Pulse Transmission by AM, FM and PM in the Presence of Phase
Distortion E. D. SUNDE 853

Farther Results on the Detectability of Known Signals in
Gaussian Noise H. C. MARTEL AND M. V. MATHEWS 423

Confocal Multimode Resonator for Millimeter Through Optical
Wavelength Masers G. D. BOYD AND J. P. GORDON 489 _

Relation Between Surface Concentration and Average Conductivity
in Diffused Layers in Germanium P. B, CUTTRISS 509

A General Method of Applying Error Correction to Synchronous
Digital Systems D. 4%. ARMSTRONG 577

On the Construction of Minimally Redundant Reliable System
Designs D. K. RAY-CHAUDHURI 595

Mode Conversion in Metallic and Helix Waveguide 4. a. UNGER 613
Winding Tolerances in Helix Waveguide H. G. UNGER 627

H. 1. ROMNES, President, Wesiern Eleciric Company
J. B. Fisk, President, Bell Telephone Laboratories

E. J. McNEELY, Executive Vice President, American
Telephone and Telegraph Company

G. E. SCHINDLER, jR., Editor
. M. FOSTER, jR., Assistant Editor
. POLOGE, Production Editor
. T. MYSAK, Technical Illustrations
. N. POPE, Circulation Manager

THE BELL SYSTEM TECHNICAL JOURNAL is published six times a year
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In pulse transmission systems, pulses modulated in various ways to carry
information may be transmitted by amplitude, phase or frequency modula-
tion of a carrier, and with each type of modulation various methods of de-
tection are possible. An important consideration in many applications is
the performance of various modulation and detection methods in the pres-
ence of phase distortion or equivalent envelope delay distortion, which may
be appreciable in certain transmission facilities. The principal purpose of
this presentation ts a theoretical evaluation of transmission impairments
resulting from certain representative types of delay distortion. These trans-
mission impairments are reflected in the need for increased signal-to-noise
ratio at the detector input to compensate for the effect of delay distortion.

The performance in pulse transmission by various carrier modulation
and detection methods can be formulated in terms of a basic function com-
mon to all, known as the carrier pulse transmission characteristic, which is
related by a Fourier integral to the amplitude and phase characteristics of
the channel. Numerical vaiues are given here for the carrier pulse transmis-
ston characteristic with linear and quadratic delay distortion, together with
the maximum transmission impairments caused by these fairly representa-
tive forms of delay distortion with various methods of carrier modulation
and signal detection. These include amplitude modulation with envelope
and with synchronous detection, two-phase and four-phase modulation with
synchronous detection and with differential phase detection and binary fre-
quency modulation.

In determining the effect of delay distortion, a raised cosine amplitude
spectrum of the pulses at the detector input has been assumed in all. cases,
together with the minimum pulse interval permitted with this spectrum and
ideal implementation of each modulation and detection method. Further-
more, optimum adjustments from the standpoint of slicing levels and sam-
pling instants at the detector output are assumed for each particular case of
delay d’stortion. These idealizations insure that only the effect of delay dis-
tortior. is evaluated and considered in comparing modulation methods, and
that this effect is minimized by appropriate system adjustments.

Signal-to-Noise Ratios in Binary FM
9 Slicing Levels and Noise Margins
10 Evaluation of Transmission Impairments
yvnehronous AM and PM
General
Synchronous AM and Two-Phase Modulation
Quadrature Carrier AM and Four-Phase Modulation
Even Symmetry Pulse Spectrum and Delay Distortion
Raised Cosine Spectrum and Quadratic Delay Distortion
Even Symmetry Spectrum and Odd Symmetry Delay Distortion
Raised Cosine Spectrum and Linear Delay Distortion
Vestigial-Sideband vs. Quadrature Double-Sideband AM
Envelope Detection vs. Synchronous Detection
with Differential Phase Detection
General
Basic Expressions
Even Symmetry Spectrum and Delay Distortion
Two-Phase Modulation
Four-Phase Modulation
Raised Cosine Spectrum and Quadratic Delay Distortion
vary Frequency Modulation (FSK)

Raised Cosine Spectrum and Linear Delay Distortion
ummary
6.1 General
3.2 Choice of Transmission Delay Parameters
3 Double-Sideband AM
3.4 Vestigial-Sideband AM and Quadrature Double-Sideband AM

PM with Synchronous Detection
PM with Differential Phase Detection
Binary FM
3.8 Comparisons of Carrier Modulation Methods
VII. Acknowledgments
Appendix. Determination of Carrier Pulse Transmission Characteristics. .
References

Binary pulse transmission by various methods of carrier modulation
has been dealt with elsewhere on the premise of ideal amplitude and
phase characteristics of the carrier channels.’ An important considera-
tion in many applications is the performance in the presence of phase
distortion or equivalent envelope delay distortion, which may be ap-
preciable in certain transmission facilities. An ideal amplitude spectrum
of received pulses can be approached with the aid of appropriate terminal
filters with gradual cutoffs, such that the associated phase character-
istic is virtually linear. Nevertheless, pronounced phase distortion may
be encountered in pulse transmission over channels with sharp cutoffs
outside the pulse spectrum band, as in frequency division carrier system
channels designed primarily for voice transmission.

The principal purpose of the present analysis is a theoretical evalua-
tion of transmission impairments resulting from certain representative
types of delay distortion in pulse transmission by various methods of
carrier modulation and signal detection. These transmission impair-
ments are reflected in the need for increased signal-to-noise ratio at the
detector input to compensate for the effect of delay distortion.

The performance in pulse transmission by various carrier modulation
and detection methods can be related to a basic function known as the
carrier pulse transmission characteristic. This basic function gives the
shape of a single carrier pulse at the channel output, i.e., the detector
input, under ideal conditions or in the presence of the particular kind of
transmission distortion under consideration. From this basic function
can be determined the envelopes of carrier pulse trains at the detector
input, together with the phase of the carrier within the envelope. The
shape of demodulated pulse trains with various methods of carrier
modulation and detection can, in turn, be determined for various com-
binations of transmitted pulses, together with the maximum transmis-
sion impairment from a specified type of channel imperfection, such as
delay distortion dealt with here.

The carrier pulse transmission characteristic is related by a Fourier
integral to the amplitude and phase characteristic of the channel. It

has been determined elsewhere’ for pulses with a raised cosine spectrum

and cosine variation in transmission delay over the channel band, and
for pulses with a gaussian spectrum with linear variation in delay. A
cosine variation in delay is approximated in some transmission facilities
and has certain advantages from the standpoint of analysis, both as
regards numerical evaluation and interpretation in terms of pulse echoes.

A somewhat similar form of delay distortion that affords a satisfactory

approximation in many cases is quadratic (or parabolic) delay distor-
tion. Quadratic delay distortion is in theory approached near midband
of a flat bandpass channel with sharp cutoffs, such as a carrier system
voice channel, and usually affords a satisfactory approximation over the
more important part of the transmission band of such channels. Linear
delay distortion is approximated when a bandpass channel with gradual
cutoffs is established to one side of midband of a flat bandpass channel
with sharp cutoffs. These and other types of delay distortion do not
lend themselves to convenient analytical evaluation of the Fourier in-
tegrals for the pulse transmission characteristic. However, at present,
these integrals can be accurately evaluated by numerical integration
with the aid of digital computers for any specified pulse spectrum and
phase distortion.

Numerical values are given here for the carrier pulse transmission
characteristics with linear and quadratic delay distortion, together with
the maximum transmission impairments caused by these limiting and
fairly representative forms of delay distortion with various methods of
carrier modulation and signal detection. These include amplitude modu-
lation with envelope and with synchronous detection, two-phase and
four-phase modulation with synchronous detection and with differential
phase detection and binary frequency modulation. In determining the
effect of delay distortion, a raised cosine amplitude spectrum of the
pulses at the detector input has been assumed in all cases, together with
the minimum pulse interval permitted with this spectrum, ideal im-
plementation of each modulation and detection method and optimum
design from the standpoint of slicing levels and sampling instants at the
detector output. These idealizations insure that only the effect of delay
distortion is evaluated and considered in comparing modulation methods,
a condition that is difficult to realize with experimental rather than
analytical comparisons.

As mentioned above, the present analysis involves a basic function
common to all modulation methods, which in general would be deter-
mined with the aid of digital computers. This approach has certain
advantages in comparison of modulation methods and from the stand-
point of optimum system design over direct computer simulation of each
modulation method. The latter direct approach may be preferable for
any specified modulation method and type of transmission impairment
and has been used in connection with a binary double-sideband AM
system with envelope detection, for cosine and sine variations in trans-

The transmission impairments caused by linear and quadratic delay
distortion, and combinations thereof, have been determined experi-

mentally for a binary vestigial-sideband amplitude modulation data
transmission system employing envelope detection."

The present analysis is concerned with certain ‘‘coarse structure” vari-
ations in transmission delay that ordinarily predominate over smaller
“fine structure” variations, except in transmission facilities where
elaborate phase equalization is used. Transmission impairments from
small irregular fine structure gain and phase deviations over the channel
band can be evaluated by methods discussed elsewhere” and are not
considered here.

In carrier pulse modulation systems the pulse trains at the transmit-
ting end modulate a carrier in amplitude, phase or frequency. In AM
the demodulated signal depends on the envelope of the received carrier
pulse train at sampling instants, in PM on the phase of the carrier
within the envelope and in FM on the time derivative of the phase at
sampling instants. To determine the performance of these various meth-
ods in the presence of transmission distortion it is necessary to formulate
the received carrier pulse trains.

The received carrier pulse trains at the channel output, i.e., the de-
tector input, can in all cases be formulated in terms of the carrier pulse

transmission characteristic, that is, the received carrier pulse in response

to a single transmitted pulse. This pulse transmission characteristic is
related to the shape of the modulating pulses at the transmitting end,
and to the amplitude and phase characteristic of the channel, by a
Fourier integral, as discussed and illustrated for special cases in the
Appendix. The general formulation of the pulse trains at the detector
input and the resultant demodulated pulse trains with various methods
of carrier modulation and signal detection is dealt with in the following

It will be assumed that a carrier pulse of reetangular or other suitable
envelope is applied at the transmitting end of a bandpass channel. The
received pulse with carrier frequency w, can then be written in the
general form [Ref. 2, Equation (2.09)|

quadrature components of the received carrier pulse and P. the resultant

envelope. The time ¢ is taken with respect to a conveniently chosen
origin, for example the midpoint of a pulse interval or the instant at
which RF. or P, reaches a maximum value.
With a carrier frequency w) rather than w,, relation (1) is modified

Po(t) = cos (wot — Wo) Ro(t) + sin (wel — Wo) Qo(t) (7)
and relations (2) through (6) are correspondingly modified by replac-
ing ¢ by the subscript 0.

a received pulse will change, provided the transmission-frequency
characteristic of the channel remains fixed, except in the limiting case
of a carrier pulse of infinitesimal duration having a flat spectrum. How-
ever, by appropriate modification of the transmission-frequency charac-
teristic the amplitude spectrum of a pulse at the channel output, i.e.,
the detector input, can be made the same regardless of the carrier fre-
quency. On the latter premise of equal amplitude spectra at carrier
frequencies wy and w,, the following relations apply |Ref. 2, Equation
(2.18)]:

PULSE TRANSMISSION WITH PHASE DISTORTION
carrier phase y, in (1) rather than the carrier phase Yo in (7). If a ear-
rier phase yp is used as reference, ¥, = 0 in (8) and (9), and
Roo = COS wlRo(t) — sin wlQo(t), (10)
Q..o = COS wlQo(t) + sin w,tRo(t). (11)
With (8) and (9) in (3), or (10) and (11) in (3):
P(t) = Py(t) = [Re(t) + Qe(t)}’. (12)

The resultant envelope of a single pulse is thus the same regardless of
carrier frequency and phase, on the premise of a fixed pulse spectrum
at the channel output as assumed above.

Let carrier pulses be transmitted at intervals 7, and let ¢ be the time
from the midpoint of a selected interval. The following designation will
be introduced for convenience

where n is the time expressed in an integral number of pulse intervals
of duration 7 and x the time in a fraction of a pulse interval.

Let a(—n) and y.(—n) be the amplitude and phase of the carrier
pulse transmitted in the nth interval prior to the interval 0 under con-
sideration, and a(n), ¥.(m) the corresponding quantities for the nth
subsequent interval. The received pulse train in the interval —7/2 <
t < T/2 is then

where the summations are between n = — x andn = @.
During the next interval, 7 to 27, the received wave is obtained by
replacing a(n) and ¥,(n) by a(n + 1) and ¥,(nm + 1) and is thus

In pulse modulation systems as considered herein it is assumed that
the modulating pulses are rectangular in shape and of duration equal to
the pulse interval. For equal phases y.(n) = y, of all the modulating
pulses, (14) then becomes

where A, is the amplitude of the transmission-frequency characteristic
of the channel at w = w, and it is assumed in the determination of R,
and @. that the phase characteristic is zero at w = w,. That is, a con-
stant transmission delay is ignored, which is permissible without loss
of generality.

When F.(t) and Q.(t) are determined from the channel transmission-
frequency characteristic by the usual Fourier integral relations, in the
form represented by (159) through (163) of the Appendix, the following
relations apply for rectangular modulating pulses of duration 7 equal
to the pulse interval:

With synchronous detection, also referred to as homodyne and co-
herent detection, the received wave is applied to a product demodulator
together with a demodulating wave cos (wot — y.). After elimination of
higher frequency demodulation products by low-pass filtering the de-
modulated baseband output becomes, when a factor of one-half. is
omitted for convenience

If w, is the bandwidth of the modulating signal, the high-frequency
output of the product demodulator will have a lowest frequency 20, —
w, , Which can be separated from the modulating wave by low-pass fil-
tering provided 2w, — w, = w,, or if w, = w, .

At sampling instants « = 0, the desired signal is a(0)R(0) and the
remaining terms in (22) represent intersymbol interference in systems
where R.(n) # Oforn = +1, +2, ete.

Owing to elimination of the quadrature components, synchronous
detection is simpler from the standpoint of analysis than envelope de-
tection, in which the demodulated signal depends on the envelope of
the received wave (21) as given by

W(x) = (>> a(n)RAx — n)\ + [> a(n)Q.(x - re’. ¢@3)
The desired signal at sampling instants x = 0 is a(0)[R2(0) + Q2(0)|)
and the remaining terms in (23) represent intersymbol interference.
2.5 Phase Modulation with Synchronous Detection

In phase modulation systems the amplitude a(n) = a = constant
and the phase y,(n) is varied from one pulse interval to the next. The
received wave (15) then becomes

In a multiphase system, the received wave is in general applied to
several product demodulators together with a demodulating wave cos
(wt — w). In the particular case of two-phase modulation a single
demodulator suffices, and the demodulator output after elimination of
high-frequency demodulation products by low-pass filtering and omitting
a factor of one-half is of the general form

where as before the summation is between n = —* andn = =.
The desired signal is represented by the term for n = 0 and is

When the phase y of the demodulating wave is so chosen that 6 = 0,
and if y.(0) = Oor as in two-phase modulation, then Uy(0) = +P.(0).

In four-phase modulation two product demodulators are required,
with the demodulating waves displaced 90° in phase. The output of the
second demodulator is then, in place of (26),

The preferable choice of the phase of the demodulating wave in the
above relations may depend on certain considerations in the imple-
mentation of modulators and demodulators. In Table I are given the
four possible combined outputs as determined by the carrier phase
y.(0) for the particular cases 6 = 0 and @ = 7/4. For convenience the
outputs for 6 = 2/4 are normalized to unit amplitude, the actual ampli-
tudes being +43+/ 2.

It will be noted that with @ = 0 the output Uy determines whether
one carrier is modulated in phase by y. = 0 or x, while the output V9’
determines whether the quadrature carrier is modulated in phase by
v. = Oor x. The two carriers can thus be modulated and demodulated
independently, without the need for circuitry to convert the two de-
modulator output to carrier phase, as would be required with 6 = 7/4.
With differential phase detection, to be discussed in the next section,
such converters would be required both with 6 = 0 and @ = 2/4. In
this case @ = 2/4 may be preferable for the reason that only two states
(—1,1) are possible for each demodulator, rather than three states
(—1,0,1) with 6 = 0.

TABLE [ DEMODULATOR OvutTPpUTS U,® AND V,° IN Four-PHASE
SysTEMS AS DETERMINED BY CARRIER PHASE y,(0) FoR DemMopu-
LATING WAVES WITH PHASES 6 = 0 AND 7/4

An alternative method of demodulation that will be considered in con-
nection with phase modulation is differential phase detection. With this
method W(x) as given by (15) is applied to one pair of terminals of a
product demodulator, and W,(.x) as given by (16) to the other pair
with a suitable phase shift 6. The demodulator output is then, with
a(n) constant as in phase modulation,

After elimination of high-frequency components present in (31) by
low-pass filtering and omitting a factor of one-half, the resultant base-
band output can be written

It will be recognized that this expression is of the same form as (27)
except that y,(0) is replaced by the phase change y.(0) — y.(1) be-
tween two successive sampling instants. In the particular case of four-
phase transmission the phase @ may be chosen as, say, 6 = 0 or 7/4, in
which case the outputs of the two product demodulators would be as
indicated in Table I for phase modulation with synchronous detection,
except that y,.(0) is replaced by the phase difference y.(0) — y.(1).

With a signal of bandwidth w, the high-frequency part of (31) will
have a lowest frequency 2(w,. — @,), since it is the product of two high-

signal represented by (32) will have a maximum frequency 2a, , so that
a flat low-pass filter of minimum bandwidth 2, is required to avoid
distortion of the baseband signal. In order that the filter also eliminate
the high-frequency components in the demodulator output, it is neces-
sary that 2(w. — w,) 2 2w, or w, 2 2, . With synchronous detection it

In frequency modulation a(n) in (14) is constant and y.(n) varies
with time so that the time derivative of [w. — y.(n)| represents a
variable frequency. Pulse transmission without intersymbol interference
over a channel of the same bandwidth as required for double-sideband
AM is in this case possible for certain ideal amplitude and phase charac-
teristics of the channels, as shown elsewhere [Ref. 1, Section 5]. The
formulation is here modified to include any amplitude and phase char-
acteristic of the channels.

It will be assumed that a space is represented by a frequency wo — @
and a mark by a frequency w) + @. Discontinuity in a transition from
mark to space can then be avoided for rectangular modulating pulses
of duration 7 provided,

In a system of minimum bandwidth k = 1, and in this case inter-
symbol interference can be avoided with a channel band no wider than
required for double-sideband AM.

When a mark is preceded and followed by a space during the nth
pulse interval, the envelope of the resultant carrier pulse is obtained
with f = (¢ + nT) in Equation (23) of Ref. 1 and becomes

P(t) = cos (wot + go)(—1)"Ro(t — nT)
(42)
oe sin ( wot + Lo) | —| "Qol(t —_ ni’).
where ¢ is the time from the midpoint of interval 0.
When ¥( —@) is the phase distortion* at the frequency w) — @, Equa-
tion (34) of Ref. 1 is modified into
E(t) = —cos (aot + go)[A(—a@) cos y — (—1)"Ro(t — nT)

y = ot + ¥(—@). (44)
When a sequence of marks and spaces is transmitted, the resultant
wave at the detector input becomes

in which the notation is in accordance with (13), 2 = ¢t/T and y =
mx + (—w).
The phase of the wave (45) is given by
sin y — uBo(.xr)
tan Wo t) = y - : 9
COS Y — Mao x)
where
mM 1/A(—@o) (49)
Expression (41) of Ref. 1 for a single pulse is replaced by the following
for the demodulated pulse train at « = ¢/T:

* As in Ref. 1, the linear component of the phase characteristic is disregarded
“nee it only represents a fixed transmission delay.

Since binary FM with frequency discriminator detection is a non-
linear modulation method, determination of the optimum signal-to-noise
ratio at the detector input for a given error probability presents a very
difficult analytical problem, at least when consideration is given to
minimum bandwidth requirements together with appropriate shaping
of bandpass and postdetection low-pass filters. In Ref. 1 these various
factors were taken into account, but the signal-to-noise ratios at sam-
pling instants were evaluated on the approximate basis of a steady
state carrier representing a continuing space or mark and a relatively
high signal-to-noise ratio. On this basis it turned out that, in the absence
of a postdetection low-pass filter, binary FM would have a disadvantage
in signal-to-noise ratio of about 4.5 db compared to an optimum bipolar
AM or phase reversal system. This would be reduced to about a 1.5-db
disadvantage by addition of an optimum low-pass filter. The analysis
further indicated that, for a specified postdetection low-pass filter, there
would be an optimum division of shaping between the transmitting and
receiving bandpass filters that would give a slight advantage in signal-
to-noise ratios over an optimum bipolar AM system. In view of the ap-
proximations involved, the above analysis does not prove that such an
advantage exists. Rather, it is probable that optimum bipolar AM has
some advantage in signal-to-noise ratio over optimum binary FM. This
is indicated by other analyses that do not assume a high signal-to-noise
ratio but introduce other approximations in that they do not consider
frequency discriminator detection or the shaping of band-pass filters or
the effect of a postdetection low-pass filter.

It is well known that an approximation is involved in assuming high
signal-to-noise ratios and thus ignoring the breaking phenomenon in
FM. Moreover, even in the absence of intersymbol interference, it is an
approximation to assume a steady state carrier over a short sampling
interval, regardless of the transmitted code, as shown below.* Referring
to Equation (202) of Ref. 1, random noise at the detector input can be
written in the form

In the absence of intersymbol interference at sampling instants,
a(0) = Oor 1, B(0) = 0, y = O and pz = 2. In this case appropriate
modification of (50) gives for the demodulated signal plus noise at
sampling instants

where r; = r:(0), gi = qi(O) and gq,’ = dq,(t)/dt for t = 0.
If r; < 1, the last equation is approximated by

where ()(0) = a(0) = 0 for space and 1 for mark, and the interfering
voltage after demodulation is

The first two terms represent the conventional approximation for a
continuing mark or space and a high signal-to-noise ratio.

0. This is not the case except for a continuing space, a continuing mark
or a mark preceded and followed by a continuing space. For other
combinations of transmitted pulses there is some contribution from the
third term. In the particular case of a raised cosine pulse spectrum, as
considered herein, the maximum effect for a random pulse train is less
than 0.15 db and can thus be ignored. For narrower pulse spectra the
effect may be appreciably greater.

In the analysis that follows, transmission impairments from inter-
symbol interference owing to phase distortion will be evaluated on the
same basis for fM as for the other modulation methods, although the
approximations involved may be somewhat greater.

As indicated by the preceding derivations, the demodulated wave is
related to the received carrier wave Wo(.) in a manner that depends on
the carrier modulation and detection method. In general the demodu-
lated wave at sampling instants may assume a number of different

amplitude or state a, of the transmitted signal and l the demodu-
lated wave at a sampling instant for an adjacent amplitude or state

ing to intersymbol interference, and also a certain sequence resulting

a.4, and of positive and negative noise voltages, the optimum level for
so > r(s r(s+1) -

The minimum margins are obtained with U' = Uys.” and with
r(s+l1 , s+1) - — ne nn 5 te ‘
= Unin in (58) and (59). The minimum margins thus be-

For sequences of marks and spaces, or other signal patterns, such
that the minimum margins for distinction between adjacent signal
states are obtained, an error will occur if the noise voltage at the sam-
pling instant exceeds M,,i, in amplitude and has the appropriate
polarity. (Polarity is immaterial except for the two extreme signal
states.) For other signal patterns the tolerable amplitude of the noise
voltage is greater. The value of 7,,;, relative to the value in the absence
of intersymbol interference thus gives the maximum transmission im-
pairment. The average impairment obtained by considering various
pulse train patterns and the corresponding values of M“’ and M“*"
as given by (58) and (59) will be less, as discussed below.

By way of illustration it will be assumed that all values of M between
M min and a maximum value M,,,. are equally probable, and that the
noise has a gaussian amplitude distribution. With a given fixed value of
M the probability of an error can be written as

where erfe = 1 — erf is the error function complement and a is a factor
that depends on the ratio of signal power to noise power.
Considering all noise margins between the limits mentioned above.
the average error probability becomes
l l »M max

lor k = 1, the latter expression conforms with (61).
The maximum error probability would be obtained by considering a
fixed noise margin equal to Win and would be

p. = 3 erfe A. (66)
The error committed in assuming .V/,,;, can be determined by writing
Lan) . aS
p. as given by (65) in the form
pe = 4 erfe (cA), (67)
where ¢ 2 1 is so chosen that (67) equals (65).
The average noise margin is then

By way of numerical illustration let A be so chosen that p, as given
by (66) in one case is 10 ‘ and in another case 10°°. The results given
in Table II are then obtained from (65) and (67).

TABLE II Ratio c = M/Myin FoR EQuaL PROBABILITY OF ALL
NorsE MarGINs BETWEEN Myin AND Mmax = k Min FOR NOISE
WITH A GAUSSIAN AMPLITUDE DISTRIBUTION

margin, whereas the actual impairment would be 1.4 db less for an error
a . 4 > ons Br
probability of 10°, about 1 db less for an error probability 10 °. For an

error probability of 10 ° or less the error committed in evaluating trans-
mission impairments on the basis of the minimum noise margin can be
disregarded. This also applies for greater error probabilities when the
transmission impairment based on the minimum noise margin is small,
in which case k < 2.

Amplitude modulation can be used in conjunction with envelope de-
tection and synchronous detection. The former method is simplest from
the standpoint of implementation, but synchronous detection, also re-
ferred to as homodyne and coherent detection, affords an improvement
in signal-to-noise ratio. Since synchronous detection is also the simplest
method from the standpoint of analysis, it will be considered here, ex-
cept for a comparison of envelope and synchronous detection for binary
double-sideband AM.

Amplitude modulation in general implies several pulse amplitudes,
and can be used with double-sideband and with vestigial-sideband trans-
mission. The particular case of bipolar binary AM with synchronous
detection is equivalent to two-phase modulation.

With amplitude modulation and synchronous detection it is possible
to transmit pulse trains on two carriers at quadrature with each other,
and under certain idealized conditions to avoid mutual interference. The
special case of bipolar binary AM on each of the two carriers is equiva-
lent to four-phase modulation.

The signal-to-noise ratio as related to error probability is discussed
elsewhere (Ref. 1, Section XVIII) for various optimized binary AM or

PM systems on the premise of ideal synchronous detection. Ideal syn-
chronous detection for AM or PM as assumed here can in principle be
approached without penalty in signal-to-noise ratio, by various methods
of implementation. For example, a demodulating wave for a product
demodulator can be derived with the aid of a resonator of sufficiently
narrow bandwidth (high Q) tuned to the carrier frequency, or the
second or the fourth harmonic thereof, depending on the particular
method and on whether two-phase or four-phase modulation is used. A
demodulating wave can also be supplied from an oscillator at the re-
ceiving end, the phase of which would be controlled by comparison with
that of the carrier of the received signal. Such phase-locked oscillator
methods have been devised for analog signal transmission by suppressed
carrier double-sideband AM?’ and vestigial-sideband AM.° With any one
of the above methods, noise in the demodulating wave would be vir-
tually absent, as would the effect of phase distortion in the channel.
Actually some penalty in signal-to-noise ratio as compared to ideal
synchronous detection would be incurred, owing to unavoidable fluctua-
tions in the amplitude and phase of the demodulating wave, resulting
from the finite bandwidth of the resonators and mistuning, or from im-
perfect oscillator control. A common property of these methods is that
a rather long time, as measured in pulse intervals, is required to establish
a demodulating wave, if the above fluctuations in amplitude and phase
are to be held within tolerable limits. This may be a disadvantage in
certain applications, which in the case of phase modulation can be over-
come by differential phase detection, in exchange for a penalty in signal-
to-noise ratio resulting from the presence of both noise and phase dis-
tortion in the demodulating wave, as discussed in Section IV.

A general formulation is given here of intersymbol interference and
resultant maximum transmission impairment as related to the carrier
pulse transmission characteristic, together with illustrative applications
to the particular cases of linear and quadratic delay distortion. The
formulation is, however, applicable to any given gain and phase devia-
tion over the channel band, provided the carrier pulse transmission
characteristic has been determined, which in general would entail Fourier
integral evaluation with the aid of computers.

THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1961
The following notation will be used:

where FR.” designates positive values of R. and R. absolute values
when R, is negative.

Let there be / different amplitude levels, between a minimum ampli-
tude @min and a maximum amplitude dnax . When a pulse of amplitude
a, = a,(Q) is transmitted, the maximum value of (69) is

ry 7 . a . 0 -
Che value of Myyin as given by (74) is smaller than /” in the absence
of intersymbol interference by the factor

in which FR, designates the absolute value of R...
The factor R.(0)/R.'(0) represents the transmission impairment ow-
ing to reduction in pulse amplitude at sampling instants. The summation

term represents transmission impairments owing to intersymbol inter-
ference.

Relation (77) applies regardless of the polarity of the transmitted
pulses and for both symmetrical (double sideband) and asymmetrical
(vestigial sideband) systems. The special case 1 = 2 and Quin = —4max
represents binary bipolar AM, which can also be regarded as two-phase
transmission,

With synchronous detection it is possible under certain ideal condi-
tions to transmit signals on two carriers at quadrature without mutual
interference. In general, however, the quadrature component in (21)
will in this case give rise to interference and (69) is replaced by

where b(n) are the pulse amplitudes in the quadrature system.
lor equal differences between maximum and minimum amplitudes in
the two systems, i.€., dmax — Gmin = Vmax — Dmin, (77) is replaced by
RO) t—1
R.(O) R.(O)

In general the phase of the demodulating carrier can be so chosen
that Q.(0) = 0, as is demonstrated later.

Expression (80) applies regardless of pulse polarities in the two
quadrature systems. The special case of two binary bipolar AM sys-
tems, i.e., 2 = 2 and dyin = Dmin = —Amax = —bmax Can also be regarded
as four-phase transmission.

When the spectrum of a received pulse at the detector input has even
symmetry about the carrier frequency, and the phase characteristic has
odd symmetry (even symmetry delay distortion), the quadrature com-

ponents Q.(n) vanish (see the Appendix). In this case (77) and (80)
are identical, so that there is no mutual interference between pulse
trains transmitted on two carriers at quadrature. In this special case it
is thus possible by quadrature carrier AM to realize a two-fold increase
in pulse transmission rate, without increased intersymbol interference.
An alternative means to the same end is to use vestigial sideband trans-
mission, as discussed below.

Let T be the pulse interval in double-sideband AM, in which case the
pulse interval in vestigial-sideband AM would be 7” = 7/2. Returning
to (10) and (11) let Ro(t) be the in-phase component in double-side-
band AM, and let Qo(¢t) = 0 for an amplitude spectrum with even sym-
metry about wy and a phase characteristic with odd symmetry. Let w,
be the carrier frequency from midband in vestigial-sideband transmis-
sion. By appropriate choice of w, it is possible to make w,7"’ = 7/2, in
which case cos w,7T’ = 0, sin wT’ = 1. The following relations are thus
obtained:

In accordance with the above relations, at even sampling points only
the in-phase components are present and are the same as in double-side-
band AM. At odd sampling points the quadrature components are pres-
ent, but are eliminated with synchronous detection and need not be
considered.

In summary, when the amplitude spectrum at the detector input has
even symmetry about the midband frequency, and the phase character-
istic has odd symmetry, relation (77) applies for double-sideband AM,
quadrature double-sideband AM, vestigial-sideband AM, as well as
special cases thereof, such as two-phase and four-phase modulation.

In the next section numerical results are given for the special case of
a raised cosine spectrum at the detector input with quadratic delay
distortion about the midband frequency.

Fig. 1 Raised cosine pulse spectrum and transmission-frequenecy character-
istic of channel for double- and vestigial-sideband AM. Curve 1: Spectrum of car-
rier pulse envelope at detector input (channel output); S(u)/S(0) = cos* (ru/4e).
Curve 2: Transmission-frequency characteristic of channel with rec se trans
mitted carrier pulses of duration 7’ = r/o and carrier at midband; A(u)/A(O) =
(ru/4m)/tan (ru/4o). Curve 3: Transmission frequency characteristic of channel
with rectangular transmitted carrier pulses of dur: ation T/2 and carrier at u =a;
A(u)A(O) = cos? (ru/4@){[(u — &)/4@]/sin [x(u — @)/4a]}.

NOTE:
ENVELOPES ARE SYMMETRICAL IN
AMPLITUDE ABOUT ZERO LINE AND
ONLY ONE SIDE OF ENVELOPE IS SHOWN

Fig. 2 — Carrier pulse transmission characteristic for raised cosine spectrum
as in Fig. 1 and quadratic delay distortion.

The shape of the transmission-frequency characteristic of the channel
required to this end depends on the shape of the transmitted pulses. It
is shown in Fig. 1 for rectangular modulating pulses, with the carrier at
midband and also with the carrier to one side of midband, as in vestigial-
sideband transmission. These characteristics, together with the optimum
division of shaping between transmitting and receiving filters, are dis-
cussed in Section XIV of Ref. 1. Even though the amplitude characteris-
tics of the detector input spectra are the same in double- and vestigial-
sideband transmission, it is necessary to use different shaping of
transmitting filters, as indicated in Fig. 1, since the rectangular modulat-
ing pulses have different carrier frequencies and different durations.

The phase characteristic is assumed to contain a linear component,
together with phase distortion component varying as the third power of
frequency from midband, which corresponds to delay distortion increas-
ing as the second power of frequency from midband, as indicated in Fig.
2. The function Ro(t/T) = Ro(x + n) for this case has been determined
by numerical integration, as discussed in the Appendix. It is given in
Table XIX of the Appendix and shown in Fig. 2. The values for « = 0,
i.e., integral values of t/T are given in Table ITI.

TABLE III Function Ro(n) FoR RAISED CosINE SPECTRUM
AND QuapRATIC DELAY DisToRTION

With exact evaluation of Ro(n) the summation }°R)(n) between
= —x andn = & should equal 1.

With the values of Ro(n) given in Table III, the values of 7, Fy ob-
tained from (78), and of main as Obtained from (77) are given in Table
IV.

TABLE IV FACTOR nmin FOR RAISED COSINE SPECTRUM AND
QuapraATIC DeLay Distortion FOR SYNCHRONOUS AM
with | AMPLITUDE LEVELS*

sideband AM and for quadrature double-sideband AM, and special cases
thereof, such as two-phase and four-phase transmission. Since the quadra-

ture component is absent the factors also apply for double-sideband AM
with envelope detection. It should be noted that 7 in all cases is the
pulse interval in double-sideband AM, which is twice the pulse interval

Fig. 3 Factor nmin for raised cosine pulse spectrum and quadratic delay dis
tortion as in Fig. 2, for AM systems employing synchronous detection and / pulse
amplitudes. Factor applies for double- and vestigial-sideband AM and quadrature
double-sideband AM.

in vestigial-sideband AM or twice the combined pulse interval in quadra-
ture double-sideband AM.

