence of
postfiltering. This suppression of background noise also implies that

Fig. 1—An idealized explanation of the effects of postfiltering, assuming a signal
with two narrowband components and a noise spectrum that is white. (a) Signal and
noise spectra at the input to the postfilter, showing signal-to-noise ratios of 15 dB and
—5 dB in signal frequency bands W, and W.. (b) Spectra of postfiltered signal and
postfiltered noise, assuming a postfilter transfer function identical to the signal spectrum
in (a). Regions W; and W, continue to have local signal-to-noise ratios of 15 dB and —5
dB as in (a), but the signals are now 45 dB and 10 dB above the out-of-band noise level.
In (a) the corresponding numbers are only 15 dB and —5 dB. The overall effect is a
reduction of perceived noise, but the price paid is a change in the relative strengths of
the signal components in W, and W2.

the residual noise spectrum after postfiltering is very similar to the
input signal spectrum itself. In speech applications, noise that is
shaped in this manner tends to be perceived as speech.

In the applications discussed, the technique realizes a broad range of
postfilter design over which the phenomenon of noise suppression
dominates the phenomenon of signal distortion.

Although speech enhancement’ is an “ancient” art, we believe that
the adaptive postfiltering technique discussed in the next section is
novel. It can be used as a very general technique for speech enhance-
ment. It can also be used very effectively in the specific context of
ADPCM noise. The coefficients of the proposed postfilter are inspired
by the coefficients of the adaptive predictor in ADPCM coding, and
are in fact very closely related to these coefficients.

Figures 2 and 3 provide block diagram descriptions of ADPCM with
adaptive postfiltering.

Figure 2 shows the decoder part of the system. Broken lines in the
figure refer to parts of the system that compute the coefficients of the
adaptive predictor and the adaptive postfilter. The coefficients used
in the postfilter are differently scaled versions of the coefficients used
in the adaptive predictor. These coefficients are already available in
conventional ADPCM. In the case of a system with Backward-Adap-
tive Prediction (APB), the predictor coefficients are updated in gra-
dient-search algorithms driven by a recent history of the input and
output of the ADPCM decoder.

A more complete block diagram of the ADPCM system appears in
Fig. 3. The quantizer and predictor assumed in this paper are both

INPUT TO CONVENTIONAL ENHANCED
ADPCM ADPCM ADPCM
DECODER OUTPUT OUTPUT
/ \ N
/ \ \

Fig. 2—Adaptive postfiltering of the output of an ADPCM decoder. The coefficients
of the postfilter are scaled versions of the coefficients of the adaptive predictor in
DPCM. In DPCM-APB, the predictor coefficients are obtained on the basis of obser-
vations of a recent history of decoder input and decoder output. The parts of the circuit
that determine coefficient values are shown by broken lines.

backward-adaptive devices, implying that no special side information
needs to be explicitly transmitted to the ADPCM decoder to enable
adaptations of quantizer step size and predictor coefficients.

This adaptive predictor we assumed is a pole-zero predictor, similar
to that in Ref. 1. As Fig. (8) shows, the predicted value x(n) of input
x(n) is a combination of two components, the outputs £,(n) and x,(n)
of an all-zero predictor B(z) and an all-pole predictor A(z). Formally,

Fig. 3—Complete block diagram of an ADPCM system with a pole-zero predictor
[defined by all-zero and all-pole components A(z) and B(z)] and a pole-zero postfilter
[defined by components A’(z) and B’(z) that are derived from A(z) and B(z)]. The
extreme case of A’(z) = B’(z) = 0 results in conventional ADPCM without postfiltering.
The case of A’(z) = A(z) and B’(z) = B(z) results in a postfilter transfer function that
is identical to the input signal spectrum, as in Fig. 1b.

where u(n) is the quantized version Q[-] of the prediction error and
y(n) is the reconstructed output:

Adaptation of the predictor coefficients a; and 6; follow the updating
algorithms’

The coefficients of the all-pole predictor are further controlled, for
stability reasons, by the following constraints:

A good starting point for designing the postfilter is the frequency
response of the inverse predictor. This is the system whose input and
output are the innovations u(n) and the reconstruction y(n). Its
transfer function, derivable from linear equations that relate u(n),
x(n), and y(n), is

2 6
A(z) = Yaz?; Biz) = Y Bz”.
j=l j=l
The speech-like transfer function of Fig. 1b is approximated if the
postfilter response is identical to the function [Y(z)]/[U(z)]. This is
because the spectrum of the quantized innovations u(n) is approxi-
mately white and that of the reconstruction y(n) is hopefully an
approximation to that of the input x(n). More generally, as in Fig. 3,
we propose a postfilter transfer function

The extreme situation of Fig. 1b is approximated when a = 6 = 1.
In practice, this approximation can be quite poor because of the effects
of nonideal predictor adaptation, usually resulting in an inverse pre-
dictor transfer function that is a flattened version of the input speech
spectrum, with poles and zeros that may also be significantly shifted
from their original locations. The case of a = 6 = 0 corresponds to
conventional ADPCM without any postfiltering. As we discuss in the
next section, intermediate designs provide different mixes of noise
suppression and speech distortion.

Figure 4 shows an illustrative spectrum of input speech and com-
pares it with the transfer functions F(z) for (a = 0.2; 8 = 1.0) and
(a = 1.0; 8 = 1.0). The latter condition simply corresponds to the
transfer function of the inverse predictor.

IV. EXPERIMENTAL RESULTS WITH ADPCM SPEECH
The speech inputs used in the experiment were the sentence-length

Fig. 4—(a) Input speech spectrum; and power transfer functions of postfilter with
scaling coefficients for (b) a = 0.2, 8 = 1.0, and for (c) a = 1.0, 6 = 1.0. The plot (c) is
merely the transfer function of the inverse predictor in the DPCM-APB system. [The
0-dB line is the same for (b) and (c) but different for (a)].

utterances “The Lathe is a big tool” and “The chairman cast three
votes,” bandlimited to 3.2 kHz in each case and sampled at 8 kHz.
These inputs will be referred to as L8 and C8, respectively.

Figures la and 1b indicate that postfiltering can result in significant
improvements in s/n. Table I further demonstrates this for the ex-
amples of 3-bit and 2-bit DPCM. The results tabulated are the values
of the s/n at the input of the postfilter and the s/n after postfiltering.
(See Fig. 3.) Table I also shows corresponding values of the segmental
s/n. In the ranges 0 < a <1 and 0 <8 <1 for the coefficient scaling
factors, the greatest gains in the s/n are obtained when a = 6 = 1.
These gains are seen to be as high as 8.9 dB for both L8 and C8. The
gains of the s/n at the input of the postfilter are always lower for the
design a = 0.2 and 6 = 1.0. But we presently note that these settings
of a and @ provide a subjectively desirable design.

Tables II and III provide the results of a subjective test involving
14 listeners, including 9 from the AT&T Bell Laboratories Acoustics
Research department and 5 listeners who had no prior exposure to
speech coding experimentation or testing. A total of eight stimuli were
included in the test. These included 4-bit ADPCM without postfilter-
ing, 38-bit ADPCM with six postfiltering conditions (including the no-
postfiltering case of a = 6 = 0), and 4-bit ADPCM with 6-kHz sampling
and a substandard speech bandwith of W = 2.6 kHz. This last condi-
tion was included to provide a 4-bit, 24-kb/s alternative to the 3-bit
ADPCM stimuli, all of which also had a bit rate of 24 kb/s. The values
of a and B used in the test were selected on the basis of a pilot test

Table |—Values of s/n at input of postfilter and after postfiltering
(see Fig. 3). Numbers in parentheses are corresponding values of
segmental s/n ratio

Table II—Number of wins in a round-robin tournament involving
eight coding conditions and four listeners, where the maximum
possible score is 196 for any given coding condition

Table !/I—Rank ordering of coding conditions by the group of 14
listeners and by a subgroup of 9 listeners from the Acoustics
Research Department

that identified the interesting ranges of these parameters from the
point of view of perceived mixes of noise suppression and speech
distortion.

In general, use of postfiltering results in an amplification of the
speech signal as suggested in Fig. 1b. The postfiltered speech stimuli
were therefore appropriately scaled down to mitigate differences in
stimulus loudness.

The subjective test involved an exhaustive pairwise comparison of
all possible stimulus pairs, with each pair appearing at random places
in the test once in each possible order of presentation. The total
number of AB comparisons was therefore 768 (8 X 7 = 56 possible
stimulus pairs for each of 14 listeners).

Table II shows, separately for inputs L8 and C8, the total number
of wins of each stimulus, with a maximum possible score of 196 for
each stimulus [a maximum score of 2-(8 — 1) for each of 14 listen-
ers]. It is seen that the worst two coding conditions stand apart from
the rest. These conditions are 3-bit ADPCM with no prefiltering and
4-bit ADPCM with 2.6-kHz bandwidth speech input (and output).
This latter condition gets a particularly low total score. Table II also
shows that the above results are not very different for the inputs L8
and C8.

of the eight coding conditions in the subjective test. Results are shown
separately for the total group of 14 listeners and the group of 9 listeners
from the Acoustics Research department. It is seen that the rankings
are not significantly different for the two populations. The best setting
of the coefficient scaling parameters is defined by

in each case, and for both L8 and C8. With input L8, 3-bit ADPCM
postfiltered as above is ranked a close second to 4-bit ADPCM speech
of equal bandwidth. In fact, the 4-bit ADPCM coder is ranked only
fourth when the results of all 14 listeners are pooled together. The
second and third ranks in this category belong to postfilters with the
design (a = 0.4; 8 = 0.8) and (a = 0.6; 6 = 0.6). The preference for the
design (a = 0.2; 8 = 1.0) has a simple interpretation. It suggests a
postfilter transfer function that mimics the approximate speech spec-
trum (the inverse predictor function) very closely at the zeros of that
spectrum (8 = 1.0), but very loosely at the poles (a = 0.2). This
suggests a condition that seeks to maximize background noise sup-
pression and minimized perceived speech distortion. In the case of
voiced speech segments, the poles tend to correspond to formant
frequencies and the value of a = 0.2 prevents an undue emphasis of
the higher-amplitude spectral peaks, a situation that was indeed
encountered in the example of Fig. 1b.

As we see in Table I, the s/n gains due to postfiltering are equally
significant for both 3-bit ADPCM and 2-bit ADPCM. Perceptually,
however, the general noise level in 2-bit ADPCM speech is such that
a useful degree of noise suppression requires the design of (a = 1.0; 8
= 1.0). With this design, the speech distortion introduced by the
postfilter is also substantial. For this reason, the case of 2-bit ADPCM
is not considered to be of sufficient practical importance to pursue
formal subjective testing. Informal testing shows, however, that the
design of (a = 1.0; 8 = 1.0) is again preferable to conventional 2-bit
ADPCM (a = 0; 6 = 0).

The specific postfiltering algorithm of Fig. 1b (a = 1.0; 6 = 1.0) was
also applied to speech degraded by additive white Gaussian noise. The
input speech was L8, the speech-to-noise ratios ranged from —3 dB to
17 dB, and all coefficients were obtained simply by simulating the
easily available case of 5-bit ADPCM, a bit rate high enough to
introduce very little quantization noise in comparison with the levels

of Gaussian noise being studied. We find the postfiltering algorithm
provides a very useful enhancement of noisy speech if the input to the
postfilter had a s/n of at least +3 dB. For lower values of s/n,
postfiltering provides noise suppression, but at the cost of substantial
distortion of the speech itself.

1. T. Nishitani et al., “A 32 kb/s Toll Quality ADPCM Codec Using a Single Chip
Signal Processor,” Proc. ICASSP (April 1982), pp. 960-3.

2. D. W. Petr, “82 kb/s ADPCM-DLQ Coding for Network Applications,” Proc. IEEE
Globecom Conf., December 1982. pp. A8.3.1-A8.3.5.

4. J. O. Smith and J. B. Allen, “Variable Bandwidth Adaptive Delta Modulation,”
B.S.T.J., 60, No. 5 (May-June 1981), pp. 719-37.

6. P. Cummiskey, N. S. Jayant, and J. L. Flanagan, “Adaptive Quantization in
Differential PCM Coding of Speech,” B.S.T.J., 52, No. 7 (September 1973), pp.
1105-18.

Nuggehally S. Jayant, B.Sc. (Physics and Mathematics), Mysore Univer-
sity, 1962, B.E., 1965, and Ph.D. (Electrical Communication Engineering),
1970, Indian Institute of Science, Bangalore; Research Associate, Stanford
University, 1967-1978; AT&T Bell Laboratories, 1968—. Mr. Jayant was a
visiting scientist at the Indian Institute of Science in 1972 and 1975 and a
Visiting Professor at the University of California, Santa Barbara, in 1983. Mr.
Jayant has worked in the field of digital coding and transmission of waveforms,
with special reference to robust speech communications. Editor, IEEE Reprint
Book, Waveform Quantization and Coding and co-author of Digital Coding of
Waveforms: Principles and Applications to Speech and Video (Prentice Hall,
1984).

Venkatasubbarao Ramamoorthy, B.E. (Electrical Engineering), 1970, The
Regional Engineering College at Tiruchirappalli, India; M.Tech., 1972, The
Indian Institute of Technology, Madras, India; Tekn.Dr., 1981, University of
Link6ping, Linképing, Sweden. Mr. Ramamoorthy was with the Indian Space
Research Organisation at Bangalore, India, prior to his joining as a staff
member at the department of Electrical Engineering, the University of Lin-
koping, Sweden, in 1974. He visited AT&T Bell Laboratories during the
summer of 1983. His current research interests include speech processing in
mobile and packet radio environments, channel and source coding, digital
modulation techniques, and development of handicap aids for children with
speech problems.

On Using ine iiakura-Saiio ivieasures for Speecn
Coder Performance Evaluation

The purpose of this paper is to discuss theoretical, as well as psychophysical,
aspects of using the Itakura-Saito type of measures for evaluating the quality
of coded speech. We present psychoacoustic interpretations of the measures
and identify their effectiveness as well as limitations within the theoretical
framework of a generalized waveform coder distortion model. The discussions
then point out some specific issues to be resolved through psychoacoustic
research effort.