In the above evaluation it was assumed that the pulses were sampled
at ¢ = 0, which is not at the peak of a pulse except for d/T = 0. For
d/T = 4 the pulse peak is nearly at (/T = x = 0.25. Sampling at
x = 0.25 gives, for ] = 2, nuin = 0.356 rather than 0.347.

The factor nmin expressed in decibels, as in Fig. 3, indicates the maxi-
mum transmission impairment, i.e., the maximum increase in signal-to-
noise ratio required at the detector input to compensate for the effect
of delay distortion. This maximum impairment would be closely ap-
proached for signal-to-noise ratios such that the error probability is suf-
ficiently small, say less than 10°‘. However, for error probabilities in
the range ordinarily considered the transmission impairment will be
less, in accordance with the discussion in Section 2.10. For example, for
d/T = 4, Ro(0) = 0.734 and (0) = 0.385. The maximum noise mar-

For k = Minax/Mmin = 3.2, the results given in Table II indicate that
the transmission impairments would be less than the maximum by
about 1 db and 1.4 db for error probabilities of 10°” and 10 * respec-
tively. Ford/T = 4and/ = 2 the maximum impairment indicated in
Fig. 3 is —20 logy Min & 9.2 db, whereas impairments of about 8.2
and 7.8 db would be expected for error probabilities of 10° and 10“, on
the premises underlying the evaluation in Section 2.10.

When the pulse spectrum at the detector input has even symmetry
about the midband frequency and the phase characteristic has a com-
ponent of even symmetry, i.e., odd symmetry delay distortion, the in-
phase and quadrature components both have even symmetry with re-
spect to ¢. That is,

With synchronous detection the phase of the demodulating carrier
would preferably be so chosen that Qo(¢) would vanish at ¢ = 0, since
this would give the maximum amplitude of the demodulated pulse at a

it is therefore convenient to modify the phase angle so that the quadra-
ture component vanishes at ¢ = 0. The modified quantities are related
to Ry and Qo by (see Appendix )

In the case of vestigial-sideband transmission at pulse intervals
i T/2, the in-phase component referred to a carrier at frequency
wy + @ is obtained from (10) and becomes

For the special case of a raised cosine spectrum and linear delay dis-
tortion the functions Rp and Qp have been determined by numerical
integration, as discussed further in the Appendix. They are given in
Table XX for certain ratios d/7, where d is the difference in delay be-

NOTE:
ENVELOPES ARE SYMMETRICAL IN AMPLITUDE
ABOUT ZERO LINE AND ONLY ONE SIDE
a ENVELOPE IS SHOWN

Fig. 4 — Carrier pulse transmission characteristics for raised cosine spectrum
as in Fig. 1 and linear delay distortion
tween the midband frequency and maximum sideband frequency as il-
lustrated in Fig. 4. The modified functions Ro and Qo are given in Ta-
ble X XI and are shown in Fig. 4. For negative values of t/ 7, Roo and Qoo
are the same as shown in Fig. 4 for positive values.

For double-sideband transmission the factors qmin given in Table V
are obtained from (77), with F. taken in accordance with (87). These
factors are shown in Fig. 5. The case | = 2 corresponds to two-phase
transmission.

Fig. 5 — Factor nmin for raised cosine pulse spectrum and linear delay distor
tion as in Fig. 4 for double-sideband AM systems employing synchronous detec-
tion and 7 pulse amplitudes.

For quadrature double-sideband AM the factors in Table VI are ob-
tained from (80) with 7, and g, taken in accordance with (87) and (88).
These factors are shown in Fig. 6. The case 1 = 2 corresponds to four-
phase transmission.

lor vestigial-sideband transmission the factor nain is determined from
(77), with #, taken in accordance with (92). The factors given in Table
VII are thus obtained.

TaBLE VI — Factor nmin FOR QUADRATURE DOUBLE-SIDEBAND
AM For / = 2 anp 3 PuLSE AMPLITUDES

Fig. 6 Factor nmin for raised cosine pulse spectrum and linear delay distor
tion as in Fig. 4 for quadrature double-sideband AM systems (solid lines) and
vestigial-sideband AM systems (dashed lines) employing synchronous detection
and 1 = 2 and 3 pulse amplitudes

TaB_LeE VII FACTOR nmin FOR VESTIGIAL-SIDEBAND AM
FoR 1 = 2 anp 3 PULSE AMPLITUDES

The factors nuin for vestigial sideband AM are compared in Fig. 6
with the corresponding factors for quadrature double-sideband AM.
With ideal transmission-frequency characteristics and ideal synchronous
detection the two methods are equivalent from the standpoint of channel
bandwidth requirements and optimum signal-to-noise ratio for a given
error probability. As shown in Section 3.4, this also applies for pulse
spectra at the detector input with even symmetry about the midband
frequency, in the presence of delay distortion with even symmetry. The
equations in Section 3.6 and the curves in Fig. 6 show that the above
two methods are not equivalent in the presence of delay distortion with
odd symmetry about the midband frequency. With linear delay distor-
tion the factor nmin is, however, very nearly the same with both methods.
For practical purposes quadrature double-sideband AM and vestigial-
sideband AM can be regarded as equivalent with any type of delay dis-
tortion that would be expected in actual facilities. This equivalence
would apply on the premise of ideal synchronous detection but not neces-
sarily with actual implementation of synchronous detection, for the rea-
son that the penalty in signal-to-noise ratio incurred in deriving a de-
modulating wave may not be the same with both methods.

In the preceding analysis ideal synchronous detection was assumed,
which permits the use of bipolar pulses. An alternative method that is
simpler in implementation is envelope detection, which, however, entails
the use of unipolar pulse transmission and for this reason has a certain
disadvantage in signal-to-noise ratio as compared to synchronous detec-

certain cases be greater with envelope than with synchronous detec-
tion, as shown below.

When both the pulse spectrum and delay distortion have even sym-
metry about the carrier frequency, so that the quadrature component is
absent, the effect of delay distortion is the same as with synchronous
detection. The results given in Table IV thus apply also for double-side-
band AM with envelope detection.

When a quadrature component is present in the carrier pulse trans-
mission characteristic the resultant demodulated wave is in accordance

Owing to the presence of both in-phase and quadrature components,
it does not appear feasible to derive a simple general expression for
Umax and Uyin*®*” similar to (72) and (73). These values can, how-
ever, be determined by examining several combinations of transmitted
pulses, as illustrated below for binary pulse transmission, a raised cosine
pulse spectrum at the detector input and linear delay distortion. Using
values of Ro and Qo given in Table XXI of the Appendix, the results
are as shown in Table VIII. Since both Ro(t) and Qo(t) in this case have
even symmetry about ¢ = 0 the maximum effect of delay distortion is
encountered for pulse trains with even symmetry about the sampling
point, i.e., a(—n) = a(n). Hence, only pulse trains with this property
need to be considered.

From Table VIII can be obtained W max” and W min’, as indicated by
asterisks, together with the optimum slicing level given by (57) and the

TABLE VIII VaLues oF (0) For RAIsED CosINE SPECTRU}
LINEAR DeLay DisTorTION FOR VARIOUS COMBINATIONS
oF MARKS 0 AND SPACES = |

TABLE LX — Factor nmin WITH BINARY AM AnD ENVELOPE
DETECTION FoR RatsED CostNnE SPECTRUM AND LINEAR
DeLAY DISTORTION

It will be recognized that synchronous detection has a significant ad-
vantage over envelope detection as regards transmission impairments
caused by pronounced linear delay distortion, for the reason that the
effect of the quadrature component is eliminated. In general, delay dis-
tortion will have a component of even symmetry and a component of
odd symmetry about the carrier frequency, in which case the quadrature
component will be smaller. The advantage of synchronous detection as
regards transmission impairments caused by delay distortion will then
be less than indicated in Table [X. The principal advantage of syn-
chronous detection is that it permits the use of bipolar transmission,
which in the case of binary systems as considered above affords about
3 db improvement in the ratio of average signal power to average noise
power for a given error probability (Ref. 1, Table VIT).

In the case of vestigial-sideband transmission a pronounced quadrature
component is present even in the absence of phase distortion. The ad-
vantage of synchronous detection over envelope detection is in this case
significantly greater than for double-sideband transmission considered
above, for the reasons that bipolar transmission can be used and quadra-
ture component is eliminated. In the absence of phase distortion and
with a raised cosine pulse spectrum at the detection input, synchronous
detection has about a 9 db advantage over envelope detector in the ra-
tio of average signal power to average noise power for a given error
probability (6 db owing to elimination of quadrature component and 3
db owing to bipolar transmission).

valuation of transmission impairments from phase distortion is more
complicated for envelope than for synchronous detection. These impair-
ments have been determined experimentally for a binary vestigial-side-
band system with an approximately raised cosine spectrum at the de-
tector input, for linear and quadratic delay distortion and combinations

» A om . n ° ° »
thereof.. They are significantly greater than determined herein for syn-

chronous detection. Hence envelope detection entails more phase equali-

zation than synchronous detection, unless a greater disparity in signal-
to-noise ratio is accepted than the 9 db applying in the absence of phase
distortion.

In phase modulation with differential phase detection, the demodulator
output would under ideal conditions depend on changes in carrier phase
between two successive pulse intervals of duration 7. In its simplest
and ideal form, the signal with two-phase modulation would be applied
to one pairof terminals of a product demodulator, while the signal de-
layed by a pulse interval 7 would be applied to the other pair. With four-
phase modulation two product demodulators are required, each with a
delay network at one pair of terminals. In addition, a phase shift of 90°
must be provided between all frequencies of the demodulating waves of
the two demodulators, as indicated in Fig. 7. Such a phase shift over a
frequency band can be realized in principle and closely approached with
actual networks.’ The modulator outputs would be applied to low-pass
filters of appropriate bandwidth for elimination of high-frequency de-
modulation products, and the output of these would be sampled at in-
terval T. The phase of the carrier would be indicated by the combined
output as discussed in Sections 2.5 and 2.6.

With the above method it is possible with ideal channel characteristics
to avoid intersymbol interference at sampling instants, without the need
for a wider channel band than required with synchronous detection.
However, the two methods are not in all respects equivalent from the

Fig. 7 Basic demodulator arrangement for four-phase modulation with dif
ferential phase detection

standpoint of bandwidth utilization. As discussed in Sections 2.5 and
2.6, with synchronous detection the carrier frequency must be at least
equal to the maximum baseband signal frequency, whereas with differ-
ential phase detection it must exceed twice the maximum baseband
signal frequency. This requirement does not impose a limitation on
bandwidth utilization with differential phase modulation provided the
midband frequency of the available channel is at least twice the lowest
frequency, or that this condition is realized through frequency transla-
tion prior to demodulation.

With differential phase detection the demodulating wave is established
without the need for a long delay (measured in pulse intervals) as re-
quired with certain other methods inentioned in Section 3.1. Moreover,
a substantial fluctuation in carrier phase can be tolerated, since only the
difference in phase between adjacent pulses need be considered. These
advantages are realized in exchange for a penalty in signal-to-noise ratio
as compared to ideal synchronous detection owing to the presence of
noise in the demodulating wave. For very small error probabilities, and
assuming ideal implementation in all respects, this impairment is about
1 db for two-phase and about 2.3 db for four-phase modulation.” Com-
parable penalties in signal-to-noise ratio as compared to ideal synchro-
nous detection may be incurred with the other methods of providing a
demodulating wave mentioned in Section 3.1, owing to small unavoidable
amplitude and phase fluctuations in the demodulating wave resulting

detection, greater transmission impairments would be expected from
phase distortion, since the effect of phase distortion, like that of noise, is
present in both the signal and the demodulating wave. The transmission
impairments resulting from quadratic delay distortion are determined
here, and compared with that encountered with ideal synchronous detec-
tion.

Other implementations of differential phase detection than assumed
herein have been used, but in principle these entail a wider channel band
than with ideal synchronous detection. For example, the two demodu-
lator inputs or outputs could be integrated over a pulse integral 7 with
the aid of a narrow-band resonator tuned to the carrier frequency, and
then be rapidly quenched before the next signal interval. When the
channel bandwidth is limited, the phase of the demodulating carrier will
then depend on the phases of the carrier during several pulse intervals.
Thus some intersymbol interference from bandwidth limitation is en-
countered even in the absence of phase distortion, and the effect of phase
distortion will be greater than that determined herein. However, exces-

sive transmission impairments from bandwidth limitation and phase dis-
tortion can be avoided by appropriate techniques, as when a large num-
ber of narrow channels are provided within a common band of much
greater bandwidth than that of the individual channels.”

In differential phase modulation the carrier would be at midband, i.e.,
we = w. With Um = V, expression (33) for the demodulated signal
becomes

The above expressions apply for the output of the single demodulator in
two-phase systems, in which @ = 0. In four-phase systems the output of
one demodulator is obtained with 6 = 6, and the output of the other
with @ = 0 = 6 + 7/2.

Examination of (97) shows that the term for m = n + 1 is independent
of the phase difference yo( —n) — Yo( —m + 1) and is given by

2
Vo= > cos [gol —n) — go( —n — 1) — 6]Py(—n)Py(—n — 1). (98)
n=— ©
Determination of transmission impairments becomes rather difficult
except for the special case in which go(n) = 0, which will be considered
in further detail below.

When the pulse spectrum has even symmetry about the midband
frequency, and the phase characteristic has odd symmetry (i.e., even
symmetry delay distortion) the quadrature component of P(t) van-
ishes, i.e., ¢(—n) = 0. In this case, Py = Ry and (96) becomes

With synchronous detection, two-phase modulation could be used in
conjunction with both double-sideband and _ vestigial-sideband trans-
mission. With differential phase detection, however, vestigial-sideband

transmission is not practicable, since severe transmission impairments
would be incurred even in the absence of phase distortion, owing to the
presence of the quadrature component. Hence only double-sideband
two-phase modulation is considered here.

Assume that in (107) a sequence of values of a,,(0) has been chosen,
for example a.sQ@) = |]. ad) = 1, asl(O) = 1, a0) 1, a(Q0) =
—1, a,(0) = 1, ete. For any other value of n than n = 0, the sequence
of a,(n) will either be identical with that for a,,(0), or all signs will
be reversed. This follows from (104), since Yo(—n) will differ from
¥o(0) by 0 or x. Hence, for n # 0, the right-hand side of (106) can be

In the absence of intersymbol interference, V as given by (99) would
be —1 or 1. In the following, the minimum possible value of V will be
determined, on the assumption that V 1 without intersymbol inter-
ference; 1.€., ao(0) -

Consider first the term Sof (0) in (99). The minimum possible value
of So is obtained from (107) by choosing a,,(0) —1 for Ry(—m) > 0
and a,,(0) = 1 for Ro( —m) < 0. The following relation is thus obtained
for the minimum possible value of Sp , on the above premise of ap(0) = 1:

where Ry designates the absolute values. In the above expression a,(0)
would be taken as a,(0) 1 if Ro —1) < O and as a,(0) —lif
Ro(—1) > 0. The term [1 a,(0)|Ro(— 1) can therefore be written alter-
natively as Ro( —1) + Ro( —1), in which case (109) becomes

of the demodulated voltage with synchronous detection, in the presence
of a mark, as given in a somewhat more general form by (73).

Having thus determined So min it follows from (108) and (110) that
the two possible associated values of S, min are given by

where the term [1 — a,4:(”)|Ro(—n — 1) in (108) has been replaced
by the equivalent representation by the first two terms in (112).

To obtain the minimum value of V as given by (99), each term in the
series must be made to have the maximum negative value. To this end
the negative sign in (112) for Uyin is chosen if Ro(n) is positive, and the
positive sign if Ro(nm) is negative. The minimum possible value of V
thus obtained with (110) and (112) in (99) is

In accordance with the discussion in Section 4.2, the demodulator
output contains a bias or pc component V» given by (103). Optimum
performance is obtained when the threshold level for distinction between
V = 1 and V = —1 is made equal to Vo. When Vo is subtracted from
both sides of (114) the following expression is obtained for two-phase
modulation :

When intersymbol interference is absent at sampling instants,
Ro(n) = 0 for n ¥ 0, and for n = 0 is Ro (0). In this case Vinin? =
Umin = [Ro (0)]’. The voltage given by (115) is smaller than in the
absence of intersymbol interference by the factor

The basic difference between two-phase and four-phase modulation is
that relation (108) does not apply for four-phase modulation. Returning
to the discussion following (107), if a sequence a,,(0) is chosen in four-
phase transmission, the sequence a,,(m) can be chosen independently.
This follows from the (104), which shows that if a,,(0) has a given value,
say a,(0) = 1, it is possible to make each a,,(n) equal to +lor —1 by
appropriate choice of ¥(—n).

For this reason the minimum value (or maximum negative value) of
the right-hand side of (106) is now, for n # 0:

Bo ie : (118)
—R (0) — >> [Rol —m) + Ro(m)].
m=1
The right-hand side of (118) is smaller than for two-phase transmis-
sion as given by (112) by —2R)(0). When this modification is intro-
duced, the following expression is obtained for four-phase modulation,
in place of (115) for two-phase modulation:

where > is given by (116).
The voltage given by (120) is smaller than in the absence of inter-
symbol interference by the factor

TABLE X MINIMUM AMPLITUDES OF DEMODULATED PULSE
TRAINS IN Two-PHASE MopULATION WITH DIFFERENTIAL
PHASE DETECTION

The function Ro(n) for this case is given in Table IIL. The values of
Umin for synchronous detection are given in Table IV for / = 2. In
Table X are given the various quantities appearing in expression (115)
for the minimum amplitudes of received pulse trains at sampling instants
with optimum slicing lead equal to the pc component V». The values
of Vinin. = Mmin. are shown in Fig. 8.

Fig. 8 Factor nmin for raised cosine spectrum and quadratic delay distortion
as in Fig. 2 for synchronous detection and differential phase detection. Curve 1:
Ideal syne hronous detection applies for two-phase and four-phase modulation
with carrier at midband and pulses at intervals 7’, and for vestigial-sideband
transmission with pulses at intervals 7'/2. Curve 2: Ideal differential phase detec
tion — two-phase modulation with pulse interval 7’. Curve 3: Ideal differential
phase detection — four-phase modulation with pulse interval 7’.

TABLE XI Minimum AMPLITUDES OF DEMODULATED PULSE
TRAINS IN Four-PHASE MODULATION WITH DIFFERENTIAL
PHASE DETECTION

* Reversal of sign indicates a reversal in sign of the demodulated pulses.

With the accuracy used herein it turns out that 2 and Vo are numeri-
cally equal but are not identical.

It will be noted that, when delay distortion is pronounced, the bias
component Vo is appreciable, and that a significant penalty can be in-
curred if the threshold or slicing level is taken as 0 rather than Vo. For
example, with d/T = 4 and 0 threshold level the minimum amplitude
of a demodulated pulse for a carrier phase y = 0 would be 0.564, and
the minimum negative amplitude for a carrier phase y = x would be
—0.12. With the optimum threshold level the minimum amplitudes are
+0.342. Hence the tolerable peak noise amplitudes would be greater
by a factor 0.342/0.12 = 2.85.

With four-phase modulation the values given in Table XI are obtained
from (120). The values of Vinsin’ = Mmin_ are shown in Fig. 8.

In the above illustrative examples it was assumed that pulses were
transmitted at the minimum interval 7’ permitted if intersymbol inter-
ference is to be avoided in the absence of delay distortion. The effect of
delay distortion may or may not be reduced by increasing the pulse
interval, that is, in exchange for a slower transmission rate. By way of

TABLE XII FuNcTION Ro(n) FoR RAISED CosINE SPECTRUM
AND Quapratic DeLay Distortion with 50 PER CENT
INCREASE IN PULSE INTERVAL

TaBLE XIII MINIMUM AMPLITUDES OF DEMODULATED PULSES
WITH 50 PER CENT INCREASE IN PULSE INTERVALS

illustration it will be assumed that the pulse interval is increased by :
factor 1.5, in which case the values of Ry are as given in Table XII.

With this modification, the various quantities are as given in Table
XIII.

In Fig. 9 values of Umin and Vinin’ are Compared with those for the
minimum interval between pulses. It will be noted that there is no signifi-
‘ant difference in the case of two-phase or four-phase modulation with
synchronous detection. With differential synchronous detection some
advantage is realized for small delay distortion in exchange for a dis-
advantage with pronounced delay distortion.

Fig. 9 Effect of pulse interval on factor nmin for raised cosine spectrum with
quadratic delay distortion (dashed curves: pulse interval 7’, as in Fig. 8; solid
curves: pulse interval 1.57’). Curves 1: Ideal synchronous detection — applies for
two-phase and four-phase modulation with carrier at midband and pulses at in
tervals 1.57 and for vestigial-sideband transmission with pulses at intervals 0.757.
Curves 2: Ideal differential phase detection — two-phase modulation with pulse
intervals 1.57’. Curves 3: Ideal differential phase detection — four-phase modula-
tion with pulse intervals 1.57’.

396 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1961
V. BINARY FREQUENCY MODULATION (FSK)

shift keying requires the same minimum bandwidth as double-sideband
AM. In the absence of transmission distortion from gain and phase
deviations, the optimum signal-to-noise ratio required at the detector
input for a given error probability is slightly greater than for two-phase
transmission with ideal synchronous detection, but would be expected
to be about the same as for two-phase transmission with ideal differen-
tial phase detection. Binary FM may be preferable to the latter method
from the standpoint of implementation and has an advantage over the
simpler method of binary AM with envelope detection from the stand-
point of signal-to-noise ratio and performance during sudden transmis-
sion level variations.

The performance of binary FM is determined here for channels with
linear and quadratic delay distortion and compared with that of the
other methods mentioned above. In this analysis ideal frequency dis-
criminator detection is assumed, in which the demodulated signal is pro-
portional to the time derivative of the phase of the received wave. This
condition may be closely approached with actual detectors when the
channel bandwidth is small in relation to the midband frequency. How-
ever, when this is not the case, ideal FM detection is only approximated
with conventional frequency discriminators or zero ‘crossing detectors.

Expression (48) for the demodulated pulse train applies for any am-
plitude and phase characteristic of the channel. In the case of a continu-
ing space, a(n) = 0 in (46) and (47) and Uo(t) = 0. With a continuing
mark, a(n) = 1 in the above expressions and

teturning to (21), it will be recognized that (122) and (123) repre-
sent the in-phase and quadrature components in a binary amplitude
modulation system when pulses of duration 7 and alternating polarity
are transmitted, i.e.,a(m) = (—1)" in (21). The fundamental frequency
of such a pulse train is @ = T/x. Let A(—@) and ¥(—@) be the ampli-

tude and phase characteristic of the channel at the frequency —a@ from
w); A(@) and ¥(w) the corresponding quantities at the frequency @
from wo. Solution of (122) and (123) for the above steady state condi-
tion of alternate marks and spaces in binary AM gives

With (124) and (125) in (50), it turns out by way of check that
Uo(a) = 1 for a continuing mark for any amplitude and phase charac-
teristic of the channel.

For pulse trains other than continuing marks or spaces, intersymbol
interference will be encountered from amplitude and phase distortion.
In the following section special cases of phase distortion will be examined
further. It will be assumed that the amplitude characteristic has the
appropriate shape so that intersymbol interference can be avoided in the
absence of phase distortion. To this end it is necessary that A(—@)
A(@) = } or» = 2 as shown elsewhere (Ref. 1, Section V). In this case
(50) becomes with Uy = U

When the amplitude characteristic of the channel has even symmetry
about wy and the phase characteristic has odd symmetry Qo(.rx + n) = 0
and (126) simplifies to

Optimum performance would be expected when a single pulse is sam-
pled at its peak, a condition which is at least closely approximated with
y = 0. This condition is met when ¢t = ¢) is so chosen that

2Zao( Xo) c=: l
lor further analysis it is convenient to introduce the quantities

where designates absolute values when (—1)"Ro(.2x ) is nega-
tive and " when it is positive.
It will be recognized that

where the last relations follow in view of (122), (124) and (128).
During transmission of a space, delay distortion will have an adverse
effect only if U as given by (133) is positive, since only in this case is
the tolerance to noise reduced. To obtain a positive value of UU, it is
necessary to have either a 2 3 or a < 0. For a space, a(0) 0 in
(129) and a value of a 2 3 can then be excluded for any reasonable

delay distortion. It will, therefore, be assumed that ay < 0. The maxi-
mum positive value of U’, i.e., the maximum adverse effect of delay dis-
tortion, is then obtained with the maximum possible negative value of
ag. This maximum value is obtained by choosing a(n) 0 in (129)

whenever (—1)"Ro(2 — n) is positive and choosing a(n) = 1 whenever
(—1)"Ro(x — n) is negative. The maximum negative value of ao(z) thus
obtained is given by (134). The corresponding maximum value of U (to)
in the presence of a space and with sampling at ¢ = ft) as defined by
(132) is

During transmission of a mark delay distortion will have an adverse
effect only if U(t)) < 1. This will be the case if ag > 1 or ay < 3 in
(133). With a(0) = 1 in (129) for a mark, the condition ag < 3 will
not be encountered with any reasonable delay distortion and only the
case a > 1 needs to be considered. The minimum positive value of
U(t) in the presence of a mark is obtained when ap is taken as the maxi-
mum positive value given by a(x) = Ro(x) + ao (x), where ay’ is
given by (135). In view of (136) it follows that, for y = 0,

With (138) in (133) the minimum amplitude of a pulse train in the
presence of a mark and at the sampling instant fy) defined by (132) be-
comes:

The optimum slicing level in the presence of delay distortion becomes
for conditions as discussed in Section 2.9,

The latter expression also applies for the difference between the slicing
level and the maximum amplitude of a pulse train at a sampling point
in the presence of a space.

Expression (141) shows that the minimum amplitude at a sampling
point is smaller than in the absence of delay distortion (ao 0) by
the factor

TABLE XIV FACTOR min FOR RAISED CosiINE SPECTRUM
AND QuapRatTic DELAY DisTorTION

In the particular case of a raised cosine spectrum of the pulses at the
detector input, as shown in Fig. 1, and quadratic delay distortion, the
function Ro(t/T) = Ro(x + n) is given in Table XIX of the Appendix.
The phase distortion ¥(—q@) in this case is given by

where d/T is defined as in Section 4.5.
In Table XIV are given ¥(—@) together with x» as obtained from
132), ao (2) as given by (134) and yin as obtained from (142). These
factors are shown in Fig. 10, together with the corresponding factor for
binary double-sideband AM as obtained from Table IV.

When both the pulse spectrum at the detector input and phase dis-
tortion has even symmetry about the frequency Wy , the following rela-
tions apply (see Appendix

The maximum amplitude of a single pulse in this case is at ¢ 0).
Optimum performance is obtained with sampling at ¢ = 0, in which case
y in (126) and (127) is given by

QUADRATURE DELAY
DISTORTION:
--——FM
AM WITH
L- SYNCHRONOUS OR
ENVELOPE DETECTION

d/T =dfyax
Fig. 10 —+ Factor nmin for binary pulse transmission by FM and double-side

band AM, for quadratic delay distortion as in Fig. 2 and linear delay distortion
as in Fig. 4

Bo’ = Bi’ (0) = >. (—1)"a(n)Qi’(n). (150)
n=--2
For the special case of a raised cosine spectrum and linear delay dis-
tortion, the functions Ry and Q» are given in Table XX of the Appendix.
The functions Ro’(n) and Qo’(n) are related to the functions R; and Q,
given in Table XXII by

The functions Ry and Qp are given in Table XV for integral values of
t/T and the functions Po’ /@ and Qo’ /a@ are given in Table XVI.
In the above case of quadratic phase distortion of the form y(u)

Owing to the several quantities ao, Bo, ao’, Bo’ , Bo’ao, ao’ Bo and Yo
involved in (126), it does not appear feasible to derive simple relations
for Umax and U yin” . However, it is possible to determine these by
examining several cases, as illustrated below ford/7T = landd/T = 2.

TaBLeE XVII VALUES OF U(0O) ror RAtseEpD CosINE SPECTRUM
AND LINEAR Deuay Distortion witH d/T = 1

For various combinations of marks and spaces, i.e., a(n) | and 0,
the results given in Table XVII are obtained.

The results in this case are given in Table XVIII. In this case U,
0.57, Umin® = 0.42, Lo & 0.5 and qmin = Umin” — Umax” & —0.15.

TaBLeE XVIII Vatues or U(O) ror Ratsep Cosine SPECTRUM
AND LINEAR DeLay DisrorTion witu d/T' 2

The factors in Table XVIII are shown in Fig. 10, together with the
corresponding factors for binary double-sideband AM with synchronous
detection as given in Table V and with envelope detection as given in

The shape of pulse trains at the detector input and output in pulse
transmission by various methods of carrier modulation and detection
has been formulated in terms of a basic function common to all modula-
tion methods: the carrier pulse transmission characteristic. This fune-
tion is related to the amplitude and phase characteristics of the channel
by a Fourier integral, which can be evaluated by numerical integration
with the aid of digital computers for any prescribed channel characteris-
tic. In this way can be determined the effect of specified channel gain and
phase deviations on the demodulated pulse train for any modulation
method, together with the resultant maximum transmission impairment.

The carrier pulse transmission characteristics are given herein for the
representative case of pulses with a raised cosine spectrum at the detector
input, for two cases of envelope delay distortion over the channel band.
In one case delay distortion is assumed to vary linearly with frequency,
and in the other case to vary as the second power of frequency from mid-
band, as indicated in Fig. 11. The resultant maximum effect on the de-
modulated pulse trains at sampling instants has been determined for
various carrier modulation and detection methods, together with the
corresponding maximum transmission impairment. The maximum trans-
mission impairment is expressed as the maximum increase in signal-to-
noise ratio required at the detector input to compensate for the effect of
phase distortion, or corresponding envelope delay distortion. The maxi-
mum transmission impairments specified here apply as the error proba-
bility approaches zero, and actual impairment will be somewhat smaller,
depending on error probability.

In evaluating the effect of phase distortion, idealized modulation and

in other respects, such as instantaneous sampling of the appropriate
instants and optimum slicing levels.

The numerical results are given in various tables and curves, summa-
rized in Fig. 12 and discussed briefly below.

In the expressions for the carrier pulse transmission characteristic the
phase characteristic of the channel is a basie function. ‘Transmission
impairments from phase distortion could be expressed in terms of some
parameter or set of parameters that would define the type of phase dis-
tortion under consideration. Alternatively, any type of phase distortion

that is, in terms of envelope delay distortion. From the standpoint of
engineering applications the latter method is preferable, since variation
in transmission delay over the channel band is more readily measured
than variation in phase, and it is ordinarily the quantity specified for
various existing facilities.

d = MAXIMUM DELAY DISTORTION OVER BAND fyax (CPS)
T = PULSE INTERVAL IN DOUBLE SIDEBAND AM AND FM(SEC)
T =1/fuax d/T=dfwsax

Fig. 11 Pulse spectrum at detector input and types of delay distortion as
sumed in comparison of modulation methods.

VESTIGIAL ANDO
QUADRATURE
DOUBLE - SIDEBAND
AM WITH SYNCHRO-
NOUS DETECTION
AND L=2,3
PULSE AMPLITUDES

AND DIFFERENTIAL
PHASE DETECTION
=—=-=— SYNCHRONOUS
DETECTION
—— DIFFERENTIAL
PHASE
DETECTION

ods for raised cosine pulse spectrum with linear and quadratic delay distortion

Linear, quadratic or any other analytically specified delay distortion
can be expressed in terms of the difference in transmission delay between

any two reference frequencies in the channel band. In the present analysis

the difference d in delay between the midband frequency and the maxi-
mum frequency fmax from midband, as in Fig. 11, has been taken as a

basic parameter. The maximum transmission impairments with various
carrier modulation methods have been given in terms of the ratio d/T’ =
Afinax , Where T is the pulse interval in double-sideband AM.

difference dinax in transmission delay between any two frequencies in the
channel band. In the case of linear delay distortion dmax = 2d, while in
the case of quadratic delay distortion dnax = d, where d is defined in
Fig. 11. A third choice might have been the difference in delay d be-
tween the midband frequency and the mean sideband frequency 4}fmax ,
in which case d = d/2 for linear and d = d/4 for quadratic delay dis-
tortion.

It will be recognized that translation from one basic delay parameter
to another can readily be made. Also, the question of whether linear or
quadratic delay distortion causes greater transmission impairments will
depend significantly on the choice of transmission delay parameters.

Maximum transmission impairments are shown in Fig. 12 for systems
employing / = 2, 3, 4 and 5 pulse amplitudes and ideal synchronous detec-
tion. With envelope detection the transmission impairments are the same
as with synchronous detection, for quadratic delay distortion and for
any type of delay distortion with even symmetry about the channel mid-
band (carrier) frequency. However, with envelope detection greater

transmission impairments are incurred in the case of linear delay dis-
tortion, and for any type of delay distortion with odd symmetry about
the channel midband frequency. The difference between envelope and

synchronous detection in the presence of linear delay distortion is illus-
trated in Fig. 12 for 7 = 2 pulse amplitudes.

As noted previously, the maximum transmission impairments indi-
cated in Fig. 12 would be encountered for extremely small error probabili-
ties. For error probabilities in the range normally considered, the maxi-
mum impairments given in Fig. 12 would be rather closely approached
when the impairments are fairly small, say less than 3 db. However,
when the maximum impairments are rather high the actual impairments
may be significantly smaller. For example, with a maximum impairment
of 10 db, the actual impairment would be expected to be about 1.5 db
less for an error probability 10 ° and about 2 db less for an error proba-
bility 10“.

Vestigial-sideband AM and quadrature double-sideband AM _ with
synchronous detection are equivalent methods as regards channel band-
width requirements and signal-to-noise ratios, in the absence of delay
distortion. Both methods may be used in preference to double-sideband
AM either (a) to realize a two-fold increase in pulse transmission rate for

a given bandwidth in exchange for a 3 db penalty in signal-to-noise ratio
or (b) to secure a two-fold reduction in bandwidth for a given pulse
transmission rate, without a penalty in signal-to-noise ratio.