A “good” speech quality measure is central to progress in the
research and development of speech processing systems. In speech
coding, for example, we need a quality measure to provide insight into
different distortions that are present in a coder output. If such a
measure existed, it would help speech researchers identify how various
kinds of distortions could be traded in order to improve the perceptual
performance of the speech coder. In an engineering context, a measure
that indicates the perceptual quality is a criterion to be optimized in
speech coder design. Without such a measure, tuning coding schemes
to achieve optimal quality is not a trivial task and the performance
cannot be conveniently evaluated.

Speech quality assessment, however, involves subjective, psycholog-
ical attributes of human perception, an area in which mathematicians

and engineers are usually not well versed. Thus, speech quality eval-
uation has never been established satisfactorily in mathematical terms.
The conventional signal-to-noise ratio (s/n), widely used in character-
izing signal transmission/reception environments, is an ineffective
measure of speech quality. Several other measurement methods and
parameters, such as the isopreference method! and the subjective
s/n,” have been proposed during the last two decades. General surveys
of classical approaches can be found in Refs. 1 and 3. Reference 2 and
its references also provide a summary of past efforts. Among these
approaches, one particular class of measures based upon the Itakura-
Saito measure has attracted engineers and scientists taking an analyt-
ical approach toward the problem. The Itakura-Saito measure and its
variations, such as the Itakura or log likelihood ratio measure* and
the likelihood ratio measure,’ have been employed in noise studies by
Sambur and Jayant;® in vocoder designs by Juang et al.’ and Wong et
al.;3 in automatic speech recognition by Itakura® and Rabiner;° and
as quality measures by Goodman et al.,!' Crochiere et al.,!? and
Barnwell et al.!°

Although successful applications of this class of measure are wide-
spread in speech processing, none of them comes close to being justified
as the speech quality measure. This paper attempts to identify the
effectiveness as well as limitations of using this class of measure for
speech quality within the theoretical framework of a generalized
waveform coder distortion model.“ We will further point out that
such limitations also exist in current automatic speech recognizers
that rely upon spectral matching. We then present some considera-
tions relating to psychoacoustic studies, aiming at a better understand-
ing of the fundamental concepts of speech quality in the presence of
spectral distortion. These considerations will help direct future rele-
vant psychoacoustic experiments for studying the dynamics of speech
perception.

Let s(i) and s’(i) be two sampled speech signals, and let x,(i) and
x/,(i) be two windowed segments, or frames, of s(i) and s’(i), respec-
tively. Segments x,(i) and x;,(i) are obtained by applying a window
function w(t), with w(t) = 0 fori <0 andi 2 N, to the speech signals
at instance n; in particular,

The windowing operation greatly facilitates using spectral represen-

tations for speech analysis because speech is considered as a quasi-
stationary signal. We denote the z-transform of x,(i) and x,(i) by
X,(z) and X;(z), respectively. The Fourier transform is obtained by
evaluating the z-transform on the unit circle, i.e., z = e’”, and thus the
notations X,,(e”) and X/(e/”) are used to designate the Fourier trans-
-form of two windowed signals, respectively. For every such pair of
spectral representations, X,(e/”) and X/(e/”), a spectral distortion
p[Xn, X/] can be defined to measure the dissimilarity between X,,(e’”)
and X/(e?*). In speech analysis, one particularly interesting distortion
measure is the Itakura-Saito measure, which is defined as

This mathematically tractable distortion measure has been success-
fully employed in vocoder designs.’ Detailed analytical properties of
the measure can be found in Refs. 4 and 5.

It has been shown in short-time Fourier analysis that a signal can
be reconstructed from a properly time-sampled sequence of short-time
Fourier transforms.’® We can, thus, further represent the two signal
sequences, s(i) and s’(1), by their corresponding short-time spectral
sequences. Using © to denote the reconstruction process,

In the above / is the underlying interval for short-time Fourier analysis
and has been dropped in the final expressions without ambiguity. Such
a representation allows us to characterize the dissimilarity between
s(i) and s’(i) in terms of distortion measures obtained from short-
time spectral representations. A distortion sequence between two
speech signals is then defined as

Extending the definition (3) to (7), then, we have a sequence of
Itakura-Saito distortions.

The Itakura-Saito distortion measure defined by (3) and (4) is in
fact the distortion measure for all-pole signal modeling; it was origi-
nally introduced as an error-matching function in maximum likelihood
estimation of autoregressive spectral models.!’ Therefore, we shall
confine ourselves to the analysis of Mth-order all-pole signal models
despite the fact that a distortion measure could be more general.
Several important results of the measure related to all-pole signal
modeling are:

when the gain terms are identical. A;,(z) takes the same form as A,(z)
in (11). In the above expressions, we have assumed that A,(z) and
A(z) have all their roots within the unit circle. Therefore,’®

For clarity, we further define the likelihood ratio measure and the
log likelihood ratio (or Itakura) measure as follows:

In defining the above two measures, A,(z) and A,(z) are the optimal
Mth-order inverse filters of X,(z) and X/(z), respectively.'® Further-
more,

Note that a, is the minimum Mth-order prediction residual energy
pertaining to signal X,,(z).

The Itakura-Saito distortion between the input and output signals
of a linear system H(e’“) can be easily calculated. Denoting the input
power spectrum as | X,(e’”)|?, we have the output power spectrum
| X7,(e%*) |? = | X(e*)H(e**) |?. Therefore,

where A,,(z), as defined above, is the optimal Mth-order inverse filter
of X,(z) and B,(z) is another Mth-order Finite Impulse Response
(FIR) filter, taking the same form as (11). We also assume that A,(z)
and B,,(z) both have all their roots within the unit circle. The input/
output relationship of the system is illustrated in Fig. 1. Since A,(z)
is the optimal Mth-order inverse filter of X,(z), E,(z) is then the
residual signal. X;,(z) is obtained by driving another all-pole filter 1/
B,(z) with such a residual signal. The distortion between X,(z) and

Fig. 1—A particular class of linear system in which A,(z) is the optimal Mth-order
inverse filter of X,(z).

= pis A,” B, ; (21)
which is determined by the two all-pole filters, and has the same
expression as the likelihood ratio measure of (14). This result gives us
a convenient means of modifying a signal in order to achieve a
prescribed distortion level from the original signal. Detailed discus-
sions in Section IV are based upon this concept. It is, however,
important to note that in eq. (21), B,(z) is not unique, and is not
necessarily the optimal Mth-order inverse filter of the output signal
X/(z). It is simply stated that within the Mth-order autoregressive
model framework, a prescribed Itakura-Saito spectral distortion can
be obtained from a given signal through proper filtering operations,
which will be convenient to realize.

Figure 2 shows a block diagram of the waveform coder distortion
model used by Crochiere et al. for an interpretation of the log likelihood
ratio measure.” This coder distortion model is composed of a time-
varying linear filter h(i), to model the “linearly correlated” distortions,
and an additive noise source q(i), to account for the nonlinear, uncor-
related distortions in the coder. Since the model attempts to split the
components of distortion, it was expected that distinctively different

Fig. 3—Measuring coder performance with the likelihood ratio in a forward manner.

perceptual effects could be meaningfully studied separately with such
a model.

Measurement of the coder performance with the likelihood ratio
measure is shown in Fig. 3, which introduces the notion of inverse
filtering. We use the likelihood ratio measure, rather than the Itakura-
Saito measure, because we try to avoid, in the following discussions,
extra complications in speech quality measurement due to amplifica-
tion or attenuation. We follow the notation of Section II, except that
the subscript indicating the frame index has been dropped, since, for
most of the subsequent expressions, signal stationarity is assumed.
We shall reinstate the frame index wherever necessary. The two
parameters, a and a’, are defined as in (17) by

where A(z) and A’(z) are the optimal Mth-order inverse filters of X(z)
and X’(z), respectively. In other words, a and a’ are the minimum
Mth-order prediction residual energies corresponding to x(i) and x’ (i)
sequences, respectively. The energy of v’(i), denoted by 8’, is then

The energy of w’(i), on the other hand, is unity due to the normali-
zation factor 1/ Va’ and eq. (23). The energy ratio of the two filtered

can be reduced to
= T | A’(e%*) |? dw
es ‘x |A(e*) |? 2m 2)
since both A(z) and A’(z) are Mth-order FIR filters, and the first
M + 1 autocorrelation coefficients of the {[x(i)]/Va } sequence are
equal to those of the impulse response of 1/A(z). Therefore, the

likelihood ratio measure of (14) can be expressed in terms of the energy
ratio of the two filtered outputs, v’(i) and u’(i), and

Alternatively, we may replace the filter A’(z) by A(z), the inverse
filter of the x(i) sequence, as shown in Fig. 4. In such a case, the
energy of u(i), denoted by y, is the likelihood ratio, and the distortion

Fig. 4—Measuring coder performance with the likelihood ratio in a backward manner.

The interpretation of the log likelihood ratio as a coder performance
measure by Crochiere et al. follows the comparison order of eq. (28).2”
More specifically, the measure they discussed was log vy, instead of
+ — 1. The difference between the log likelihood ratio measure and
the likelihood ratio measure may be insignificant in terms of measure-
ment. However, the likelihood ratio measure of eq. (14) appears to
correspond more closely to the Itakura-Saito measure in representing
the distortion relationship between the input and output signals of a
particular class of linear systems. This was shown in eq. (21).

We now express the measures within the coder model. Referring to
Fig. 2 and denoting the Fourier transforms of h(i) and q(i) by H(e?“)
and Q(e’*), respectively, we have

For simplicity we assume that H(z) does not have poles and zeros on
the unit circle. From (24), the likelihood ratio distortion measured
from {x(i)} to {x’(z)} is thus

the following way:
1. Additive noise distortion, p,, is defined when there is no corre-

In the above, the superscripts, f and b, denote the forward and
backward measurements, respectively.

2. Correlated spectral distortion, p,, is defined when the additive
noise component vanishes, i.e.,

The above decomposition of the measure into additive noise and
correlated spectral distortions provides a helpful means in cross-
verification between the measure and many known perceptual attri-
butes. In the following we shall discuss the merits as well as limitations
of the above measure in measuring the perceptual quality of waveform-
coded speech signals. Such discussions point to some necessary psy-
chophysical experiments for a closer link between objective and sub-
jective measures.

The key contribution of the uncorrelated additive noise, q(t), ap-
pears in the integral terms in (33) and (34). Let us consider (34), where
the integrand involves the inverse filter A(z) for the input speech
signal.

where P, is a constant, when A(z) is the optimal (Mth-order) inverse
filter of the g(i) sequence. In other words, for a given noise power, the
integral is minimized if the noise has the same spectral shape as the
input speech, within the Mth-order autoregressive signal modeling
framework. This appears to be in very good agreement with the results
of auditory masking that has been proposed as a method for improving
the perceived quality of digitally encoded speech.’®”° The same obser-
vation can also be made on (33), where the integrand involves A’ (z)
instead of A(z). A’(z) is the optimal Mth-order inverse filter of the
encoded output sequence x’(t). If q(t) is truly uncorrelated with x(i)
(recall that | H(e/”) | = 1 here) and has the same spectral shape as
x(i), then A’(z) is, in fact, identical to A(z). However, when exact
shaping of noise spectra is not achievable (as in most practical coder
systems), (83) and (34) lead to significantly different distortion mea-
surements since a’ involves A’(e/”), which demonstrates attributes of
Q(e’”). The following example illustrates the difference between the
forward and the backward measurements.

Consider two signals, one being tonelike and the other being white
noise. These two signals are represented in terms of second-order all-
pole models as 1/A,(z) and 1/A,,(z), where

The two roots of A;,(z) are 0.9 e*/*, which indicate a resonance at
a/4 normalized frequency or at 1000 Hz when the sampling frequency
is 8000 Hz. These two all-pole models have corresponding reflection
coefficient vectors k, and k,,:'°

Clearly, if measured in the forward direction, when an input tonelike
signal is being distorted into white noise, the distortion is higher than
vice versa. The result is reversed if the distortion is measured in the
backward direction; that is, distorting an input noise signal into a
tonelike signal will result in a more serious objective distortion mea-
surement than distorting a tone-like signal into white noise. Previous
studies in auditory masking demonstrated a similar asymmetry of
masking between tone and noise.”"~” In particular, it has been reported
that noise masks a tone more effectively than a tone masks noise. A
1-kHz tone masked by noise that is one critical band wide typically is
inaudible at a signal-to-masker ratio of —4 dB, while the corresponding
ratio for noise signal masked by tone is approximately —24 dB. In
other words, it is easier to perceive noise in a tone than it is to perceive
a tone in noise. For an objective measure to consistently predict the
perceived quality, we thus would require that such a measure show
higher distortion when the input tone is corrupted by noise and that
it show lower distortion when input noise is distorted by an additive
tone signal. Despite the slight difference between masking and distor-
tion, forward measurements of (33) thus appear to be more justifiable.
More rigorous psychoacoustic studies are obviously very important in
carefully resolving this measurement direction issue.

Compared to additive noise, correlated spectral distortion has not
been as well studied in the past, but it is a key factor affecting the

perceived quality. One well-known example is that “telephone speech”,
which is essentially bandlimited to the range of 200 to 3200 Hz, is
considered to be of poorer quality and of lower intelligibility than the
unfiltered original speech. Since correlated spectral distortion can be
a result of the filtering operation, we shall discuss it using linear
filtering concepts.

Linear systems can be categorized into time-invariant and time-
variant systems. Accordingly, correlated spectral distortion can be
time-invariant or time-variant as demonstrated in eq. (19), where the
correspondence between the filtering operation and the distortion
measure was established. The above-mentioned bandpass filtered
speech signals, such as telephone speech, have essentially a time-
invariant spectral distortion (here we are not considering tone noise,
clicks, or channel variations, etc.), while Linear Predictive Coding
(LPC) vocoders involve many time-variant spectral distortions, as will
be discussed shortly.

The use of the Itakura-Saito type of measure for time-invariant
spectral distortions, such as (3), (14) and (15), appears to be justifiable,
at least within the short-time frame boundary where stationarity is
reasonably assumed. This can be seen from the application of the
likelihood ratio measure in vector quantization for voice coding.”* In
fact, the code words designed for vector quantization using (14) are
substantially consistent with the vowel triangle of Peterson and Bar-
ney from an acoustic-phonetic point of view.® It also has been shown
that the log likelihood ratio measure usually leads to a better recog-
nition rate in speech recognition schemes.’°”> (Note that the log
likelihood ratio and the likelihood ratio measure make no significant
difference in most speech recognition applications. The only theoret-
ical difference is in template generation where minimization of some
criterion, such as the average distortion or maximum distortion, is
required.) For interests in psychoacoustic studies, however, it may be
desirable to further translate the measurement into a perceptual scale
that better interprets the relative perceived quality. (The complication
here is the possible sound dependence on a perceptual scale. Consider
the following example. Suppose X has been distorted, resulting in Y
and Z. We can confidently say Y sound is closer to X sound than Z
sound is, if p[X, Y] < pLX, Z]. However, we are not sure that Y is
perceptually closer to X than Z is to W, even if p[X, Y] < p[Z, W].)

Beyond the short-time stationary segment level, the time-variant
distortion is a more important and complicated factor to consider in
speech processing. Spectral distortion measures are defined for every
pair of spectral representations. A natural extension of the distortion
measure for measuring dissimilarity between time-varying signals is
thus the distortion sequence is expressed by (7). Previous experiments

and several reported results that help illustrate the effect of time-
variant spectral distortions upon speech quality are in order.

Voice coding results in time-variant spectral distortion. The key
contribution to the time variation of distortion in voice coding such
as LPC is a result of parameter quantization, although the parameter
analysis procedure itself may also introduce some time-variant distor-
tion because of frame alignment, change in excitation, etc. The effect
of such distortion thus can be best explained in performance compar-
ison of different parameter quantization schemes.

The experiment in Ref. 7 that compared the distortion performance
of vector and scalar quantization for LPC voice coding provides
important insights in this regard. In order to conduct the so-called
equal average distortion comparison in the experiment, speech signals
were vocoded at a lower bit rate with vector quantization and at a
higher bit rate with scalar quantization. Subjective comparison of
these two sets of synthesized signals of equal average distortion showed
that the vector quantization synthesis samples sounded smoother and
more pleasant, and were considered of better quality. Substantial
background warble was perceived in the scalar quantization samples.
Differences in spectral continuity, distortion contour, and some sta-
tistics of the distortion process {p,} between the two sets of synthesis
samples were then reported to explain the difference in the perceived
synthesis quality. It was concluded that a coder that preserves more
spectral continuity, achieves smoother distortion contour, and pro-
duces less divergent distortion statistics is better than a coder with
otherwise different distortion performance, even though they yield the
same average distortion. Vector quantizers appear to produce “better”
distortion sequences than do scalar quantizers in LPC voice coding.
The importance of considering the distortion as a process or sequence
(instead of just an average distortion) and of looking into the spectral
continuity (a mathematical definition of which has yet to be obtained)
was thus highlighted.

The concepts of time-variant distortions and spectral continuity
also raise a possible explanation for the experimental results of Tri-
bolet et al.?° Here, performances of four different types of waveform
coders at three different bit rates were compared. An average noise-
to-signal measure [eq. (2) of Ref. 26], 4,, which was derived through
the concepts of log likelihood ratios, was used as an objective measure
to predict the subject performance. As seen from Figs. 5 and 6 (dupli-
cated from Figs. 7 and 9a of Ref. 26), the main failures of the likelihood-
ratio-derived measure are in predicting the performance of all coders,
at 9.6 kb/s [in particular, Sub-band Coder (SBC) at 9.6 kb/s] and
Adaptive Differential PCM (ADPCM) coder with a fixed predictor at
24 kb/s. At 9.6 kb/s all coders perform subjectively worse than objec-

Fig. 6-—Objective noise-to-signal measure, 4,, averaged over 16 articulation bands for
the 12 coders in Fig. 6.

tively predicted. At 9.6 kb/s, SBC is objectively very close to the
Adaptive Transform Coder (ATC) but turns out to be subjectively
even worse than the ADPCM with a variable predictor. At 24 kb/s,
ADPCM with fixed predictor is objectively much worse than ADPCM
with variable predictor, but they in fact are subjectively very close.
These failures can be attributed to the fact that 4, does not correctly
consider the correlated spectral distortion, and more importantly, it is
only an average over the entire speech sample, revealing no informa-
tion on possible perceptual degradation due to time-variant spectral
distortions. The outcome that all coders perform subjectively worse
than objectively predicted at lower bit rates is probably a result of
increased sporadic spectral distortions and reduced spectral continuity
along the time axis. Sub-band coding schemes inherently preserve less
spectral continuity at lower bit rate, and thus it is possible that
relatively more quality degradation is perceived at 9.6 kb/s with SBC.
Finally, the ADPCM coder with adaptive, variable predictor poten-
tially introduces more spectral discontinuity, due to quantization of
the predictor parameters, than does the ADPCM with a fixed, un-
quantized predictor.

To illustrate this, plots of the log spectral (eighth-order all-pole)
difference between the original and the reconstructed speech signals
are shown in Fig. 7. Coders used in Fig. 7 are ADPCM with fixed
predictors and adaptive predictors, respectively. More spectral discon-
tinuity is observed in the adaptive predictor case, particularly in the
low frequency region. Therefore, even though adaptive predictors yield
higher prediction gain than fixed predictors,”’ this objective advantage
has been subjectively offset by the perceptual sensitivity to time-
variant distortions, particularly at higher bit rates, where the effect of
additive noise becomes relatively less significant. As a result, the
subjective performance gap between the two coders is substantially
reduced.

Similar limitations apply to automatic speech recognition schemes
that use one single average or accumulative figure to represent the
dissimilarity between the spectral sequences of the input speech and
the stored reference template. In parallel with the concept of meas-
uring speech quality with the segmental s/n, recognition schemes
usually resort to segmentation and time warping in order to obtain
better distortion or distance measurements for more accurate recog-
nition decisions. Nevertheless, segmentation schemes produce hard
segmental boundaries, instead of natural, soft transitions, and are
never completely reliable. The original problem of measuring the
dissimilarity between time-varying signals thus has never been entirely
solved.

Fig. 7—Log spectral difference between the original and reconstructed signals: (a)
with a fixed, unquantized predictor; (b) with an adaptive, quantized predictor.

The above considerations clearly point out the necessity of psycho-
physical experiments for developing a better speech quality measure.
Specifically, with regard to using the Itakura-Saito type of measures,
issues to be further studied are: the measurement direction, the feasi-
bility of characterizing subjective quality by distortion sequences, and
the incorporation of some transitive functions into the distortion
measure to account for spectral continuity. In light of the analytical
features of the Itakura-Saito type of measures, research on these issues
appears to be vitally important to an analytical speech perception
model.

It is beyond the scope of this paper to propose and discuss in detail
the psychophysical experiment procedures necessary to answer al] the
questions above. It is, however, appropriate to address one of the
difficulties in psychoacoustic experiment designs here. In addition, we
shall propose to consider a class of transitive functions to be used in
defining the spectral continuity measure.

Figure 8 illustrates the modification procedure. The speech signal is
first inverse filtered by A,(z) to obtain the residual E,,(z), which then
drives the chosen filter 1/B,(z) to form the desired signal.

a prescribed value can be made simple if we have a good-sized vector
code book, as designed in vector quantization.’ The search for 1/B,,(z)
is then quantum-selectively finite, although there are theoretically
infinite number of all-pole filters. Also, the test stimuli designed
according to (45) are free from excitation variations, such as funda-
mental frequency changes, that are better considered separately.
4.2 Spectral continuity
As discussed above, spectral continuity is an important factor af-
fecting the perceived quality of speech signals. Speech signals carry
distinctive time-frequency or spectral transition patterns. Phonetic
manifestation in articulated speech signals could be very fast, like
/str/ in “strange”, or sustainingly slow, like /i/ in “eat”. To avoid
complications due to such an inherent nonuniform spectral change,
Ref. 7 used the model error spectral sequence {A,(w)}, defined by
An(w) = log

| An(e’*) |? | A,(e?*) |?’
where A,,(z) is a quantized version of A,,(z), to illustrate the difference
of the ability of various quantization schemes in preserving spectral
continuity. The rationale was based upon the fact that the ultimate
spectral continuity to be retained is the inherent spectral transition
pattern, and that if a coder produces spectral distortion that is inde-
pendent of time, that is,

then the time-variant spectral distortion is completely eliminated.
While {A,,(w)} adequately explained the spectral continuity differences,

more rigorous alternatives are necessary for, at least, the following
reasons: (1) the variation in A,(w) along the frequency axis, w, as well
as the time axis, n, is often so substantial that it is difficult to use
only (47) to define a spectral continuity measure; (2) it was never
concluded that the change in A,,(w) along the time axis, if regarded as
an indication of spectral smoothness, is indeed perceptually independ-
ent of the spectral transition pattern of the speech signal.

Before we can completely characterize the spectral continuity along
both the frequency and time axis, we would like to propose to tenta-
tively consider two transitive functions that indicate the spectral
changes in a speech signal as a function of time. The notion of eq.
(21), measuring the distortion between two all-pole spectra, is empha-
sized in defining such transitive functions. Denoted by ¢,(k), the
forward transitive function is defined by

where ); is a time constant and, A,(z) and Az-,(z) are the optimal
Mth-order inverse filters of X;,(z) and X;-,(z), respectively. ¢,(k)
measures the all-pole spectral change in the speech signal in a forward
manner, i.e., it measures the distortion resulting from replacing the
current spectral envelope with previous spectral envelops. Character-
istic changes in excitation, such as the pitch inflection, are not actively
considered in ¢;(k), although they may affect the estimation of all-
pole spectral models. One interpretation of measuring the transition
in speech by the distortion between all-pole models instead of speech
spectra is that we try to keep the current excitation signal unchanged,
as if it were present in the previous segments as implied by eq. (21).
We also assume that the time constant );, accounting for short-time
auditory memory,” is independent of the particular sound that is
articulated and perceived.
Similarly, we define the backward transitive function ¢,(k) as

Note that if the distortion measure were symmetrical and if dy = As,
the two transitive functions would be identical. The appropriateness
of these functions remains to be studied.

The transitive functions are to be regarded as part of the speech
signal. When a speech signal is distorted because of processing or
encoding, the corresponding transitive functions are distorted also.
The distortion, or noise, in the transitive functions thus provides a
measure of the time-variant spectral distortion that affects the spectral
continuity in the original signal. Further research effort, of course, is

necessary to verify the suitability of these functions or to develop a
better spectral continuity measure. We feel that the concept in (49)
and (50) provides a good starting point.

While the Itakura-Saito distortion measure and its variations have
been widely employed and are considered promising in characterizing
speech quality,’ limitations in such measures still exist and have been
identified within the theoretical framework of a generalized waveform
coder distortion model in the above discussion. This type of measure
is inherently nonsymmetric and therefore, in measuring distortions, a
proper measurement direction needs to be determined. Subjective
quality evaluation involves perceptual response to various degrees of
distortion that has to be considered as a time function or a stochastic
process. The feasibility of describing the subjective quality by finite-
order statistics of the distortion process is to be studied. Furthermore,
evidence shows that speech spectral continuity is also a key, if not the
most important, factor affecting the subjective quality and thus, the
speech spectral transition pattern should be regarded as a vital part
of the speech signal. An even more fundamental and difficult task is,
then, the incorporation of the spectral transition patterns into the
rather static measurements of the Itakura-Saito distortion. Psychoa-
coustic studies are necessary to resolve these issues.

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Our Capacity for Processing Information,” Psychol. Rev., 63 (1956), pp. 81-97.

A Packei/Circuit Switcn
By Z. L. BUDRIKIS* and A. N. NETRAVALI*
(Manuscript received July 29, 1983)

We propose a switch, suitable for an integrated local communications
network, that will support packet switching and circuit switching, with a wide
range of bit rates. Key components are two serial memories; a multiplicity of
access units, each capable of writing and reading uniformly formatted, ad-
dressed information; and a programmed controller. Circuit switching is
achieved when the controller repeatedly allocates memory slots, following call
setup. Data communications can proceed concurrently without setup, compet-
ing for unused slots. We give an example of a 10,000-telephone-line switch
carrying a similar load of other traffic. The switch would delay voice by less
than 5 ms and could be interfaced to the existing telephone system. We
indicate a method of fault detection and isolation that will limit the impact of
a failure on a serial memory to an arbitrarily small group of connected lines.
We define an index for measuring failure impact and use it to derive most-
favorable fault-isolating partitions.

The telephone system is by far the world’s largest communications
network. It was primarily designed for voice, but its role widens
continuously, as it adapts to new requirements. Presently it is changing
to accommodate data communications.

Already the network extensively caters to data communications, but
not yet as well as it might. Although internally the telephone system

* AT&T Bell Laboratories. On leave from the Department of Electrical and
Electronic Engineering, University of Western Australia, Nedlands. ‘AT&T
Bell Laboratories.

Copyright © 1984 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.

is rapidly becoming a vast interconnected computer system, proffered
data are still largely carried as analog signals externally. That will
change, however, as special provisions for data come on-line. As more
of the plant, including switches, becomes digital, it will be possible to
offer, on a selective basis, switched digital telephone channels usable
for 56-kb/s data throughput. Also, packet-switched data services will
widen in scope and access. Packet-switched data services are overlay
networks that use the digital transmission facilities of the telephone
network but bypass its switches, of which many are still analog. The
packet networks eventually may become totally interconnected, just
as the voice network, and also may become integrated with it.