The maximum transmission impairments shown in Fig. 12 are for the
same bandwidths as in double-sideband AM with a two-fold increase in
the pulse transmission rate. In this case transmission impairments from
quadratic delay distortion are no greater than in double-sideband AM,
and this applies for any type of delay distortion with even symmetry
about the channel midband frequency.

With linear delay distortion, or any delay distortion with odd sym-
metry about the channel midband frequency, transmission impairments
are not identically the same for vestigial-sideband AM and quadrature
double-sideband AM. However, the difference is not significant in the
ease of linear delay distortion, as indicated in Fig. 12. For practical
purposes the two methods can be regarded as equivalent for any type of
delay distortion actually expected, as regards channel bandwidth require-
ments and signal-to-noise ratios for a given error probability, assuming
ideal synchronous detection.

With linear delay distortion the transmission impairments with the
above two methods are significantly greater than for double-sideband
AM as indicated by comparison of the curves in Fig. 12 for the two meth-
ods for / = 2 and 3 pulse amplitudes. This assumes that the pulse trans-
mission rate is twice as great as in double-sideband AM.

When the pulse transmission rate is the same as in double sideband
AM but the bandwidth is halved, delay distortion over the channel band
is reduced. In this case vestigial-sideband AM or quadrature double-
sideband AM affords an advantage over double-sideband AM in the
presence of delay distortion with even symmetry about the channel mid-
band frequency, but not necessarily when delay distortion has odd sym-
metry. With linear delay distortion the ratio d/T is halved, and in this
case there is a slight disadvantage compared to double-sideband AM,
for 1 = 2 pulse amplitudes. However, with the type of delay distortion
ordinarily encountered vestigial-sideband AM and quadrature double-
sideband AM would afford some advantage in signal-to-noise ratio over
double-sideband AM for equal pulse transmission rates and with ideal
synchronous detection.

band AM with equal amplitudes but opposite polarities of the trans-
mitted pulses. The curves in Fig. 12 for double-sideband AM and / = 2

pulse amplitudes apply also for two-phase transmission, for the reason
that the transmission impairments for a given peak-to-peak difference
between pulse amplitudes is the same regardless of polarities.

Two-phase modulation can also be used in conjunction with vestigial-
sideband transmission. The curves in Fig. 12 for vestigial-sideband AM
and 1 = 2 pulse amplitudes also apply for two-phase vestigial-sideband
modulation.

lour-phase modulation is equivalent to bipolar AM on each of two
carriers at quadrature with each other. The curves in Fig. 12 for quadra-
ture double sideband AM and / = 2 pulse amplitudes also apply for the
special case of four-phase modulation.

The maximum transmission impairments with double-sideband two-
phase and four-phase modulation and synchronous detection are shown
separately in Vig. 12 for comparison with PM with differential phase
detection.

In phase modulation systems differential phase modulation (described
in Section 4.1) may be used in place of synchronous detection. Differen-
tial phase detection has been implemented in various ways, which in gen-
eral involve some transmission impairments from channel bandwidth
limitations, even with a linear phase characteristic. Such transmission
impairments from channel bandwidth limitation is avoided with the
implementation assumed herein (Section 4.1), and only the effect of
phase distortion is evaluated. Transmission impairments from delay dis-
tortion will be greater with this method than with synchronous detection,
as illustrated in Fig. 12 for double-sideband two-phase and four-phase
quadrature systems and delay distortion. Transmission impairments from
linear delay distortion have not been determined for this case.

requires the same bandwidth for a given pulse transmission rate as binary
double-sideband AM. Maximum transmission impairments with these
two methods are compared in Fig. 12. It will be noted that with quad-
ratic delay distortion the impairments are smaller with FM than with
AM employing either envelope or synchronous detection. In the case of
linear delay distortion, the transmission impairments are greater with
FM than with synchronous AM, but are somewhat smaller than with
AM employing envelope detection.

The transmission impairments given in Fig. 12 for kM apply without
a postdetection low-pass filter for noise reduction, and may involve
somewhat greater approximations than for the other modulation meth-
ods. Approximately the same impairments from phase distortion would
be expected with an appropriate low-pass filter.

Signal-to-noise ratios at the detector input for a given error probability
and various methods of carrier modulation are ordinarily compared on
the premise of ideal amplitude versus frequency characteristics of the
channels, and a linear phase characteristic. The curves in Fig. 12 indicate
that transmission impairments resulting from phase distortion depend
significantly on the carrier modulation method. The optimum method as
regards signal-to-noise ratio will thus depend on the type and degree of
phase distortion encountered in a particular application. For example,
two-phase modulation with synchronous or with differential phase detec-
tion may have a slight advantage in signal-to-noise ratio over binary
frequency shift keying in the absence of delay distortion. However, the
advantage in signal-to-noise ratio would be expected to be with fre-
quency shift keying in application to channels with pronounced quad-
ratie delay distortion or other types of delay distortion with essentially
even symmetry about the carrier frequency.

In comparing the performance of various methods of carrier modula-
tion it is necessary to consider other factors than signal-to-noise ratios
and channel bandwidth requirements as discussed here. Among them
can be mentioned the adverse effects of sudden or gradual level and phase
variations and the complexity of instrumentation.

The writer is indebted to A. P. Stamboulis for pointing out some errors
in the original equation (50) and for showing the presence of the third
term in (56) and to C. IF. Pease for numerical evaluation of integrals in
the Appendix with the aid of a 704 digital computer.

of the carrier pulse transmission characteristics for any carrier frequency

can be determined from those for any other carrier frequency w» , for
example the midband frequency of the channel. Basic Fourier integrals
are given here for the carrier pulse transmission characteristics for a
reference or carrier frequency wo . In addition, special integrals are given,
applying for a raised cosine pulse spectrum with linear delay distortion,
quadratic delay distortion and the type of delay distortion introduced
by flat bandpass filters with sharp cutoffs. For these three cases the
carrier pulse transmission characteristics have been determined by nu-
merical integration and are tabulated here.

The shape of Ro(t) and Qo(t) depends on the shape of the transmitted
carrier pulse and on the transmission-frequency characteristic of the
channel. If the carrier pulse is assumed of sufficiently short duration, the
spectrum will be essentially flat over the channel band, so that the shape
of the received spectrum is the same as that of the amplitudé character-
istic of the channel. The functions Ry and Qo are then obtained from ex-
pression given elsewhere (Ref. 2, Section 2) in terms of the amplitude
characteristic A(u) of the channel, where u is the frequency measured
from the carrier frequency wo , as indicated in Fig. 13. In the more general
case of carrier pulses of any shape and any channel transmission-fre-

quency characteristic, the functions Ry and Qo are obtained by replacing
in the above expressions A(uw) with the spectrum So(u) of the pulse

AMPLITUDE
CHARACTERISTIC OF
SPECTRUM AT CHANNEL CARRIER FREQUENCY
OUTPUT ~-

Fig. 13 — Amplitude characteristic So(u) and phase characteristic Wo(u) of
pulse spectrum at channel output (i.e., detector input) for carrier at frequency w

The various quantities in the above expressions are as shown in Fig. 13.
It will be recognized that the upper limit w in (160) and (162) can for
practical purposes be replaced by ~, since So( —wo) & 0.

4.2 Even Symmetry Spectrum and Delay Distortion
Let the spectrum at the detector input have even symmetry about
wy and the phase distortion odd symmetry, in which case
Sof — a) So{u), (164)
Vo(—u) = —WVol(u). (165)

With (164) and (165) in (159) through (163), the following rela-
tions are obtained when the upper limit wo is replaced by « :

When the phase characteristic has a component with even symmetry
about the frequency wo , so that

lor reasons discussed elsewhere (Ref. 2, Section 5) it is desirable in
pulse systems to employ raised cosine pulse spectra, as shown in Fig. 1
and given by
‘yy

where @ is the mean frequency from midband.
The corresponding carrier pulse transmission characteristic obtained
from (159) through (163) with Wo(w) = 0 is

Pulses can in this case be transmitted without intersymbol interference
at intervals 7’ such that

It will be assumed that the phase characteristic contains a linear com-
ponent, which can be disregarded, and a distortion component given by

where ¢ is a constant. The corresponding delay distortion is then quad-
ratic or parabolic, as given by

NOMLMOLSI(] AVIAG] OLLVuaVvaA’d aN
WIUMLOAIS ASTAG ANISOY) GASIVY, UOA CL /))"Yy ANV (f/))°Y SNOLLONOG XIX Wavy,

where
16 d
b=—-. (177)
32° 7
The ratio ¢/T is the time measured in pulse intervals and the ratio d/T
the maximum delay distortion measured in pulse intervals, with d de-
fined as in Fig. 2 or Fig. 11.
In certain cases, as in connection with pulse transmission by frequency
modulation, the time derivative of Ro(t) is involved. This derivative is
given by

The functions Ro(t) and R,(t) obtained by numerical integration of
(176) and (179) are given in Table XIX. The function R)(¢/T) is

shown in Fig. 2.
\.6 Linear Delay Distortion
It will be assumed that the phase distortion component is given by
Wola) cu, (180)
which corresponds to a linear delay distortion given by
Wo’ (ua) Pou.
In this case expressions (169) and (170) give

NOMLUOISI(] AVIA(] UVANITT ANY
WOAUNLOAdS ASTAG ANISOD) aasivy woa (f/7)°%) axv (f/2)°Y SNOIONA A XX Wavy

The values of Rp and Qo obtained by numerical integration of (182)
and (183) are given in Table XX.

It will be noted that Qo(0) # 0. From the standpoint of analysis, it
may be convenient to modify the phase such that Qo(0) = 0. The modi-
fied values are given by

The modified values are given in Table XXI. The functions Ro(t/T)
and Qo(t/7) are shown in Fig. 4.

The time derivatives of Ro(t) and Qo(¢) are of interest in connection
with frequency modulation and given by

The functions R; and Q; obtained by numerical integration are given
in Table XXII.

NOLLUOLSI(Q] AVIAC] UVANITT GNV
WAULIdG ASTAG ANISOD aasIvyY Yoda (,f,/2)°) GNV (L/2)Y SNOMONOY — [XX AMV

(LAI) *) “C(L/'Y— = CL/I-)'e
NOILYOLSI(] AWIAC] UVANI'T aNV
WAULIddg ASTAG ANISOD aasivy wod (7/7)'7) ANV (7/2) SNOMLONO GT WXX OMavVy

Let a bandpass filter have an amplitude characteristic Ay between
—w. + wo and wo + w, and A, outside this band. When the bandwidth
2w, is small in relation to the midband frequency wo , the phase charac-
teristic is closely approximated by

The corresponding envelope delay distortion is D(w) = dyo(u)/du and
delay distortion relative to the midband frequency becomes

It will be noted that the first term in (196) represents quadratic delay
distortion, which is approximated for u/w, < 1.

Let the pulse spectrum at the detector input have a raised cosine
shape, as given by (171), in which case the maximum radian frequency
to each side of midband is 2. With a phase characteristic as given by
(193), the carrier pulse transmission characteristic is in this case ob-
tained with (171) and (193) in (166) and becomes

Carrier pulse transmission characteristics for raised cosine pulse
spectrum and phase distortion resulting from flat filters with sharp cutoffs.
Taste XXIII— Function R,(t/T) For Ratsep CosingE SpectruM
AND PHASE Distortion RESULTING FROM FLAT FILTERS
WITH SHARP CUTOFFS
k = W/W, 1.05
Ao/Ai

. Sunde, E. D., Ideal Binary Pulse Transmission by AM and FM, B.S.T..J..
38, 1959, p. 1357.
2. Sunde, E. D., Theoretical Fundamentals of Pulse Transmission, B.S.T.J.,
33, 1954, pp. 721; 987.
3. Gibby, R. A., An Evaluation of AM Data System Performance by Computer
Simulation, B.8S.T.J., 39, 1960, p. 675.
Fowler, A. D. and Gibby, R. A., Assessment of Effects of Delay Distortion in
Data Systems, Comm. & Elect., no. 49, 1959, p. 918.
Costas, J. P., Synchronous Communications, Proc. I.R.E., 44, 1956, p. 1713.
). Rieke, J. W. and Graham, R.S., The L-3 Coaxial System Television Termi
nals, B.S.T.J., 32, 1953, p. 915
Darlington, 8., Realization of Constant Phase Difference, B.S.T.J., 29,
1950, p. 94
Cahn, C. R., Performance of Digital Phase Modulation Communication Sys-
tems, Proc. I.R.E. Trans., CS-7, 1959, p. 3
. Cahn, C. R., Combined Digital Phase and Amplitude Modulation Communi
cation Systems, I.R.E. Trans., CS-8, 1960, p. 150.
Mosier, R. R. and Clabaugh, R. G., Kineplex A Bandwidth Efficient Binary
System, Comm. « Elect., no. 34, 1958, p. 723.

Further Results on the Detectability of
Known Signals in Gaussian Noise

The detection of a completely known signal which may or may not be
present in a finite sample of gaussian noise is considered from two points
of view. The first examines the performance of a maximum likelihood de-
tector operating on a finite set of discrete measurements of the stimulus as
the set becomes large. The stimulus is either signal plus noise or noise alone.
Examples are presented for signals in bandlimited noise, using as measure-
ments either equispaced amplitude samples or derivatives at one instant in
time. For both, the detectability grows without bound as the number of meas-
urements is increased. The second point of view bases detection on a con-
tinuous measurement (linear integral operator) which maximizes the de-
tectability. Solutions have been obtained when the noise has a rational power
spectral density. The detector utilizes a cross-correlation between stimulus
and signal which is well known and a mechanism, designated extrapolation
detection, which involves evaluation of derivatives of the stimulus. The con-
tribution of the derivative measurements to the detectability is examined as
the noise approaches bandlimited noise and is found in many cases to
grow without bound.

The problem under consideration here is the detection of a completely
known signal which may or may not be present in a finite sample of
gaussian noise. That is, we imagine a situation similar to Fig. 1 in which
a stimulus is made up of either signal plus noise or noise alone and we
ask, given 7’ seconds of this stimulus, how accurately can we decide
whether or not the signal is present. The noise is thought of as having
been produced by a stochastic process and thus the question is really
one of statistical hypothesis testing.

This particular problem has been treated rather extensively,’7 and
certain questions, even controversies, have arisen. These concern what

constitutes a proper description for the stimulus, under what circum-
stances can the stimulus be characterized by a finite number of samples,
and under what conditions is perfect detectability obtained, i.e., when
is it always possible to detect the presence or absence of the signal.
Peterson, Birdsall and Fox’ have described the stimulus as being Fourier
series bandlimited and by so doing have obtained quite different results
from the other authors, who for the most part consider stationary gauss-
ian noise. In many cases, finite-duration stimuli have been character-
ized by a finite number of samples usually chosen so they are independ-
ent, and maximum likelihood detectors operating on these samples have
been developed. This has led to the equivalent of a correlation detection
process in which the test statistic is the integral of the product of the
stimulus and a function derived from the signal. Such detectors always

out by an argument involving analytic continuation that many signals
can be perfectly separated from noise provided the noise is considered to
have a bandlimited spectrum. Clearly some mechanism in addition to
correlation detection is inherent in Slepian’s result, and indeed he points
out one such detector.

The results of Peterson, Birdsall and Fox have been used extensively
for comparison with the performance achieved by humans and other
animals, and questions as to the validity of such comparisons originally
motivated this investigation. However, it seems very doubtful if the
mechanisms which will be developed can have anything to do with per-
ception. In addition, we have chosen to work with stationary gaussian
noise rather than Fourier series bandlimited noise, the former being :
much more satisfactory characterization of real noise.

Two different attempts to better understand the questions cited above
have been undertaken. The first examines the performance of a maxi-
mum likelihood detector operating on a finite set of discrete measure-
ments of the stimulus as the set becomes very large. The results show
cases where the detectability grows without bound. Thus, the charac-

terization of the stimulus by a finite set of measurements is incomplete.
However, in some cases, a law of diminishing returns operates so that
the rate of increase in detectability slows as the number of samples is
increased.

The second study bases detection on a continuous measurement
(linear integral operator), which is the solution of an optimizing integral
equation. The test statistic so obtained has two parts, one similar to
correlation detection, the other based on measurements of the deriva-
tives of the stimulus. The contribution of this latter term is usually the
smaller of the two, but, where the noise spectrum approaches a band-
limited form, it may grow without bound. In addition, it may be im-
portant if the stimulus is very short.

Both maximum likelihood detection with a finite number of samples
and the integral equation for the continuous statistic have been pre-
viously presented. The new contributions arise from the more complete
solutions which have been obtained. The most significant result is un-
doubtedly the solution of the integral equation in closed form so that
its characteristics and particularly its asymptotic properties for many-
pole noise can be seen. The derivative detector, which will be termed
extrapolation detection, was apparent from this solution.

In this section we will derive the maximum likelihood detector for de-
tecting a known signal in gaussian noise from a finite number of samples
of the stimulus and apply this detector to two specific problems involv-
ing bandlimited noise. Each sample results from some linear operation
on the stimulus and the samples need not be independent. The deriva-
tion of the detection equation differs only slightly from previously pub-

which are the principal new results. In the problems the behavior of
the detector is studied as the number of samples becomes large, first
when the samples consist simply of amplitude measurements of the
stimulus and second when the samples are a set of derivatives at one
point in time.

426 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1961
is either a gaussian noise N(¢) or that noise plus a known signal S(t)
and is observed for the interval 0 Ss ¢ < T. The n samples,

on which the detection is made are obtained by n linear operations
L, , Le, -++,L, on the stimulus

and N; will be gaussian random variables which may be completely
characterized by their matrix 8 of correlation coefficients,

Bi; = EXN.N>,
and by their means which for simplicity will be assumed to be zero,
E<N > = 0.

The density function of the Y; samples when the stimulus is noise
alone may then be written

where ,°°*,Y¥, are the dummy arguments of the density function
corresponding to Y,,---,Y, and | 8| is the determinant of 8, with all
sums going over the range | to n unless especially indicated otherwise.
The density function of Y, for signal plus noise is simply

A maximum likelihood detector says that signal is present if test
statistic L(Y,,---,Y,) is greater than some threshold a and will maxi-
mize the conditional probability of detecting a signal when it is present
for a given conditional probability of indicating signal for noise alone.

lor signal plus noise gsy is also gaussian with the same variance but
with mean
Exgsy> = >> Bij 'SiS;.

The density functions for ¢ are pictured on Fig. 2. The effectiveness of
this detector as indicated by the signal detection probability at a given
false alarm rate can be characterized by a single number d, which is the
ratio of the squared mean of the signal plus noise distribution to the
variance of either distribution. The larger d is, the more completely
separated are the distributions on lig. 2 and the higher will be the de-
tection probability. This number d is then

This form is usually preferable for computations since it involves the
solution of n linear equations rather than the inversion of an n X n
matrix. In addition this form more closely resembles the integrals which
will appear when continuous statistics are considered.

To summarize, a statistic ¢ which operates on a set of n correlated
samples and which is equivalent to a maximum likelihood statistic has
been developed. Signal is indicated if ¢ is greater than some threshold.
¢ is formed as a linear sum of the samples, it has a gaussian distribution,
and it has the same variance for both noise alone and signal plus noise
vases. The performance of the detector may be characterized by a single
number d = [E<¢gsy>]’'/E<gy’>, the larger the d, the better the perform-
ance.

The argument presented by Slepian’ indicates that theoretically, be-
cause of the analytic nature of the noise, a sinusoid can always be de-
tected in spectral bandlimited noise. However, this result says nothing
about how fast the detectability increases with the complexity of the
detector. In this section an example is examined in which the stimulus
is time sampled with n samples equally spaced over the interval 0 S
t < T and detectability is computed as a function of n. In addition to
the general behavior of this function, it is of special interest to note
whether any peculiarities occur at n = 2WT (the Nyquist rate), W
being the noise bandwidth, since this is the maximum number of in-
dependent samples which may be formed. The correlation function of
the noise is
_ sin 2*Wr
— QeWr ’

no = 2WT.
Unfortunately, no analytic way for either inverting this matrix or solv-
ing (5) is known, hence the detectability was computed numerically.
This computation was carried out on an IBM 704 machine for a signal
with frequency centered in the noise band

A being the amplitude and an,/4n being an arbitrary phase chosen for
computational convenience. The normalized results of a solution of (5)

the number of samples n/n, and of the stimulus duration in terms of
the number of independent samples n, . The curves exhibit a knee, not
at n = n, but for n a bit larger than n, . Detectability continues to in-
crease but the rate of increase becomes imperceptible. The curves are
all carried out to a matrix of size 128 X 128, which is the limit of the
capacity of the computer program. Double precision arithmetic and a
sufficient error analysis were used to insure the accuracy of the results.
The increase in detectability beyond n = n, is essentially equivalent to
that which would be obtained by increasing T to T + 2/W and sampling
at the Nyquist rate. Heuristically we can say that, by adding extra
points inside the interval, it is quite easy to predict N (¢) two independent
sample times beyond each end of the interval, but very hard to predict
further. In an unpublished proof Slepian has shown that the quadratic
form for d given by (3) does become infinite for bandlimited noise as n
becomes infinite. However, the present example indicates it increases
at an exceedingly slow rate. Clearly a statistic which improves more
rapidly is desirable, and such is evaluated in the next section.

The solution for the optimum integral operator detector carried out
in the next section produced a statistic involving derivatives of the
stimulus. This result suggests trying derivatives for bandlimited noise,
particularly since all derivatives of a bandlimited stimulus exist. Con-
sequently, the detectability achieved by n samples, which are the stim-
ulus and its n — 1 derivatives evaluated at one point in time, is studied.
This quantity, as will be seen, has the pleasant characteristics of being
analytically rather than only numerically determinable and of increasing
uniformly with n rather than exhibiting the knee curves of the time
samples. A curious property is that the duration of the stimulus is no

Fig. 3 Normalized results of a solution of (5) and (6), with d/A2as a function
of number of samples n/n, and of stimulus duration in terms of number of inde
pendent samples n,
longer a factor in detectability since, theoretically at least, any number
of derivatives can be measured from as short a sample as desired.

The correlation coefficient may be written
a+ Oe
| ; ers ree
Br = G(w)(—jw)’ (gw) da,
or . a
_ $ oa
where G(w) is the power spectrum of the noise

If bandlimited noise with a flat spectrum from —1 to +1 rad/second
and unit rms amplitude is selected, then (7) yields

0) if r + sis odd.
A solution for (5) and (6) with these coefficients can be effected, since
the determinants involved are reducible to a form with a solution attri-
buted to Cauchy. The answer can probably be written on a large enough
sheet of paper for signals having simple derivatives such as sinusoids,

for n odd.
The asymptotic behavior of d for large m can be seen by substituting
Stirling’s approximation
a! & V/2n exp |—a + (log a)(a + 3)}

Equations (8) and (9) exhibit the behavior of a statistic in which d
increases linearly with the number of samples, each sample being a
derivative. A similar behavior will be shown for rational noise where
one term in the detectability depends linearly on the number of deriva-
tives which exist and form part of the statistic. The bandlimited noise
differs from the rational noise in that all its derivatives theoretically
exist and the detectability can be made, at least theoretically, as good
as desired by making m large enough. Obviously, in any practical case,
the number of derivatives which can be estimated is limited. In addition,
the characterization of the random process as gaussian undoubtedly
fails for high enough derivatives.

Equations (8) and (9) are derived only for a signal which is a con-
stant. However, a similar dependence on m would probably occur for
sinusoidal signals.

The prominence of derivatives as an effective statistic for both band-
limited and rational noise gives a possible indication why detectability
based on equally spaced time samples increases so slowly. These, being
uniformly distributed, give poor estimates of derivatives. A more effec-
tive distribution might well be n, independent samples spaced uniformly
over the interval and the rest of the samples clustered as closely as pos-
sible about two points at each end of the interval. Such arrangement is
suggested by statistics for the rational noise case.

The preceding section discussed the detection of a known signal in
bandlimited noise using a finite number of samples of the stimulus as a
statistic. In this section we consider the detection of a known signal in
gaussian noise using as the statistic a continuous measure of the stimu-
lus over an interval 7 in length. The noise is now taken to have a ra-

sented at the ratio of two polynomials in w. Such noise can be thought
of as resulting from the passage of ideal white gaussian noise through a
finite linear lumped-element filter, although it need not actually have
been produced in this way. For the purposes of the analysis, it is con-
venient to think of the situation as shown in Fig. 4. White gaussian
noise is passed through a filter whose transfer function is H(s), (Laplace
transform of its impulse response ) and to this may or may not be added
the known signal S(t). 7 seconds of the combination form the stimulus
Y(t). The problem is to decide from an examination of the stimulus
whether or not the signal was present.

The detection scheme in this case is essentially an extension of the
finite sampling procedure. One asks for that linear integral operator
which will extract from the stimulus a statistic giving the maximum
detectability. Thus, the statistic is obtained from

-
g= [ Y(t)Z(t) dt, (10)
“0
where Z(t) is that function of time which maximizes the detectability.
Because the noise is gaussian of zero mean and the signal (when present )
is simply added to the noise, the statistic g¢ again has a gaussian proba-
bility density function whose mean value is zero or not zero according
to the absence or presence of the signal and whose variance is the same
with or without the signal. Thus it is reasonable to again define the de-
tectability measure d as

d= (11)
The optimization problem is thus to find Z(t) which maximizes d or,
that which is equivalent, to find Z(t) which minimizes E(gw) while
holding E(¢sy) constant. This latter form is a straightforward calculus
of variation problem and its solution, the details of which are omitted,
leads to the following integral equation for Z(t):

The discussion up to this point has not required that the noise have a
rational spectral density. Unfortunately, it does not appear possible to
earry (13) any further without actually solving (12) for Z(t), and this
has only been done in certain special cases. In particular, if the noise
spectral density is the reciprocal of a polynomial, the solution for (12)

an be exhibited in some detail; and furthermore if the signal is a sine
wave, an exponential, or a constant the detectability can be expressed
in a surprisingly simple form.

If the noise has a spectral density G(w),
o
G(w) Rir)e *" dr,
° x
which is rational and contains only poles (2N in number), it can be
written in the form
. l
G(w | = (14)
ay — Aw + ayw? — 2.00 aie doy"
Such a noise could have been produced by passing white noise of unit
spectral density through a filter whose transfer function H(s) has N
poles,
l | -
H(s) = . (15)

and the poles can be placed in evidence by writing the denominator
polynomial P(s) as

where the y’s are (possibly ) complex numbers giving the pole locations
and each has a negative real part. In terms of H(s), the spectral density
can be written
G(w) H (jw)

-, yw and one constant by , or even the magnitude and phase of the
transfer function //(s) for real frequencies. The particular set of param-
eters to be used will be chosen to simplify the final answer.

One characteristic of N-pole noise is that its first NV — 1 derivatives
exist, while the Nth and higher do not. Because of this it is clear that
a necessary condition for finite detectability of a signal S(t) is that its
first N — 1 derivatives be continuous in the interval 0 to 7. If this con-
dition is not satisfied; that is, if among the N — 1 derivatives of S(t) a
discontinuity occurs, then the detectability is infinite. This is clearly
true, because one could simply differentiate the stimulus enough times
to produce a step function in the interval and this could always be found
by measuring the change in the differentiated stimulus just before and
just after the time of the step.

Using this N-pole noise, it is possible to exhibit explicit solutions to
(12) and (13). Unfortunately, strictly speaking, (12) does not have a
solution unless S(t) and its derivatives up to order N — 1 satisfy a
certain set of boundary conditions (boundaries at 0 and 7’). If S(t) does
not satisfy this set of boundary conditions, and in general for an arbi-
trary signal it will not, then (12) has a formal solution if Z(t) includes
delta functions and their derivatives to order N — 1 at the end points
of the interval (approached from inside the interval). The details of
this argument are presented in Appendix B, where it is shown that the
solution to (12) is

where the superscript (n) indicates n-fold differentiation with respect
to time, and the a’s and §’s are given by

Among the several other ways of writing d, one which is convenient
is the following (partly operator notation ):

N—1
P;( ) = 2» Dyes"
k=i
and p is the derivative operator d/dt. The derivatives of S(t) and U(t)
at 0 and T are to be interpreted as the limit of the value of the deriva-
tives approached from inside the interval.
The form of Z(t) in (17) is quite interesting. The first part contributes
a function of time which is similar to the conventional cross-correlation
result. One simply multiplies the stimulus by this function and integrates
the product. In the second part, the delta functions, when used with (10)
to form the statistic, represent evaluating the stimulus and its first
N — 1 derivatives at the ends of the interval. The derivatives at the
ends give information about the stimulus outside the interval. Essen-
tially they allow prediction or estimation of the stimulus outside the in-
terval, and this information is to be added to that from straight cross-
correlation. As N becomes larger the noise spectrum drops off faster at
high frequencies and more derivatives of the stimulus are used (more

derivatives of the noise exist ); effectively, the stimulus can be predicted

further outside the interval. Usually, this will mean that the signal can
be detected better (see examples below ).
3.2 Damped Sinusoidal Signal

As a particular example, consider the case in which the signal is a
damped sine wave of arbitrary phase,

Since the detectability is of primary interest, specific values for the co-

efficients of the delta functions will not be calculated. The details of the
calculations are carried out in Appendix C, where it is shown that

With given signal parameters and noise filter, specific values of detect-
ability can be calculated from this expression.

As the number of poles in the noise filter increases, P”( —a)/P*(a) 0,
assuming the poles are bounded away from the imaginary axis and that
a > 0. In this case d becomes

If as the number of poles is increased the pc gain of the filter is kept
constant (or allowed to increase), then P’(a) increases without bound.
This can be seen by thinking of P(a@) in factored form, which for con-
stant pe gain looks like

and noting that | (a — y;)/y:| > 1. Thus, for fixed signal, more poles
mean more detectability. A similar result obtains if a < 0.

A noise filter of particular interest is a Butterworth filter, that is, one
whose poles are uniformly distributed on a semicircle in the left-half
plane. Such a filter gives noise whose spectrum is maximally flat low-
pass and approaches ideal bandlimited noise as the number of poles
increases. In this case, the approximate behavior of d for large N can

be calculated by taking the poles as smeared out on a semicircle of radius

A sketch of B versus @/ wo is shown in Fig. 5. Clearly B is greater than
one and the detectability grows exponentially for large N.

For an undamped sine wave (a = 0), (22) can be put in a more con-
venient form by using the magnitude and phase of the noise filter trans-
fer function, H(s), which can be written

The angle @(w) then is the phase lag of the noise filter, a function of
frequency. In these terms (22) becomes
a ee
i= T + 20(w) —

If w7’ > 1, that is, if the time is long so that there are many cycles
of the sine wave in the interval, then the last term in (25) can be neg-
lected. In conventional circuit analysis, 6 is generally considered the
time delay of a network; thus, the detectability includes a term pro-
portional to twice the time delay of the noise filter. Roughly, this says
that the derivatives at the ends of the interval allow extension of the
stimulus a distance equal to the time delay outside each end.

It is clear that the 6 term grows without bound as the number of
poles bounded away from the imaginary axis is increased. In the par-
ticular case of noise with a maximally flat spectrum {Butterworth H(s)],
this growth can be shown more explicitly. The contribution to 6 from a
single pair of poles located at —aye*” is

wo At + 1 + 2A? cos 28’ wo
To add up the contributions from N poles on a semicircle would lead to
a rather complicated expression, but an approximation for large NV can
be obtained by imagining the poles smeared out on the semicircle, so
that the sum can be evaluated as an integral. Then

This shows clearly that, for large NV, 6 increases directly in proportion
to N. As a sidelight, the proportionality constant, plotted in Fig. 6, is
larger if the signal frequency is near the band edge. The apparent in-
finity for @ = a is a mathematical fiction; it resulted from smearing the
poles. For any finite N, 6 is finite; thus, the curve in Fig. 6 really should

W/W,
Fig. 6 — Proportionality constant.
be rounded over at the peak. For signal frequencies outside of the noise
band, the detectability becomes large simply because the 1/G(w) term
multiplying everything in (25) becomes large. Even straight cross-

shows clearly that the detectability increases as the number of poles
bounded away from the imaginary axis is increased.
For N-pole Butterworth noise of bandwidth wo, (27) becomes (ex-

Here again the detectability grows directly in proportion to N for
large N.

We have presented solutions to some problems involving detection of
the presence of known signals in gaussian noise. Thus, we are concerned
with what a statistician would term hypothesis testing. Two general
classes of detectors are studied, the first a maximum likelihood detector
operating on a finite number of samples of the stimulus, the second an
optimum integral operator treating the stimulus as a continuous fune-

tions, which differ little from ones previously given, but rather in the
specific solutions to these equations.

In the finite sampling case, detectability of a sinusoid or constant in
bandlimited noise is computed for the cases where the samples are
equally spaced time samples spread over a finite duration and where the
samples are measurements of successive derivatives at one point in time.
As the number of samples increases, detectability increases without
bound for both cases. However, for the time samples the rate of increase
is very slow for a large number of samples while for derivatives the rate
becomes a linear function of the number of samples.

For optimum linear integral detection a general solution is presented
for arbitrary signals in noise with a rational all-pole spectrum. The solu-
tion in closed form is sufficiently tractable so that the asymptotic be-
havior of certain simple signals can be evaluated as the number of poles
in the noise becomes very large. ‘The solution puts in evidence two differ-
ent detection mechanisms, one involving integration of the product of
the stimulus with a function derived from the signal, the other involving
measurement of the derivatives of the stimulus. The first is denoted
correlation detection, the second extrapolation detection. Usually, the
term arising from correlation detection is the more important. However,
if the stimulus is very short or if the noise spectrum has a great number
of poles, the extrapolation term may become relatively large. For signals
such as a sinusoid it grows without bound as the number of poles in-
creases.

What are the implications of these solutions on previous detection
results? Probably they have very little bearing on the perception prob-

lem which engendered the study, since it seems unlikely that animal
sense organs embody the mechanisms implied by the solutions or that
the characterization of exactly known signals in gaussian noise is appro-
priate. Both the solutions and the character of the stimuli differ signifi-
cantly from the Fourier series bandlimited case treated by Peterson,
Birdsall and Fox. In particular, the extrapolation detection does not
appear in their universe. Also, we feel that the characterization of the
noise as described by a correlation function is, to say the least, more
suited to the present style of engineering and, to say the most, a much
more satisfactory model of most detection situations.

The practical impact, if any, of the detectors developed here would
seem to inhere in situations where short pieces of valuable signals must
be detected and a great quantity of computing equipment is available.
Such might be the case for some space communication problems.