In-house, or proprietary, telephone networks can benefit from the
changing character of the overall network more immediately. Already
available are switches and other components that permit an all-digital
network that will accommodate on one facility both voice and data.
As good as this already is, we are proposing a switch that could make
the private network even better. Eventually it might even influence
the entire system.

Currently available switches provide only circuit-switched connec-
tions. This gives fixed-capacity channels on a continuous basis,
whereas much of data comes in bursts. Thus, computer communica-
tions are characterized by very long call durations with only low
average, but in many instances very high, peak rates. Given the option,
direct memory transfers could proceed in some instances at rates of
many megabits per second. This is far too high for a switched and
continuously held circuit.

It is true that the needs of bursty traffic can be catered to by what
already is available, namely by some packet-switched networks. But
that introduces a separate communications network for data, with the
consequences of proliferating wiring plans, divided responsibilities,
and probable long-term dyseconomies. It is better for one facility to
serve all communications, and to do so without imposing mismatches.

We propose a switch and, more generally, a new switch architecture
that support within one switching fabric both circuit- and packet-
switched connections. This would largely avoid mismatches in respect
to bursty data traffic, while preserving unity in communications.

The cardinal components of the switch (see Fig. 1) are a pair of
Serial Memories (SMs), a Central Controller (CC), and Accessing
Units (AUs). The memories do not recirculate and both ends (head
and tail) of each terminate on the central controller. The AUs are
connected to read-and-write taps along the SMs, an AU having one
connecting tap to each memory. The two taps of an AU form a
symmetrical pair: the tap to the second memory is as many places
from the tail end as that to the first is from the head. Thus, each AU

Fig. 1—Block schematic of switch. Access Units (AUs) communicate via two Serial
Memories (SMs) on behalf of client Stations (Sts). Central Controller (CC) reserves
slots for circuit-switched communications.

can reach every other AU by either one, or the other, memory. It can ©
reach, and be reached by, the central controller by either memory. All
writing is logical OR.

An AU acts as an agent of a client station (St) (e.g., telephone,
facsimile terminal, computer) and mediates communications between
it and other stations by way of corresponding AUs. Communications
are carried on by write-and-reads in memory/time slots of uniform
length and format. Each slot consists of a data field and several control
fields. Collectively, the control fields provide synchronization, “Slot
busy” indication, source and destination addressing, and slot pleading.
A circuit-switched communication is carried on in regularly recurring
slots, which are appropriately premarked by the central controller.
For a packet-switched communication an AU simply uses the next
available slot.

The capacity of a connected circuit can vary over a wide range, from
a small fraction of a single (64-kb/s) telephone channel up to a large
multiple of that capacity. It is settled by negotiation with the CC at
the time of circuit setup, and need not be the same on different
occasions. The capacity available to a packet-switched communication
depends on the prevailing competition and can be any portion of the
total switch capacity. The latter is a function of size and would be just
several megabits per second for a 100-line switch and several hundred
megabits per second for a switch that supports 10,000 lines.

A simple realization of the serial memories would be by clocked
shift registers. The shift registers can be bit-paralleled to any degree
needed to keep the clock rate low. The memories and all access units
can be located centrally at the controller, with all connections to the
switch then forming a single star. But it is also possible to segment
the memories and form the network in clusters. The segments of
memory would be connected serially to each other and to the central
controller by transmission lines, forming two contrary rings, and the
clusters again would form star topologies.

All elements of our proposal are well established and tried. Central,
or stored program, control in circuit switching is over twenty years
old.'! The idea of switching by time slot interchanges is even older,”
followed shortly by its realization through read-and-writes in computer
memory.** Packet switching is more recent,° but is also well estab-
lished both in local and wide-area networks.®®

In essence, our scheme is an adaptation of seemingly diverse pro-
cedures, so that they may coexist. Time division slots are enlarged
from what is usual in circuit switching, so that they can carry the
control information essential to packet switching. Unlike normal
packet-switched schemes, packets are of a single fixed length so that
they can also be circuit-switched. Instead of separate time and space
division stages, common in current telephone switches, we have a
combined space/time fabric, abstracted from ring and bus networks,
with a particular debt to Fasnet.? This makes packet switching possible
without controller intervention. Finally, the controller maintains cir-
cuit connections by repetitive slot allocations, which is only marginally
different from what takes place in a time division stage of a standard
switch.

Also, our proposal is not first in its suggestion that voice and data
be integrated on a common network.’ But it appears to be first in
suggesting a common switch for circuits and packets as the basis for
that integration. With few exceptions,” prior suggestions have been
to treat voice as data and to packet-switch it both in local and wide-
area networks. However, these proposals have attendant delays that
have to be addressed.

The point is important since, in the global telephone system, trans-
mission delays can limit the quality of many possible connections. In
the case of our switch, the delay of voice signals can be kept to less
than 5 ms. It depends only on the clock rate, the size of switch, and
the size of slots. Since delay considerations have an overriding sway
on system choices, we discuss them in Section II. In Section III we
give further details of our proposal.

In Section IV we address the question of reliability in our switch.
We do this because our proposal may be seen as being particularly

vulnerable, since all its communications are to take place via two
serial memories to which all AUs have writing privileges. We introduce
a scheme for sectional detection and isolation of faults applicable to
our switch. We show that this would limit the impacts of faults in our
case to those that would prevail in switches that have much more
dispersed and/or redundant architectures.

A communication system is expected to deliver messages to the
destination in a timely fashion. The permitted delay is different in
character for data and for real-time signals. We review the two cases
separately.

Within limits, the exact times of arrival at the destination of the
different parts in a data stream generally are unimportant. It is usually
required that the sequence in the stream be preserved and that the
average delay does not exceed some specified value. When data are
presented for transmission at a fluctuating rate and there is not
sufficient transmission capacity to cope with the peaks, the flow is
smoothed by buffering. Waiting times in buffer stores are the predom-
inant cause of delay.’°

In the transmission of real-time signals, the delay should be a
constant and not greater than a specified value. Given fluctuation in
transmission rate, there will be a time-varying delay W,(t) in a buffer
at the sending end. A further delay W,(t) must be deliberately intro-
duced in a buffer at the receiver,!® so that the total delay could stay
constant:

If, at some time, W, exceeds D, then, at the same time, the buffer
at the receiver will become empty and there will be a break in the
received signal. Hence, there is no point in storing more at the
transmitter than the amount of data that represents the total designed
delay. If the rate \ of the real-time data is constant, as in Pulse Code
Modulation (PCM) voice, then waiting times are directly related to
amounts of stored data. The buffer-store capacities, N, and N,, that
need to be provided at the two ends are equal, given by

If X is not constant, then the required capacities are still equal and are
found by substituting the maximum value of ) in eq. (2).

It is important to note that, given fluctuations in data and/or
transmission rate(s) and buffer stores to smooth them, the relevant
delay for real-time signals is the maximum, i.e., designed, value, not
the statistical average. How much larger that designed delay is to be
than the average depends on the actual fluctuations in rate(s) and the
relative tolerance to lost quality by signal discontinuities and by delay.

The basic configuration of the switch was shown in Fig. 1 and
outlined in the Introduction. The functions of the AUs and the CC
will be defined in more detail when we discuss protocols in the next
subsection. It will be seen that there are considerable differences in
the tasks of an AU that is mediating a circuit-switched, as compared
to packet-switched, communication. Further differences in speed and
buffer requirements may be identified between, and within, those two
categories.

Clearly, there is a choice between designing a number of special-
purpose AUs and designing a single universal AU. Further choices
concern sharing of, and multitasking by, access units. Should AUs be
placed in a common pool and shared by a larger group of stations?
That would entail further switching outside the main switch to mediate
connections between AUs and stations. Should an AU be multitasked,
serving simultaneously different stations? That would make the AU a
more complex device. Figure 2 illustrates a switch that incorporates
both sharing and multitasking.

Our inclination is towards universal AUs, one to each station, and
towards neither sharing nor multitasking. True, this calls for the
largest number of AUs, and not the least complex, at that. But it has
the advantage of uniformity and, in the light of technology trends, of
likely overall economy.

The next choice concerns the serial memories. They may be active,
made in semiconductor, or also, reverting to earlier technologies,
passive, e.g., acoustic or electromagnetic delay lines. Passive compo-
nents are attractive because they promise more reliability. However,
our purpose would be better served by clocked shift register memories
in an arrangement as, say, shown in Fig. 3. This makes for easier
synchronization and permits bit-paralleling to hold down clock rates.

Fig. 2—Different options in AU tasking. All AUs could be of one type, each serving
a single station (right); AUs could be shared by a larger group of stations requiring
selector switching outside the main switch (center); or an AU could be multitasked,
serving more than one type of station (left).

Reliability is a matter of overall design and implementation. In Section
IV we discuss an architecture-related aspect of reliability, namely
isolation of faults to limited sections.

Finally, we have the question of overall network topology. Three
different arrangements are shown in Fig. 4. Figure 4a shows the
traditional topology of a central switch and star network. In Fig. 4b a
completely distributed arrangement is shown in which the serial
memories wend their way past every station. This would make it
similar to a local area ring network and would be possible only with
passive lines as the memories. A compromise between the above two,
and an interesting topology for a PBX that has to serve an extended
area, is shown in Fig. 4c. The serial memories are cut into sections,
and each section is placed close to the group of stations that it serves.

Fig. 3—Shift register realization of serial memories. Access points are at the inputs
of clocked unit delays; the writing is through OR gates.

The lines connecting the individual sections and the central controller
could be optic fibers, which would carry the total information streams
serially even in a large switch.

In the context of our proposal, a packet used in packet-switched
communication is made up of five control fields and data, as shown in
Fig. 5. The same format could be used in circuit-switched packets. But
for these, at least one of the two address fields is unnecessary. Its
space may either be added to the data field, or it could be used as a
separate channel, a companion to the main channel.

. RCVR—address or password of AU intended to receive packet
. DATA—data field

The roles of all the fields, except RQST and SYNC, are self-evident.
RQST is used by packet-switching AUs and we will see its function
presently when we discuss data communications. The SYNC field is
written by the central controller to ensure slot and frame synchroniz-
ation. Although both synchronizations could be achieved with just one
bit per slot, a field of two bits will make them more secure. Altogether,
the following numbers would be of the right order: BUSY and RQST
one bit each, SYNC two bits, the addresses 14 bits each, and DATA
192 bits, for a total packet of 224 bits, or 28 bytes.

Fig. 4—Network topology options: (a) central switch, stations connected by lines; (b)
switch completely distributed with AUs at individual stations and connected by the
serial memories realized as buses; (c) switch distributed in clusters, the seral memories
within clusters realized by shift registers and between clusters by transmission lines.

propagated memory block as consisting of the same number of bits
and divided among respective fields. Note, however, that it is not
necessary that the different parts of a packet be placed into a single
slot or block. There may be interleaving of packet parts to any extent
that is desirable.

Thus, it is conceivable that in order to alleviate the pressure of time
for the signaling from receiver to transmitter within the AU, the

Fig. 5—Slot format. Typically, BUSY, RQST and SYNC would be one-bit fields, the
address fields could be two bytes each and the data field 24 bytes.

BUSY field could refer to the state of occupancy—not of the slot that
it is riding in, but of the following slot. Similarly, other fields could be
advanced or retarded, and not necessarily by only one slot, nor, indeed,
just as a complete field. Thus, the DATA field could be broken into
single bytes or even bits, and the fragments made to follow the header
as an arbitrarily dispersed tail, provided only that all packets are
fragmented and dispersed identically.

The extreme fragmentation of packets, as just alluded to, may seem
an attractive way of restoring smoothness to data flow for circuit-
switched communications. Indeed, almost complete smoothness is
possible for any one chosen rate. But it would be at the expense of
considerable complication for all other communications, particularly
the packet-switched and the circuit-switched that have higher rates
than the one singled out for favorable treatment. We will dismiss it
from further consideration and turn to describing procedures.

Assume that AU addresses are in numerical order along the two
memories, ascending in the direction of propagation along one and
descending along the other. We will call the memory with ascending
addresses the forward channel, and hence the other the reverse chan-
nel.

Suppose that an AU has to communicate to another AU of higher
address. It must send a message, or packet(s), on the forward channel.
To do so, the dispatch processor of the AU will follow the data dispatch
routine of Fig. 6. This can be understood more easily with the help of
the state diagram of Fig. 7. For the sake of description, this diagram
relates to an exclusive forward channel dispatcher, although in practice
a single dispatcher would service both directions.

When idle, the dispatcher is normally in the “Go” state and monitors
the sending buffer (for the forward channel), checking whether it
contains a packet for transmission. If it does, it reads the BUSY field
of the next block on the forward channel and at the same time writes
a “ONE” in that field so as to seize the slot, should it be available. If
it is not, i.e, BUSY was already “ONE,” then it will write “ONE” in
the next RQST field on the reverse channel and wait for the next
BUSY field on the forward channel. It will repeat reading and writing
of BUSY on the forward channel and sending RQSTs on the reverse
channel until a “ZERO” BUSY occurs. It will then write in the related
SNDR, RCVR, and DATA fields, so dispatching a packet.

Having sent a packet, the dispatcher moves to the “One packet sent’
state. If the sending buffer has at that moment one or more further
packets for dispatch, then the dispatcher will behave exactly as in the
“Go” state and send off the next packet, thereby moving to the “Two

packets sent” state. But if there is no packet in the sending buffer on
entry to the “One packet sent” state, then the dispatcher will proceed
to the “Halt” state. It will remain there until the next “ZERO” is
written in the RQST fields on the reverse channel, whereupon it will
revert to the “Go” state. Similar conditions apply on entry to the “Two
packets sent” and further states, until the dispatcher has sent in a
contiguous sequence M packets and entered the “M packets sent”
state. From this it must proceed unconditionally to “Halt.”