A number of unsolved problems arise directly from the work. For a
finite number of time samples of the stimulus, the optimum distribution
in time of these samples is unknown. Spectra with zeros as well as poles
have not been treated with anything near the elegance of the pure pole
situation. Only very specific classes of signals have been studied. It
would be of interest to establish which signals give unbounded and which
give bounded detectability as the number of poles in the noise increases.
Finally, only the case of signals known exactly has been examined. The
far more difficult area involving signals with random parameters is al-
most untouched so far as practical solutions are concerned.

The authors would like to express their appreciation for the advice
and assistance of D. Slepian and J. L. Kelly, Jr., which greatly furthered
this work.

In the main body of the paper it was shown that, for samples which
are derivatives, detectability in terms of d can be determined from (5)
and (6)

S; is the 7 — 1 derivative of the signal evaluated at ¢ = 0 and 8,;, the
correlation coefficient of the noise derivatives, is

for flat bandlimited noise with unit rms amplitude.
Mquation (5) may be written out in matrix form for odd n as

This equation may be simplified by separating into two equations and
multiplying by minus one in appropriate places to remove minus signs.
Two forms occur, one for even n, the other for odd n. For‘n odd,

The determinants of these matrices can be evaluated by applying a rule
attributed to Cauchy. In general, the rule says that a determinant whose
ijth element is

In addition, all cofactors of the matrices are also of Cauchy form.
Hence, it is possible to invert the matrices by the method of cofactors
and thus solve the equations. Such solutions are quite complex for arbi-
trary signals. However, an especially simple answer can be obtained for

Z, may be determined by the well-known method for solving equations
as the ratio of two determinants,

APPENDIX B
In this appendix we give a general solution to the integral equation

the great advantage of being an explicit function of R(t) and S(t) rather
than involving the solution of a set of linear equations. In addition, it
possesses the aesthetically pleasing property of not involving analytic
continuation of S(¢) outside the interval 0 < ¢ S 7. The noise spectral

, | \ 9
G(w) Q(p) >} Ay). (33)
| P) \p=ie k=0
If we think of Q(p) as an operator with p interpreted as d/dt we see
that

pm I
Q(p)[R(t)] ()(p) a _— 6(t),
Lx (jw) 2r
where 6(¢) is the Dirac delta function. Operating formally on both sides
of (32) with Q(p) vields
N

ZAt Q(p){S(t)| >, GxS*” (t), cick. (3)
The subscript has been added to Z to indicate that this may be only part
of the answer and the superscript (mn) indicates n-fold differentiation
with respect to time. If (32) had a Z(t) solution which was continuous,
then (35) would be that solution. But the fact that (35) is continuous
(as it would be if S(t) and its derivatives were continuous) does not
prove that it is the complete solution. In fact, one can readily verify that
(35) is not the complete solution by inserting it back in (32) and seeing
if (32) is satisfied. It turns out that (35) is indeed part of the answer,
and the remaining part is found by just this process of inserting (35)
back in (32) and finding what is missing. If we imagine for the moment
that S(t) is extended in some arbitrary way outside the interval (so
that it is Fourier transformable and the function and its derivatives go

what is equivalent, by finding an exact differential expression for the
integrand. We first note that Q(p) can always be factored,

N
U(t) = P(—p)|S(0)] > (-1)‘o S(t).
The exact differential that we need is obtained by clairvoyance.

Now, since P(p) has the left-half plane zeros of Q( p), the Fourier trans-
form of P(p)[R(t)| will have only right-half plane poles and thus

The third integral on the right of (36) is evaluated in a similar way,
using now (39) with U. replaced by U, and by by (—1)"b. , and noting
that P(—p)R(t) 0 for? < 0. In this way we get

It is interesting to note that (40) and (41) depend only on values of
S(t) inside the interval 0 < ¢ S 7, so that the way in which S(?¢) was

extended outside the interval does not matter. To summarize this, we
find

It is now clear that, for Z, to be the complete solution to (32), the double
sum in (42) must be zero for all ¢ in the interval. This is equivalent to
the following boundary conditions on S(t):

If the signal is such that these conditions are not satisfied, then (32)
g ;
has a solution only if Z(¢) includes delta functions and their derivatives,

If this is used in (32), the delta functions bring out FR and its derivatives
evaluated at ¢ and ¢ — 7’, and the a’s and §’s can be directly identified

This can be put in a slightly different form which may be more con-
venient by again partially integrating. Using another exact differential
obtained by clairvoyance, which is

and observing that when this is inserted in (46) the terms evaluated at
T cancel, we get

In this form the summation only involves derivatives at / 0, which
in some cases simplifies the algebra of a solution.
APPENDIX C

As a particular example, we calculate the detectability d for the case
in which the signal is an exponentially damped sine wave,

and the asterisk denotes complex conjugate. Using this in the expression

for detectability (20) or (49), we find that the second term—eall it
da — becomes
N—1
ds = : Be S'(O){[((—1)'Pi(p)P(—p) + Pil —p)P(p)|S(t)} 0

Three special cases are now considered, the pure exponential, the
pure sine wave, and a constant (pc) signal. For a pure exponential
signal, \ — —a where a is real in (55), giving

lor a pure sine wave signal, \ — jw. The second term in (55) requires
a little special treatment, but it is easily shown that

Now, (jw) is simply the reciprocal of the transfer function of the noise
filter at the frequency w; that is,

P( jw) = 1/H( jw) ———.,
V G(w)
where @(w) is the phase lag of the noise filter. Using this expression,

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. Middleton, D., An Introduction to the Theory of Statistical Communication,
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). Peterson, W. W., Birdsall, T. G. and Fox, W. C., The Theory of Signal De
tectability, I.R.E. Trans., PGIT4, 1954, p. 171.
. Slepian, D., Some Comments on the Detection of Gaussian Signals in Gaussian

G. E. Schindler, Jr., was appointed editor of the Bell System Tech-
nical Journal, effective January 1, 1961. Mr. Schindler studied chemi-
cal engineering at the Carnegie Institute of Technology, received the
bachelor of science degree from the University of Chicago, and re-
ceived the master of arts degree in English literature and languages
from the University of Pittsburgh. After additional graduate work at
the University of Chicago, Mr. Schindler joined Bell Telephone Labo-
ratories in 1953. He was editor of the Bell Laboratories Record from
1957 to 1959, and most recently was with the Public Relations depart-
ment of the A. T. & T. Co.

A_ theoretical investigation has been undertaken to study diffraction of
electromagnetic waves tn Fabry-Perot interferometers when they are used as
resonators in optical masers. An electronic digital computer was programmed
to compute the electromagnetic field across the mirrors of the interferometer
where an initially launched wave is reflected back and forth between the
mirrors.

It was found that after many reflections a state is reached in which the
relative field distribution does not vary from transit to transit and the ampli-
tude of the field decays at an exponential rate. This steady-state field dis-
tribution is regarded as a normal mode of the interferometer. Many such
normal modes are possible depending upon the initial wave distribution.
The lowest-order mode, which has the lowest diffraction loss, has a high
intensity at the middle of the mirror and rather low intensities at the edges.
Therefore, the diffraction loss is much lower than would be predicted for a
uniform plane wave. Curves for field distribution and diffraction loss are
given for different mirror geometries and different modes.

Since each mode has a characteristic loss and phase shift per transit, a
uniform plane wave which can be resolved into many modes cannot, properly
speaking, be resonated in an interferometer. In the usual optical inter-
ferometers, the resolution is too poor to resolve the individual mode resonances
and the uniform plane wave distribution may be maintained approximately.
However, in an oscillating maser, the lowest-order mode should dominate
if the mirror spacing ts correct for resonance.

A confocal spherical system has also been investigated and the losses are
shown to be orders of magnitude less than for plane mirrors.

‘ rm 1 . - .
Schawlow and Townes have proposed infrared and optical masers
using Fabry-Perot interferometers as resonators. Very recently, Mai-

bility of stimulated optical radiation in ruby. In these experiments two
parallel faces of the ruby sample were polished and silvered so as to
form an interferometer. The radiation due to stimulated emission
resonates in the interferometer and emerges from a partially silvered
face as a coherent beam of light.

In a maser using an interferometer for a resonator, a wave leaving one
mirror and traveling toward the other will be amplified as it travels
through the active medium. At the same time it will lose some power due
to scattering by inhomogeneities in the medium. When the wave arrives
at the second mirror some power will be lost in reflection due to the
finite conductivity of the mirror and some power will be lost by radia-
tion around the edges of the mirror. For oscillation to occur, the total
loss in power due to density scattering, diffractive spillover and reflection
loss must be less than the power gained by travel through the active
medium. Thus diffraction loss is expected te be an important factor,
both in determining the start-oscillation condition, and in determining
the distribution of energy in the interferometer during oscillation.

While it is common practice to regard a Fabry-Perot interferometer as
being simultaneously resonant for uniform plane waves traveling parallel
to the axis and at certain discrete angles from the axis, this picture is
not adequate for the computation of diffraction loss in a maser. It is true
that, when the interferometer is operated as a passive instrument with
uniform plane waves continuously supplied from an external source, the
internal fields may be essentially those of uniform plane waves. In an
oscillating maser where power is supplied only from within the inter-
ferometer, the recurring loss of power from the edges of a wave due to
diffraction causes a marked departure from uniform amplitude and phase
across the mirror.

The purpose of our study is to investigate the effects of diffraction on
the electromagnetic field in a Fabry-Perot interferometer in free space.
The conclusions can be applied equally well to gaseous or solid state
masers provided the interferometer is immersed in the active medium,
i.e., there are no side-wall discontinuities.

Our approach is to consider a propagating wave which is reflected back
and forth by two parallel plane mirrors, as shown in Fig. 1(a). [This is

equivalent to the case of a transmission medium comprising a series of
collinear identical apertures cut into parallel and equally spaced black
(perfectly absorbing) partitions of infinite extent, as in Fig. 1(b).}] We
assume at first an arbitrary initial field distribution at the first mirror
and proceed to compute the field produced at the second mirror as a
result of the first transit. The newly calculated field distribution is then
used to compute the field produced at the first mirror as a result of the
second transit. This computation is repeated over and over again for
subsequent successive transits. The questions we have in mind are: (a)
whether, after many transits, the relative field distribution approaches a
steady state; (b) whether, if a steady-state distribution results, there
are any other steady-state solutions; and (¢) what the losses associated
with these solutions would be. While it is by no means obvious that

tem which has no side-wall boundaries, it will be shown that such solu-
tions do indeed exist.*

Fig. 1 The Fabry-Perot interferometer and the transmission medium analog.

* Schawlow and Townes! suggested the possibility that resonant modes for a
parallel plate interferometer might be similar in form to those for a totally en
closed cavity

We shall use the scalar formulation of Huygens’ principle to compute
the electromagnetic field at one of the mirrors in terms of an integral of
the field at the other. This is permissible if the dimensions of the mirror
are large in terms of wavelength and if the field is very nearly transverse
electromagnetic and is uniformly polarized ir one direction. Later, we
shall show that these assumptions are consistent with the results of our
solutions and therefore are justifiable. We shall also show that other
polarization configurations can be constructed from the solutions of the
scalar problem by linear superposition.

where wu, is the aperture field, k is the propagation constant of the me-
dium, F is the distance from a point on the aperture to the point of ob-
servation and @ is the angle which R makes with the unit normal to the
aperture. We now assume that an initial wave of distribution u, is
launched at one of the mirrors of the interferometer and is allowed to be
reflected back and forth in the interferometer. After q transits the field
at a mirror due to the reflected field at the other is simply given by (1)
with u, replaced by u,4: , Which is the field across the mirror under con-
sideration and u, by u, , which is the reflected field across the opposite
mirror giving rise to Ugii.

It is conceivable that after many transits the distribution of field at
the mirrors will undergo negligible change from reflection to reflection
and will eventually settle down to a steady state. At this point the fields
across the mirrors become identical except for a complex constant;

where v is a distribution function which does not vary from reflection to
reflection and y is a complex constant independent of position coor-
dinates. Substituting (2) in (1) we have the integral equation

in which the kernel of the integral equation, K, is equal to (jk/4rR)
(1 + cos @)e *“. The distribution function v, which satisfies (3), can

be regarded as a normal mode of the interferometer defined at the mirror
surface, and the logarithm of y, which specifies the attenuation and the
phase shift the wave suffers during each transit, can be regarded as the
propagation constant associated with the normal mode.

We have studied and obtained numerical solutions for several geo-
metric configurations of the interferometer. These are (a) rectangular
plane mirrors, (b) circular plane mirrors and (c) confocal spherical or
paraboloidal mirrors.

When the mirror separation is very much larger than the mirror di-
mensions the problem of the rectangular mirrors reduces to a two-
dimensional problem of infinite strip mirrors. This is shown in Appendix
A. The integral equation for the problem of infinite strip mirrors, when
a’/bd is much less than (b/a)’, is

quation (4) is a homogeneous linear integral equation of the second
kind. Since the kerriel is continuous and symmetric [A (22, 21)
K(x, , 22)|, its eigenfunctions v, corresponding to distinet eigenvalues
y, are orthogonal in the interval (—a,a); that is (Ref. 5, p. 413),

It should be noted that the eigenfunctions are in general complex and are
defined over the surface of the mirrors only. They are not orthogonal in
the power (Hermitian) sense as commonly encountered in lossless systems.
Here, the system is basically a lossy one and the orthogonality relation is

one which is generally applicable to lossy systems, such as lossy-wall
waveguides.

The eigenfunctions are distribution functions of the field over mirror
surfaces and represent the various normal modes of the system. The nor-
mal modes for rectangular plane mirrors are obtained by taking the
products of the normal modes for infinite strip mirrors in « and y direc-
tions; that is,

We designate this as the TEM,,,,, mode for the rectangular plane-mirror
interferometer. In view of (5) we see that the normal mode distribution
functions v,,, are orthogonal over the surface of the rectangular mirror.
The logarithms of the eigenvalues represent propagation constants
associated with the normal modes. The propagation constant for the

The real part of the propagation constant specifies the loss per transit

and the imaginary part the phase shift per transit, in addition to the
geometrical phase shift, for the normal modes.

for circular plane mirrors (lig. 3) when a°/b\ is much less than (b/a)°,
are given by
virg~) = R,(r)e ’"* (n integer ),
where P,,(7) satisfies the reduced integral equation
2a

problem of infinite strip mirrors, (9) is a homogeneous linear integral
equation of the second kind with a continuous and symmetric kernel.
Its eigenfunctions corresponding to distinct eigenvalues are orthogonal
in the interval (0,a); that is,

Therefore, we see that the distribution functions v,,,(7,¢) = Rin(re
corresponding to distinct eigenvalues y,, are orthogonal over the sur-
face of the mirror; that is,

A number of geometries other than plane parallel mirrors have been
suggested, and it is believed that most of these can be studied using the
same iterative technique. One of the geometries we investigated is that

have identical curvatures and their foci are coincident, as shown in
Fig. 4. One of the possible advantages of such a system is the relative
ease of adjustment, since the mirrors are no longer required to be parallel
as in the case of the parallel plane system. Another is that the focusing
action of the mirrors might give rise to lower diffraction losses.

A spherical mirror with a small curvature approximates closely a
paraboloidal mirror. In the case of confocal spherical mirrors, the condi-
tions that its curvature be small is equivalent to saying that the separa-
tion between mirrors is large compared to the dimensions of the mirrors.
It is shown in Appendix C that the solutions to the integral equation
for confocal paraboloidal mirrors, when a’/bd\ is much less than (b/a)’,
are given by

Again, we see that (13) is a homogeneous linear integral equation of
the second kind with a continuous and symmetric kernel. Therefore,
general remarks concerning the normal modes of circular plane mirrors
given in the foregoing section are also applicable to confocal spherical
or paraboloidal mirrors.

An IBM 704 computer was programmed to solve the integral equa-
tions for the various geometries of the interferometer by the method of
successive approximations. As mentioned previously, this is analogous
to the physical process of launching an initial distribution of wavefront
in the interferometer and letting it bounce to and fro between the
mirrors.

was employed for the computation, using an initial excitation of a uni-
form plane wave at the first mirror. A total of one hundred increments
was used for the numerical integration. After the first transit the field
intensity (electric or magnetic) had the amplitude and phase shown in
Fig. 5. In these and subsequent amplitude and phase distributions the
curves are normalized so that the maximum amplitude is unity, and the
phase at that point is zero. The large ripples are due to the fact that the
initial wave front contains 6.25 Fresnel zones as seen from the center of
the second mirror. Therefore, in passing from the center to the edge of
the second mirror there is a change of 3 X 6.25 Fresnel zones, and this
agrees with the number of reversals in curvature seen in the amplitude
distribution.

With subsequent transits, these ripples grow smaller, the amplitude
at the edge of the mirror decreases, and the relative field distributions
approach a steady state. By the time the wave had made three hundred
bounces, the fluctuations occurring from bounce to bounce were less
than 0.03 per cent of the final average value. The amplitude and phase
for the 300th bounce are also shown in Fig. 5.

We regard this field distribution as an iterative normal mode of the
interferometer. In other words, if this distribution is introduced as an
initial wave at one mirror it will reproduce the same distribution at the
other mirror. Indeed, this is what the computer is verifying when we
compute the 301st bounce.

Once the solutions have reached a steady state, we can pick any point
on the wavefront, say the center of the mirror, and examine how the
absolute phase and amplitude change from bounce to bounce. In this
way we determined that the power loss of this mode is 0.688 per cent
per transit and the phase shift per transit has a lead of 1.59 degrees.

Fig. 5 — Relative amplitude and phase distributions of field intensity for in
finite strip mirrors. (The initially launched wave has a uniform distribution.

Since phase shift is measured relative to the free-space electrical length
between the mirrors (360 b/ degrees), this means that the mode has an
effective phase velocity which is slightly greater than the speed of light,
just as for a metal tube waveguide.

Fig. 6 — Fluctuation of field amplitude at z = 0.5 as a function of number of
transits. (The initially launched wave has a uniform distribution.)

point (x = 0.5a) approaches its steady-state normalized value after a
start from a uniform plane wave. After the 100th transit the plot ap-
pears to be a damped sine wave. We interpret this damped oscillation
as the beating between two normal modes having different. phase veloci-
ties. The mode with the lower attenuation, of course, survives the
longest, and this is the one shown in Fig. 5. We regard this as the dom-
inant mode of the interferometer. We believe the other mode which
beats with the dominant mode to be the next-higher order, even-sym-
metric mode. Prior to the 100th transit, the curve is irregular, indicating
that a number of still higher order modes are present which are damped
out rapidly.

small compared to (b/a)*, the actual dimensions of the mirrors and their
spacing are no longer important, the only parameter of importance being
the Fresnel number N a /b\. This is approximately equal to the
number of Fresnel zones seen in one mirror from the center of the other
mirror, and as pointed out earlier, it determines the number of ripples
in the field distributions. Amplitude and phase distributions for the

RESONANT MODES IN A MASER INTERFEROMETER 465
dominant mode obtained by solving (27) are shown in Fig. 7 for differ-
ent values of N. The larger the NV, the weaker is the field intensity at the
edge of the mirror, and the smaller is the power loss due to spill-over.
The plot of power loss per transit as a function of N is approximately a
straight line on log-log paper and is shown as the lowest line in Fig. 8.
The phase shift per transit as a function of N is given by the lowest
line in Fig. 9.

A uniform plane wave excitation can never give rise to a mode with
odd symmetry. In order to investigate the possibility of modes of this

Fig. 7 — Relative amplitude and phase distributions of field intensity of the
lowest order even-symmetric mode for infinite strip mirrors.

Fig. 8 — Power loss per transit vs. V = a?/bd for infinite strip and circular
plane mirrors.

type, the problem was re-programmed for an initial wave for which the
field intensity over one-half the strip (0 to +a@) was equal but opposite
in sign to the field intensity over the other half of the strip (0 to —a).
Steady-state solutions did indeed result, and odd-symmetric normal
modes therefore exist. The amplitude and phase distributions are shown
in Fig. 10 for several values of NV. The amplitude is zero at the center,
as expected. While shown for only one half of the strip, it is the same in
the other half, but with a reversal in sign. Note that for the same values
of V, the amplitude at the edge is higher than for the dominant mode.
The spill-over loss should be higher and this is confirmed by the loss
curve in Fig. 8 labeled ‘infinite strip odd-symmetric mode.”’ The cor-
responding phase shift curve is shown in Fig. 9.

The feasibility of obtaining the normal mode solutions for the infinite

investigate the modes for plane circular mirrors. The first case considered
was that for uniform plane wave excitation of the system. Once again,
the polarization was assumed to be everywhere parallel to the same axis,

with n = O|. That is, the amplitude and phase of the field intensity is
the same for all points at the same radius from the center. The transverse
field distributions for the lowest order mode of this type are shown in
Fig. 11 for various values of NV. The loss and phase shift are shown in
Figs. 8 and 9 under the title “circular dise (dominant mode, TEM).”
One hundred increments along the radius were used for the numerical
integrations involved.

Next we examined modes of the odd-symmetric type for circular
plane mirrors. The equation we used was (9) with n = 1. Fig. 12 shows
amplitude and phase distributions for the lowest order mode of the odd-
symmetric type for circular plane mirrors. Again the loss and phase

Fig. 9 — Phase shift per transit (leading relative to geometrical phase shift)
vs. N = a?/bd for infinite strip and circular plane mirrors.

Fig. 10 — Relative amplitude and phase distributions of field intensity of the
lowest-order odd-symmetric mode for infinite strip mirrors.

Normal modes with higher orders of angular variation (n 2 2) and
radial variation (m 2 1) have greater losses and phase shifts than those
of TEM and TEM, modes. The mode with the least attenuation is

therefore the lowest order, of TEMoo mode, which we designate as the
dominant mode for circular plane mirrors.
3.4 Confocal Spherical Mirrors

Before (13) was programmed for solutions on the computer a more
general method for solving the problem of the confocal spherical mirrors

Fig. 11 — Relative amplitude and phase distributions of field intensity of the
dominant (TEMoo) mode for circular plane mirrors.

Fig. 12 telative amplitude and phase distributions of field intensity of the
TEM)» mode for circular plane mirrors

was tried — a procedure that can be used to solve problems involving
mirrors with rather arbitrary but small curvatures. In this method the
field at each mirror is caleulated using the equation for circular plane

mirrors and then a phase distribution corresponding to the curvature of
the mirror is added to this field before it is used in the next iterative com-
putation. The results from this general method of solution and from
solving (13) are in perfect agreement.

The problem of confocal spherical mirrors has also been solved by
Goubau’ and Boyd and Gordon.’ The results of their analyses are in
good agreement with our computed results.

Amplitude distributions of the field intensity for TEM) and TEM)
modes are shown in Figs. 13 and 14. The phase distributions are all
uniform over the surface of the mirrors and therefore are not plotted.
The loss and phase shift per transit are given in Figs. 15 and 16. We
note some rather remarkable differences between these solutions and
those obtained for circular plane mirrors. First, the field is much more
tightly concentrated near the axis of the reflector and falls to a much
lower value at the edge than is true for plane mirrors; also the am-
plitude distribution does not have ripples in it, but is smooth. Second,

Fig. 13 — Relative amplitude distribution of field intensity of the dominant
(TEMoo) mode for confocal spherical mirrors. The relative phase distribution on
the surface of the mirror is uniform.

Fig. 14 — Relative amplitude distribution of field intensity of the TEMhio
mode for confocal spherical mirrors. The relative phase distribution on the surface
of the mirror is uniform.

the surface of the reflector coincides with the phase front of the wave,
making it an equiphase surface. Third, the difference between the phase
shifts for all the normal modes are integral multiples of 90 degrees.
Fourth, the losses may be orders of magnitude less than those for plane
mirrors.

The result that the mirror surface is an equiphase surface should not
be surprising, but can be deduced from integral equation (13). If we

Ms » n+l . 7 .
associate the factor j""" with y, the kernel becomes real. Since the eigen-

values and eigenfunctions of a real symmetric kernel are all real,’ we
see that the field distribution is of uniform phase over the surface of the
mirror. Furthermore, since (j"*'y,) is real, the phase shift for the normal
modes belonging to a set of modes with a given angular variation must
be an integral multiple of 180 degrees and the difference between the
phase shifts for the normal modes with different angular variations but
the same radial variation is an integral multiple of 90 degrees; that is,
the phase shift is equal to [180m + 90(n + 1)] degrees. Therefore, if
the mirrors are adjusted for the resonance of a particular normal mode,

half of the totality of all the modes are also resonant. However, the
resonant mode with the lowest loss would persist longest in the resonator.
Just as in the case of plane parallel mirrors, the mode with the lowest loss
is the TE Moo mode.

The results of machine computation have shown that a two-mirror
interferometer, whether of the plane or concave mirror type, can have
100

Fig. 15 — Power loss ‘tod transit vs. N = a?/bd for confocal spherical mirrors.
(Dashed curves for circular plane mirrors are shown for comparison).

Fig. 16 — Phase shift per transit (leading relative to geometrical phase shift)
vs. V = a?/b\ for confocal spherical mirrors. (Dashed curves for circular plane
mirrors are shown for comparison

normal modes of propagation which are self-perpetuating or self-repro-
ducing in the distance of one transit. We use the term mode of propaga-
tion rather than mode of resonance to emphasize the fact that these
steady-state solutions are the result of multiple transits whether or not
the plate separation happens to be adjusted for resonance. An analog of
the plane mirror interferometer is a transmission medium consisting of a
series of periodic collinear apertures, as was shown in Fig. 1. The same

solutions apply, and here it is clear that the reproduction of a normal
mode field at successive apertures does not depend on any critical rela-
tion between b and X.

In Fig. 17 is shown the way in which a number of square-plate modes
can be synthesized from the infinite-strip modes. Diagram «a shows
schematically the field distribution for the dominant square-plate mode
obtained as the product of the field distributions of two even-symmetric
strip modes crossed at right angles and with polarization as shown.
Since the eigenvalue for the square plate is the product of the eigen-
values for the two strips, the phase shift per transit is the sum of the

phase shifts for the two strips and, if the loss is small, the loss per transit
is essentially the sum of the losses for the two strips. Diagram B repre-
sents an odd-symmetric square-plate mode formed by taking the prod-
uct of an even- and an odd-symmetric strip mode; B’ is the same mode
but with the polarization rotated 90°; c is a circular electric type of
mode formed by adding two modes of the type B. This addition is per-
mitted because the two components are degenerate. It follows that the
circular electric mode c is degenerate with B and has the same loss and
phase shift per transit. By taking the difference between the same two
B modes as shown, the mode pD is obtained, resembling the TE. mode in
circular waveguide. We give all the patterns B, B’, c and p the same
designation, TEM, (or TEM), since they are composites of the one
basic mode type. Similar syntheses can be performed for circular mirrors,
either plane or concave. It is interesting that degeneracies of this type
are common for the interferometer because the electric vector F is at
liberty to be parallel or perpendicular to the mirror edges. In a metal
waveguide they are uncommon because the polarization of / at the
boundaries is restricted.

The dominant mode and a number of higher-order modes for square
and circular mirrors are depicted in Fig. 18, in which electric field vectors
are shown. This classification of modes applies to plane as well as con-
focal spherical mirrors. In the case of rectangular mirrors, the x axis
may be taken along the longer dimension, in which case the first sub-

script always denotes the number of field reversals along the longer di-
mension.

In formulating the problem we have assumed that the waves were
almost transverse electromagnetic. The solutions for the flat mirror are
consistent with this assumption. At the edges of the mirror there is a
phase lag of approximately 45 degrees relative to the center, but this is
only one-eighth of a wavelength out of many wavelengths for the mirror
diameter. Thus the curvature of the wavefront away from the transverse
plane is exceedingly small, and the assumption appears justified. For
higher-order modes such as Bb’ of Fig. 17, it is clear that the field lines
must have longitudinal components. This is illustrated by an edge view
in Fig. 19. However, provided the width of a cell c is much greater than a
half-wavelength, the longitudinal field intensity should be negligible
compared to the transverse. Only for very high-order modes should this
approximation begin to fail. Because the low-order modes of importance
are essentially transverse electromagnetic, they are designated as TEM
modes.

The plane mirror modes have a phase which is not constant over the
mirror. This does not mean that it is impossible to space the mirrors for
resonance of the entire field pattern. Actually, the phase delay for one
transit is the same for every point on the wavefront. Therefore, if the
plates are separated by the distance b plus an additional amount for the
phase shift per transit of the mode desired, that mode should resonate in
|
|

the interferometer. Other modes should not be resonant for this separa-
tion because they have different phase shifts per transit.

Since the field configurations of many of the normal modes of the
interferometer are very similar to those of metal tube and parallel-plane
waveguides, it is not surprising to find that simple waveguide theory can
be used to predict certain characteristics of the interferometer modes.
One of these characteristics is phase shift per transit. For instance, the
field distributions of the normal modes for infinite strip mirrors are very
similar to those of the TE modes of parallel-plane waveguide; also, by
adding two orthogonally polarized TEM), modes for circular plane mir-
rors, one obtains a field configuration which is very similar to that of the
circular electric (TE) mode of circular waveguide (Fig. 20). Thus the
amount of phase shift per transit computed for these modes of the
interferometer agrees well with the phase shifts obtained for TE modes
of parallel-plane waveguide and TE»; mode of circular waveguide. This is
illustrated in Fig. 21. We see that agreement becomes better for larger
values of V. This is because the similarity between field configurations
becomes closer for larger values of NV.

If we regard a uniform plane wave as being resolvable into a set of
normal modes, there can be no such thing as a resonance for a uniform
plane wave. Why then does it appear that there is such a resonance in
passive optical interferometers? It is because for the usual optical case
a /bd is in the thousands. The phase shifts per transit are extremely
small, hence the mode resonances lie very close together in frequency.
At the same time, the reflection coefficients of the best optical mirrors
are so poor, and the Q of the interferometer is so low, that the resonance

Fig. 20 — TE modes in a parallel-plane waveguide and circular electric mode in
1 circular waveguide

Fig. 21 — Comparison of computed phase shifts based on waveguide theory and
on interferometer theory.

line width contains hundreds of normal mode resonances. Thus the uni-
form plane wave undergoes very little decomposition when resonated.
Nevertheless, in the case of an active interferometer, the decomposition
may be complete.

We now make use of the formula for the Q of a resonant waveguide
cavity to compute the Q of an interferometer system. The Q of a reso-
nant waveguide cavity is given by

where a@ is the attenuation constant of the waveguide and A, is the guide
wavelength. For the interferometer we assume that @ is zero and that A,
is equal to A, the free-space wavelength. The voltage reflection coeffi-

where 6, is the power loss in reflection and 6g is the power loss due to
spill-over. When these losses are very small, Q reduces to

The resonance line width at half-power points given as the change in
electrical length of the resonator, Ag, is

Let us consider an interferometer having circular plane mirrors with
2a = lem, b = 20cm, = 5 X 10 °cmand a reflection loss of 5, = 0.02.
In this case N = a'/b\ = 250. Extrapolating the loss and phase shift
curves of Figs. 8 and 9, we obtain diffraction loss 6, = 9 X 10° and
phase shift for the dominant (TEM) mode gg = 0.11 degree. The dif-
fraction loss is thus negligible compared to reflection loss, which limits
the Q to a value of 1.25 X 10°. The phase shift for the next higher order
(TEM) mode is 0.30 degree and therefore it is separated from the
dominant (TEM) mode by 0.19 degree or 0.0033 radian. The resonance
line width, as given by (19), is 0.02 radian. Thus we see that TEMoo
and TEM, modes are not resolved. As the mirror separation is reduced
or mirror size increased, more and more normal modes will become
unresolved and a uniform plane wave will suffer less decomposition when
resonated.

When an interferometer is filled with an active medium, the medium
can compensate for the mirror losses and yield an enormously increased
(). Under these circumstances, the modes may be clearly resolved, and
their Q’s will be determined by the diffraction losses. If the gain of the

medium is increased until it compensates for mirror losses plus the dif-
fraction loss of the lowest order mode, that mode will become unstable
and oscillation can result. All higher-order modes will be stable and
have positive net loss. If the gain of the medium is further increased,
then many modes may become unstable. In starting from a quiescent

condition, spontaneous emission can initiate a large number of charac-
teristic waves in the interferometer. These may then start to grow, but
the dominant mode will always grow faster and should saturate first.
At saturation the steady-state field distribution will be considerably
altered. The relative field at the edges of the mirrors should increase,
thereby increasing the relative power loss. This can be described as a
coupling of power into other modes as a result of the nonlinearity of the
medium. No attempt has yet been made to analyze this situation. The
linear theory is at present of most interest because it allows the com-
putation of the starting conditions for oscillation.

With the development of the normal mode picture of interferometer
operation and the computation of the losses for these modes, we may
now ask if there is an optimum geometry for a maser interferometer
which will permit oscillation for the lowest possible gain in the medium.
We know that the power gained from the medium can be increased by
increasing length. For very great lengths corresponding to the far-field
region (NV < 0.1), the power gained from the medium increases more
rapidly than transmission loss as length is increased, and there must
always be some length beyond which oscillations can occur. However,
these lengths are too great to be of practical interest. In the near-field
region (N > 1), represented by the curves of Fig. 8, the diffraction
loss increases more rapidly than the medium gain. Therefore, if the
reflection loss is sufficiently small, an optimum length may exist which is
most favorable for oscillation.

To be more specific, let us consider a circular plane mirror interferome-
ter. From Fig. 8 we find that the loss for the dominant mode may be
represented by the expression

In order to find the optimum value of b to give a maximum Q, (20) and
(18) are substituted in (17) and the resulting equation is differentiated
with respect to b. For the optimum ), the diffraction loss is 2.5 times the
reflection loss, and not equal to it, as might be supposed. Moreover, this
result is general and holds for all modes and all shapes of plane mirrors
represented in Fig. 8, provided the optimum falls on the straight-line

portions of the loss curves. Since the power supplied by the medium is
proportional to the stored energy in the interferometer, while the power
loss of the passive interferometer is just w/Q times the stored energy,
oscillation is most likely to occur when Q is a maximum. Fig. 22 il-
lustrates the way the interferometer dimensions affect Q. If a given mir-
ror diameter is chosen (as represented by the dashed line a), there is
clearly an optimum distance 6 which will produce a maximum Q (inter-
section of lines A and B). However, if the distance b is held constant,
there is no optimum value for a. The larger a, the higher will be Q,
although it will approach a limiting value beyond which there is nothing
to be gained by further increase of a.
As an example, let us assume a case where

5, = power reflection loss = 0.001.
The optimum proportions require that 6, be 0.0025, and for this, b is
$35 em and the resulting Q is 7.8 X 10°. The length of 435 em is prob-
ably impractically large for a maser. If 6 is reduced to a more reasonable
value of 50 cm, the Q will drop to 3.14 X 10°, which is the limiting value
due to reflection loss. (The value assumed here for 6, is already much
lower than can be obtained from evaporated metal films and would

cillate, the active medium would have to have a power amplification
factor in excess of 1.00002 per centimeter of path.

In the case of confocal paraboloidal mirrors of 2 em diameter, the
optimum length turns out to be 8900 cm. If the diameter is reduced to
0.5 em, the optimum length is still 530 cm, and for these proportions Q
is 3.1 X 10”. It is clear that with confocal mirrors the diffraction losses
are negligible for any reasonable proportions of the interferometer.

One question of importance is whether there is an optimum set of
dimensions which will discriminate against unwanted modes. It has
sometimes been suggested that by making the mirror diameter small
relative to the mirror spacing, ‘‘slant rays’’ will be more rapidly lost from
the system. However, from Fig. 8 it can be seen that the ratios of the
losses for the several modes is independent of N provided N is greater
than 1. Thus, if diffraction losses predominate, there is no way of dis-
criminating against unwanted modes by juggling dimensions. The limit-
ing amount of discrimination is merely governed by the ratio of the
losses for the different modes, which is independent of the dimensions.
However, if reflection losses predominate, the discrimination between

Fig. 22 — Interferometer dimensions for constant Q. (Circular plane mirrors,
reflection loss = 6, = 0.001.)

lower-order modes would be almost nonexistent and it would be ad-
vantageous to increase mirror separation and/or decrease mirror dimen-
sions so as to make diffraction losses predominate. In the case of the
confocal mirrors, the loss ratios between modes are not constant (Fig.
15) although, for values of N larger than those shown, they may become
so. At any rate, for values of N close to unity, a small amount of in-
creased discrimination against higher order modes can be obtained by
making the mirrors larger.

Diffraction studies carried out on the IBM computer have led to the
following conclusions:

1. Fabry-Perot interferometers, whether of the plane or concave
mirror type, are characterized by a discrete set of normal modes which
can be defined on an iterative basis. The dominant mode has a field
intensity which falls to low values at the edges of the mirrors, thereby

causing the power loss due to diffractive spillover to be much lower than
would be predicted on the assumption of uniform plane wave excitation.

2. Uniform plane waves are not normal modes for a flat-plate inter-
ferometer. Consequently, interferometer resonances do not exist for
“slant rays,” 1.e., plane waves traveling at an angle with respect to the
longitudinal axis.

The losses for the dominant mode of the plane mirror system are
so low that for most practical geometries performance will be limited by
reflection losses and scattering due to aberrations. For confocal mirrors
the diffraction losses are even lower.

There are no higher-order modes with losses lower than the dom-
inant (lowest-order) mode.

The ratio of diffraction losses between the modes investigated for
the plane mirror system is independent of the interferometer dimensions
in the range of interest. Therefore, if diffraction losses predominate, there
is no way of proportioning the interferometer so as to favor any one
mode.

The computer technique we employed is general and versatile. It can
be used for studying mirrors having rather arbitrary but small curva-
tures. With little modification, the same technique can be used to study
the effects of aberration and misalignment.

The geometry for rectangular plane mirrors parallel to the xy plane is
shown in Fig. 2. According to (1), the iterative equation for computing
the field at <i wartace of mirrors is

Actually, the stringency of this requirement can be relaxed somewhat for
ao sr-order modes in which field intensities near the edges of the mirror are rr ather
low. We have made check computations for the case a?/b\ = 5 and (b/a)? = 25
and have found that the results based on the exact equation and on the smpeeni

and the factor e ”” is absorbed in y.
Here, the kernel of the integral equation is separable in x and y. If

it is possible to separate (23) into two equations, one involving «x only
and the other involving y only; that is,

el® 4 iia
r Kh ly y2)*/26b Qf
K, = — « = . (25d )
av Nb
The product of the eigenvalues y, and y, is equal to the eigenvalue y in
(23).

It remains to be shown that (25a) through (25d) represent integral
equations for infinite strip mirrors. Let us consider a pair of infinite
strip mirrors of width 2a and separated by b. The iterative equation
for computing the field at the mirrors can be derived from (1).

186 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1961
The corresponding integral equation is

APPENDIX B
Circular Plane Mirrors
Assuming approximately plane waves propagating normally to the

circular plane mirrors (Fig. 3), the iterative equation for computing the
steady-state field distribution can be written as

¥2""r, dg, dr; ,
which is valid for (a°/b\) « (b/a)”.*
The integral equation corresponding to (30) is
sa alr

*Comments in Appendix A regarding the stringency of this requirement are
also applicable herein

is absorbed in y. Making use of the relation
Tes) l [ et (rirelbeos(¢y #2
Qa “0
and integrating (31) with respect to g; , it is seen that

For confocal spherical mirrors of circular cross section (Fig. 4), the
iterative equation corresponding to (29) is

K(r2, ¢2371,¢1
and where the factor e ”’ is absorbed in y. Just as in the case of circular
plane mirrors, it can be shown that

Collins, R. J., Nelson, D. F., Schawlow, A. L., Bond, W., Garrett, C. G. B.
and Kaiser, W., Coherence, Narrowing, Directionality and Relaxation
Oscillations in the Light Emission from Ruby, Phys. Rev. Letters, 5,
1960, p. 303

Hildebrand, F. B., Methods of Applied Mathematics, Prentice-Hall, Englewood
Cliffs, N. J., 1952

Connes, P., Increase of the Product of Luminosity and Resolving Power of
Interferometers by Using a Path Difference Independent of the Angle of
Incidence, Revue d'Optique, 35, 1956, p. 37

Goubau, G., and Schwering, F., On the Guided Propagation of Electromag
netic Wave Beams, URSI-IRE Spring Meeting, May 1960, Washington,
np. GC

Boyd, G. D., and Gordon, J. P., Confoeal Multimode Resonator for Millimeter
through Optical Wavelength Masers, this issue, p. 489.

Multimode resonators of high quality factor will very likely play a sig-
nificant role in the development of devices, such as the maser, which operate
in the millimeter through optical wavelength range. It has been suggested
that a plane-parallel Fabry-Perot interferometer could act as a suitable
resonator. In this paper a resonator consisting of two identical concave
spherical reflectors, separated by any distance up to twice their common
radius of curvature, is considered.

Mode patterns and diffraction losses for the low-loss modes of such a
resonator are obtained analytically, using an approximate method which
was suggested by W. D. Lewis. The results show that the diffraction losses
are generally considerably lower for the curved surfaces than for the plane
surfaces. Diffraction losses and mode volume are a minimum when the
reflector spacing equals the common radius of curvature of the reflectors.
For this case the resonator may be termed confocal. A further property of
the concave spherical resonator is that the optical alignment is not extremely
critical.

Schawlow and Townes’ proposed that coherent amplification could
be achieved in the infrared through optical regions of the frequency
spectrum by maser techniques. At such frequencies multimode resonators
are necessary to achieve reasonable dimensions and high Q. They and
Prokhorov’ and Dicke’ have suggested as a resonator two plane-parallel
reflecting planes, known as a Fabry-Perot interferometer, or etalon."

In Fabry-Perot resonators the major factors contributing to the Q
(i.e., resolving power) are reflection losses and diffraction losses. Reflec-
tion losses result from absorption in the reflectors, and from transmission

through them. At optical frequencies a very good layered dielectric
reflector? can have a 993 per cent reflection coefficient. Diffraction
losses result from the finite aperture of the reflectors and from im-
perfections in their “flatness.”

Fox and Li have shown in the accompanying paper’ that modes, in
the sense of a self-reproducing field pattern, exist for an open structure
such as a Fabry-Perot interferometer. They also have recognized that
the diffraction losses of a plane-parallel Fabry-Perot are very much less
than those obtained by assuming a uniform intensity distribution over
the reflector and the Fraunhofer far field diffraction angle. They have
made numerical self-consistent field calculations based on Huygens’
principle to determine the actual diffraction losses and mode patterns.

In interferometry using a Fabry-Perot resonator, one normally ex-
cites a system of plane waves traveling at certain discrete angles to the
axis. Constructive interference at each of these discrete angles, as is
appropriate to ring order, wavelength and spacing, results in a pattern
of concentric bright rings. Schawlow and Townes indicated that each
ring of the interference pattern is not a pure mode of the resonator but
an infinite sum of such modes, each representing a different field pattern
over the reflector. This idea has been given much substance by the work
of Fox and Li.

The plane-parallel Fabry-Perot is not necessarily ideal, however, as a
high-frequency multimode resonator. A resonator formed by two
spherical reflectors of equal curvature separated by their common
radius of curvature is considered in detail in this paper. The focal length
of a spherical mirror is one-half of its radius of curvature. Therefore the
focal points of the reflectors are coincident and the resonator is termed
confocal. G. W. Series, Fox and Li* and Lewis’ have also suggested the
confocal resonator. Lewis has recognized that it would have lower
diffraction losses than the plane-parallel Fabry-Perot and has described
the analytic solution presented here.

they have an axis and thus lose the advantage of ease of adjustment.
Il. RESONATOR QUALITY FACTOR
Resonator quality factor, or Q, is defined as

Consider an interferometer consisting of two reflecting surfaces separated
by a distance d which is large compared to the wavelength in the medium
\. By considering waves bouncing back and forth between the surfaces,
one may derive an approximate Q as

where the power reflection coefficient per bounce is r = 1 — a. Resolving
power is thus synonymous with Q within the small loss approximation
of (2).

If diffraction losses are small compared with reflection losses, then
resonator Q is proportional to the spacing bet ween the reflecting surfaces.
lor a given reflector aperture size, the resonator Q will continue to
increase with the spacing d between the reflectors until the diffraction
losses become roughly comparable with the reflection losses. Further
increase in spacing then decreases the Q because of increasing diffraction

All resonator dimensions are assumed large compared to a wavelength;
the modes and diffraction losses of the confocal resonator are therefore
obtainable from a self-consistent field analysis using Huygens’ principle."
A confocal resonator is considered, with identical spherical reflectors
of radius b, as shown in Fig. 1. Assume the field to be linearly polarized
over the 7” surface in the y direction and given by Eofm(2’)g,(y’),
where £5 is a constant amplitude factor and f,,(.2”) and g,(y’) are the
field variations over the aperture. At point P?(.r,y) on the other surface,
one computes the electric field by summing over contributions from

P’P and the normal to the reflector surface at 2’, and k is the propaga-
tion constant of the medium between the reflectors. Note that /

2a/d, where \ is the wavelength in the medium. The electric field in the
xz plane is approximately zero. The reflector is assumed square and of
dimension 2a, which is small compared to the spacing 6 (since the con-
focal spacing is under consideration d = 6), and thus @ is very nearly
zero. The medium is assumed to fill all space.

The normal modes or eigenfunctions of the confocal resonator are
obtained by requiring that the field distribution over .x’y’ reproduce
itself within a constant over the ry aperture, and thus F, Evfim( x) gn(y),
where EF om0,/25. The proportionality factor ¢,0, is generally com-
plex, giving both amplitude and phase changes. The resulting integral
equation is

The distance p varies only a small amount for small apertures and thus
may be replaced by the separation b except in the exponential phase
term. For x and y small compared to b one can show that

where w = 2° + y’. The third term makes a negligible contribution to
the phase when a’/bA < b’/a’. Note that in this approximation one
cannot distinguish between spherical and parabolic surfaces. In terms
of some dimensionless variables

ve
. ° 11 ° . ° ° ‘
Slepian and Pollak” have considered the following integral equation:

rus. . ‘ . . . xx’
rhis is a homogeneous Fredholm equation of the second kind with «
as the kernel. It is often referred to as a finite Fourier transform. They

where Son(e,n) and Rom (e,L) are respectively the angular and radial
wave functions in prolate spheroidal coordinates as defined by Flammer,””
and where 9 = X//c = x/aand n = Y/Vc = y/a respectively for
F’,,(X) and G,(Y). There is an infinite number of eigenfunctions and
corresponding eigenvalue solutions to (9) for any value of c. Flammer™
gives values of these functions for ¢ S 5 and Slepian and Pollak"
have computed the eigenvalues x,, for the important region of ¢ > 5.
The eigenfunction solutions of (8) are thus the spheroidal wave
functions So»,(¢,v/a)NSo,(¢e,y/a). The eigenfunctions are real; therefore,
the reflecting surfaces are of constant phase. The eigenvalues are
Oo mon XmXnt ¢ - (12)

The phase shift between the two reflecting confocal surfaces equals the
phase angle of o,¢0,. For resonance the round-trip phase shift must

494 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1961
equal an integer g times 27. From (11) and (12), one finds therefore

The confocal resonator is seen to have resonances only for integer values
of the quantity 4b. If 4b, is odd, (m + n) must be even, likewise if
tb AX is even, (m + n) must be odd. Note that considerable degeneracy
exists in the spectrum; increasing (m + n) by two and decreasing q
by unity gives the same frequency. The degenerate modes are or-
thogonal over the reflector surface since they satisfy the integral (5)
with different eigenvalues. The modes have negligible axial electric
and magnetic fields and thus will be designated by TEM,,,¢, where
m and n equal 0, 1, 2,--- , and refer to variations in the x and y di-
rections, while g equals the number of half-guide wavelength variations
in the z direction between reflectors.

The fractional energy loss per reflection due to diffraction effects is
given by

rhe function | — | x, |° versus ¢ is shown in Fig. 2 for m OG: Boe:
It can be shown that Fig. 2 also gives the diffraction losses for an infinite
cylindrical reflector strip of width 2a and radius of curvature b. The
diffraction losses for various TEM,,,, modes are shown in Fig. 3. Note
that TEM,,, and TEM,,, (u # v) have the same diffraction losses;

, and TEM)», are so nearly
equal that they can be plotted as one curve. As indicated previously,
these last two types of modes cannot both be resonant at the same
frequency. Note that the losses are primarily determined by the higher

of the transverse mode numbers m, n, regardless of the field polarization.

with circular reflectors are also shown. The diffraction losses for the
confocal resonator are seen to be orders of magnitude smaller than for
the plane parallel resonator. Fox and Li have also obtained numerical
results for the confocal resonator with circular cross section of radius a.
These are in good agreement with the results presented here, allowing
for the fact that in this paper the reflectors have a square cross section
of width 2a.

Fig. 2 — Kigenvalues of integral equation; also the diffraction losses of an in
finitely long evlindrical reflector of width 2a.

If one approximates the diffraction loss curve by a function ap
SS yaks
flector radius a that the resonator Q is a maximum as a function of the

confocal spacing 6 when the reflection loss equals [2.30B(a?/bA) — 1]
times the diffraction loss. For the THMo9, mode, A 10.9 and B

The diffraction loss for the plane-parallel case assuming a uniform
field and phase distribution and a diffraction angle of 6 \/2a is also
shown. This diffraction angle corresponds to the first Fraunhofer mini-
mum in far field theory. For a square (or circular) reflector of side 2a
the diffraction loss is approximately

Fig. 3 clearly demonstrates the inadequacy of the assumption of uniform
intensity distribution.

Though the eigenvalues given by (12) must be known accurately the
eigenfunctions are only of approximate interest. Flammer’ shows that,
in the approximation of n° < 1 (near the center of the reflector), (10)
becomes

The inode shape is thus approximately a Gaussian times a Hermite
polynomial H,,(X ). The gamma function is arbitrarily chosen as normal-
ization such that F,,(X = 0) = +1 for m even:

F3(¢,n) (2en> — l)e *”™
The approximation involved in (17) fails away from the center of the
reflector. For reasonably large values of c, however, the field is weak
there, and of little interest. The diffraction losses were previously ob-
tained from (15). Curves representing (18) for various values of ¢ are
shown in Fig. 4. The dotted curves for ¢ = 5 are the true eigenfunctions
’ : . ’ 12
Son(¢e,n) as obtained from llammer.

The exponential dependence of the electric field on en, which is
independent of the reflector half-width a, leads one to define a “spot
size” at the reflector of radius w w, , Where w ve + y, at which
the exponential term falls to ¢

The only effect of increasing the reflector width 2a is to reduce the
diffraction losses; the spot size is unaffected.

If one allows the reflectors to be somewhat lossy or partially trans-
parent, then the resonator Q is reduced over that implied by diffraction
losses alone. The field distribution, i.e., the mode pattern, is not seriously
affected so long as the losses are small and fairly uniform over the plates.

Fig. 4 Approximate field amplitude variation versus normalized radius for
various modes. The exact dependence given by the angular prolate spheroidal
function Som (ec, 7) is shown by dashed lines.

The electric field patterns derived thus far have all been linearly
polarized. Fox and Li’ have recognized that, by superimposing the
TEM, mode linearly polarized in the x direction and the TEM,
mode linearly polarized in the y direction, the lowest-order circular
electric mode can result, and it has the same diffraction losses as the
linearly polarized TEM, mode. Many other polarization configurations
can be obtained in this manner.

The field over the confocal aperture has been obtained in the preced-
ing section. The field over an arbitrary plane z = 29, as in Fig. 1, is also
obtainable by a straightforward application of Huygens’ principle as
stated in (4). The arbitrary plane zo) may be placed outside the confocal
geometry as well as inside provided one takes into account the trans-
mission loss of the reflector. The field distribution over the confocal
surface is given by F,,(c,7/a)G,(c,y/a). For large ¢ the spheroidal
functions may be approximated by the Gaussian-Hermite functions.
The integral can be evaluated in the limit of c > «.

Within these approximations, the traveling wave field of the confocal
resonator resulting from the field at one of the reflectors is given by

When the reflecting surface is made partially transparent, as will be the

will be a traveling wave as given in (20) reduced by the transmission
coefficient of the reflector. Within the resonator, the field will be a

standing wave. The transverse standing wave is as given in (20) except
that the exponential phase function is replaced by the sine function.

The surface of constant phase which intersects the axis at 2) as ob-
tained from (20) is given approximately by

neglecting the small variation in ¢ due to variation in z. This surface is
spherical, within the approximations of this paper, and has a radius of
curvature 6’ given by

At & = +1 it coincides with the spherical reflector as expected. Also note
that the symmetry or focal plane ( = 0) is a surface of constant phase.

The field distribution throughout the resonator is given by the modulus
of (20). The complete field distribution within the confocal resonator is
shown schematically in Fig. 5 for the low-loss TEM oo, mode.

The field distribution over the focal plane is less spread out than over
the spherical reflectors. The field spot size over the spherical reflectors
was defined by (19). In any arbitrary plane zp the exponential term in
the field distribution falls to e * at a radius

To obtain the radiation pattern angular beam width of the TE Moo,
mode spherical wave, one takes the ratio of the spot diameter from (20)
or (24), as—— ~, to the distance from the center of the resonator. The
beam width between the half-power points is given by

Since the surfaces of constant phase of the confocal resonator are
spherical, it is apparent that (20) also represents approximately the
field distribution between two spherical reflectors of arbitrary spacing.
That is, any two surfaces of constant phase may be replaced by re-
flectors. The frequencies at which such a resonator will be resonant. will
of course be determined by satisfaction of the phase condition.

Fig. 5 Field strength distribution within the confoeal resonator for the
TEM oo, mode.

Consider two identical spherical reflectors of radius of curvature b’
spaced a distance d. The only restriction is that b’ 2 d/2. The confocal
geometry of spacing b of which this resonator is a part is [set & = d/b in

The spot size at the reflectors in the nonconfocal resonator may be
immediately obtained from (24) with & = +d/b. It is

Note that the factor [2(d/b’) — (d/b’)*| achieves a maximum of unity,
as a function of 6’, when 6’ d. Thus, for a given spacing between re-
flectors, the spot size is a minimum for the confocal resonator.

502 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1961
section of dimension 2a’ on the assumption that this loss is equal to that

of its equivalent confocal resonator with reflector dimensions scaled up
by the ratio of their spot sizes. The equivalent confocal resonator has

Thus the confocal geometry gives minimum spot size and minimum
losses for a given spacing. If one defines the mode volume as the spot
size at the reflector times the spacing, it is clear that the minimum mode
volume also results from the confocal geometry. The mode volume, so

when the diffraction losses derived from the ‘equivalent’ confocal
geometry are small, that is, when the reflector dimension a’ is somewhat
larger than the spot size. In an exact solution for the nonconfocal case
one should again start from the integral (4), and clearly the field dis-
tribution and losses so derived will depart from that obtained from the
equivalent confocal case if the confocal field is not substantially all
intercepted by the nonconfocal reflectors. Conversely, so long as the
spot size is small compared to the reflector dimension a’, one expects the
field distribution and losses to be very nearly correctly given by the
equivalent confocal solution.

The phase shift between the two reflecting nonconfocal surfaces may
be obtained from (20). The condition of resonance may then be shown
to be

In the nonconfocal case 4¢d/ is no longer necessarily an integer at
resonance. It is more important, though, that the modes are no longer
degenerate in m + n. The spectral range or mode separation for the
nonconfocal resonator is given by

Note that in the confocal case the set of modes mnq 00g, Olq are
maximally split in frequency, whereas if the parameter in parenthesis
equals 3 (when d/b = 0.414) then the mng = 00g, Olg, 11g, 12¢ modes
are maximally split in frequency.

where m and n are small integers and g a large integer. In the confocal
case (b = d) note that equations (31) and (33) reduce to (14).

The theory of this section does not extend to the limit of plane-parallel
reflectors, i.e., infinite radii of curvature. Let the spacing d remain fixed
while b’, and consequently [by (26)] the confocal radius b, approaches
infinity. The spot size, as seen from (27), keeps increasing with 6’, and,
as has been noted above, this results eventually in the breakdown of
the whole idea of an equivalent confocal resonator. The relations for
the nonconfocal resonator are valid as long as the reflector aperture
radius a’ is somewhat larger than the field spot size radius given by (27).
That is, one must require

For comparison purposes, consider the resonances of a rectangular
° . : ‘ a bh ]
conducting box in the manner of Schawlow and Townes. Let the
dimensions be 2a X 2a X b:

where q, r, and s are integers. Modes where q > r,s can be thought of
physically as waves bouncing predominantly back and forth between
the reflecting end plates of the rectangular box. The spectral range or
mode separation is given by

Removing the conducting side walls causes large diffraction losses for

the r = 0 or s = O modes since they have a strong field at the edge of
the reflectors. Large r or s modes represent waves traveling at a con-
siderable angle to the normal between the reflectors and thus these
modes have such large diffraction losses that they are eliminated as
resolvable resonant modes. Modes with r, s = 1,2,---, have small dif-
fraction losses, and are approximations to the actual modes which can
exist in the resonator without the conducting side walls. Fox and Li’s*®
work shows that for a°/b\ greater than unity the mode separations of a
plane-parallel Fabry-Perot are given approximately by (36), the ap-

The mode separation corresponding to Ar or As = 1 has, to the writers’
knowledge, never been resolved at optical frequencies due to the large
values of a°/b\ and low values of reflectance used. Calculations show,
though, that for reflectance coefficients of about 0.99 and a°’/b\ & 4,
such that diffraction losses are comparable with reflector losses, the
resonances should be resolvable.

The mode separation due to Ag = | is easily resolvable and is given by
A(1/X) = 1/2b. This is the spectral range as normally stated for the
plane parallel Fabry-Perot interferometer. It corresponds to changing
the number of half wavelengths between the reflecting surfaces by one.

The confocal resonator is resonant for integer values of 4b/A. The
mode separation due to Aq l is A(1/X) | /2b. The modes are de-
generate in frequency in that for a given integer 4b/\ all TEM,,,,, modes
are resonant such that m + n remains even or odd according to whether
1b/X is odd or even. The modes of the plane-parallel Fabry-Perot are
not degenerate, except for rsq and srq. A possible advantage of this de-
generacy of the confocal modes will be discussed in the next section.

A type of solid state optical maser has recently been demonstrated by
Maiman”™ and by Collins et al.’ It consists of a fluorescent crystal ma-
terial (ruby) a few centimeters in length and a few millimeters in diame-
ter. The crystal material should be optically homogeneous. The ends of
the crystal are optically flat and parallel. The ends are silver-coated for
high reflectance. One of the reflecting surfaces must be slightly trans-
parent, as the output of the optical maser is obtained through the
reflecting surfaces. Thus far, silver has been used to provide the reflec-
tion, but for ultimate performance multiple-layer dielectrics’ should be
used to obtain low transmission loss as well as high reflectance. The
pump power enters the fluorescent erystal from the side.

flection losses, then the resonator Q is proportional to the spacing be-
tween the reflecting surfaces. Consider a confocal resonator and a
plane-parallel resonator each of spacing b and of equal Q. The energy
distribution in the former is more concentrated on the axis and thus the
confocal resonator has a smaller effective mode volume. The volume of
maser material required will thus be less for the confocal than for the
; plane parallel resonator. For maser oscillation the required excess density
of excited states depends only on the cavity Q and in no other way upon
the resonator shape.’ The pump power is proportional to the volume of
maser material times the density of excited states divided by the natural
lifetime of the excited state. Thus, assuming equal Q, the confocal
resonator with its smaller volume of material requires less pump power
than the plane parallel resonator by the ratio of their cross-sectional

and pump power with regard to the use of optical fibers in maser
applications.

The minimum volume of maser material is limited by diffraction losses.
If diffraction losses are to be considerably less than | per cent for the
lowest-order mode so as to be small compared to achievable reflection

Ifb = 4emand \ = 10‘ em, then the rod of maser material should be
approximately 0.4 mm in diameter. A rod of larger diameter would waste
pump power in that the field of the confocal resonator would be very
weak outside this minimum diameter of material.

The analysis of the confocal resonator assumes a uniform dielectric
material between the spherical reflectors. For reasons of minimizing the
pump power, it is necessary to use a small diameter of maser material.
Therefore, to prevent internal reflection of energy from the sides of the
maser material, it may be advisable to grind rough the sides of the rod
of maser material or to immerse it in a surrounding medium of equal
dielectric constant. If this is not done, the energy assumed lost due to
diffraction effects would not escape and the electric field pattern will not
be as computed herein. A more important effect of internal reflection
from the side walls would be to increase the Q of the transverse modes
which would increase the spontaneous and stimulated emission power
to these undesired modes, and thus increase the over-all pump power
required.

The natural linewidth of the material used in an optical maser will,
for reflector spacing d of a few centimeters, be large compared to the

mode separation determined by integer changes in r,s for a_plane-
parallel resonator. Hopefully, the natural linewidth of the maser ma-
terial will be less than the mode separation corresponding to integer
changes in g. Thus, there is the possibility that a plane-parallel resonator
optical maser may frequency wander between low-order r,s modes.

If the diffraction losses are comparable to or exceed the reflection
losses for the lowest mode then, as can be seen from Fig. 3, the ratio of
the Q’s of the lowest two modes of the confocal resonator exceeds con-
siderably the ratio of the Q’s of the lowest two modes of the plane-
parallel resonator. By the lowest order mode is meant m = n = 0, and
r = s = 1, respectively, for the confocal and plane-parallel resonator.
Therefore, maser oscillation is more likely to take place in only the
lowest-order mode of the confocal than of the plane-parallel resonator.
This greater loss discrimination between modes may be one of the
significant advantages of the confocal resonator.

In the confocal resonator optical maser, if the maser oscillation wan-
ders between modes the output beam pattern will change, just as in the
plane-parallel resonator, but the frequency will remain fixed due to the
mode degeneracy. Thus, the observed linewidth of the maser output
may be narrower for the confocal resonator.

The required accuracy on the confocal condition to achieve degeneracy

the mode splitting of the near-confocal resonator equals the mode separ-
ation of the plane parallel resonator. To achieve a significantly smaller
mode separation in the near-confocal resonator than the plane parallel
resonator would require proportionately greater accuracy in the radius
of curvature and spacing of the curved surfaces.

The plane Fabry-Perot requires accurately parallel reflecting surfaces.
The confocal resonator requires only that the axis of the confocal reso-
nator approximately coincide with the axis of the rod of maser material.
The axis of the confocal resonator is the line passing through the two
centers of curvature. The resonator axis must intersect the two reflect-
ing surfaces near their center. Define the effective aperture radius as
the distance from the point of intersection of the axis of the confocal
resonator with the reflector surface to the nearest edge of the aperture.
The diffraction losses will be approximately determined by this distance.

If the minimum diameter of maser material is used, then the axis of
the confocal resonator must coincide with the material axis. Increasing

the diameter of the maser material wastes pump power but relaxes the
tolerance on the resonator axis.

It is well to note that a single spherical reflecting surface and a plane
reflecting surface spaced by approximately half the radius of curvature
will have similar properties to the confocal resonator and may be ad-
vantageous if it is desired to bring the output through a plane surface.

A confocal multimode resonator formed by two spherical reflectors
spaced by their common radii of curvature has been considered. The
mode patterns and diffraction losses have been obtained. The confocal
spacing of the reflectors is found to be optimum in the sense of minimum
diffraction losses and minimum mode volume.

The diffraction losses are found to be orders of magnitude smaller
than those of the plane-parallel Fabry-Perot, as obtained by Fox and Li.”
It is more important, though, that a greater diffraction loss diserimina-
tion between modes occurs, and thus oscillation in other than the lowest-
order mode is less likely for the confocal resonator, assuming that
diffraction losses are comparable to reflection losses.

The modes of the confocal resonator are degenerate, in that one-half
of all the possible field pattern variations over the aperture are resonant
at any one time. This degeneracy is split if the resonator is nonconfocal.
The splitting is comparable with that of the plane-parallel resonator

different from the common radius. The mode volume and diffraction
losses are insensitive to the confoeal condition.

The required volume of maser material is smaller for the confocal
resonator than for the plane-parallel resonator, and thus the required
pump power is less. The confocal resonator is relatively easy to adjust
in that no strict parallelism is required between the reflectors. The only
requirement is that the axis of the confocal resonator intersect each
reflector sufficiently far from its edge so that the diffraction losses are
not excessive.

The example of a confocal resonator mentioned here was taken at
infrared-optical wavelengths; however, such resonators may be useful
down to the millimeter wave range by virtue of their low loss. In this
connection, recent work of Culshaw’’ on the plane-parallel Fabry-Perot
at millimeter wavelengths is of importance.

The writers have been informed that Goubau and Schwering”” have
recently investigated diffraction losses of parabolic reflectors and that
their results agree with the work presented here.

Fruitful discussions with A. G. Fox, W. D. Lewis, T. Li, D. Marcuse,
S. P. Morgan and G. W. Series are sincerely appreciated. Mrs. F. J.
MacWilliams performed the computations.

Schawlow, A. L. and Townes, C. H., Phys. Rev., 112, 1958, p. 1940
Prokhorov, A. M., J.E.T.P., 34, 1958, p. 1658.
3. Dieke, R. H., U.S. Patent 2,851,652, September 9, 1958.
. Meissner, kK. W., J. Opt. Soe. Am., 31, 1941, p. 405; 32, 1942, p. 185.
5. Heavens, O.S8., Optical Properties of Thin Films, Butterworths, London, 1955.
). Fox, A. G. and Li, T., this issue, p. 453; Proce. LR.E., 48, 1960, p. 1904.
. Lewis, W. D., private communication.
Connes, P., Revue d'Optique, 35, 1956, p. 37; J. Phys. Radium, 19, 1958, p.
262
Jenkins, F. A. and White, H. E., Fundamentals of Optics, 3rd ed., MeGraw
Hill, New York, 1957
. Silver, S., Microwave Antenna Theory and Design, M.1.T. Radiation Labora-
tory Series, Vol. 12, McGraw-Hill, New York, 1949.
1. Slepian, D. and Pollak, H. O., B.S.T.J., 40, 1961, p. 43.
12. Flammer, C., Spheroidal Wave Functions, Stanford Univ. Press, Palo Alto,
Calif., 1957
3. Slepian, D. and Pollak, H. O., private communication
. Maiman, T. H., Nature, 187, 1960, p. 493.
5. Collins, R. J., Nelson, D. F., Schawlow, A. L., Bond, W., Garrett,
and Kaiser, W., Phys. Rev. Letters, 5, 1960, p. 303
Snitzer, E., J. Appl. Phys., 32, 1961, p. 36.
. Culshaw, W., I.R.E. Trans., MTT-7, 1959, p. 221; MTT-8, 1960, p. 182
. Goubau, G. and Schwering, F., U.R.S.1.-I.R.E. Spring Meeting, Washington,
May 1960

Relation Between Surface Concentration
and Average Conductivity in Diffused
Layers in Germanium

In this paper an expression is derived for calculating the average con-
ductivity of a diffused layer in semiconductor material as a function of the
surface concentration of the diffused impurity and the background impurity
concentration. Curves are presented depicting the relationship among these
parameters for the case of germanium. Included are curves for both diffused
impurity types for the complementary error function, gaussian, exponential
and linear impurity distributions.

In the design of semiconductor devices in which junctions are pro-
duced by solid state diffusion of impurities, it is of great value to know
the relationships which exist between the surface concentration of the
diffused impurity, Co, the background impurity concentration, Cg , and
the average conductivity of the diffused layer, ¢ These relationships
can be calculated from a knowledge of the resistivity as a function of
impurity concentration for material uniformly doped with a single
impurity. Such calculations are presented in this paper.

For convenience, assume initially that impurity atoms are 100 per
cent ionized. Therefore, the conductivity at a point in a diffused layer
in semiconductor material can be given by

primarily a function of the total number of ionized impurities present.
If it is assumed that both ionized impurity types scatter a majority
carrier identically, then the mobility in (1) may be considered to be a
function of (C + Cz).
The conductivity of material doped with a single impurity can be
expressed, again assuming 100 per cent ionization of impurities, as
*

C-€¢
oO o* = . o ( | )
C + Cz,
A log-log plot of the resistivity of single impurity doped material as a
function of the impurity concentration can be approximated by a set of
intersecting straight lines each having an equation of the form

integrating (6b) over values of « from the surface (2 = 0) to the june-
tion (2 = x;) and dividing by the junction depth, x; . Thus,

‘0, t) + Cal "O(Cy . xv) — ('p| dv. (7)
The values assigned to B and @ at any point on the interval are deter-
mined by the value of [C(Cy, 7) + C,] at that point. Equation (7)

generally requires numerical integration, using experimental values for
B and a.

lig. | shows the variation of resistivity, p, of single-impurity doped
germanium as a function of the impurity concentration, NV. Points in the
range of 10 < N < 2 X 10" for n-type material and in the range of
10'* < N s 6 X 10" for p-type material were taken from Prince.’
Points in the range of 2 X 10'"° < N < 10” for n-type material and
6 X 10°" < N S 10” for p-type material were taken from Hall effect
measurements of Tyler and Soltys.” Hall effect measurements give the
resistivity as a function of carrier concentration. However, direct meas-
urements of resistivity as a function of impurity concentration by

a be) ri - 3 ¢ . . .
lrrumbore and Tartaglia” for p-type material agree with the results of

Tyler and Soltys, thus justifying the assumptions made, at least for the
case of p-type material. Five straight-line approximations were made to

ach curve giving equations of the form of (5). Values of B and @ and
the range of validity of each set of values are shown in Table I.