Fig. 7—State diagram of data dispatcher. The dispatcher goes temporarily into HALT
state whenever it has no more packets to send or has already sent M packets since the
ae oe It goes from HALT to GO as soon as the RQST bit on the reverse channel
is :

M is a parameter that may vary with AU. It represents priority
standing: The larger its value, the less sensitive the AU is to pleadings
for slots by other AUs that are downstream from it. It is normally set
in relation to the rate of the station that the AU serves.

The task of receiving is less involved but no less time consuming,
and an AU will have a separate processor for it. A routine that it could
follow is given in Fig. 8. This is set out on the assumption that the
SNDR and RCVR fields of a packet would precede the DATA field by
one slot.

An AU serving a real-time device has to act in two distinct modes,
one in setting up or tearing down a circuit and the other in transmitting
and receiving the real-time signals when the circuit is set up. We

outline the procedure, limiting our attention to telephony. Other
devices requiring circuit connections would be served similarly.

Looked at from the telphone, the AU would appear as the line
selector of the standard switch. When the telephone is taken off hook,
the AU would supply dial tone. As the number is dialed, it would be
stored by the AU, which, on completion, would assemble a packet for
transmission to the CC. The DATA field of that packet would disclose
the fact that a telephone link is being sought, and the numbers of the
calling and called stations. The sending procedure for the packet could
follow the routine of Fig. 6, even though a simpler routine is possible
since no “Halt” state is necessary.

The CC would process received requests using a routine that could
be as in Fig. 9. First, the CC would check the total switch capacity
already committed to circuit traffic, and from this it would decide
whether the setting up of the further circuit is permitted. If it is not
permitted, then the CC would inform the originating AU, and that
would terminate the processing. If setting up the circuit is permitted,
then the CC would determine which AU serves the called station and
check whether it is engaged. If it is engaged, then the CC would inform
the originating AU accordingly. If it is not engaged, then the CC would
tag both AUs as engaged and send messages to both AUs and inform

them of each other’s addresses. The two addresses would also be
inserted at appropriate places in ring buffers to cause the necessary
premarking of slots by writing of BUSY and SNDR on the correct
channels at the right frequencies. This would complete the setting up
of the two-way circuit. Given a setup circuit, the dispatch and reception
of the real-time data would follow the routines of Fig. 10.

Fig. 10—Flowchart of (a) real-time signal dispatch, and (b) real-time signal reception.

missions: The receiver is given the sender’s address and recognizes it
for the duration of the call. Apart from saving one address field for
other use, there is a further bonus in that more than one receiver can
be given the same SNDR address and simultaneously receive the same
real-time signal. This leads to the possibility of a simple arrangement
for broadcasting to designated outlets for, say, a public address system.
If, furthermore, the AU’s receiver capability was enlarged to noting
several SNDR addresses and taking in packets with those markings,
then a telephone conference facility, with voice signal summation at
each receiver, would be possible.

The setting up, tearing down, and maintaining of calls to subscribers
outside the switch’s own area would have to interwork with equipment
in other offices. But there is no particular problem about this. The
AU serving a trunk would interface with the outside system, sending
and responding to signals in conformity with existing specifications.
But, in other respects, it would not be different from an AU serving a
local subscriber. Data out of, and into, the local switch area could also
be carried by circuit-switched trunks, with suitable interfacing to a
wider-area data network. The role of an AU providing that interfacing
would then amount to that of a gateway processor.

The bit rate required along the SMs is related to the total peak load
for which the system is designed, multiplied by a factor that accounts
for efficiency. Assuming a telephone voice signal sampled at 8 kHz,

represented by 8 bits per sample and an allowed delay due to packe-
tization of 3 ms, a packet may contain 24 samples or 192 bits of data.
The overheads are mainly in the addresses: Assuming a 10,000-voice-
line switch and a total number of AUs not exceeding 16K, the SNDR
and RCVR fields could be 14 bits each. As we already noted, BUSY
and RQST need be only 1 bit each, and SYNC 2 bits. The total
overhead will then be 32 bits and the packet length will be 224 bits.

Another criterion by which the overall size of a packet can be
decided is efficiency. Since the allowable delay for voice is binding,
the best size indicated for maximum efficiency will be of interest only
if it is smaller than that already decided.

It is a reasonable simplification to suppose that all offered traffic
divides into two categories: very short bursts, and prolonged streams.
Furthermore, it is reasonable to assume that the number of packets-
per-second from the very short burst will be independent of packet
size. Thus, such very short bursts would be produced by single ASCII
characters from, and echoed to, computer terminals, when carried in
individual packets. On the other hand, circuit-switched traffic and
data file transfers are examples of streamed flow.

Consider the total bit rate, R, that results from traffic consisting of
b, short bursts per second, and an aggregated stream flow of S bits per
second. If the packet has h bits of header and x bits of DATA, then

In a system serving a business, one might provide for a busy-hour
voice traffic of 10 ccs (hundred call seconds) per telephone. In the
switch, this will divide equally between the two memories. With 10,000
telephones, the aggregate stream S, on each SM due to voice would
then be

A reasonable assumption for the present is that all other traffic would
amount to 20 percent of the total, or in our example it would be a
further 22 Mb/s.

For the sake of illustration, assume that the very short burst rate,
b, is 20,000 packets per second. If each of these carries only one 8-bit
byte, then the net traffic from them is 160 kb/s, a negligible amount
within the assumed 22 M/bs. But the gross traffic may be much larger,
depending on packet size. Hence, the decision for best size of DATA
field, which, with the numbers already invoked, follows from eq. (4):

For xp to be less than 192 bits, decided by delay considerations, the
very short burst traffic would have to be 4.8 times larger than was
assumed. But the assumed rate is already large, and therefore it is
unlikely that efficiency considerations would indicate a smaller packet
than given from delay.

Given packets of 224 bits and the numbers cited above, the rate, R,
in each memory follows from eq. (3) as 134 Mb/s. If 8-bit bytes are
propagated in parallel, then the required clock rate is 16.75 MHz, a
none too demanding frequency for present technology. The packet
rate, which is of greater relevance to AU and CC speeds, would be 598
kHz.

A switch would be designed for a given ultimate size and given an
appropriate clock rate from the start. But it would not be necessary
to give it immediately the full complement of AUs, nor, indeed, full
lengths of memories. AUs could be added without any disruption and
memory sections with only a minor pause.

Availability of communications services is extremely important and
has prompted switch designers to adopt the very highest standards of
reliability.*'° Thus, it is accepted practice to have two identical central
controllers, one being a “hot” standby that can take over at any
instant. This and other common practices would also apply to our
switch. The features by which our switch is rendered most vulnerable
in respect to reliability are its serial memories, which carry all messages
and are accessed by all AUs. Below we consider the general question
of disruptive impact by failures and suggest a measure for it. Then we
introduce a fault detection and isolation scheme that would make the
robustness of switching by serial memories with multiple read-and-
write taps comparable to that of much more redundant architectures.

In switching equipment, including ours, failures are unequal in
likelihood and in disruptive consequence. We introduce the notion of
expected failure impact. Let 7; be the probability that component C;
will fail during the course of one year; let the expected repair time for
it be 7;,; and the number of potential communication connections that
are unavailable while C; is in the failed state be v;,. We define U;, the
expected per annum failure impact (EPAFI) of C;, as

We assume that failures are statistically independent and disregard
the probability of another component failing during the repair time of

an existing failure. EPAFI values then are additive, and U, with respect
to an assembly of N components, is

U _ > U;. (6)
i=l
We consider the expected failure of the SMs and all the AUs connected
to them. Suppose that there are altogether N AUs, each with (1) a per
annum rate 7; of failing in a way that affects only one subscriber and
takes time 7, to repair, and (2) a rate 72, which disrupts communica-
tions on the memory past the failed AU and takes 7 to repair. Also,
let the memories have a rate 73 of failing at each of the 2N connecting
points, with expected repair times 73.
- With N mutually communicating AUs in the system, the number of
potential two-way communication links is N(N — 1)/2. If an AU
failure of the first kind occurs, then vy; = (N — 1) of these are disrupted.
When the failure of the AU is of the second kind, or when a memory
fails, then the number of disrupted links is much larger and depends
on the actual location of the failure. One can calculate an average
number on the assumption that all potential failure locations are
equally likely. It is found to be approximately (N”)/3. Hence the total
expected impact due to failures of AUs and memory links is

We propose to divide the memories into sections and have a fault
detector at the end of each section. Further, each section would have
a bypass and, in case of a detected failure, a switch would be actuated
to pass on to the next section the data stream at the output of the
bypass (Fig. 11). Thus, effectively, the consequence of the fault is
isolated to one section. A possible realization of the switch is shown
in Fig. 12.

A Fault Detector (FD) would compare the data streams at the
outputs of the memory section and the bypass, and it would decide
that failure has occurred when the evident modification to the stream
in passage through the memory section violates existing constraints.
The particular constraint of the several that exist in our case and
which we use is the following: There may never be a change of any
field that is already nonzero. Detecting any such illegal changes will
catch failures both in AUs and the memories. The detection will, of
course, rely on the output from the bypass being a flawless replica of
what entered that section. If necessary, redundancies and error control
could be implemented on the bypasses to make that more sure.

Fig. 11—Fault isolation using bypasses. Whenever the Fault Detector (FD) detects
constraint violations, it switches input to next section to output of the bypass from the
previous section.

With the memories divided into sections and with fault detection
and isolation in place, failure impacts will be reduced. Unless the fault
is in the fault detector itself, then if each memory is divided into m
sections, only the N/m stations within a section will be affected by a
fault. The disruption depends on which section and which point within
the section is involved. Again assuming equal likelihoods for locations,
we can derive the average number of potential connections that are
disrupted and find this, to a good approximation, amounting to 0.75
(N2/m).

Suppose that fault detectors have their own failure rate 7,4 for
failures that produce an open line and z; for switching off a section
when it should not be. Further, suppose that these have expected
repair times 74 and 75, OF ws = 7474 and ws = 7575. On average, these
events disrupt, respectively, N?/3 and N?/2m potential connections.

The total EPAFI value, with division into m sections and fault
detection and isolation, is then

The first and last terms in eq. (8) are much smaller than the others
and may be neglected. The value of m that results in minimum U,, is

Fig. 12—Realization of two-way switch for fault isolation. FD outputs F and F are
complementary, signifying FAULT and NO FAULT.

and the failure impact, with the first and last terms in eq. (8) neglected,
comes to

This should be compared with the expected failure impact without
sectional detection and isolation, as found in eq. (7). The improvement
ratio is

Further improvement is possible by instituting super sections by which
a number of consecutive sections would be bypassed and again fault-
tested and isolated, as shown in Fig. 13. Indeed, one can take the
hierarchy of protection to any number of levels.

With just one level of protection and, say, N = 10,000, and the
different failure rates and repair times comparable to each other, the
optimum number of sections would be around 185, or 55 AUs to one
section. The improvement over no protection would be by a factor of
40. A second level of protection would increase the improvement by a
further factor of around 5. Asymptotically, as the hierarchy of protec-
tion is taken to higher levels, the functional dependence of EPAFI on
N becomes quadratic, which is the relationship that applies when the
effect of a failure is confined to a single AU.

We have proposed a switch architecture that supports circuit- and
packet-switched communications. Both kinds of communications can
proceed at widely varying rates: Circuits can be set up with different
capacities, selectable as a binary fraction or multiple of a basic capac-
ity, while packet-switched communications share in the pool of the
total switch capacity that is not in use at any given time. Thus, the
proposed switch could cater efficiently in mediating real-time signals

and data. Specifically, it could be a PBX that, apart from voice, could
provide other circuit- and packet-switched services.

The possibility for the two modes is brought about by having
enlarged time slots that can include addressing information, and then
by making these of fixed length so that they can be made available
regularly. Further, the switching is performed by access units that
write and read on serial memories on which synchronization can be
maintained without interruptions and information transfers can occur
without collisions.

We have proposed that data packets be fixed at 192 bits, or 24
samples of pulse-code-modulated voice. This limits the delay due to
packetization to 3 ms. The total delay, which includes propagation
along the memories, will then be less than 4 ms, even in a very large
switch.

It is recognized that a switch used in telephony should conform to
very high standards of reliability. We have proposed a scheme of fault
detection and isolation applicable to memories as in our switch. This
would substantially overcome any added vulnerability due to the serial
nature of the signal paths. However, other issues (e.g., overall system
reliability) not addressed by us remain to be resolved. In summary,
our proposed switch offers the possibility of integrating voice and data
in a way that would preserve the quality and reliability of voice
communications and therefore, in turn, could be integrated with the
telephone system at large. We believe that, provided no compromises
need to be made, very real benefits flow from having all communica-
tions mediated by a common facility. It is possible that our proposed
switch could meet such objectives.

The authors would like to thank M. G. Hluchyj for helpful discus-
sions during early phases of this work.

. H. E. Vaughan, “Research Model for Time-Separation Integrated Communication,”
B.S.T.J., 38 (July 1959), pp. 909-32.

. H. Inose et al., “A Time-Slot Interchange System in Time-Division Electronic
Exchanges,” IEEE Trans. Commun. Syst., CS-11 (September 1963), pp. 336-45.

. A. G. Fraser, “Datakit—A Modular Network for Synchronous and Asynchronous
Traffic,” Proc. ICC (June 1979).

. R.M. Metcalfe and D. R. Boggs, “Ethernet: Distributed Packet Switching for Local
Computer Networks,” Comm. ACM, 19 (July 1976), pp. 395-404.

. G. T. Hopkins and P. E. Wagner, “Multiple Access Digital Communications
Systems,” U.S. Patent 4,210,780, issued July 1, 1980.

. J. O. Limb and C. Flores, “Description of FASNET—A Unidirectional Local-Area
Communications Network,” B.S.T.J., 61 (September 1982), pp. 1413-40.

10. T. W. Forgie and A. G. Nemeth, “An Efficient Packetized Voice/Data Network
Using Statistical Flow Control,” IEEE Commun. Conf. III, 1977, pp. 38.2.44—48.