Fig. 2 shows data of resistivity as a function of electron concentration
for n-type germanium as reported by Tyler and Soltys,) Moody and
Strauss,* Furukawa,’ Zhurkin et al.° and Spitzer.’ Also shown in Fig. 2
is a portion of the n-type curve from Fig. 1. As can be seen, this curve
represents a reasonable average of the arsenic data, for which case the
present calculations are intended.

Equation (7) was evaluated on the IBM 704 computer for various
impurity distributions, and the results were checked by hand calculation
of several points. Seven values of background concentration were used,
and four points per decade of surface concentration were evaluated. The
results are shown graphically in Figs. 3 through 10 on pages 514 through
521 for the various distributions as follows:

Since the conductivity of perfectly compensated material is not zero,
(7) will be in error by some small amount. However, for values of Cy
and Cs, such that (Cy — C,) 2 10 n; this error will be negligible. All
values of Cy and C’, used in these calculations fulfill this requirement.

The author wishes to thank J. C. Irvin and W. H. Jackson for many
helpful and enlightening discussions, and special thanks are due Mrs.
G. N. Alfandre whose able programming of the IBM 704 made solution
of (7) possible.

3. Trumbore, F. A. and Tartaglia, A. A., Resistivities and Hole Mobilities in Very
Heavily Doped Germanium, J. Appl. Phys., 29, 1958, p. 1511.

1. Moody, P. L. and Strauss, A. J., Electrochemical Society Meeting, Chicago,
May 2, 1960; also private communication.

Expressions are presented for the magnetization and pull relations of the
mating flat magnetic reeds used as contacting members in reed relays. The
results of an experimental study expressed in dimensionless form give the
force or pull between the reeds in terms of their dimensions, the gap between
them, and the flux density. Since the attainable flux density is limited by
saturation, the pull expression leads to conditions which must be satisfied
by the reed and gap dimensions to provide desired levels of contact and
retractile force.

The ampere-turn sensitivity of a relay using mating magnetic reeds
depends on the flux required and on the reluctance of the magnetic path
through the reeds. Expressions are given for the reluctance in the case of
an air return path and in that where the air return ts partially replaced by
a shielding member.

Expressions are also given for the operate time of such reed relays. This
depends on the concurrent flux development and motion of the reeds. The
time of flux development varies inversely with the power input to the coil,
but the motion time cannot be less than that required when the flux density

A sealed reed relay comprises one or more sealed contacts assembled
with a coil and shielding and supporting members. The distinctive fea-
ture is the construction of a sealed contact.’ In its simplest form, as
shown in Fig. 1, it consists of a glass envelope in which are sealed a pair
of magnetic reeds, which serve as contacting members, actuated by the
magnetic field induced in them by the coil current.

In another form,’ there is only one moving reed, the other magnetic
member being short and nearly rigid. This construction, using two of
these short members to provide both back and front contacts, has been

used with mercury-wetted contact surfaces.°* Both forms have been
used in single- and multiple-contact relays, both neutral and with
permanent magnet bias used to give polar or locking performance.’
Recent work has included development of a miniature sealed contact.”

As an aid to the design of sealed contacts, and to the understanding
of their characteristics, an analysis has been made of their performance
relations, including the dependence of the pull, contact force, speed and
sensitivity on the dimensions of the reeds. This analysis is presented
here in the form applying to the simple case of equal mating reeds, but
is, with some modification, applicable to other forms of magnetic reed
contacts.

The major controlling factor in the performance of sealed contacts is
the flux-carrying capacity of the reeds, as limited by the saturation
density of the reed material. The treatment given here therefore starts
from the relation between the reed flux and the attractive force at the
gap, which must deflect the reeds and provide the contact force. This
relation is determined essentially by the reed dimensions alone. The
sensitivity, as measured by the ampere turns for operation and release,
depends upon both the flux and the reluctance of the magnetic circuit,
and hence upon the dimensions of the coil and shielding members, as
well as upon those of the reeds. Relations are given for the estimation
of reluctance and hence of sensitivity. Finally, expressions are given
for the estimation of the speed of operation, which depends upon the
times of field development and of reed motion, with the latter establish-
ing a lower limit to the attainable time of operation.

Using the notation of Fig. 1 for the reed dimensions, the attractive
force F between the reeds is given by Maxwell’s law as
a
Rrab’

where ¢q is the flux in the gap where the reeds overlap. The total flux
® is the sum of ¢g, and the fringing flux which passes from one reed to
the other by air paths around the gap. As shown below in the discussion
of reluctance, approximate estimates may be made of the fringing flux,
but the relations involved are not adapted to a simple direct treatment
of the relation between the pull / and the total flux &. A direct experi-
mental study was therefore made to determine the approximate form
of this relation.

In this study, pull and flux measurements were made of four different
sets of reeds, having the dimensions listed in Table I. Each set of reeds
was assembled with overlap values, a, of 25, 50 and 100 milli-inches, and
the pull and flux measured for values of gap x in the range from 1 to
10 milli-inches over the range of coil energization from 0 to 300 ampere
turns.

These measurements were all made with the reeds supported in a
brass fixture which could be adjusted to make the overlap surfaces
parallel and to set the overlap and the gap at desired values. The coil
used had inside and outside diameters of 0.39 and 0.86 inch, and was
1.64 inches long. It was centered over the gap in making the measure-
ments, and was provided with a central search coil located on its inner
diameter. Measurements were also made of the flux in the coil alone,
and this air-core flux was subtracted from the flux readings to give the
reed flux ®.

If the reeds are long compared with the overlap, as in practice and in
these measurements, the pull is independent of the reed length. If the
permeability of the reed material is high enough for the reeds to be

essentially equipotential surfaces near the gap, the pull F is a function

of &, x, a, b and h only. For dimensional consistency, therefore, the pull
must be given by an equation of the form
SrabF of
ea f 4 (2)
¢? .
This is a convenient dimensionless form, since the left-hand term must,
from (1), approach unity for « = 0. (In evaluating this term, consistent
TABLE I — DIMENSIONS OF REEDS TESTED

units must be used: in C.G.S. units, ® is in maxwells, a and b in centi-
meters, and F in dynes.) From direct plots of the measured values of
F and # there were read values of F for & = 10,000 bh (i.e. for a flux
density of 10,000 gauss). For each set of reeds and each overlap, the

lustrated in Fig. 2, which shows the results for h = 10 milli-inches, b
100 milli-inches and three values of a. As the points for all three values
of a fall on the same curve, the right-hand side of (2) is independent of
a/b, and reduces to a function of 2/a and h/b only. The experimental
relation of Fig. 2 is linear, and similar linear plots, independent of a/b,
were obtained for the other three sets of reeds. Thus the functional
relation (2) was found to be of the form

where k is a function of h/b. The observed values of k for the four sets
of reeds were plotted against the corresponding values of h/b and a
linear plot was obtained, conforming to the equation

of 10,000 gauss. Similar plots were made for other values of density,
and substantially the same relation was found to apply for densities
ranging from 20 per cent to 80 per cent of the saturation density, which
was about 15,000 gauss for the material used in these tests.

Within this range of reed flux density, therefore, all the observed
values of pull conformed approximately to (3), with k given by (4).
These expressions may therefore be used to estimate the pull of mating
magnetic reeds at densities up to 80 per cent of saturation, when each
reed is long compared with the overlap. The expressions are approxi-
mate, and minor deviations, particularly in the value of k, are produced
by changes in coil length and in the return path and shielding con-
figuration.

A further, and more important, deviation from these relations occurs
in the closed gap condition, x 0, where the pull is given by (3) as
© /8rab. For release, this pull equals the retractile force of the reeds,
shown in Fig. 3 as sX. Thus the flux at which release occurs should

variation, with the upper limit close to this computed value. This cor-
responds to the fact that surface irregularities or lack of parallelism of
the mating surfaces concentrate the closed gap flux over a smaller area
than ab, and increase the pull over that given by (3) for x = 0. The
actual closed gap pull is therefore variable, and in general higher than

lor operation, the pull curve of the reeds must exceed their stiffness
load, shown in Fig. 3 as the force s(X — x). Here X is the open gap
separation, and s is the combined stiffness, or half the stiffness of one
reed if the two are alike. (For unequal reeds, 1/s is the sum of the
compliances of the two reeds.) Let Fo’ be the closed gap pull, &°/82ab,
for the minimum value of ® required for operation. Then from (3) the
pull for this value of ® is given by

sX 4CX
If CX = 1, the tangency condition no longer applies, the pull and
load are equal at « = 0, and Fy’ = sX.
After operation, the flux exceeds the just-operate value. Let Fo be

the contact force is equal to Fy — sX, and the retractile force, tending
to open the contact on release, is sX. The contact behavior varies with
the contact force, so that one design requirement is that the contact
force should exceed some specified minimum value F,,

A second requirement is that the ratio of retractile force to the contact
force should exceed some minimum value n (of the order of unity) to
minimize the danger of sticking. The area of actual intimate contact,
and hence the force required to break cold welds over this area, varies
directly with the contact force. In formulating this requirement, the
operated pull is taken as given by Fy. As stated above, the actual op-

erated pull is variable and in general exceeds Fo. Hence the actual
contact force is usually in excess of its computed value, and the actual
value of the retractile force ratio n less than its computed value.

A satisfactory design must meet the requirements for operation,
contact force and retractile force through the range of dimensional
variation occurring in manufacture. The present discussion is confined
to the case where only variations in the gap X need be considered. Let
the minimum and maximum gap values be X; and X,, and let ¢,
X./X,. The final flux must be below saturation, and for purposes of
estimation may be assumed to correspond to a density B”, of the order
of 90 per cent of the saturation density. For operation to occur without
excessive reed reluctance, the operating density B’ should be of the
order of 80 per cent of saturation. Let ec. = Fo/ Fo (B”’/B’)*. Then
the operate condition (7) will be satisfied for all values of X if

The contact force requirement is satisfied for all values of X if
Fo F, + sXo ‘

The retractile foree requirement that sX > n( Fo — sX) is satisfied
for all values of X if

rom the preceding three equations, the design requirements can be
written as
1X,
(1+ CX.)

The reed dimensions must be chosen to satisfy these three equations
for given values of X,, ¢, fe and n (ey being approximately fixed by
the reed material used). The dimensions to be selected are a, b, A and
whatever other dimension determines s: the free length / if the reeds
are of uniform cross section. If the section ratio bh is tentatively taken

as fixed by manufacturing considerations, the reed dimensions can be
evaluated as follows:
The value of C is computed from (8), and since a k/C and k is
given by (4), the required value of a is thereby determined.
Since the value of ® for Fy is bhB”,
—
F, bh B . (11)

Then, with Fo evaluated from (9), and a, B” and b/h known, the thick-
ness h (and hence the width b) can be determined from (11). The stiff-
ness s is given by (10). For a reed of uniform section b & h the length /
can be determined from the required value of s. As (11) determines the

requirements will be that for the largest value of b/h consistent with
manufacturing considerations.

The relations outlined above allow for variations in the gap Y, but
not in the other dimensions. In practice, allowance must also be made
for variations in the reed thickness A. This can be done by applying the
treatment outlined above to the case where A has upper and lower
limits, leading to expressions similar to (8), (9) and (10).

Using the relations given above, reed dimensions have been computed
to give the three values of minimum contact force F- shown as param-
eters for the curves of Fig. 4, over the range of gap dimensions shown.
These illustrative cases were computed for the values of b/h, Xo/X,, n
and flux density shown in the legend of this figure. The values of flux
density are those applying to iron-nickel alloys of about 50 per cent
nickel. These have a Young’s modulus value of about 25 X 10° psi,
which was taken as applying in computing the length required to meet
the stiffness requirement.

The computed dimensions shown in lig. 4 are the reed thickness h,
length /’ and overlap a. The overlap varies inversely with the gap, as
shown by (8) when bh, n, ¢ and c are fixed, as in these computed cases.
The thickness and length both increase as the gap and the required
contact force are increased. The gap is the major factor controlling the
length and hence the over-all size of the sealed contact. The choice of
gap is fixed in part by voltage breakdown requirements, and in part by
manufacturing considerations which determine the variation in gap for
which allowance must be made. The comparisons of lig. 4 are somewhat

the ratio ¢, is usually larger for small gaps than large ones, in which case
the reduction in reed thickness and length resulting from reducing the
gap is less than that shown in Fig. 4.

In the computations for Fig. 4 the retractile force ratio n is taken as
unity. This is the computed minimum ratio of retractile force to contact

force. As discussed previously, the actual contact force may be increased
by flux concentration in the closed gap, and the actual ratio may be
materially below the computed minimum, which serves therefore as a
basis for comparing one design with another, rather than as an absolute
measure of the liability to sticking. From (9), increasing n increases Fy ,
and thus increases the thickness A and the required length /’.

The reed length is denoted /' in Fig. 4, as that giving the desired stiff-
ness [2s from (10)] for a uniform reed of length /’ and stiffness given
by Ebh®/4l’°. As shown in Fig. 1, convenient construction for the seal
requires the use of a cylindrical section extending from the seal for a
length cl, with the flat section extending beyond this for a length J.
The cylindrical and rectangular sections are of equal area. The stiffness
of such two section cantilevers are given in Fig. 6-30 of Peek and Wagar,’
from which can be derived the following expression for the ratio /'/l of
two cantilevers of equal stiffness, one having the dimensions shown in

Thus if cl and / are known, /’ can be evaluated and used to determine
the stiffness. Conversely, if /’ is determined from the required stiffness,
as in Fig. 4, the corresponding flat reed length / can be determined for a
given value of cl.

The flux @ between the reeds required for operation can be estimated
by means of the relations given in preceding sections. The corresponding
value of coil magnetomotive force 5, or 44rNJ, is given by ®&b, where
® is the reluctance of the magnetic circuit. Estimating the sensitivity,
or the value of ¥ for operation, therefore requires some method for
estimating the value of the reluctance @.

Most reed relays have essentially an air return path, whose reluctance
is only slightly reduced by the can or shield placed over the core. For
an air return, the flux through the reeds is analogous to the current
through a leaky transmission line, as discussed in Section 9-2 of Peek
and Wagar.’ Referring to Fig. 5, let ¢ be the flux in the reed at a distance

Fig. 5 Notation for analysis of air return magnetie field.
between the point x and the plane of symmetry. The flux dg leaving the
reed over the length dv is given by
de
¥ (13)

where p is the permeance per unit length of reed of the air path from x
to the plane of symmetry. The rate of change in f is given by

lor continuity, the expressions for g and dg/dx at x L,/2 given
by the two forms of (15) must be equal, from which expressions are
obtained for the coefficients C and D in terms of A and B. The co-
efficients A and B in turn are determined by the boundary conditions,
which require that ¢ be zero at x I./2 and that for a gap flux ¢,
at « = 0, the potential f —ggRe/2 atx 0, where ®, is the gap
reluctance. Then, from (13), the boundary conditions become

As the boundary condition for x 0 shows, the flux ¢ in the reed
initially increases with x, and hence the point of maximum flux ® is at
some positive value of « where dg dr 0. Applying this condition to
the expression for ¢ given by (15), the maximum flux ® is given by

This maximum flux ® is the total flux between the reeds. The reluc-
tance ®, or ¥ @, is given by gl, ®&. Substituting from the preceding
equations, this expression for ® may be written in the form

where Q RepLs , and r¢,g q¢ is given by (19). Thus the ratio ®pLe is a
function of the three ratios Ly, Le, al, and Q, or Rypl. . For the limit-

ing case where the reed reluctance per unit length r is so small that a
approaches zero, the hyperbolic functions in (19) may be replaced by
their series expansions and the higher power in @ neglected. Then for

Fig. 6 shows RpLe for the case L,/L. = 0.5 plotted against Reple
for various values of al. . The increase of ® with increasing al. measures
the effect of the reed reluctance. Aside from the reed reluctance, ®&
comprises the air return reluctance in series with the gap reluctance as
shunted by the parallel air path across the gap. The curves show how
this shunt path limits the increase in ® with increasing Rez .

The effect of increasing L/L. is to increase ®, as can be seen from (21
except for large values of ®_ (and hence of Q). Thus the reluctance is
reduced and the sensitivity increased by using a short coil concentrated
over the gap.

In deriving the preceding equations, p and r have been taken as
constants. The estimation of the air path permeance of bar magnets is
discussed in a classic paper by Evershed, who showed that satisfactory
estimates can be obtained by using the constant value of p applying to
a uniformly magnetized ellipsoid. This is the function of the ratio L/D
shown in Fig. 7. In applying this to mating rectangular reeds, L is taken
as the over-all length L., and D as 2(b + h)/r.

The assumed constancy of r, the reluctance per unit length, is at best
an approximation, since r is inversely proportional to the permeability of
the material, which varies with the flux density, so that r varies along
the length of the reeds. If, however, r is small, the error introduced by
this approximation is minor.

The effect of the permeability on the reluctance and sensitivity, as
related to the size and capability of mating reed contacts, is illustrated
by the computed results of Table II. Here the dimensions h, b, a and |
have been taken as those shown in Fig. 4 for minimum gap values of
ft and 10 milli-inches and contact force values of 2 and 10 grams. The
operate flux values, as in Fig. 4, are for a density of 12,000 gauss. The
over-all length L». has been taken as 3.75 times the computed reed length.
The permeance p is taken from Fig. 7. The critical gap reluctance has
been obtained by adding a 2 milli-inch air gap allowance for the closed
gap to the critical gap x’ computed from (6).

With ®gpLl, thus computed for each of these two cases, values of als
have been computed for two values of permeability u, 1000 and 5000.
With r given by 1 wbh, corresponding values of al, have been computed,
and corresponding values of pl, read from Fig. 6, taking the coil

Contact force, Fc (grams)
Operate flux, & (maxwells)
Length, L». (em)
Permeance factor, p

length L, as half of L.. There were thus obtained the values of relue-
tance ® shown in Table II, and thus finally the operate values, given
(in abampere-turns) by ®/4r.

In the range of magnetic reed dimensions and contact and retractile
forces illustrated by these computations, al, is small and the reed
reluctance minor for » of the order of 5000. The materials used for these
reeds have permeabilities of 5000 and higher at densities of the order
of 12,000 gauss, or at about 80 per cent of the saturation density. The
permeability falls rapidly as the density approaches saturation, and is
of the order of 1000 at about 90 per cent of the saturation density.
Hence a design requiring an operating density over 80 per cent that for
saturation may have a materially increased reluctance and consequent
loss of sensitivity.

As noted in the discussion of the pull relation, the computed value of
flux at which release occurs is given by ~/ 8xabsX. Since this is necessarily
a lower value of flux than that for operation, the density is lower and
the permeability higher than in operation, so that in the release case the
reed reluctance is minor, and the preceding equations can be used to
estimate the reluctance for the closed gap condition. Since flat metallic
pole faces in contact have an effective air gap xo of about 2 milli-inches
(0.005 em), ®&g for this case is given by xo/ab. Then the abampere-

turns for release are given by ®&&/ 42, where ® and ® are the values
applying in release. As previously stated, the actual release flux is
variable, and in general less than its computed value, and there is there-
fore a corresponding variation in release sensitivity.

The analysis of the two preceding sections is useful in providing an
understanding of the field distribution in the reeds and coil and of the
way in which the leakage field shunting the gap varies with the gap,
the flux density, and the reed and coil dimensions. It is, however, limited
to the case of an air return, and does not apply to a configuration such
as that shown in Fig. 8, where sleeve members are used to couple the
reeds to the return path provided by the cover. In such cases, the
reluctance can be approximately estimated in terms of the lumped con-
stants of the magnetic circuit shown in the figure. A similar treatment

As indicated in Fig. 8, the coil magnetomotive force is taken as de-
veloping an air flux g, , which only affects the inductance, and a reed
flux @, the maximum or total flux ® of the preceding discussion. The path
of @ comprises the reed reluctance R, in series with the sleeve gap re-
luctance 2Rs and the parallel combination of the gap reluctance ®,
and the shunting leakage reluctance ®,. The path, of course, includes
the shielding return members, but these are: negligible in reluctance
compared with the rest of the path.

The reed reluctance can be estimated from the reluctance per unit
length r = 1/ubh. The sleeve reluctance can be estimated by an ap-
proximation obtained from Fig. 9-8 of Peek and Wagar,’ from which

As shown in the air return case, the leakage reluctance ®, varies with
the gap reluctance, and cannot readily be expressed in a simple form.
An approximate expression for the parallel combination of ®,g and ®, ,
however, can be derived from the experimentally observed pull relation
of (3). The pull given by this relation must equal that given by the
general pull equation for a variable reluctance @&,

If Ry is the parallel combination of Rg and KR, , this expression for
F'/® can be equated to that given by (3), and the resulting expression
for d®y ‘dx integrated to give the increase in ®y from its value for « = 0.
This zero gap value may be taken as .x»/ab, with x 0.005 em, as in
the Table II estimates of ®, . Thus ®,y is given approximately by

where, as before, k is given by (4).
Then the total reluctance ® for the magnetic circuit of Fig. 8 is given by

with ®y given by (23) and @R» taken as 2Rs, as given by (22). This
expression can be used to estimate the reluctance for other relay con-
figurations, using expressions for @» corresponding to the return path
coupling and configuration. It may be applied as an approximation to
the air return path case, with ®, taken as the air return reluctance. For
negligible reed reluctance, this is given by (21) for ®, and hence Q
equal to zero, so that ®» is given by
Ss
p(2L. — Ly)

While these expressions for the reluctance are approximate, they are
simple in form, and are convenient to use, in conjunction with operate
and release flux estimates, in estimating operate and release sensitivity.

The operate time of magnetic reed contacts is that required for field
development and reed motion. These two occur together, since reed

motion starts as soon as the field and the resultant pull start to develop.

If the power input is high enough for the field to develop rapidly, the
reeds may be saturated before any significant motion has occurred. In
this case the times of field development ¢; and reed motion f. are additive
in determining the operate time. The value of f: for this case constitutes
a lower limit to the operate time, since no shorter motion time is possible
than that for full magnetization of the reeds. An expression for this
minimum motion time may be obtained by the following approximation.

The pull curve is given by (3), where @ has its maximum value, which
may be taken as about 90 per cent of saturation, as in the capability
estimates. This pull exceeds the just-operate pull shown in Fig. 3, and
hence always exceeds the stiffness load s(X¥ — x). The area between
the pull curve and the load line represents the kinetic energy supplied
to the reeds, which may be denoted 27, so that 7 is the kinetic energy
supplied to each of the equal reeds. Writing Fy , as before, for the closed
gap pull for maximum flux, the kinetic energy is given by

While the acceleration is variable, the motion time is nearly the same
as that for uniform acceleration, for which the kinetic energy 7 for each

27
where T is given by (26). To illustrate the magnitude of the quantities
involved for mating magnetic reeds, Table IIT gives the computed values
of t, for the two cases of Fig. 4 in which the minimum gaps are 4 and 10

gap is taken as having its maximum value X»2, and Fy as corresponding
to a density of 13,500 gauss, as before. As the reeds deflect as cantilever
beams, the effective mass m of each reed is taken as one quarter of its
actual mass, assuming a density of 8.2 grams/cm.° The values of C, or
k/a, are those applying for the dimensions given in Fig. 4 for these two
cases.

These estimates show that for the range of reed and gap dimensions
covered in Fig. 4, the motion times lie in a relatively narrow band. These
times are small compared with those of most other devices which open
and close contacts in a metallic circuit: they are large compared with
the switching times of many electronic devices.

To estimate the total operate time, an expression is required for the
time of flux development. As shown in Chapter 4 of Peek and Wagar,’
an adequate approximation can be obtained from the exponential rela-
tion for flux development with the constant reluctance (and hence
inductance ) for the initial open gap condition. For magnetic reeds, eddy
current effects are negligible except for flux development so fast that 4
is negligible compared with &. The flux and coil current develop to-
gether with a time constant L/R. The time ¢; at which the ampere-
turns equal the just operate value (VJ )o is given by

where v (NI)o/ (NJ) and L’ is the single-turn inductance, or 42, @,
where ® is the open gap reluctance for the average flux linked by the

coil. If both numerator and denominator in this expression are multi-
plied by J’, NJ is written as (NJ )o/v, and t: added to ¢; to give the total
operate time ¢, the resulting expression is

As shown in the text reference cited above, the function of v appearing
in brackets has a minimum value of 2.5 for v = 0.715, and departs from
this minimum, by less than 5 per cent for 0.6 < v < 0.8. Thus for mini-
mum operate time, (VJ) ) NJ should lie in this range, and (28) re-
duces to

where nL” is written for L’. If there are n sealed contacts (pairs of
mating reeds) used in a single relay coil, the single-turn inductance is
nL”, where L” is the single turn inductance of one sealed contact.

If the total time ¢ is materially larger than ¢, , the two terms of (29)
are not strictly additive, but the approximation has been found to give
adequate agreement with experimental observations.

In (28), the term L’(NJ),¢ is proportional to @(N/J)o (where ® is
the operate flux) and hence to the field energy required for operation.
Thus the electrical energy input for operation, which is proportional

to J*Rt, , is proportional to this field energy. As previously noted, (28)
also shows that f; varies with v, or /)/7, and that the time is a minimum
for a given power input if the coil circuit is designed to give Jo/J a
value of about 0.7. It also follows from (29) that increasing the steady-

state power reduces the operate time until it approaches the lower
limit tf , the minimum time for reed motion.

To use (29) for preliminary estimates requires estimation of the
operate ampere-turns (NJ), and the single-turn inductance nL”. (NJ)o
can be estimated by the procedures described in the sections on relue-
tance estimates. To estimate the single-turn inductance requires estimat-
ing the reluctance for the average field linked by the coil. In general,
this is a larger reluctance than that for the maximum reed flux, to which
the expressions given in preceding sections apply. The maximum flux is
limited by reed saturation, and hence controls the capability.

For a tightly coupled magnetic circuit, as in Fig. 8, the maximum
and average flux are nearly the same, and the same reluctance expres-
sions apply, except that some allowance may be made for the air field

an air path return relay, the average flux linked by the coil is materially
less than the maximum value. The analysis of this case given above can
be used to give expressions for the average flux, but these are complex
in form, and are not included here.

The agreement of experimental observations with (28) and (29) is
illustrated in Figs. 9 and 10. Fig. 9 shows direct measurements of operate
time plotted against the coil cireuit constant N°/R for various values
of steady state power input. The dotted curves are for constant values
of ampere-turns NJ. [(NJ)° = ['R-(N*/R).| The sealed contacts in this
relay had a value of (NJ)o of about 100 ampere-turns, for which the
minimum times should occur for a value of NJ of about 140, which is
near that for the observed minima.

The minimum times for this two-switch assembly are plotted against
n FR in Fig. 10, together with those for three other cases with one,
three and four switches. In agreement with (29) the plots are linear,
and have a common intercept &, the minimum motion time. These

times were measured to the end of the short chatter interval following
initial closure, so that the indicated value of ft includes this chatter
interval. The slopes of the lines measure the relative value of the single

to the considerable air core inductance in this coil, which had space for
four switches (sealed contacts). This results in a decrease in the ap-
parent value of L” as the number of switches increases.

The analysis given in this paper is intended for use in development
studies of sealed magnetic reed contacts. It can be used to estimate
the gap and reed dimensions needed to meet design objectives with

sensitivity. In development studies, models based on the initial estimates
can be made and measured, and the measurement results used in con-
junction with the analysis as a guide in further work.

In the performance of mating magnetic reeds, the major controlling
quantity is the flux between the reeds. This flux, which determines the
pull, is limited by saturation and hence by the reed cross section. This
total reed flux comprises the gap flux proper and the leakage flux shunt-
ing the gap. As shown in the general analysis of the air return reluctance,
this leakage field varies with the gap, reed and coil dimensions, and the
relation between the pull and the total reed flux has a corresponding
variation. Hence the simple pull equation derived from experiment ap-
plies rigorously only for the experimental coil length and return path
conditions used in deriving it, and minor deviations oceur in applying
it to other cases.

Corresponding deviations occur for the reluctance given by the ex-
pression derived from the pull equation, and for the other approximate
reluctance equations. All these approximations, however, are adequate
for preliminary estimates, and may be applied with greater accuracy in
detailed studies by using values of the constants that are experimentally
determined for the case in question. While the general analysis can be
used to estimate the air return reluctance, and the treatment can be
extended to give expressions for the pull and the inductance, it is limited
to the air return case, and is too complex for convenient application.
It serves, however, to show the character and extent of the deviations
from the simpler approximations resulting from the distributed char-
acter of the leakage field.

As the times of flux development and reed motion depend respectively
on the inductance corresponding to the coil flux linkages and on the
pull for maximum flux, timing estimates based on the simple approxima-
tions given here are also subject to some minor variation. This can be
reduced in development studies by using experimentally determined
values of the constants.

size, but are approximately applicable to unequal reeds provided both
are long compared with the gap overlap. In such cases the effective
stiffness of the reed combination must be taken as the reciprocal of the
sum of the compliances of the two reeds.

J. Jones, using apparatus designed and constructed by J. L. Agterberg.
The author is also indebted to W. B. Ellwood, O. M. Hovgaard and
other associates for information used in preparing this article.

. Ellwood, W. B., U.S. Patent 2,289,830.
Ellwood, W. B., Glass-Enclosed Reed Relay, Elect. Engg., 66, 1947, p. 1104.
3. Brown, J. T. L. and Pollard, C. E., Mercury Contact Relays, Elect. Engg., 66,
1947, p. 1106.
Hovgaard, O. M., Capability of Sealed Relay Contacts, A.I.E.E. Trans., Pt.
I, 75, 1956, p. 466
Hovgaard, O. M. and Perreault, G. E., Development of Reed Switches and
Relays, B.S.T.J., 34, 1955, p. 309.
Hovgaard, O. M. and Fontana, W. J., A Versatile Miniature Switching Capsule,
Proc. 1959 Electronic Components Conf., p. 32.
Peek, R. L., Jr. and Wagar, H. N., Switching Relay Design, D. Van Nostrand
Co., New York, 1955.
. Evershed, 8., Permanent Magnets in Theory and Practice, J.I.E.E., 68, 1919,
p. 780.

The problem of protecting apparatus against lightning surges from con-
nected transmission facilities has become more complex with the use of
solid state devices in apparatus design. Consideration of the protection re-
quirements for such apparatus has indicated that existing information con-
cerning the incidence and characteristics of lightning surges is insufficient
to develop optimum protection measures. A recently completed field investi-
gation provides additional information in this specific area.

The results of this field investigation and supplemental laboratory surge
tests indicate that, in well-shielded underground cable pairs, electrical
surges do not exceed approximately 90 volts peak, and that transistorized
apparatus capable of withstanding such surge amplitudes needs no further
protection. In aerial and buried cable, however, transistorized apparatus
requires protection up to the full sparkover potential of 3-mil protector gaps,
t.e., to about 600 volts peak. A firm basis for testing and evaluating tran-
sistorized apparatus from the lightning surge voltage standpoint is presented.

The 3-mil air gap carbon block protector, which has a maximum spark-
over value of 600 peak volts, is the basic protection device employed by
the Bell System for the protection of communication apparatus against
extraneous potentials. Prior to the introduction of transistors and mini-
aturized circuitry, it was the general practice in apparatus design to
provide a withstand level for both metallic* and longitudinal? potentials
greater than 600 peak volts so as to coordinate directly with 3-mil
protector gaps. This customary design objective of providing an in-
herent withstand capability exceeding the operating value of standard
protector gaps is not always feasible in the case of apparatus employing

solid state devices. Furthermore, lower voltage protection cannot be
attained satisfactorily by simply reducing protector-gap spacing below
3 mils, since excessive protector maintenance would be introduced. To
meet the lower voltage requirements of apparatus employing transistors,
therefore, it is necessary at present either to modify the circuitry so
that surge currents appearing in the more susceptible components are
limited in magnitude, or to introduce an additional stage of protection
employing semiconductor diodes supplementing the gaps. These pro-
tection measures may introduce significant additional expense and, in
some cases, produce adverse effects on transmission characteristics.

It became apparent that selection of optimum protection measures
to meet the exacting requirements of transistorized apparatus neces-
sitated a more complete knowledge of the incidence and characteristics
of lightning surges in the range below the operating level of standard
protector gaps. Recognizing this, a field investigation was undertaken
to supplement existing information in this area. The results of this
recently completed field study and conclusions drawn from analysis of
the data are presented in this article.

Since it appeared, at the time the investigation was undertaken, that
the present practice of employing 3-mil protector gaps as basic apparatus
protection would continue into the foreseeable future, all circuits used
for purposes of observation were equipped with such protectors. The
area of study therefore was intentionally restricted to surges up to about
600 peak volts as limited by protector operation.

Observations of lightning surges appearing in trunk pairs in aerial
and buried cable were recorded by means of automatic cathode ray
oscillographs. The plant locations selected were in areas known to ex-
perience heavy thunderstorm activity. Surges were also monitored by
means of peak amplitude recording devices in urban underground cables.
Because of the shielding provided by buildings and buried piping facili-
ties, the exposure of cables to lightning in this situation was relatively
low.

Information of engineering value secured includes the probability
distribution of voltage magnitudes and the rise and decay time char-
acteristics of surges in the lower voltage range specifically under study.
Such data have been used as a basis for selecting waveshapes suitable
for laboratory testing of the energy and power handling capabilities of
transistorized apparatus.

Field data on lightning surge characteristics were obtained from types

1. Low exposure, typified by underground plant in well-shielded
urban areas.

2. High exposure, typified by aerial and buried cables in suburban
and rural areas.

The study of surge activity in underground cable was conducted on
spare trunk pairs in Baltimore, Maryland; Pontiac, Michigan; and South
Orange, New Jersey. Table I gives a brief description of the route and
make-up of these facilities.

The waveshapes of lightning surges in aerial and buried cable were
studied with cathode-ray oscillographs arranged to monitor continuously
the pair selected for observation and to record automatically on 16-mm

measurements were made of open-circuit longitudinal surge voltages and
any resultant metallic voltages appearing across a representative resis-
tive termination. Spare H88-loaded trunk pairs in aerial cable were

TABLE I DESCRIPTION OF UNDERGROUND TRUNKS MONITORED
WITH SURGE COUNTERS

South Orange to West Orange, 100-pair underground cable with
N. J. aerial complements
South Orange to Summit, N. J. 155-pair 100°) underground cable
with H8S8 loading

Pontiae to Birmingham, Mich. Underground cable with aerial
complements and FI88 loading

Baltimore to Towson, Md. 900-pair underground cable with
aerial complements and H88&
loading

TEST
‘ LOCATION ™ 5 MILES > peak eed
(a) CENTRAL ENTRA
OFFICE 22 GAGE, H88 LOADED, LEAD-SHEATHED AERIAL CABLE
75 TRUNK PAIRS

TEST
UNDERGROUND | LOCATION
, fo "e: pain CENTRAL.
51 PAIRS
2 MILES OFFICE
FAR END w«« 10.5 MILES > y
CENTRAL
OFFICE 100 PAIR AERIAL PIC-ALPETH CABLE
75% OF CABLE iS OVERBUILT BY ONE
CROSSARM OF OPEN WIRE ]
5) PAIRS
ATLANTA

CENTRAL
OFFICE 19 GAGE, JUTE-PROTECTED, LEAD-SHEATHED,
BURIED QUADDED CABLE
(c) 27 QUADS

Fig. 1 Characteristics of test cable at (a) Mt. Freedom, N. J., (b) Buford,
Ga., (¢) Griffin, Ga.; (d) arrangement of measuring equipment at test locations

studied in Buford, Georgia, and Mt. Freedom, New Jersey. The buried
cable studies were conducted in Griffin, Georgia, on a spare nonloaded

Supplemental laboratory surge tests were also conducted on a one-
mile test cable to augment the information on the behavior of under-
ground cables with aerial complements and extensions.

The incidence and characteristics of the lightning surges recorded and
the resulting protection. considerations will be discussed in the order of
the degree of plant exposure.

The field study of surge activity in well-shielded urban underground
plant, covering the first two categories, revealed that in no case did
voltages attain the 90-volt triggering value of the lowest stage of the
counters. During the five-month observation period, a total of 44 thun-
derstorm days was reported by the U.S. Weather Bureau for these
areas. Several of these storms were known to be quite severe, with
their centers located over the test cables. The counters were tested
periodically during the study period to ensure proper operation. The
lack of surge activity recorded during this study reveals the shielding
benefits enjoyed by urban underground plant. In such areas, buildings
and buried metallic piping systems divert and dissipate lightning strokes
that otherwise might directly involve the telephone cables. Furthermore,
duct runs usually contain two or more cables, the sheaths of which are
bonded at each manhole. Surge current will, therefore, divide between
the cables, and the voltage induced in any one cable will be propor-
tionately reduced.

On the basis of these field studies, it appears that apparatus capable
of withstanding surges in the order of 90 to 100 volts peak will not
require lightning protection when associated with all-underground cable
pairs (trunk or loops), whether such pairs are in an all-underground
cable or one with well shielded aerial complements. The significant point

is that this conclusion holds only for well-shielded plant, the shielding
being provided by closely spaced buildings, extensive power distribution
and buried metallic piping systems, and other telephone cables in the
same conduit run.

The question now arises as to the magnitude of surges that may ap-
pear in a 100 per cent underground trunk complement through coupling
with underground pairs extended aerially in cable having greater ex-

Information on the following specific points relative to such surge
coupling is useful in estimating the secondary exposure likely to be
experienced by apparatus connected to the 100 per cent underground
trunk pairs:

1. The ratio between an open-circuit longitudinal voltage surge on a
disturbing circuit (underground pair associated with an aerial ex-
tension) and the resultant voltage appearing in a disturbed circuit (100
per cent underground trunk pair).

2. The resultant magnitude and waveshape of longitudinal current
appearing in the disturbed circuit.

To secure this information, supplemental laboratory surge tests were
conducted on a 50-pair, one-mile test cable. Longitudinal impulses
(approximately 50 by 250 microseconds*) were applied to one or more
cable pairs acting as the disturbing circuit. Several cases were investi-
gated: energizing a single pair, then 5 pairs in parallel and finally 25
pairs in parallel. Measurements were made of longitudinal open-circuit
voltage and short-circuit current in the disturbing circuit. Measurements
were then made of longitudinal open-circuit voltage and short-circuit
current in a disturbed pair. Pairs both adjacent and remote from the
disturbing circuit were investigated.

The ratio of longitudinal open-circuit voltage appearing in a disturbed
pair to the longitudinal open-circuit voltage in the disturbing circuit
varied, depending on test conditions, from about 0.47 to 0.85. The lower
value was obtained when only a single pair was energized and the larger
value when the surge was applied to 25 pairs paralleled at the generator
end. Although the magnitude of the open-circuit surge voltage appear-
ing in the disturbed circuit was significantly lower than that in the
disturbing circuit, the waveshapes of the two were essentially the same.
In the cases investigated it was found that the longitudinal short-circuit
current in the disturbed pair was approximately 3 milliamperes per volt
appearing in the disturbing circuit. The short-circuit current in the

* That is, 50 microseconds rise time to crest and 250 microseconds from origin
to point where wave has decayed to half of crest value

disturbed pair assumed the shape of a square pulse about 20 micro-
seconds in duration. From a protection standpoint, this radical reduction
in the duration of the induced current wave in the disturbed circuit is
probably the most significant bit of information secured in these labora-

surges, usually fail as a result of overheating of their junction or junc-
tions — the heating effect being related to the magnitude and duration
of the junction current. Therefore, the possibility of failure from longi-
tudinal current of semiconductor components in apparatus associated
with a disturbed pair is much reduced because of the relatively short
duration of the coupled surge.

It is of further interest to note the effect of grounding the disturbing
circuit at the far end, such as would occur with protector operation.
This condition was investigated by grounding one of the conductors
constituting the disturbing circuit, and it was found that the open-
circuit longitudinal voltage on the disturbed pair was reduced approxi-
mately 50 per cent and the short circuit longitudinal current 30 per cent
below the values that would obtain if the conductor of the disturbing
circuit had not been grounded. This indicates the order of beneficial
shielding enjoyed by the disturbed circuit when protector blocks operate
on the disturbing circuit.

Additional laboratory surge studies were made employing a one-mile
test cable to determine the protection requirements for apparatus con-
nected to underground pairs extended aerially. These tests revealed
that surges having typical waveshapes will propagate longitudinally on
a cable pair for one mile with little attenuation. Therefore, underground
cable pairs extended aerially or in buried plant should be considered as
exposed to lightning, unless protection is applied at the underground
junction to limit surges from the exposed extensions.

The lightning exposure of buried and aerial cable is sufficiently severe
te require supplementary protection for transistorized apparatus as-
sociated with this type of plant. The most useful way of defining pro-
tection requirements in this situation is in terms of a simulated lightning
surge which such apparatus must withstand in laboratory tests. Simu-
lated surges, of course, must be based on surge conditions in the field.
Consequently, derivation of suitable test surges requires knowledge of
the distribution of waveshapes and peak voltages of lightning surges
appearing in the plant.

The data on longitudinal surges in aerial cable at Buford covered a
period of six months, during which time six thunderstorm days occurred
and 103 oscillograms were obtained. Additional data on aerial cable
were secured at Mt. Freedom, with 105 oscillograms being recorded
during three thunderstorm days. The buried cable studies conducted at
Griffin covered a period of six months, during which 36 thunderstorm
days occurred and 1120 oscillograms of longitudinal surges were ob-
tained. Since these cables involved very little unexposed plant, the
surge magnitudes and waveshapes recorded at the central offices should
also be reasonably representative of surge conditions along the cable
route.

Comparison of the recorded waveshapes of longitudinal surges in-
duced in aerial and buried cables indicates that the two types of plant
do not differ significantly in their response to lightning surges. The data
also confirm that load coils have little or no effect on the waveshape of
longitudinal surges. This observation is based on the similarity of the
surges recorded on H88-loaded aerial cable and those recorded on non-
loaded buried cable. These surges were found to be essentially impulses,
as exemplified in Fig. 2. Longitudinal impulses can conveniently be
characterized on the conventional basis of crest magnitude, time to crest
and time from origin to the point at which the wave has decayed to
one-half of crest value.

Both the rise time to crest and decay time to half-crest value of
lightning surges observed in cable exhibited log-normal distributions.

follow an exponential distribution similar to the lightning stroke cur-
rents that produce these voltages.

Distribution of surge rise times for the three test locations are pre-
sented in Fig. 3. The median rise time measured at each location was
approximately 100 microseconds, but the dispersion about the median
value varied widely among locations, probably due to the difference in
cable lengths involved. As a surge propagates along a cable pair, there
is some modification in waveshape. Consequently, slightly greater dis-
persion should be expected in longer cables. This is borne out by the
measurements plotted in Fig. 3.

The peak voltages recorded in aerial and buried cable in the range
below protector block operation (voltages less than 400 volts) were
found to be exponentially distributed. This distribution provides a basis
for determining the probability of any given surge exceeding a particular
value. The derivations of these probability functions for aerial and

buried cable are given in the Appendix. These probability functions,
used in conjunction with the average number of surges induced in the
test cables per thunderstorm day, provide an order-of-magnitude esti-
mate of the number of surges per thunderstorm day exceeding any given
amplitude. The similarity between the peak voltage distributions for
aerial and buried cable makes it feasible to develop a single plot of the
estimated number of surges per thunderstorm day as a function of
voltage.

Fig. 4 presents both the distribution that would be expected if no
protector blocks were associated with the test pair and the modified
distribution reflecting the operating characteristics of standard 3-mil
air gap carbon protectors. This information is useful in the design of
reliability tests for apparatus vulnerable to repeated low-amplitude
voltage surges, and is used in the selection of suitable test surges, as
discussed later.

B S |
MT. FREEDOM oe OA \ GRIFFIN
S5-MILE AERIAL CABLE } 8-MILE BURIED CABLE
3-SIGMA LIMITS ~._ |__| 3-SIGMA LIMITS
70 {£LSEC ;2900 KLSEC ‘ : = 69 SEC ;2300 “SEC

The distributions of decay times of lightning surges appearing in
cable plant are presented in Fig. 5. The median values and dispersions
differed slightly among the three test locations. Median values ranged
from 350 to 450 microseconds. Three-sigma limits ranged between 40
to 70 microseconds on the low side and between 2300 and 2900 micro-
seconds on the high side. The variation of median values is possibly due
to differences in the relative exposure of the test cables.

In Buford and Griffin, the cables traverse relatively level terrain and,
therefore, should have fairly uniform exposure along their lengths. Close
correlation is noted between the median decay time values for these
cables. In Mt. Freedom, however, the cable is placed on hilly terrain,
which results in a higher exposure for some sections of the cable. With
the majority of the surges being induced into one section of the cable, a
shift in the median value results.

Metallic surges were recorded on the same trunk cable pairs used at
Buford and Griffin to record longitudinal surges. Measurements were
made across a 940-ohm resistive termination. Simultaneous operation

of the automatic recording oscillographs observing longitudinal and
metallic voltages was arranged in order to permit correlation between
each longitudinal surge and any resultant metallic surge.

In the absence of conductor insulation breakdown or protector opera-
tion, the circuit balance of cable pairs is sufficiently good that metallic
surge voltages should be much lower than longitudinal surge levels,
which is confirmed by the measurements secured in this study. Of all
longitudinal surges exceeding 60 volts* only about 10 per cent produced
metallic voltages exceeding 10 volts peak. However, when protector
operation occurred, metallic surge potentials of significant magnitudes
were produced. An approximate breakdown of those metallic surges
which exceeded 10 volts peak is presented below on the basis of wave-
shape and magnitude:

1. Twenty per cent were low-amplitude, high-frequency oscillations
having maximum peaks of about 35 volts and frequency components
ranging from 10 ke to approximately 50 ke.

2. Twenty-five per cent were impulses ranging in amplitude up to
about 60 volts peak, which were probably caused by protector operation
on adjacent pairs.

3. Fifteen per cent ranged in amplitude from 120 to 200 volts peak.
The oscillograms of the associated longitudinal surges definitely indi-
cated protector operation on the test pair although the surge amplitudes

were considerably lower than the level normally required for protector

block operation. The protector blocks on the test pair were changed peri-
odically during the study period, but these low-voltage operations (16 in
all) occurred with the same set of blocks. However, visual inspection of
these protector blocks by local personnel did not reveal anything un-
usual in their appearance.

4. Forty per cent of the metallic surges exceeded 350 volts peak.
These surges were all associated with protector operation on the test
pair.

Examination of the oscillograms of these high-amplitude metallic
surges and the corresponding longitudinal surges illustrates some interest-
ing aspects of protector block operation. Fig. 6(a) shows one type of
single-block operation where the discharge is continuous for the duration
of the surge. Fig. 6(b) illustrates another type of single-block operation
in which clearing and restriking of the are discharge occurs. This situa-
tion is the result of circuit regulation when the longitudinal surge po-
tential is just sufficient to initiate gap sparkover.

* Threshold value of automatic recording oscillographs measuring longitudinal
surges

Fig. 6 Lightning impulse voltages on a nonloaded buried cable pair: (a)
with protector operation on one conductor; (b) with multiple protector block
operation on one conductor.

A brief explanation of surge voltage relationships during single-block
operation as indicated by the oscillograms follows:

At point a [Fig. 6(a)], the gap associated with one conductor operated
and remained operated for the duration of the surge. The longitudinal
voltage dropped to about one-half of peak value as the oscilloscope is

560 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1961
connected to the midpoint of a 940-ohm termination (see Fig. 1).*
The metallic surge shows the true voltage unbalance, which is the dif-
ference between the are discharge voltage on one conductor (approxi-
mately 50 volts) and the longitudinal surge voltage on the other con-
ductor.

At the first point a [Fig. 6(b)], the gap associated with one conductor
operated and remained operated for approximately 350 microseconds.
The metallic surge began at point A and continued for the same dura-
tion. At the second point a the protector gap ‘‘cleared” and then operated
again. When it cleared, the longitudinal scope recorded the true longi-
tudinal voltage* and the metallic voltage dropped to zero. When the
block restruck, the longitudinal scope again read one-half true longi-
tudinal voltage and once more there was a metallic voltage.

Metallic surges resulting from operation of both blocks on a pair are
shown in Figs. 7(a) and 7(b). In the first case, one block operated
initially, then cleared, and then the other block operated. This sequence
of operation produced a metallic surge having impulses of both polarities.
In the second case, both blocks operated, but unbalances were produced
by nonsimultaneous clearing and restriking of the two gaps.

Further explanation of surge voltage relationships resulting from a
longitudinal surge of sufficient magnitude to operate both blocks is as
follows:

At the first point a [Fig. 7(a)], the protector on one conductor op-
erated, in this case, the one on the “tip” conductor. The recorded
longitudinal voltage then dropped to one-half of the actual value and
the metallic surge began. At the second point a, the protector on the
“ring” conductor operated and the longitudinal surge voltage dropped
to essentially zero. At the third point a, the protector on the “tip”
conductor cleared. This is evidenced by the reappearance of metallic
voltage of reversed polarity.

Fig. 7(b) illustrates a phenomenon that commonly occurs during the
operation of protector gaps on telephone circuits. The discharge is not a
continuous process but is punctuated by a random restriking of the are
discharge. In this case there was sufficient longitudinal potential to
operate both blocks, but apparently some differences in electrode con-
ditions caused nonsimultaneous clearing and restriking of the are dis-
charge which produced metallic potentials. It is only during the brief

* In effect, the longitudinal scope always reads the true open circuit voltage
until one block operates; then it reads one-half the true longitudinal voltage. The
metallic oscilloscope always reads the total voltage difference between the two
conductors of the pairs.

Fig. 7 Lightning impulse voltages on a nonloaded buried cable pair: (a) with
protector block operation first on one conductor, then on the other; (b) with
multiple striking and restriking of protector blocks.

periods when the two gaps are either both discharging or not discharging
that metallic potentials are reduced to negligible values.
IV. SELECTION OF SUITABLE WAVEFORMS FOR LABORATORY TESTING

The similarity of longitudinal waveforms observed in buried and aerial
plant under loaded and nonloaded conditions makes it practicable to

employ common waveforms for laboratory testing of apparatus intended

for use with all of these types of plant. In the past, a 10 by 600-micro-
second surge was selected on the basis of limited field data. The supple-
mental surge data obtained in this study makes it possible to select
waveshapes for laboratory testing which more closely simulate surges
produced in the telephone plant by natural lightning.

Analysis of the recorded data obtained during this study provided
the distributions of rise times, peak voltages and decay times presented
in Figs. 3, 4 and 5. Using these distributions as a basis, it is possible to
select suitable laboratory test surges.

Since the severity of a surge is dependent on its peak voltage and its
decay time, it is necessary to establish whether decay time is independent
of peak voltage before computing the joint probability of occurrence of
a surge with a given amplitude and decay time. Accordingly, new decay
time distributions were developed from the recorded data for two voltage
ranges; voltages below 225 volts and voltages above 225 volts. Fig. 8
presents the results of this analysis. It will be observed that there is a
slight correlation between the decay time and voltage. This is the result
of the manner in which the surge current in the sheath induces voltage
onto the cable pairs. The capacitive coupling between the sheath and
the core results in an integration of the sheath current. Thus, surges of
longer duration will tend to produce higher voltages on the cable pairs.
In determining the joint probability of exceeding a given amplitude and
a given decay time, this correlation, as indicated by the field data,
should be included by using the probability distribution of decay times
associated with higher voltages (upper curve in Fig. 8).

The parameters of laboratory test surges should be so selected as to
evaluate apparatus properly for its dielectric strength and its energy
and power handling capabilities. The effect of these factors on the
parameters of test surges is discussed below.

Surges suitable for test purposes should have a peak amplitude of at
least 600 volts to provide a minimum test of apparatus dielectric strength,
since 3-mil protector gaps associated with apparatus assure protection
only against surges in excess of this value. Furthermore, when the

more detrimental than energy content and, given two surges of equal
content, more power will be delivered by the surge with the higher
amplitude.

The energy content of a surge is dependent on its peak voltage and
waveshape (i.e., rise time and decay time). Test surges should have
short rise times for two reasons. A short rise time will provide a more
severe test of inductive circuit elements and, for a given decay time, the
shorter the rise the higher will be the total energy of the surge. Ac-

mately the lower 3-sigma limit of the rise times recorded in the field.
In those instances where the energy-handling capability of apparatus
is the controlling factor, the reliability of a surge testing program will
depend on the degree of assurance that the energy content of the test
surge will not be exceeded in the field. The total energy of an impulse
with a short rise time is proportional to the decay time and the square
of the voltage. The energy content of any arbitrarily assumed test surge
can be exceeded in the field in two ways: surges of lower amplitudes but
appreciably longer decay times, or surges of higher amplitude and only
somewhat shorter decay times. This second classification can be elimi-
nated, however, by selecting test surges having peak amplitudes of 600
volts or greater, as the standard 3-mil air gap protectors associated with
equipment will not permit surges in excess of 600 peak volts. To deter-

mine the probability of exceeding the energy of a particular test surge,
the joint probability of two factors must be calculated: (a) the prob-
ability of obtaining a particular voltage and (b) the probability that a
surge of this voltage has a sufficiently long decay time such that its
total energy exceeds the energy content of the test surge. These joint
probabilities must be summed for all surges with realizable combinations
of voltages and decay times which exceed the energy of the test surge.

Mathematically the procedure is as follows: The energy content of a
lightning impulse* is:

Assuming a test surge of amplitude V» and decay time dy, the prob-
ability [P|] that a surge of somewhat lower amplitude V, will have

Impulse = rapid rise to crest implitude followed by a slow exponential fall

having greater energy content than the test surge requires the summa-
tion of [P] for all possible values of V. These probabilities need be
summed for voltages only above 400 volts, since lower voltages would re-
quire associated decay times far longer than observed or expected in
cable plant. This follows for two reasons: In order for a small amplitude
surge to have an energy content equal to that of the test surge, its decay
time must increase as the square of the voltage ratio of these surges.
This would require existence of decay times much longer than 2,000
microseconds, a condition generally contrary to all test observations,
and contrary to the observed correlation between peak voltages and
decay times which indicated that lower voltage surges have smaller
decay times.

Therefore, the total probability (27) of exceeding the energy of a
particular test surge is:

[P(590 — 600)|[P(d = d,)] A,
( Pr) 2
The probability of obtaining a voltage in the range Av may be deter-
mined from Fig. 4. For example:

The probability of obtaining a decay time greater than d; may be
read directly from the upper curve of Fig. 8.

This type of calculation must be repeated for each test surge desired.
In this study, test surges with peak amplitudes of 600, 700, 800 volts
and a number of different decay times were examined. By providing
several waveshapes with equivalent energy content: (a) a better in-
dication may be obtained as to the adequacy of the thermal time con-
stants of vulnerable apparatus components, (b) more latitude is provided
for determining the dielectric strength margin of apparatus, and (c)
flexibility is provided in the selection of laboratory surge forming circuit
constants. The above calculation indicates the probability, per thunder-
storm, of exceeding the energy content of a particular test surge. For
engineering purposes, however, it is more practical to develop curves
of apparatus energy-handling capabilities versus trouble expectancy in
years rather than in thunderstorm days. Reference to isoceraunic charts
indicates thunderstorm incidences varying from 5 to 90 thunderstorms
per year for various sections of the country, with the higher incidences
in the Southeast. However, even when a thunderstorm is reported in
the general area of a particular cable it is not necessarily close enough
to induce surges having magnitudes of the order under discussion. It is
felt that an average of only 25 thunderstorm days per year in the higher
storm incidence areas are likely to produce significant surges in the cable
plant. This factor, together with the calculated probability per thunder-
storm of exceeding various energy levels, provides the basis for the curves
presented in Fig. 9. These curves indicate the probable surge trouble
rate of apparatus tested with surges having parameters that will just
cause failure.

The curves in Fig. 9 provide a means of determining the probable
lightning surge-handling capabilities of apparatus in two ways. [irst,
where the acceptable lightning trouble rate has been established by
system requirements, the parameters of the appropriate test surge may
be read from the curves. A second approach may be employed in the
case where it is desired to determine the probable trouble rate with re-
gard to a specific piece of apparatus. The procedure would be to estab-
lish, by tests, the withstand level of the apparatus in question by em-

the withstand point, the corresponding estimated trouble rate may be
read from the curves.

The waveshapes presented in Fig. 9 will concurrently test apparatus
for its dielectric strength, its energy-handling capabilities and its ability

Fig. 9 — Relationship of probable trouble rates to test surges having param
eters that will just cause equipment failure

basis of longitudinal surge data, it is felt that they will provide a reason-
able test of the surge-handling capabilities of apparatus, both longi-
tudinally and metallically. Since the total energy contained in a metallic
surge must be somewhat less than the total energy of the associated
longitudinal surge, the recommended test surges, when applied metal-
lically, will provide an added safety factor. However, this additional

safety factor cannot be specifically evaluated from the limited metallic

The main detrimental effect of lightning surges on semiconductor
devices is excessive heating of their junction or junctions, the heating
effect being related to the energy content of the surge. Since semicon-
ductor devices are used in circuits having different frequency response
ranges, it is desirable to determine the energy versus frequency distribu-
tion of lightning surges.

Analysis of the energy versus frequency distribution was performed
on a 25 by 160-microsecond surge, as it represented one of the shortest
duration surges observed on cable plant in this study, and therefore con-

cent of the total energy of this wave is contained in the frequency band
up to 3,500 cycles per second. For comparison with a longer duration
surge, a similar analysis was performed on a 10 by 1000-microsecond
surge, which indicated that 90 per cent of the energy of this surge is
contained in the frequency band up to 660 cycles per second. Plots of
the cumulative per cent energy as a function of frequency for both of
these surges are given in ig. 10.

In view of the limited frequency spectrum of the energy content of
longitudinal surges, it is desirable to determine the energy versus fre-
quency distribution of those metallic surges which are oscillatory. The
oscillograms of metallic surges showed the shortest time interval be-

Fig. 10 — Cumulative distribution of energy vs. frequency for two sample surges.

tween polarity reversals to be 80 microseconds. An energy-frequency
spectrum analysis for such a wave indicates that 90 per cent of the en-
ergy is contained in frequencies below 7 ke. It therefore appears that
most of the energy of metallic surges appearing across a resistive termina-
tion is likely to be in the frequency band below 7 ke. This does not, how-
ever, preclude the need for lightning protection of apparatus operating
at carrier frequencies. If reactive components are present in the metallic

termination, the resulting metallic surges may have considerable energy

Lightning surges were recorded in trunk pairs in aerial and buried
cable at several locations known to experience heavy thunderstorm ac-
tivity. Surges were also monitored in trunks in well-shielded under-
ground cables in urban areas. Observations included measurement of
longitudinal surge voltages (from conductor to sheath or ground) and
metallic surge voltages (between conductors of a pair). Supplemental
laboratory measurements of surge characteristics in simulated under-
ground plant were made using test cable.

In the underground cables monitored, no surges appeared in excess
of 90 volts (the minimum sensitivity of measuring equipment ) although
a total of 44 thunderstorm days occurred during the observation period
at the three test locations. Based on this field experience and supple-
mental measurements made on test cable, it was concluded that appara-
tus capable of withstanding surges up to 90 volts peak should not re-
quire lightning protection if connected to well-shielded all-underground
cable pairs. This category includes underground trunks in cables with
well-shielded, aerial subscriber complements such as block cable. How-
ever, underground pairs extended aerially, or in buried plant, should be
considered as exposed to lightning, unless protection is applied at the
underground junction to suppress surges from the exposed extensions.

In the aerial and buried cable plant studied, about 1400 surges were
recorded, ranging in peak amplitude from 60 volts (minimum sensitivity
of equipment) up to 450 volts, the value at which carbon block pro-
tectors operated. About 90 per cent of the recorded surges were longi-
tudinal; the remainder were metallic. Analysis of this data helped es-
tablish the relationship between the parameters of specific test surges
and the probable lightning trouble rate of exposed transistorized appa-
ratus. This information will facilitate appropriate laboratory testing of
apparatus for specific levels of reliability.

With the recording technique used in this study, only those surges
in the range above 60 volts and less than the operating value of carbon
block protectors were registered. The 3-mil gap, carbon protector blocks

nominal value of 500 volts. Due to manufacturing and field duty varia-
tions, the operating value of these carbon blocks may vary from approxi-
mately 400 to 600 volts. Protector block operation, therefore, affected
the distribution of peak voltages above 400 volts. The recorded data
established the distribution of surge voltages between the limits of 60
volts and approximately 400 volts. This distribution was extended to
600 volts by considering the effects of block operation in the 400 to 600
volt range as discussed later.

In the aerial cable plant at Buford, Georgia, a total of 103 surges was
recorded during the six-month study period. In the buried cable plant
at Griffin, Georgia, a total of 1120 surges was recorded for the same
period of time. Histograms of the distribution of these surges as a fune-
tion of voltage are presented in Figs. 11 and 12 for the two types of

Fig. 12 — Amplitude distribution on buried cable.
plant. These distributions are of an exponential form and can be closely
approximated by

Smoothing of the raw data (fitting of an exponential by the method
of least squares) resulted in the plots on Fig. 13. The constants for the
exponential distribution on aerial cable are a = 0.012 and ¢ = 40. The
constants for the buried cable are a = 0.0094 and ¢ = 330.

The large difference in the calculated values of ¢ for the two types of
plant merely reflect the difference in the sample sizes and will not affect
their probability distributions. It will be noted that the values caleu-
lated for the constant term a for the two types of plant varies over 20
per cent, but it is recommended that the smaller value, calculated for
buried cable, be used for both types of cable installation for two reasons.

which permitted a more accurate determination of the shape of the ex-
ponential distribution. Also, since our objective is to provide suitable
surge voltages for laboratory testing of apparatus, it is safer to use the

distribution with the smaller value of a which predicts a greater per-
centage of higher voltage surges.

Having determined the distribution of surge voltages, it is possible to
estimate the probability that V is less than or equal to some value 7’:

Using the previously established value of a, the probability distribu-
tion of surge voltages was then plotted (Fig. 14). This, however, gives

the distribution of all surges to the cable plant, while the voltages re-
corded in the field were truncated, with only values above 60 volts be-
ing recorded. The desired distribution is, therefore, the conditional
probability that a voltage peak exceeds T volts given that it is greater
than 60 volts. This may be accomplished by shifting the distribution
function to the left until the probability of exceeding 60 volts is equal
to one.

This expression provides the probability distribution presented in
Vig. 15.
To estimate the number of surges per thunderstorm day which exceed

PROBABILITY (PER CENT) THAT SURGES EXCEEDING 60 VOLTS
WILL ALSO EXCEED ABSCISSA

Fig. 15 — Probability distribution of all surges exceeding 60 volts in buried
and aerial cable

ber of surges induced in a cable per thunderstorm day. An estimated
value was computed by counting the total number of recorded surges
over the entire study period and dividing by the number of thunder-
storm days occurring in the area. Very good correlation was established
between observed thunderstorm days (on film) and those reported by
local weather stations.

A total of 36 thunderstorm days was recorded in the vicinity of the
buried cable at Griffin and six thunderstorm days were recorded in the
vicinity of the aerial cable at Buford. These 42 thunderstorm days re-
corded for the two test locations accounted for a total of 1220 surges,
for an average of 29 surges per thunderstorm day.

The actual number of surges induced in a cable, however, is approxi-
mately 2.5 times this value or 73 surges per storm. This results from the
fact that multiple surges occur in approximately half of all lightning
strokes, and that the average number of surges in a multiple discharge

second.” The sweep speed of the test oscillographs (full scale deflection

Fig. 17 — Sparkover distribution characteristics of 3-mil carbon block protectors

A General Method of Applying
Error Correction to Synchronous
Digital Systems

A general method is presented for applying error correction to synchronous
binary digital systems to improve reliability. It includes the familiar scheme
of triplication and “‘vote taking’’ as a special case. In principle, the method
permits the system to operate continuously, even when a fault is present or
maintenance is being performed. An efficient maintenance routine, including
rapid repair of faults, is an essential adiunct to the scheme if the potentially
large increase in reliability made possible by error correction is to be realized.

The percentage redundancy needed to realize the scheme decreases as the
complexity of the system to which it is applied increases, but may amount
to triplication of equipment even for moderately large systems. The paper
describes some error-correcting codes to implement the scheme, discusses
error-correcting circuits in a general way, indicates how to estimate the
redundancy, and presents a formula for determining the reliability improve-
ment obtainable with a particular maintenance routine. In a companion

This paper describes a general method of applying error correction to
svnchronous digital data systems. It includes, as a special case, the well-
known scheme of triplication with vote taking.? Since the scheme employs
error-correcting codes, it is capable of detecting errors as well as correct-
ing them. Hence, maintenance personnel can be alerted as soon as a
fault occurs. Also, it has the property of enabling the system to which
it is applied to continue to function correctly even when faults are
present and maintenance is being performed, provided all the faults
are confined to any one of the several subunits which comprise the sys-

the error-correcting capabilities of the scheme, the system may be kept
in continuous operation for much longer periods than could the equiva-
lent system without error detection and correction.

In this regard, it is estimated that the “mean life’’* of the system
with error correction can be made several thousand times as long as
that of the equivalent nonredundant system, provided faults are re-
paired sufficiently soon after their occurrence. Such potentially vast
increases in reliability depend of course on the availability of rapid
diagnostic and fault-repair facilities. Conversely, in the absence of
maintenance the mean life of the redundant system will in general be
less than that of the nonredundant system. Hence the scheme is not
usefully applicable to a system which must operate in an environment
where rapid fault repair is impossible — in such situations some other
method of building in reliability, such as microlevel redundancy,’ would
be necessary.

In comparison with triplication and vote taking, our procedure will
permit more precise localization of faults. Also, for large systems it
should result in less over-all equipment redundancy. For small systems,
however, an equipment advantage may not always be realized. Since
the triplication scheme is fairly well known, we shall start by deserib-
ing it, but from a slightly different point of view, which shows how it
appears as a special case of our procedure.

Fig. 1 shows a system, A, with m inputs and n outputs, and two exact
replicas of the system, B and c. Corresponding output wires from a, B
and ¢ are fed to ‘“‘majority”’ circuits, or vote takers, each of whose out-

* The mean life of a svstem is here defined as its mean time to failure, assum
ing it is in perfect condition at the start

puts agrees with the majority, i.e., with any two or all three, of the
inputs which are in agreement. Thus, the system corrects for errors
which are confined to the outputs of any one of systems A, B or c.

We may consider the outputs from A to carry information bits, and
those from B and ¢ to carry check bits, which generate Hamming?
single error-correcting codes, in the following manner. The matrix be-
low displays the output bits from a, B and ¢ in a matrix consisting of
three rows, each row having n entries:

Alternatively, the matrix may be thought of as displaying n columns,
each with three entries consisting of one information bit and two check
bits. For example, the first column contains the information bit A; , and
check bits B; and C;. Two parity checks are constructed from this
column; bits A; and B, satisfy the parity relation

where ® represents the sum modulo 2. Bits A; and C; satisfy the parity
relation

These relations merely state that, when the complete system is operating
correctly, both B; and C; will have the same value as A; .

This coding has the ability to detect any single error in column 1,
and moreover tells us which bit is in error, so that corrections can be
performed. Therefore, in particular, this scheme permits the correction
of any pattern of errors which is confined to a single row of the matrix.
Since faults which are confined to one of the systems A, B, or C can
cause errors on the outputs of that system alone, this error-correcting
scheme will permit the over-all system to operate correctly even when
any one of the three systems comprising it is faulty, or is disabled for
maintenance purposes.

Obviously, this particular coding is inefficient, because two check bits

Suppose system a of Fig. 1 is designed so that it breaks down into a
number, say r, of electrically independent subunits, each subunit carry-
ing not more than p of the n system outputs, as shown in Fig. 2.

A fault or faults that is confined to any one subunit can at most cause
errors on the outputs of that subunit. Therefore, consider the following
matrix, in which the outputs of each subunit are displayed in a separate
row, with p entries per row. There are (q — r) additional subunits shown
in Fig. 2; these provide k check bit outputs:

nore a Care? outputs from subunit r
| aes oe oe
rows X X --- X outputs from subunit gq

Since faults in a single subunit affect only a single row of the matrix,
we may, for example, apply Hamming single error-correcting codes on a

per-column basis. Thus, if r = 4, we need only three check bits per
column to provide Hamming single-error correction, and the code re-
dundancy is much less than in the triplication scheme (+ instead of $).