11. N. F. Maxemchuck, “A Variation on CSMA/CD That Yields Movable TDM Slots
in Pleased Voice/Data Local Networks,” B.S.T.J., 61 (September 1982), pp.
1527-50.

12. G. J. Coviello et al., “System Design Implications of Packetized Voice,” IEEE
Commun. Conf. III, 1977, pp. 38.3.48-53.

13. T. Bially et al., “Voice Communication in Integrated Digital Voice and Data
Networks,” IEEE Trans. Commun., COM-28 (September 1980), pp. 1478-90.

14. D. H. Johnson and G. C. O’Leary, “A Local Access Network for Packetized Digital
Voice Communication,” IKEE Trans. Commun., COM-29 (May 1981), pp. 679-
88.

15. L. Kleinrock, Queuing Systems, Vol. 1: Theory, New York: Wiley-Interscience, 1975,
Vol 2: Computer Applications, New York: Wiley-Interscience, 1976.

16. Z. L. Budrikis, J. L. Hullet, and D. Q. Phiet, “T'ransient-Mode Buffer Stores for
Nonuniform Code TV,” IEEE Trans. Commun. Technol., COM-19 (December
1971), pp. 913-22.

17. E. T. Klemmer, “Subjective Evaluation of Transmission Delay in Telephone Con-
versations,” B.S.T.J., 46 (July-August 1967), pp. 1141-7.

18. R. W. Downing, J. S. Nowak, and L. S. Tuomenoksa, “No. 1 ESS: Maintenance
Plan,” B.S.T.J., 43 (September 1964), pp. 1961-2019.

19. W. P. Karas, “Reliability and Maintainability Improvements Through Distributed
Controls in Communication Systems,” NTC Record 1981, pp A4.4.1-7.

Zigmantas L. Budrikis, B.Sc., 1955, and B.E. (Hons I, Electrical Engineer-
ing), 1957, University of Sydney; Ph.D., 1970, University of Western Australia;
P.M.G. (now Telecom Australia) Research Laboratories, 1958-1960; Aeronau-
tical Research Laboratories, Fishermen’s Bend, 1961; Electrical Engineering
Faculty at University of Western Australia, 1962—. Mr. Budrikis has had a
number of visiting appointments: University of California at Berkeley, 1968;
AT&T Bell Laboratories, 1972, 1973, 1981, 1983, 1984; TU Munich, 1977. He
is interested in problems in communications, man-machine interfaces, and
foundations of electromagnetism. Fellow, IE Australia; member, IEEE, Optical
Society of America, New York Academy of Science.

Arun N. Netravali, B. Tech. (Honors), 1967, Indian Institute of Technology,
Bombay, India; M.S., 1969, Ph.D. (Electrical Engineering), 1970, Rice Uni-
versity; Optimal Data Corporation, 1970-1972; AT&T Bell Laboratories,
1972—. Mr. Netravali has worked on problems related to filtering, guidance,
and control for the space shuttle. At AT&T Bell Laboratories, he has worked
on various aspects of digital processing and computing. He was a Visiting
Professor in the Department of Electrical Engineering at Rutgers University
and the City College. He is presently Director of the Computer Technology
Research Laboratory. Mr. Netravali holds over 20 patents and has had more
than 60 papers published. He was the recipient of the Donald Fink Prize
Award for the best review paper published in the Proceedings of the IEEE
and the journal award for the best paper from the SMPTE. Editorial board,
Proceedings of the IEEE; Editor, IEKE Transactions on Communications;
senior member, IEEE; member, Tau Beta Pi, Sigma Xi.

In most time-shared computer systems a program is processed by the central
processing unit for, at most, a fixed period of time called a time slice, or
quantum. If the program requires more processing after it has received its
quantum, it is placed at the end of a run queue. This procedure is repeated
until the program has finished executing. To the user who submitted the
program the two most important performance measures of such a system are
the mean and variance of the program’s total elapsed time of execution. This
total elapsed time is often referred to as the “response time”. In this paper we
investigate the effect of the quantum size on the mean and variance of the
response time.

The round-robin queue has been studied by several authors as a
model of time-shared computer systems. In a time-shared system, the
arrivals of requests for service as well as the service times may be
thought of as random variables. From the user’s point of view, the two
most important measures of performance in such a system are the
mean and variance of the response time. The round-robin discipline
implicitly favors jobs with shorter service times, in the sense that the
mean response time is approximately a linear function of the service
time.’ Thus far, however, the variance has proved to be intractable in

the case of a general service-time distribution. In the case of exponen-
tial service times, Muntz has found the Laplace transform of the
waiting-time distribution.”

The round-robin model can be described as follows: New arrivals
join the end of the queue, and all jobs in the queue are served on a
first-come first-served basis until they have completed their service
requirement or have received one quantum of service. When a job has
completed service, it leaves the system, and the next job in the queue
begins service immediately. If a job requires more service after receiv-
ing its quantum, it rejoins the queue. In this paper we assume that the
arrival process is Poisson but the service times are governed by a
general distribution. Overhead due to switching between jobs can be
included by adjusting the service requirement. For simplicity of ex-
position we assume that the quantum is constant. Variable quantum
sizes also yield to our method of analysis.

1. What value of the quantum minimizes the mean sojourn time of
a given class of jobs?

2. For a given class of jobs what is the variance of the sojourn time
and what quantum minimizes the variance in the sojourn time?

Here “sojourn time” refers to the total amount of time that a job is
in the queueing system, both in the queue and in service. To answer
the second question, we use a new light traffic-heavy traffic interpo-
lation (which we will call the RS interpolation) developed by M.
Reiman and B. Simon, which makes use of a “light traffic derivative”.

The mean waiting time in a round-robin queue has been studied by
several authors.*° By “waiting” time we mean the total amount of
time that the job spends in the queue (but not in service). In this
section we describe the authors’ analysis and set the notation that will
be used throughout the paper. As mentioned above, we assume that
the arrival process is Poisson with rate )\ and that the service times of
_ newly arriving jobs are independent, identically distributed random
variables with distribution function F and density f. Let gq denote the
quantum size. We say that a job in the system is type j if its service
time requirement as a newly arriving job is between (j — 1)q and jq,

The following are additional notations:
m is the mean service time.
p is the traffic intensity, \m.
p; is the probability a newly arriving job is type j.

m, is the mean amount of time that a type (i, 7) job in the queue
will occupy the server the next time it receives service.

M;; is the second moment of the amount of time that a type (i, /)
job in the queue will occupy the server the next time it receives
service.

w; is the mean amount of time that a job must wait in the queue
in the ith time in the queue.

W is the mean amount of time that an arbitrary job must wait in
the queue before it completes its total service-time requirement.

Using Little’s Law, which takes the form Q, = \p;w; in this case, we
can eliminate Q; from (1) and (2). One can easily verify that the
matrix form of the equations for the a; is

where w is the vector whose ith component is w;, and M and b are a
matrix and a vector, respectively, that are independent of p. Finally,
the mean waiting time, W, is given by

Using the above results we can compute the mean waiting time W?
of a particular class of jobs if we are given the service-time distribution
F, of arrivals of that class:

Figure 1 suggests that, at least in some fairly typical cases, the
quantum size that minimizes the mean waiting time of a preferred
class of jobs is neither very big [which is, in essence, First-In First-
Out (FIFO)] nor very small (processor-sharing), but is just big enough
to let all the preferred jobs complete service without having to feed
back.

In this example, F; is uniform between one and three, F, is uniform
between four and eight, and F = 0.75 F, + 0.25 F». In addition, \ =
0.1, so p = 0.3. Note that gq = 3 minimizes W! over all q, and in
addition, the mean waiting time of the complementary class decreases
with increasing q.

Since the expression for W' is complicated, it is quite difficult to
find the optimal quantum qo without significant computational effort.
The following observation (from numerical experiments) yields a
simple method for finding qo.

Observation: The value of g that minimizes [dW'(0)]/(dp) is close
to the value of g that minimizes W’ (for arbitrary values of p, 0 < p<
1).

Fig. 1—Expected sojourn times as a function of the quantum for (a) type II jobs and
(b) type I jobs.

treatment of the method. Let W,(p), 0 < p <= 1, be the nth moment of
the steady-state waiting time distribution as a function of p, the traffic
intensity. Let W,,(p) = (1 — p)"W,(p) when 0 < p <1, and let W,,(1)
= lim(1 — p)"W,,(p). W,,(1) is well-defined and finite by the argument

Note that W,,(0) = W,,(0) is zero and W,,(1) can be calculated as the
heavy traffic limit using a diffusion process. One can interpolate
between light and heavy traffic to get the approximation formula:

This idea is, of course, well known. The novelty of the RS interpo-
lation is that it makes use of the derivative of W,,(p) at p = 0, which
we will denote by W,(0). It is clear that W/,(0) = W/(0). The RS

and let F; be the service-time distribution of J;, i = 1, 2. The following
theorem allows us to calculate W/(0).

The proof of a more general version of this result is contained in
Ref. 3. We now present a formula for W;,(0) using the above theorem.
A straightforward calculation yields

The queueing system under consideration in this paper is a special
case of the multiclass feedback queue analyzed in Ref. 6. Here we
follow the development in Ref. 7 for the readers’ convenience.

where V(t) = L(t) — t and L(t) is the total amount of work entering
the system in [0, t]. (We assume the system is empty at t = 0.) Define
a sequence of systems whose parameters, and queue-length and so-
journ-time processes are indexed by n = 1, and consider the normalized
processes

over some finite interval, which we normalize to [0, 1] for convenience.
For a heavy traffic limit we assume \") — m7! as n > 0. We have the
result of D. C. Igelhart and W. Whitt (1971).

Here s is the variance in the service times, and a is the variance in
the interarrival times. RBM(d, o”) is one-dimensional reflected
Brownian motion with drift d and infinitesimal variance o’.

in probability.
Theorem 3: Let
Y(t) = sup | Ap-Qy(t) — Aj LPO) |,
where the supremum is taken first over alll <i Sj and1 1’ sj’ and

converges in probability to zero.
Using these results one can prove the following.

jz
1sisj
Thus, from p™ — 1 and Theorem 3 one can prove Theorem 5.
Theorem 5: Under the conditions of Theorem 2,
om” = iT1O

Since we are interested only in the stationary behavior of the
queueing system, we observe that the stationary density of RBM
(a, 8), (a <0) is

2| a| 8 exp(2a6'x).
So the kth moment of the stationary distribution of RBM(a, 8) is
R(2| a] B*)*
fork = 1,2, ---.

Let w'(p), 0 < p <1 be the kth moment of the amount of time a
job waits in the queue the ith time in the queue, and let

In this section, we present numerical examples of two sorts. Figures
3 and 4 demonstrate the accuracy of the interpolation when applied
to the mean sojourn time. Here we are comparing the interpolation
with the exact formula given in Section II. In addition, these examples
tell us that for some typical load situations, the quantum that mini-
mizes the sojourn time for the type I jobs does not seriously degrade
sojourn time for the type II jobs.

Figures 5 through 10 present the approximation to the variance in
the waiting time for some typical choices of service times using the
RS interpolation to approximate the second moment W,.(p) of the
waiting time. An interesting thing here is the similarity between the
mean and variance regardless of the service-time distribution. Notice
that the qualitative properties of the mean and variance of the sojourn
time as a function of the quantum size are quite sensitive to the
service-time distribution. In the case of exponential service times it
appears that the waiting time gets smaller as the quantum gets smaller,

whereas in the deterministic and uniform cases it is best to allow all
the favored jobs to finish in one quantum.
The following notations are used in Figs. 3 through 10.
rho is the overall traffic intensity of the example queue.
p is the proportion of jobs that are type I.
G1 is the service-time distribution of the type I jobs.
G2 is the service-time distribution of the type II jobs.
IeXP(X) means an exponential distribution with mean X.
D means a deterministic distribution with mean X.
U(X,Y) means a distribution that is uniform between X and Y.

I would like to thank Marty Reiman and Burt Simon for many
useful suggestions.

. R. Muntz, “Waiting Time Distribution for Round-Robin Queueing Systems,” Symp.
on Computer-Communications Networks and Teletraffic, Polytechnic Institute
of Brooklyn (April 4-6, 1972), pp. 429-39.

. M. Sakata, S. Noguchi, and J. Oizumi, “An Analysis of the M/G/1 Queue Under
Round-Robin Scheduling,” Oper. Res., 19, No. 1 (March-April 1971), pp. 317-85.

. E. G. Coffman, Jr. and M. I. Reiman, “Diffusion Approximations for Computer/
Communication Systems,” in Mathematical Computer Performance and Reliabil-
ity, : rete T. J. Courtois, and A. Hortijk, eds., New York: North Holland,
pp. 33-53.

D. W. Igelhart and W. Whitt, “Multiple Channel Queues in Heavy Traffic,”
Advance. Appl. Probab., 2 (1970), pp. 150-77, 355-64.

Philip J. Fleming, B.A. (Mathematics), 1974, Wayne State University; M.A.
and Ph.D. (Mathematics), University of Michigan, Ann Arbor, Michigan, 1977
and 1981, respectively; AT&T Bell Laboratories, 1982—. Before joining AT&T
Bell Laboratories, Mr. Fleming was Assistant Professor of Mathematics and
Statistics at Case Western Reserve University in Cleveland, Ohio, from 1981
through 1982. Mr. Fleming joined AT&T Bell Laboratories as a Member of
Technical Staff in the Systems Design and Exploratory Development depart-
ment. His work has been in the area of computer performance modeling and
analysis, queueing theory, and cryptography as it relates to computer security.
He is currently a member of the Operating System Architecture department.
Member, AMS and ORSA.

An analysis of an integrated voice-data network with Demand Assignment
Time Division Multiple Access (TDMA) is presented using the following
model: (1) voice calls that cannot be serviced are blocked, whereas requests to
transmit data messages are queued; (2) no traffic boundaries are assumed, i.e.,
any new traffic arrival may be assigned to any unassigned time slot; (3)
message lengths are exponentially distributed with the mean voice message
length assumed to be much larger than the mean data message length; (4)
traffic requests are generated according to two independent Poisson processes;
and (5) time slot assignments are made instantaneously and no priorities are
assumed. Such a model applies to a single-channel TDMA network in which
voice and data traffic arrivals are serviced on a first-come first-served basis.
An approximate analysis, based upon physical insight, is presented that yields
the blocking probability for voice messages, the mean number of queued data
requests, and the mean value of the peaks of the data queue process. Compar-
isons with simulation results indicate that the analytical results are very
accurate. Performance curves are presented and compared with analogous
results for TDMA networks that handle only one traffic type.

The popularity of integrated voice-data networks has motivated
numerous analyses of associated network queueing models.’’ This
paper presents an analysis of a voice-data network using Demand
Assignment Time Division Multiple Access (DA/TDMA). This work

* AT&T Bell Laboratories; present affiliation Bell Communications Re-
search, Inc.

Copyright © 1984 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.

evolved from a study of a multichannel DA/TDMA protocol that
handles different traffic types, and that has received much attention
in the context of satellite communications.®® These networks typically
have a few hundred megabits of capacity and can be used to carry
real-time digital voice traffic in addition to data traffic. Analysis in
Ref. 10 indicates that a large multichannel TDMA network (i.e., a
network with more than 100 communicating traffic nodes) behaves
much like a single-channel TDMA network with an equivalent number
of time slots per frame. We therefore concentrate on the simpler
single-channel network and attempt to characterize its performance
when used to handle both real-time voice and data traffic. The results
in this paper carry over to the multichannel case when the number of
traffic nodes in the network is large.’°

A TDMA protocol divides the broadcast channel into a series of
time slots of identical width. A prespecified number of time slots forms
a TDMA frame that continually repeats itself. A demand assignment
protocol assumes that when a traffic source has a message to transmit,
it must first send a message to a central controller indicating that it
wishes to transmit a message to a specified destination address. The
central controller assigns specific time slots to each received request
on a noninterfering basis. Only one time slot per frame is assigned to
each request. Once a time slot is assigned, it remains assigned to the
same traffic source for the duration of the message. We therefore
assume that each data message consists of a variable number of
packets, where the length of a packet is the number of bits per time
slot. Voice calls are assigned a dedicated time slot for the duration of
the call. (Full-duplex voice traffic actually requires two time slots per
frame, one for each direction.) Figure 1 shows an example in which
there are four time slots per frame. The numbers in each slot specify
the source and destination addresses. The controller has assigned slot
1 to the source-destination pair 1-2. Since the message generated by
source 1 requires more than one time slot, source 1 also uses the first
time slot in the succeeding frame. The number of time slots per frame
and number of bits per time slot are design parameters that can vary
from system to system. Notice, however, that an additional constraint
might be that the number of frames per second and number of bits

per time slot must be selected so that each time slot represents a 64-
kb/s circuit, which is necessary to provide toll-quality PCM voice
transmission.

As the source traffic intensity increases, so does the probability of
not being able to assign a new voice or data request because all slots
have already been assigned. New voice requests that cannot be as-
signed are blocked, whereas unassigned data requests enter a queue.
As slots become free, requests from the queue are then assigned. The
netwerk model analyzed here is different from the models used in
Refs. 1 through 7 in one or more of the following respects: (1) Each
channel time slot can be assigned to either traffic type. In particular,
no moving boundaries’®’ are assumed that partition the time slots in
each frame into a section reserved for voice traffic and a section
reserved for data traffic. (2) The duration of each message measured
in number of time slots, or equivalently, the number of TDMA frames,
is an exponentially distributed random variable. Furthermore, the
mean voice message length is assumed to be at least an order of
magnitude larger than the mean data message length. (3) No priorities
are assumed, so that traffic is serviced on a first-come first-served
basis.

To simplify the analysis, approximations based upon physical in-
sight are used. Results are expressions for voice blocking probability,
mean number of queued data requests, and the mean value of the
peaks of the data queue process as functions of the number of time
slots and traffic parameters. Comparisons with simulation results
indicate that these analytical results are quite accurate.

The TDMA network is modeled as a c-server queueing system,
where c is the number of time slots per TDMA frame. We assume
voice and data traffic requests to be generated according to two
independent Poisson processes with respective arrival rates \, and dq.
We therefore implicitly assume an infinite source model. Service times
(or message lengths) in both cases are assumed to be exponentially
distributed, with service rates yu, for voice messages and pa for data
messages. In particular, the mean number of time slots required for a
voice message is 1/y,. Notice that in practice the service distribution
must be discrete, since messages can only last an integral number of
time slots. The continuous distribution assumed here is a good ap-
proximation to a “discrete” exponential distribution as long as the

Each voice or data arrival demands one time slot per frame for the
duration of the message. The rates (i.e., bits per second) at which both
voice and data messages access the channel are therefore identical.
Voice messages that cannot be assigned a time slot immediately are
blocked (i.e., disappear), whereas unassigned data requests enter a
queue. To simplify the analysis, we assume that time slot assignments
occur instantaneously, rather than at the beginning of the next frame.
In particular, as soon as a request is generated, it is immediately
assigned, assuming unassigned slots exist. This approximation is rea-
sonable as long as the average message lengths are much greater than
one time slot. Given that there are queued data requests, they are
assigned as soon as time slots are relinquished by traffic already
assigned. Voice messages that arrive while data requests are queued
are therefore blocked. As either the voice or data traffic intensity
increases, the data traffic tends to grab enough time slots to empty
any queue that may appear, thereby depriving voice traffic of available
time slots. One therefore expects that under high traffic intensities,
voice blocking probability is quite high, whereas the mean data queue
length (i.e., number of queued requests) is moderate.

An exact analysis of the model just described can be performed by
extracting the associated embedded Markov chain. In this case, the
two-dimensional state defining the embedded Markov chain is (d, v),
where d and vu represent the number of data and voice messages,
respectively, in the system. Transition probabilities are easily deter-
mined in terms of the traffic parameters and number of time slots,
thus a solution for the steady-state probability distribution p(d, v) can
be theoretically obtained. If we assume that the data message queue
can be arbitrarily large, however, the number of states in the Markov
chain becomes infinite. As the dimension of the problem increases,
the amount of numerical computation required to obtain p(d, v)
increases, which in turn causes further propagation of round-off errors.
The exact analysis just outlined was attempted."! However, it was not
successful due to finite word-length effects. The approximate analysis
in the next section is therefore proposed as a simple alternative.

We start by deriving an approximate expression for voice blocking
probability. In steady state, the minimum number of time slots per
frame needed to ensure that the system remains stable (i.e., the number
of queued data requests does not become infinite with probability one)
is

where [x | denotes the largest integer less than x. If the number of
slots is less than this amount, then the system would be unstable even
with no additional voice traffic. If the number of slots is greater than
this amount, then the system must be stable since unassigned voice
requests disappear, and because voice requests cannot preempt queued
data requests. All queued data requests must therefore be assigned
before any voice messages can be assigned.

The number of time slots per frame available for data messages is a
random process that varies according to how many time slots are
assigned to voice traffic. Assuming that the mean service time for
voice messages (1/u,) is orders of magnitude greater than the mean
service time for data messages (1/ua), the voice “state”, i.e., number
of voice-occupied slots per frame, varies much more slowly than the
data state, which is the total number of data messages present in the
system. It is therefore a good approximation to assume that the time
spent in each voice state is much longer than the time it takes the
number of data requests present in the system to reach steady-state
behavior. This “steady-state approximation” is the basis for the anal-
ysis that follows. Using this approximation, it follows that if the
number of voice messages in the system is greater than or equal to

the normalized data traffic intensity conditioned on the number of
voice-occupied slots, Ag/[(c — Vo)]ua, is greater than one, and data
requests become queued with probability one. Since the number of
queued data requests is assumed to reach steady-state behavior, this
queue cannot be emptied until a voice message relinquishes a time
slot. The steady-state approximation therefore implies that the num-
ber of voice messages in the system is never greater than U9. Computer
simulations of the queueing model considered have verified that the
probability of the voice state v becoming greater than Up is indeed very
small when yp, is much less than yg. The “competition” of voice and
data traffic for available time slots can therefore be expected to
produce intermittent queue “spikes,” representing times at which the
voice state v = Up. During this time, the data queue process experiences
a “transient instability.”

Given that v time slots are assigned to voice traffic, the probability
that an incoming voice message is blocked is equal to the probability
that the number of data requests in the system, d, is greater than or
equal to c—v. Using the steady-state approximation, this is simply

the steady-state probability that a queue exists in an M/M/c—v
queueing system and is given by”

where p(v) is the probability that v time slots are assigned to voice
traffic.

Consider now a blocking system with up servers and one Poisson
input. Given v < Uo, let ¢, denote the probability that a new arrival
can be served (i.e., even though all servers are not busy, a newly
arriving request is blocked with probability 1 — ¢,). Assuming expo-
nential service times, the probability that v servers are busy is known

and is given by (2). Substituting (2) and (5) into (4) therefore gives
the desired result. Notice that because the arrival processes are Pois-
son, the blocking probability Ps is equal to the probability of being in
a blocking state (i.e., all time slots are busy), which is equal to the
probability that data requests are queued.

An analogous argument can be applied to compute the mean number
of queued data requests. Denoting this queue length as g, we have

where E(q|v) is the mean number of queued data requests given v
assigned voice messages. Assuming py < pg implies that E(q|v) is

approximately equal to the mean queue length for an M/M/c —v
queueing system,’” i.e.,

where Po is given by (3). If v = Uo, this expression no longer applies,
however, since the system becomes unstable. To approximate E(q| vo),
note that when v = Uo, the mean data queue length increases approx-
imately at rate \g — (c — Uo)ug until a voice message relinquishes its
time slot. At this point the queue starts to empty at rate (c — Up + 1) ug
— qg. As the queue empties, more time slots may be relinquished by
voice messages, causing the queue to empty at a faster rate. Suppose
that we assume

where to is the duration of the queue spike and q(t), 0 St S to, is the
queue length as a function of time (given that v increased from Up — 1
to Up at t = 0). Furthermore, we assume that E[q(t)] is piecewise linear
(fluid flow approximation).* Then it is shown in Appendix A that

Substituting (5), (8), and (10) into (7) therefore gives the approximate
mean queue length.

The expressions for voice blocking probability and mean data queue
length presented thus far have been found to be quite accurate when
compared with simulation results. To gain further insight into the
behavior of the system, however, we now attempt to characterize the
transient instabilities, or queue spikes, which occur when v = Uo.

Figure 2 illustrates the problem under consideration. The queue
spikes appear whenever the voice state is equal to ug. Not shown are -
queues that occur when the voice state v is less than uo. The height of
each spike is denoted as dy, the duration of each spike is denoted as
Tw, and the time between spikes is denoted as T’. Figure 2 is not meant
to indicate a typical sample function for the data message queue
process. Significant queues appear when v < Up; however, the purpose
of the following analysis is to determine whether the transient insta-
bilities shown in Fig. 2 cause serious performance degradation.

We begin by computing the distribution of the peak value of each
spike. Let p(d,t) denote the probability that d data messages are
queued at time t, given that v = vp. At t = 0 we assume d = 0. The
following equations can be derived in a straightforward manner,”

Solving (14) gives the probability that the maximum queue length is
equal to d given the time until the first voice departure is t. (Recall
that as soon as v decreases from Up to Up — 1, the mean queue length
decreases.) We know, however, that the time until the first voice
departure is exponentially distributed with parameter upp,, so that

is the Laplace transform of p(dy,t). Equations (14a) and (14b)
are standard “birth-death” equations.’2 The Laplace transform,
Qu(s, d), can be computed directly from (14) and is given by

The distribution of the maximum of the queue spike is therefore
geometric with parameter re(Upu,). Consequently,

A sample path of the number of queued data requests versus time is
shown in Fig. 3. (Every 10th sample is plotted.) Queues periodically
build and empty, which suggests that as an approximation, the mean
value of the peaks of these buildups is given by (23). This approxi-
mation becomes more accurate as yp, decreases relative to pg. Note,
however, that (23) gives the mean value of the peaks of the queue
process with initial conditions v = vp and d = 0. It is possible that the
mean value of the peaks of the queue process, assuming v < Ug, is
larger than that predicted by (23). [In some cases, the conditional
mean queue length given by (8) for v = Up — 1 is in fact larger than
E(dy).] A better approximation to the mean value of the peaks of the
queue process can be obtained by computing the conditional means
assuming v = 0, 1, --+ , Uo, and then using the distribution p(v) given
by (5) to form a weighted average. As the present analysis is concerned
with evaluating the performance degradation caused by transient
instabilites, this computation was not performed.

We now compute the mean duration of the queue spike. Denoting
this quantity as 7,,, it is apparent that

where 7,,; is the mean time it takes the queue to reach its maximum
value given v = Uo, and 7». is the mean time it takes the queue to
empty. From the previous discussion,

Let 77, denote the mean time it takes to reach state d = 0 (i.e., no
data message queue) given an initial state of d queued data messages
and v voice-occupied time slots. We have the following transition
equation,

and 1/[c(v)] is the mean amount of time spent in state (d, v) before a
state transition occurs, and

are, respectively, the probabilities of going to states (d — 1, v), (d +
1, v), and (d, v — 1) from state (d, v). Equation (27) is a two-dimen-
sional difference equation that is nonlinear in v. Notice, however, that
we desire

is the partial z-transform of 73,. An iterative method for computing
D(1/[re(vony)], Vo — 1) is discussed in Appendix B.

To compute the mean time between unstable periods, we first define
T., as the expected value of the first passage time it takes to go from

an initial voice state v < Up to voice state Up. The mean time between
unstable periods is then

where p*(v) is the probability that the voice state is v at the end of a
queue spike (i.e., when d returns to zero). The computation of p*(v) is
similar to the computation of 7,,. In particular, let p(v|v1, d) denote
the probability of v voice-occupied time slots at the end of an unstable
period given an initial state (v,, d). The following transition equation
is easily obtained,

In analogy with (27), (33) is a two-dimensional difference equation
that is nonlinear in v. As before, we desire

D(z, vj) = Y 2@p(v|u, d). (35)
d=0
An iterative method for computing D(z, v | v,) is discussed in Appendix
B.

The mean time between unstable periods, given the initial starting
state, can be approximated by again assuming voice traffic service
times are very long relative to data traffic service times. For each voice
state, we assume that the data traffic exhibits steady-state behavior.
This leads to the following difference equation,

(37a)
is the mean time spent in voice state uv before going to state v — 1 or
v + 1, ¢, is the “entrance” probability for an incoming voice message
and is equal to the probability that a data message queue exists, and

are, respectively, the probabilities of going from voice state v to state
v — 1 and from voice state v to state v + 1. In Appendix C we show

Therefore, computation of T, and p*(v) by way of (38) and the
method given in Appendix B, respectively, yields the mean time
between unstable periods. The relative frequency, or probability, that
the system is in an unstable state (v = uo) is approximated by

where 7,, and T are given by (25) and (32), respectively. Notice that
this expression should be approximately equal to the value of p(v9)
obtained using (5).

This completes the presentation of analytical results. These results
are used in the next section to evaluate the performance of an inte-
grated voice-data TDMA network, and to demonstrate the improve-
ment over analogous TDMA networks that handle only one traffic

measured by the number of time slots. In all cases, half-duplex voice
traffic is assumed. We expect the full-duplex case, where each voice
message requests two time slots, to exhibit similar behavior. Denoting
the voice traffic intensity as p, = \,/(cu,), and the data traffic intensity
as pa = Xa/(cug), where c is the number of time slots, the total
normalized offered load is defined as

Initial traffic parameters are selected to produce a preselected value
of r, and the input traffic intensity p is varied between zero and one
by multiplying both \, and dg by a constant. The service rates used
are pu, = 0.001 and pg = 0.025, which corresponds to a mean voice
message length of 1000 time slots and a mean data message length of
40 time slots. In practice, the mean voice message is much greater
than 1000 time slots; however, the analytical results in the last section
become more accurate as p, decreases relative to pg.

Figures 4a, 4b, and 4c show voice message blocking probabilities
computed by means of (4) versus the offered load for systems with 20,
100, and 500 time slots, respectively. In each case, curves are shown
for three different traffic blends. A few randomly selected points from
Figs. 4a, 4b, and 4c were compared with results obtained by a computer
simulation of the queueing model under consideration. In each case
the approximate result and the computer-simulated result were nearly
identical. Also shown are plots of blocking probabilities versus load
produced by systems that handle only (half-duplex) voice traffic and
which have rc time slots, where c is the number of time slots in the
voice-data network. These curves are computed directly from (5),
where ¢; = 1 forj < rc.

At high traffic intensities, blocking probabilities for the integrated
systems are consistently higher than those for the analogous single
traffic systems. In this region, data queues often form, causing data
messages to grab time slots relinquished by voice messages. In contrast,
at lower traffic intensities, data message queues are less likely, so that
voice messages often have access to additional time slots. At low traffic
intensities, integrated systems therefore always exhibit superior per-
formance when compared with analogous voice-only systems. In each
comparison, there appears to be a unique traffic intensity, p*, where
both systems give the same blocking probability. Notice that as the
traffic blend r decreases, p* increases. This is due to the fact that the
probability of a data message queue existing at a fixed, normalized
traffic intensity decreases as the number of time slots allocated for

Fig. 4—Voice message blocking probability versus offered load for integrated systems
with: (a) 20 slots and single traffic systems with 2, 10, and 18 slots; (b) 100 slots and
single traffic systems with 10, 50, and 90 slots.

data traffic increases. As r decreases, voice messages therefore often
have access to additional time slots not used by data traffic. At a fixed
traffic intensity, as r decreases, the blocking probability produced by
the integrated system should therefore decrease, relative to the block-
ing probability produced by the analogous voice-only system. A final
observation is that as the traffic intensity decreases, blocking proba-

g.4(c)—Voice mess age. blocking probability versus offered load for integrated
Ree with 500 slots and single traffic systems with 50, 250, and 450 slots.

bilities obtained using the integrated systems become insensitive to
the traffic blend. This is in contrast to the analogous voice-only
systems, which result in a much wider variation in blocking probabil-
ities as the number of time slots is varied.

Figures 5a, 5b, and 5c show plots of mean data message queue length
(number of queued data requests), computed by means of (7), (8), and
(10), versus normalized offered load for systems with 20, 100, and 500
time slots, respectively. Curves are again shown for three different
traffic blends. Also plotted is the mean number of queued data requests
produced by a system handling data traffic only with c time slots.
Close agreement was again found between randomly selected points
from these figures and computer simulation results. At high traffic
intensities, the variation between curves is caused by the different
traffic loads, at which the mean data queue length approaches infinity.
In particular, the single traffic curve has its asymptote at p = pg = 1.
In contrast, because queued data messages can grab relinquished voice-
occupied time slots, the integrated traffic curves have asymptotes at
pa = 1, which corresponds to p = 1.1, p = 2, and p = 10 for r = 0.1,

0.7 0.8 0.9 1.0 1.1
OFFERED LOAD
Fig. 5—Mean data message queue length versus offered load for integrated systems

Fig. 5(c)—Mean data message queue length versus offered load for integrated systems
and a single traffic system with 500 slots.

r = 0.5, and r = 0.9, respectively. Mean queue length in these high
traffic intensity regions is not of interest, however, since the corre-
sponding voice blocking probability is near unity.

Figures 6a, 6b, and 6c show plots of E(dy) given by (23) versus
offered load, which indicates the mean value of the maximum of queue
buildups. The discontinuities in Fig. 6a are caused by discontinuous
changes in the voice instability state up as a function of traffic load.
As an example, for the case c = 20 and r = 0.5 shown in Fig. 6a, Uo
changes from 14 to 13 as the traffic intensity p increases from 0.7 — «
to 0.7, where « is small. Discontinuities were observed in all curves
shown in Fig. 6; however, in most cases these discontinuities were
hardly noticeable. In particular, as the number of slots c increases,
E(dm) becomes less sensitive to changes in Up. A comparison of the
results in Figs. 6a, 6b, and 6c with simulated sample paths of the data-
message queue process indicates that the results presented here are
typically about 10 to 25 percent smaller than the actual peaks observed,
indicating that the peaks that occur when v < up are often greater
than those that occur when v = Ug. As p, decreases relative to ua,

Fig. 6—Approximate mean value of queue peaks versus offered load from eq. (23) for
integrated systems with: (a) 20 slots and (b) 100 slots.

Fig. 6(c)—Approximate mean value of queue peaks versus offered load from eq. (23)
for integrated systems with 500 slots.

however, E(dy) given by (23) should give a more accurate indication
of the mean value of these peaks.

For each point computed in Figs. 4 through 6, the probability of
being in an “unstable” state p(vo), the mean duration of the unstable
state (7,), and the mean time between unstable states (TJ) were
calculated by means of the results in Section 3.1. A few representative
points are listed in Table I. The probability of being in an unstable
state and the resulting queues that form are typically too small to
cause significant degradation in system performance.

Results obtained from the analysis of an integrated voice-data
TDMA protocol indicate that voice message blocking probability in
the integrated system is insensitive to the blend of traffic at low input
traffic intensities. As the traffic intensity increases, the blocking

Table |!—Mean duration of unstable state (7,, in number of frames),
mean time between unstable states (T in number of frames), and the
probability of being in an unstable state [ p(vo)] for a few
representative cases (assuming r = 0.5)

probability of the integrated system increases, relative to the blocking
probability of the analogous voice-only system. The traffic intensity
at which the two blocking probability curves intersect is a function of
the traffic blend. Mean queue length in the integrated system displays
a wide variation with traffic blend at high traffic intensities, due to
the variation in traffic intensity at which instability occurs. At offered
loads of 0.7 to 0.8, very good performance can be achieved (i.e., a
blocking probability <0.01 and a mean queue length near zero) with
moderately sized systems (~100 slots/frame). Finally, the data mes-
sage queues that form during the unstable transients are moderate for
the cases examined, and the frequency at which these transients occur
is in most cases quite small. As the number of time slots per TDMA
frame increases, results presented here show significant improvements
in system performance. This is an important observation since most
practical networks are much larger than those considered here (i.e.,
greater than 1000 time slots per frame).

The author thanks S. M. Barta and B. E. Simon for many helpful
conversations concerning this work.

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We wish to show (10) using the approximation (9). In addition, we
assume that E[q(t)] is piecewise linear. (This would be true, for
instance, if the data message queue length q(t) were allowed to assume
negative values.) At time t = 0 we assume that the state of the system
is (Uo, C — Uo). Let t,(i) denote the mean time it takes i + 1 voice
messages to relinquish their time slots. Then

A plot of E[g(t)], 0 s t S ty), assuming three voice departures (ip =
2) is shown in Fig. 7. If we use (9), it follows that

1 igt1
E(q|v0) =— X Aj, (52)
; to j=0
where A; is the area of region R;. It is apparent that
1 qu _ 1 da (¢ — Vo)Ma
Ao == = - —_, 53
© 2 t(0) 2 (Vote)? me
and that
1 nace
Ain = 3 [to — ty(to) Gi,
~92
Jip
a a ATE 54
2[(c — Vo + lo + 1)ua — dal ee)
Finally, regions R,, ---, R;, are trapezoids with upper-boundary

We are interested in computing D{1/[re(vou,)], vo — 1} and D{1/
[ro(Vopy)], v| vi}, where D(z, v) and D(z, v|v;) are given, respectively,
by (81) and (35) and re(vouy) is given by (18b). Multiplying both sides
of (27a) by 2? and summing from d = 1 to infinity gives, after
algebraic manipulation,

enn {o(v) + Vo7(v) — 4(c — v)padaf, (58)
where k = 1 corresponds to “+”, k = 2 corresponds to , and o(v) is
given by (28). Notice that the roots r;(Uo) given by (58) are identical
to the roots r;(Vouy) given by (18). For convenience, we therefore refer
to the roots r;,(Vo“y) as Te(Uo), for k = 1, 2. From Rouche’s Theorem,”
it follows that re(v) <1 andr,(v) = 1. The z-transform, D(z, v), must
be analytic outside the unit circle, and hence 7,, in (56) must be
selected to cancel the pole at 1/[r2(v)]. In particular, s(v, z~') has roots
1/[ri(v)] < 1 and 1/[re(v)] > 1, so that

As an example, suppose that we assume the data message queue
empties with probability one before k + 1 voice messages relinquish
their time slots. Once the voice state becomes v = Up — k — 1, the voice
service rate nu, = 0. Substituting these values for u, and v in (56) gives

Equation (62) constitutes an initial condition for (56), which can be
iterated numerically using (59). In particular, assuming no more than
k voice messages can relinquish their time slots after the queue begins
to empty, D(z, vo — k — 1) for z = 1/[re(vo)] and z = 1/[re(vo — kR)] is
calculated from (62). The value of 71,,-» is subsequently computed
from (59) and is used in (56) to compute D(z, vo — k) for z = 1/[re(vo)]
and z = 1/[re(vo — k + 1)]. Equation (59) is subsequently used to
compute 71,,,-r+1, Which is used to compute D(z, up — k + 1), and so
forth until D{1/[r2(vo)], vo — 1} is computed.

To compute D(z, v| vi) given by (35) at z = 1/[re(vo)], we multiply
both sides of (33a) by z~¢ and sum from d = 1 to infinity to get

, (64)
where 6;; is the Kronecker delta. Using the condition (33b) gives the
boundary condition

Because D(z, v|v;) must be analytic outside the unit circle, p(v | v;, 1)
is selected to cancel the pole at z = 1/[r2(vo)]. This implies that

++, Uo — Rk — 1, where k is the maximum number of voice departures
allowed, the boundary condition (65) is first used in (66) to get

where j is initially one and ranges from one to k + 1. This expression
is evaluated at z = 1/[re(vo — j)] and substituted into (66) to obtain
p(Vo ~ J | Vo —j + 1, 1), which is used in (64) to compute D(z, vp — j| vo
—j+1) at the appropriate values of z. This procedure continues until
the value of D{1/[re(vo)], vo — j | Vo — 1} is obtained, whereupon j is
incremented and the procedure starts over again. In this way the
values D{1/[re(vo)], vo — J | Vo — 1} for j = 1, 2, --- ,k +1 are generated
systematically.

Xv (v ell (es 1)!y, Av m=0 \Av (v — m)\yy
where y, is given by (39). Using the initial condition,
= sa 1
=T,-T)=-——, 12
Bal 1 0 eR (72)

Michael L. Honig, B.S. (Electrical Engineering), 1977, Stanford University;
M.S. and Ph.D. (Electrical Engineering), 1978 and 1981, respectively, Univer-
sity of California, Berkeley; Bell Laboratories, 1981-1982; AT&T Information
Systems, 1983. Present affiliation Bell Communications Research, Inc. At
AT&T Bell Laboratories and AT&T Information Systems Mr. Honig worked
on modulation, coding, and echo cancellation of voiceband data signals,
performance analysis of local area networks, and office information systems.
He is now a member of the Communications Principles Research group in the
Applied Research Area of Bell Communications Research, Inc. Member, IEEE,

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On the Application of Energy Contours to the Recognition of Con-
nected Word Sequences
L. R. Rabiner

Spatial Filtering Radio Astronomical Data: One-Dimensional Case
H. E. Rowe

1982/83 End Office Connection Study: ASPEN Data Acquisition
System and Sampling Plan
J. D. Healy, M. Lampell, D. G. Leeper, T. C. Redman, and
K. J. Vlacich

1982/83 End Office Connection Study: Analog Voice and Voiceband
Data Transmission Performance Characterization of the Public
Switched Network

magazine :: Bell System Technical Journal :: BSTJ V63N08 198410 Part 1