We now wish to show how it is possible to break down a system into
electrically independent subunits. Digital systems may be classified into
two types: those which perform only combinational logic (have no
memory), and those which perform sequential logic (have memory).
The latter type is of more interest, but it is useful to deal with the former
first. We assume throughout that the data on the input wires are not
in error, and that faults in the system do not cause errors on the input
wires.

Suppose then that the r subunits in Fig. 2, which produce the 7
system outputs, consist entirely of combinational logic. It is evident
that the system can be broken down into such subunits because, for
example, each output can be realized by designing a separate combina-
tional logic circuit which generates the appropriate Boolean function of
the m input variables. Alternatively, some savings in logic elements
may be possible by designing multifunctional logic circuits, each generat-
ing only the p outputs of a single subunit.

To provide check outputs, additional subunits are needed, and are
designated (r + 1) through gq in Fig. 2. To design these, it is necessary
to be able to express each check output as a Boolean function of the m
input variables. This can be done because the structure of the error-
correcting code will specify each check output to be the sum modulo 2
of some set of information outputs, and since the latter are known
functions of the inputs, we can therefore express the check outputs
directly as functions of the inputs. We may, of course, work with truth
tables instead of functional representations.

In the case of sequential logic, a complication is introduced which
may be explained with the aid of Fig. 3. In this figure, a sequential

* Some authors replace the memory elements by unit delay elements. See for
example, Fig. 1 of Unger.® His paper deals with asynchronous circuits, whereas
we are treating synchronous circuits of the type designated ‘‘PP’’ by Cadden.®

(MEMORY )
Fig. 3 — A possible configuration for a finite-state sequential system.
The combinational logic generates two sets of outputs:

The s outputs of the memory unit, in conjunction with the m system
inputs, comprise the inputs to the combinational logic unit.

Suppose that unit | is designed as r electrically independent subunits.
In general, the (m + s) inputs to unit 1 will feed all r subunits. A fault
in a single subunit will cause errors on the output of that subunit, and
these will feed back via the memory unit to the inputs of some or all
of the other subunits. Hence, in a few cycles of operation it is possible
that the outputs of all subunits will be in error because of a fault in
just one subunit. This situation can be remedied by applying error
correction to some or all of the s feedback wires in addition to the n
system output wires. These additional corrections should be made be-
tween the outputs of unit 2 and the inputs of unit 1, in order to correct
errors caused by faults in unit 2 as well as in unit 1.

Alternatively, it is possible to design the system so as to avoid cor-
recting the internal feedback wires, and yet insure that a fault affects
not more than p of the n system outputs. For example, instead of break-
ing down the system into r subunits, one could replicate the system r
times and utilize only outputs 1, 2, --- , p, from the first replica, out-
puts p + 1, p + 2,--- , 2p, from the second replica --- and outputs
n—p+1,n— p+ 2,---,n, from the rth replica.

No doubt this alternative realization could be achieved without using
r complete replicas of the system. However, the necessary design pro-
cedures are not well formulated and the resulting equipment redundancy
is difficult to estimate. In contrast, the design procedure for the first
mentioned method is straightforward, its redundancy is easier to esti-
mate and, at least with present devices and techniques, it appears to
result in considerably less over-all redundancy. Therefore, in the re-
mainder of the paper we shall assume that the first method is to be used.

Accordingly, the correction of a sequential system requires that unit |
of Fig. 3 be designed as r independent subunits and that it be augmented
by a combinational logic unit which generates k check outputs, where
k is large enough to provide the necessary parity checks for correcting
(n + s) wires (we assume that all s feedback wires may require correc-
tion). As in the purely combinational case, each check bit can be ex-
pressed as a sum modulo 2 of an appropriate subset of the n outputs of
unit | and the s outputs of unit 2, and since these are known Boolean
functions of the (m + s) inputs to unit 1, each check bit output ean
likewise be expressed as a Boolean function of these same inputs.

It is of course necessary that the check bit logic circuits also be de-
signed as independent subunits with not more than p outputs per sub-
unit.

Before discussing specific codes, we wish to establish lower bounds on
the number of check bits, /, needed to fulfill our error-correcting re-
quirements. Specifically, referring to the matrix above for the outputs
from r subunits of system A, we ask what minimum value of / is required
to permit correction of every possible pattern of errors in any single
row of the q rows.

Actually, two lower bounds are applicable. The first bound, which is
also the larger of the two when q > (2? + 1), p being the number of en-
tries per row, is easily derived as follows: Observe that the number of
possible error patterns in a single row is (2? — 1), if we exclude the no-er-
ror pattern. Therefore, the total number of error patterns in all g rows is
q(2” — 1). Obviously, & must be large enough to permit as many ‘‘parity
failure”? patterns as there are error patterns. This requires that / satisfy
the inequality

where the square bracket denotes the smallest integer which is
= logs (q2” — q + 1).
The second lower bound, which is larger than the first when
(2” + 1), and which is therefore of greater practical significance, |
k 2 2p.

can be chosen out of a set of 2” binary words of lengths pg, and which
fulfill the specified error-correcting requirements. Its derivation § is
relegated to the Appendix.

Surprisingly, it was found not too difficult to construct codes for
most values of p and q in the range 2 < p < 10,3 < q < 9 which
achieved the appropriate lower bound.

Subsequent to the work described here, Ray-Chaudhuri! developed a
general theory of minimally redundant codes for this application. How-
ever, it will not be out of place to exhibit here some of the codes pre-
viously derived, since they are also minimally redundant, and since the
error-correcting equipment required to implement either them or the
Ray-Chaudhuri codes is of the same general character and complexity.
Three families of codes* are exhibited below in matrix form, correspond-
ing to three values of p, as follows:

* These codes were constructed by George Allen at Bell Telephone Labora
tories in Summer, 1959

The set of digits beside each bit position indicates the parity groups
which that bit enters. For example, in family 1, the digit set attached
to the last bit in the last row is 123, indicating that this bit enters parity
check groups 1, 2 and 3. As a further illustration, the bits in family 1
that are labeled 1, 14, 13, 124 and 123 enter parity group 1; their sum
modulo 2 is zero when there are no errors. The bit positions which carry
only a single digit are check bit positions. They are denoted by X’s and
the information bits are denoted by O’s. The matrix displayed for family
lisa 2 X 5 matrix; however, if a 2 * 4 or a2 X 3 matrix is desired,
one omits respectively the last row or the last two rows. Similar remarks
apply to families 2 and 3.

Several remarks should be made at this point. First, observe that in
the original matrix we represented the check bits as being located en-
tirely in the last (¢ — r) rows. However, this is not necessary; the check
bits may appear along with information bits in some or all rows, as is
the case in the three matrices above. The arrangement is dictated by
the structure of the code, but bits may be permuted within each row
with impunity.

Secondly, it is possible to delete information bits from any row with-
out destroying the utility of a code; in such cases, the deleted bits will
be omitted from the parity checks in which they would normally par-
ticipate.

Finally, we observe that the three matrices above provide only for
values of p S 4andq S 9. If for any reason we wish to form matrices
with p > 4, or q > 9, this may be done by building up the over-all
matrix, either vertically or horizontally, or both, from several of the
above matrices. Thus a wide variety of equipment arrangements can be
accommodated. However, if we build up vertically, we will sacrifice
minimal code redundancy. For example, if we form a matrix with 3
columns and 18 rows by using two matrices of family 2, we shall have

A discussion of error-correcting circuits is included here to indicate
roughly the amount of equipment involved in error correction, and to
provide a basis for a maintenance routine which is proposed later.

It was explained in Section II that, with the present scheme, it is
necessary to apply error correction either to the nm system outputs of a
purely combinational system, or to the (nm + s) outputs and feedback

connections of a sequential system. To do this, k parity check bits must
be generated, where k is determined by the structure of the error-cor-
recting code employed. Therefore, the inputs to the error-correcting
circuits consist of (V + k) wires, where VN = n+ s (s 0 for a purely
combinational system) and / wires carry the check bits. The output of
the error-correcting circuits consists of NV wires which carry the cor-
rected versions of the corresponding NV inputs.

We shall want to distinguish between the correcting circuits and the
circuits which are corrected or correctable. The latter comprise the set
of g subunits which generate the (V + k) outputs; for example the

The error correcting-circuits may be considered to perform the fol-
lowing three functions:

2. From the pattern of parity check failures, determine which of the
(NV + k) input wires are carrying erroneous bits.

3. Correct that subset of the V wires which are carrying erroneous
bits. Erroneous check bits do not require correction.

Circuits to perform the above tasks may be realized in several ways,
and with varying degrees of redundancy to assure reliability. At one
extreme, the error-correcting circuits could be nonredundant, in which
case an efficient preventive maintenance routine would be required to
insure that they perform for long periods without error. Alternatively
they could be built with microlevel redundancy, in which case preventive
maintenance would again be necessary but would be appliec less fre-
quently. A third alternative would be to make some or all of the error-
correcting circuits ‘“‘self-error-detecting.”’ Those parts which were self-
error-detecting would be subjected to maintenance only when a fault
was detected; the parts which were not self-error-detecting would require
preventive maintenance.

As a fourth alternative, it might be attempted to make the error-
correcting circuits completely self-error-correcting. However, a simple
heuristic argument can be given which indicates that it is impossible to
achieve this goal.

Fig. 4 shows a block diagram of a proposed error-correcting circuit,
designed according to the third alternative above. Box 1 in Fig. 4 con-
tains the units which perform functions | and 2 above. Box 2 performs
function 3 above, and also an error-sensing and alarm function. For

simplicity, sets of wires in this figure are represented by single directed

lines with an associated symbol indicating the number of wires. Box |
is designed to be self-error-detecting, but box 2 is not. A detailed deserip-
tion of the operation of the circuits in Fig. 4 does not contribute sig-
nificantly to an understanding of the over-all scheme, and so is omitted.

The maintenance routine is as follows. If a fault occurs in some sub-
unit of the system, unit bp under control of box | corrects the resulting
errors. Simultaneously, box | operates alarm 1. The faulty subunit is
then located and repaired as quickly as possible. In principle, the svstem
can continue to operate correctly even when the faulty subunit is being
replaced or repaired, provided a fault does not develop in another sub-
unit or in the error-correcting circuits during the repair of the original
fault.

If box 1 fails, alarm 2 is operated, and possibly alarm 1 also, and
simultaneously relay E is switched to the bypass position, thus causing
essentially no interruption in system operation. Box | must also be
repaired quickly, because if a fault occurs in the system during the
repair of box 1, the system will fail, since it is not now being error cor-

rected. If units c or p fail, they do not operate an alarm since they are
not provided with error detection. The probability of failure of these
units must, therefore, be minimized by a preventive maintenance routine.
To minimize the outage times of units ¢ and pb, two copies of each could
be provided, one operating and one standby. They would be interchanged

to the units currently in the standby condition. In this way, the system
would not lose its error correcting capability during preventive main-
tenance. Relay E must be switched to the bypass position to permit
continuous operation while unit ¢ or b is being replaced. We assume that
the relay switching time is short enough so as not to interrupt system
operation.

We now wish to determine a quantitative measure of the reliability
improvement of the maintained error-correcting system over that of the
nonredundant system. A useful measure is the ratio of their respective
mean lives; namely, R, = L,/L;, where L, is the mean life of the re-
dundant system and L; is the mean life of the nonredundant system.
In general, the derivation of Ry, is quite complicated, but by making
suitable simplifying assumptions we can obtain an approximate formula
which is useful. These assumptions are:

1. All components have an exponential survival probability function
and the same mean life, which is taken to be the time unit. Therefore,
the survival probability of any component is exponential (—?).

3. Failure of any component in the nonredundant system causes that
system to fail.

5. The times taken to repair faulty circuits are assumed constant.
These and other parameters are now defined:

A; = time that unit c¢ or unit D is removed from circuit when
being exchanged with its standby,

T = time interval between successive replacements of unit ¢ or D
by its standby

ERROR CORRECTION FOR SYNCHRONOUS DIGITAL SYSTEMS — 589
values of the above parameters, we can obtain the following formulas:

As an application of (3) and (4), consider a system having parameter
values as follows:
time unit = mean component life 1000 years,
A, = As = one-half hour } X 10-7 units (time to repair a
system subunit or box 1),
A; = 8 seconds = 2 X 10°'° units (time to replace unit ¢ or
unit D by its standby),
T = one month = 10~* units (maintenance interval for units
cand pb),
nz = 333 components,
666 components,
6
r= 4,
Substituting these values in (3) and (4) results in Ry equal to 2900.

That is, the mean life of the redundant system is 2900 times that of the
nonredundant system. Actually, this figure could be improved if we
reduced the repair times A; and A, from one-half hour to, say, five min-
utes. Such a reduction would be possible if the system were built of small
modular packages and a highly automated diagnostic routine were
available to locate faults in the order of a minute or less. Thus, a fault
could be pinpointed to one or two particular packages, and these pack-
ages could be replaced immediately by good standby packages, thus
permitting correction of the fault in minutes. The faulty packages could
then be tested and repaired in more leisurely fashion, and this latter
time would not be chargeable to A; or Ae .

Therefore, it appears that, with an efficient maintenance routine, the
mean life of the error-correcting system can be several thousand times
that of the nonredundant system.

in simple terms, but the contributing sources can be delineated and
roughly evaluated. They are:

(b) the cireuit redundancy which results from designing the system
as q independent subunits;

The redundancy contribution of item (a) can reasonably be estimated
to be in the ratio k N to the amount of equipment in the original non-
redundant system. That is, if the amount of equipment in the non-
redundant system is treated as one “unit”? of equipment, the amount
of equipment needed to generate the check bits would be roughly k/N
“units.”’ As indicated by the codes in Section II], values of kN as
small as 3 are achievable for V = 18. The assumption underlying this
estimate is that the amount of circuitry required to generate each check
output is the same as the amount required to produce each original
system output.

The redundancy contribution of item (b) is believed to be insignificant
compared to the other contributions, especially for large systems, pro-
vided optimal design techniques are employed. The contribution of
item (c) is by far the largest, and is the most difficult one to estimate.
It depends on the type of logic technology employed, on the amount of
time delay that the error-correcting function is permitted to introduce,
and to some extent on the particular error-correcting code used. The
following estimates may suggest an order of magnitude for item (¢), in
the particular case of the correcting circuits proposed in Fig. 4, and
assuming the use of diode logic. Based on “paper” designs of these
circuits, the author estimated that the correcting circuits might require
roughly 60 to 70 “equivalent”? diodes per wire corrected (the number of
wires corrected is V). Transistors were counted as equivalent to two
diodes, resistors etc. were not counted.

This estimate assumes that units A and a’ of Fig. 4 both employ
c — 1) EXCLUSIVE OR circuits per parity check over ¢ bits, and that
unit B of Fig. 4 is realized with two-stage logic. Thus, if these error-
correcting circuits were to be applied to a system which, in its non-
redundant form, was realizable with 30 to 35 “equivalent”? diodes per

wire corrected, it is evident that the correcting circuits would comprise

In general, therefore, it is to be expected that the scheme in question
may introduce an amount of redundancy equivalent to at least triplica-
tion of the original equipment. However, it has the potential of being

generation of the check bits alone results in triplication and the correct-

In principle, it appears that the redundancy penalty might be made
to decrease monotonically as the systems to which error correction is
applied becomes increasingly complex, provided the following two as-
sumptions are valid:

(a) as the number N of corrections is increased, the coding efficiency
also increases; that is, k/N becomes smaller;

(b) the amount of equipment per correction in the original system
increases faster than the amount of equipment per correction in the
correcting circuits.

Assumption (a) is realizable, but (b) cannot be verified. Indeed, (b)
may be plausible only provided the correcting circuits use an increasing
number of logic stages, which can be expected to result in an increase
in time taken to perform corrections; that is, an increasingly severe
speed penalty is imposed.

In this regard, the parity check circuits referred to earlier in this
section require 2 X [logec] logical stages. (The square bracket denotes
the smallest integer which is equal to or greater than logec.) For the
special codes described in Section III, the number of bits per parity
check, ¢, is typically equal to g, the number of subunits in a system,
and gq must increase in order to increase the coding efficiency. It there-
fore follows that greater coding efficiency can be achieved only at the
expense of greater delay in the corrector, or more complex correcting
circuits, or both, and a compromise must be reached.

Finally, a remark should be made concerning the impact of this scheme
on the over-all design of a sequential system when the method which
requires both feedback and output corrections is used. To minimize
the number of corrections necessary, the number of feedback and output
wires should be kept to a minimum. The designer usually is able to
exercise some control over both. In particular, there are roughly as
many feedback connections in a sequential system as there are binary
memory elements; therefore it would be desirable to minimize the
number of memory elements. At present, large systems are frequently

designed with many more memory elements than necessary, presumably
because this results in simpler design procedures. It may, therefore, be
desirable to find methods which lead to designs having nearly minimal
numbers of memory elements, in order to make our error-correcting
procedure more attractive.

applicable to synchronous digital systems, and which includes the
triplication and vote-taking scheme as a special case. It permits systems
to which it is applied to operate continuously even when faults are
present and maintenance is being performed. The scheme can lead to
very large increases in system reliability, but only if augmented by a
maintenance routine which effects rapid repair of faults.

Two types of error-correcting codes have been discussed, Hamming
codes and special codes. The Hamming codes are universally applicable,
but are not minimally redundant in this application. The special codes
are minimally redundant but not universally applicable, in that they
have not been developed for a large range of values of p and q.

The equipment redundancy required to implement the scheme may
be equivalent to at least triplication for moderately large systems, but
should be less for more complex systems. It is not specifiable in simple
terms and can be determined accurately only by carrying through the
detailed design of the specific systems. Such detailed applications have
not yet been made.

We shall derive this bound by showing that an upper bound on the
number of code words of length pg which satisfy our error-correcting
criterion is 2°?” words. This implies that the maximum number of
bits which can be assigned values arbitrarily is (¢ — 2)p bits. The re-
maining bits must be check bits; therefore, a lower bound on the number
of check bits is gp — (q — 2)p = 2p bits.

[t is useful to think of the pq bits which comprise a code word as being
arranged in a single row, with each successive block of p bits being re-
placed by a single symbol, D;, which can take on any one of the 2”

different values. In this alternative representation, a typical g-symbol
word would be

a:
In terms of this representation, our error-correcting criterion requires
that an error in any one of the g symbols be correctable. This implies
that admissible code words must differ in more than two symbol posi-
tions. For, consider the following two words which differ only in the first
two symbol positions:

It is possible for an error in the first symbol of word | and in the second
symbol of word 2 to cause both to become, for example, the word

Hence, we cannot determine whether word 1 or 2 was the correct word:
therefore, admissible code words must differ in more than two symbol
positions.

tion S into disjoint subsets S; ,7 = 1, 2,3, --- , such that two elements
of S belong to the same subset if they are identical in the last (¢ — 2)

symbol positions. Thus there are as many subsets as there are trun-
cated words D,D, --+ D
tains 2?” elements.

Now arbitrarily choose a (q-symbol) word from subset S; to be a code
word. Then no other words from subset S; can be chosen as code words,
because any two words from S; differ only in the first two-symbol positions.
By the same argument, at most one word can be selected as a code
word from S., ete. Therefore, there cannot be more code words than

2. Von Neumann, J., Probabilistic Logies and the Synthesis of Reliable Organ
isms from Unreliable Components, in Shannon, C. Ek. and MeCarthy, J., eds.,
Automata Studies, Annals of Math. Studies, No. 34, Princeton Univ. Press,
Princeton, N. J., 1956, pp. 44-98 (particularly Section 8.3).

3. Moore, E. F. and Shannon, C. E., Reliable Cireuits Using Less Reliable Re
lays, J. Frank. Inst., 262, 1956, pp. 191; 281.

5. Unger, H., Hazards and Delays in Asynchronous Sequential Switching Cir
cuits, I.R.E. Trans., CT-6, 1959, p. 12.

Several authors have considered the possibility of increasing the reli-
ability of large and complex binary digital systems by introducing some
redundancy in the system. In a companion paper, Armstrong’ proposes a
scheme for applying error correction to a synchronous digital system. In
this paper we develop a general mathematical theory for generating mini-
mally redundant error-correcting codes for the scheme in question. This
results in what are called *‘minimally redundant reliable systems.” The
problem of constructing minimally redundant reliable systems whose out-
put is free of error when there is a fault in at most one block of the system
is completely solved. An example is considered in detail showing how the
mathematical theory can be actually applied.

In complex binary digital systems employing a large number of blocks
of electrical equipment it often is difficult to ensure a sufficient level of
reliability of each single block of equipment. An attempt to attain the
desired degree of reliability by improving the reliability of each block
may prove to be uneconomical. On the other hand, by introducing some
redundancy in the system, it is possible to construct highly reliable
complex systems, even though each single block is not as highly reliable.
Moore and Shannon,” Tryon,’ Von Neumann,’ Lofgren’ and Armstrong!
have considered the problem of constructing reliable system designs.
In this paper a general mathematical theory has been developed for the
construction of minimally redundant reliable system designs, based on
the scheme outlined by Armstrong.’ This theory is closely related to the
theory of error-correcting codes. The problem of constructing minimally
redundant system designs whose outputs will be free of error whenever
there is fault in at most one block of the system is completely solved in
this paper.

Suppose there are m binary input variables X, , X., ---, X,,. Let B,,
denote the set of 2” m-place binary sequences. Every set of values of
the m binary input variables will be regarded as an element of B,, . Any
mapping of B,, into B, will be called a Boolean function of the m input
variables X,, X2,---,X,,. For the sake of brevity, the collection of m
input variables will be denoted by X. Let

be pk Boolean functions of the m binary variables X,, X.,---,X,.
Our problem is to construct a system which will synthesize the pk
Boolean functions with a high degree of reliability. Thesystem uses blocks
of electrical equipments each of which can synthesize p Boolean func-
tions. For the sake of brevity, a collection of p Boolean functions, will
be called a Boolean p-function. Thus f; = (fia, fie, +++, fip) isa Boolean
p-function. Any Boolean p-function is a mapping of B,, into B,. Each
block of our system synthesizes a Boolean p-function. Fig. 1 is a sche-
matic diagram for the original nonredundant system.

The blocks act as units in the system. If there is a fault in a block,
then some or all the p outputs of the block are erroneous. In other words,
in the case of a fault a block will synthesize the corresponding Boolean
p-function wrongly. Let V,' denote the set of 2” binary p-tuples. Then

t= 1,2---,k. Let f = (fi, fo, --:,f,). Then f can be regarded as a
mapping of B,, onto V,*. We shall define the addition of p-tuples as the
usual mod 2 addition. For example, if p = 3, a; (OO1) and as (101),
then a; + a (100). Let @ and a’ be two elements of V,* given by

is defined to be the element (a; + ay’, --+, a + a’). The p-tuple
(00 --- 0) will be called the null element of V,'. The weight w(a) of
the k-vector @ is defined to be the number of nonnull elements among
a, @,-**:, a. For any particular value X’ of the input variables
f(X’) is a vector in V,". Suppose there are faults in ¢ (t < k) blocks.
Then ¢ of the functions fi; , fo, -- +, will be synthesized wrongly. Hence
the output will be the vector f( X’) + «, where « = (6, @,°°:,e) isa
vector in V,* with weight ¢. While designing a system to synthesize the
Boolean function f, one might require that whenever the number of

faulty blocks is ¢ or less, the output is error-free. One can achieve this by
introducing some redundancy in the system, i.e., by synthesizing (/ + r)
Boolean p-functions and adding a logical corrector unit to the system.

Suppose ¢; , ¢2, °**,¢, are n Boolean p-functions and C is a mapping
of V," onto V,. We shall consider ¢ (g1,¢2,°**,¢n) asa function

from B,, to V,". For every value X’ of X, ¢(X) is an element of V,”.
Suppose the functions g and (' possess the property P’ stated below:

lor every vector ¢ belonging to V," with w(e) not exceeding ¢ and
every value X’ of the input variable X,

The functions ¢ and C enable us to construct a system which will
synthesize the k Boolean p-functions f; , fe, «++, f, free of error whenever
the number of faulty blocks in the system is ¢ or less. The n Boolean
p-functions ¢1 , g2, +++, ¢, can be considered as a collection of np Boolean
functions of m input variables. Therefore we can easily obtain the
logical design of a system which will synthesize these np Boolean func-
tions. This system will be called the encoder subsystem. Similarly, the
function C can be considered as a collection of pk Boolean functions of
np binary input variables, and therefore we can obtain a system which
will synthesize these pk functions. This system will be called the cor-
rector subsystem. The np outputs of the encoder subsystem will be the
inputs of the corrector subsystem. Now it is easily seen that, because
of the property P of the functions ¢ and C, whenever the number of
faulty blocks in the encoder subsystem is ¢ or less and the corrector unit
is free of error, the pk outputs of the corrector subsystem will be

the above discussion, the following two definitions given below are
meaningful.

Definition 1: The functions ¢ ,¢2,°-:,¢, and C possessing the
property P stated in (1) will be called a reliable system design of order
tand redundancy r = n — k for thek Boolean p-functions f; , fo, «++, fi.

Definition 2: A reliable system design of order ¢ and redundancy 7»
for the & Boolean p-functions f; , fo, «++, fe will be called minimally re-
dundant if the redundancy r of any other reliable system design of order
t for the same functions is not less than ro .

In the present paper we have given a method of obtaining a minimally
redundant system design of order | for any set of k Boolean p-functions
for arbitrary / and p. System designs of higher order will be given in a
subsequent paper.

We have used the redundancy r as a measure of the extra amount of
equipment which has to be used for making the system reliable. And
hence we seek the system which has minimum possible value of the re-
dundancy r. It should be pointed out that we assumed that the corrector
subsystem does not make any error at all. Therefore, to make the whole
development practically feasible, it is imperative that either the amount
of equipment necessary for the corrector subsystem is small in com-
parison to the amount of equipment necessary for the whole system, or
that other steps be taken, such as are suggested in Ref. 1, to ensure
reliability of the corrector system. We have not used any mathematical
criterion to incorporate this requirement in the development of the
theory.

(00,01,10) and d(a,e’) = 2. It can be easily checked that the distance
defined above satisfies the three conditions of a distance function. We
have seen that Boolean p-functions as defined in Section II can be con-

shall assume that all Boolean p-functions appearing in our discussion
are nondegenerate. In the following we have s = 2” andn = k + r.

Theorem 1: A necessary and sufficient condition that there exists a
reliable system design of order ¢ and redundancy r for the k Boolean
p-functions f, , fe, «++, f; is that there exists a subset A of V,," contain-
ing s* elements such that d(a,a’) = 2+ 1; aa’ © A’, a # a’.

Proof: Necessity. Suppose there exists a reliable system design of
order ¢. Let the encoder functions be ¢ = (¢),¢0,: + -,¢g, ) and the corrector
function be C. For every value X’ of the input variable Y, ¢g( YX’) is a
vector of V,,". Consider the set

Using the fact that the Boolean p-functions f; , fo, +-+,f, are nonde-
generate functions, it follows easily that the set A contains at least s°
vectors of V,". Consider two distinct vectors a and a’ of the set A. If
possible, suppose d(a,a’) S 2t. Since d(a,a’) S 2t, we can find a vector
eof V,”" such that a + «€ = a’ + cand w(e) S ¢. Since w(e) S t, we
have

Mquation (3) contradicts the assumption that @ and a’ are distinct
vectors of A. This completes the proof of necessity.

Sufficiency. Suppose A is a subset of V," containing s° elements and
having the property that d(a,a’) 2 2t+ 1; aa’ © A, a ¥ a’. We set
up a one-to-one correspondence between the s* vectors of V,* and the

‘ . a . . . ‘ a ; ,
[fi NX’) fol X"),- ++ fe X’)| is a vector of V,° and there is a corresponding
vectora of V," belongingto A. The encoder function ¢ 5025 °**, Gn)
is defined by

where a is the vector of V," belonging to A and corresponding to the
vector f( X’) of V,°. The corrector function C is defined in the following
manner. Let y = (y1,¥2,°**, Yn) bean arbitrary vector of V,”". First

we choose a vector @ belonging to A such that d(y,a) S d(y,a’),
aa’ © A, Let 8 = (8), 8, +++, Be) be the vector of V," which corresponds

Thus C is a mapping of V," onto V,". It is easy to check that the en-
coder function ¢ and the corrector function C defined above possess the
property P stated in Section II. This completes the proof of sufficiency.

Theorem 2: If there exists a reliable system design of order ¢ and re-
dundancy r for k Boolean p-functions, then

where n = k + rands = 2”.
Proof: From Theorem 1, it is necessary that there exists a subset A of
V,,” with the property that

d(y,a) & t. It follows easily from (7) that, for any two distinct vectors
a and a’ of A, the sets S, and S,’ do not have any common element.
Let S, | denote the set of elements of V,”" which have distance k from
a, k = 0,1,2,---,¢. Obviously S, is the union of the (¢ + 1) sets ig .

Theorem 2 follows from (8).
Theorem 2 gives a lower bound on the redundancy r of a reliable

system design of order ¢ for k Boolean p-functions. Theorem 2 is actually
a generalization of a result of Hamming.°

Let n.(r) denote the maximum integer n for which there exists a
reliable system design of order ¢ and redundancy r for k = n — r Boolean
p-functions. For ¢ = 1, the inequality (6) becomes

If there exists a reliable system design of order ¢ and redundancy r for k
Boolean p-functions, then n,(r) 2 k + r.
Lemma 1:

Proof: Suppose n,(r) = n. Then there exists a reliable system design
of order ¢ and redundancy r for k = n — r Boolean p-functions. Hence
by Theorem | there exists a subset A of V," containing s* elements
with the property that d(a,a’) 2 2¢ + 1; aw’ © A, a # a’. To every

of V,"*'. Thus we have a subset A of V,""' containing s* elements and
also possessing the property that d(a@,a@) = 2¢+ 1; aa © Aja ¥ &.
Hence, by Theorem 1, we can obtain a reliable system design of order ¢
and redundancy n + 1 — k = r + 1. It follows that

n(ir+t1)2k+r4+1.=n4+1 = nr) +1.
Theorem 3: If for a reliable system design of order ¢ and redundancy
r for k Boolean p-functions we have
n(r—1)—-(r-1) <k Snir) — 7, (9)
then the design is minimally redundant.

Proof: If possible, suppose the system is not minimally redundant.
Then there exists a reliable system design of order ¢ and redundancy

The inequality (10) contradicts the inequality (9); hence the theorem
is established.

In this section we shall consider a particular subclass of system de-
signs called the linear system designs. To define the linear system de-
signs, we have to use the theory of finite fields. Let K be the finite field
of characteristic 2 containing s = 2” elements and x denote a primitive
element of AK. Any binary p-tuple (apo, a, +++, @,-;) will be made to
correspond to the element ajo + qx +--+ + 4a, w’*' of K and
vice versa. An element a = (a, a@:,°*:,a@,) of V," now will be con-
sidered as an n-vector with elements in K. The weight w(a@) of a@ is

Obviously V," is a vector space over K. Consider a system design for k
Boolean p-functions. Suppose the encoder function is

Definition 3: A system design for k Boolean p-functions is said to be
a linear system design if the subset A of V," defined by (11) is a vector
space over K.

Lemma 2: A necessary and sufficient condition that a reliable linear
system design for k Boolean p-functions is of order ¢ is that the weight
of any nonnull vector of the set A defined in (11) is not less than
(2 + 1).

By definition d(a,a’) = w(a + a’). Since A is a vector space, a + a’
is an element of A and also a + a’ is a nonnull element of A. Hence it
follows that (12) will hold if and only if w(a@) 2 2¢ + 1 for every non-
null element @ of A.

Definition 4: A matrix M with elements in K will be said to have the
(P,)-property if no ¢ rows of the matrix are linearly dependent.

Theorem 4: A necessary and sufficient condition for the existence of a
reliable linear system design of order ¢ and redundancy r for k Boolean
p-functions is that there exists a matrix WV with n = (k + r) rows and r
columns with elements in A which possesses (/’,,)-property.

Let A denote the vector space orthogonal to the vector space generated
by the r rolumn vectors of W/. A contains at least s‘ elements. It would
be sufficient to show that the weight of any nonnull vector @ of A is at
least (24 + 1). If possible, suppose A contains a nonnull vector with
weight less than (24 + 1). For simplicity of writing assume that the
vectora = (a, a2, +++, a@,0,---,0) belongsto A where a; , a2, +++, ax,

Equation (14) implies that the first 2¢ vectors of the matrix 1/7 are
linearly dependent which is a contradiction. This completes the proof
of sufficiency. Necessity can be proved by exactly similar arguments.
The reader acquainted with the literature on error-correcting linear
codes would recognize from Theorem 4 that a reliable linear system de-
sign of order ¢ for k Boolean p-functions exists if and only if a (error
correcting linear code in s( = 2”) symbols with n places and /& informa-
tion places exists. Lemma | and Theorem 4 given above are not new
results; they were proved by Bose‘ and Zierler® in a different form. We
have included short proofs for these results for the sake of completeness.

In this section we shall give methods for constructing minimally re-
dundant linear system designs of order | for / Boolean p-functions for
any arbitrary value of k and p.

To prove (15) we shall construct a matrix 7 with r columns and n =
(s" — 1)/(s — 1) rows which has (P2)-property. We shall denote the ele-
ments of K by 0,l,a2, ++ ,a,-1, where 0 is the null element and 1 is the
multiplicative identity. Consider the matrix 7 given by
M,
M = (16)
I,
where /, is the identity matrix with r rows and r columns and J, is a

It can be easily checked that the matrix V7 has (/?.)-property; 1.e., no
two rows of VV are linearly dependent. This completes the proof of
Theorem 5.

It should be observed that Theorem 5 enables us to construct mini-
mally redundant system designs of order | for any arbitrary values of k
and p. For given k, we find out the integer r for which

mir —1)—(r—1) <k Snr) —--..
If n,(r) — r = k, we construct the matrix M with r columns and n,(r)
rows as defined in (16) and then obtain the system design as illustrated

in the proof of Theorem 4. If n(7) — r > k, then we delete n(r) —
(k + r) rows from M, and thus obtain a matrix WM with (P2)-property
which has r columns and (k + r) rows. From Theorem 3, is follows that
the resulting system design will be minimally redundant. Now we shall
give explicitly the encoder function of the minimally redundant system
designs of order 1. Let

where /, is the identity matrix with & rows and & columns and J/)’ is
the transpose of the matrix /,. It can be verified that the k column
vectors of B are orthogonal to each of the r column vectors of 4/7. The
k column vectors of B generate the vector space A and every nonnull
vector of A has weight greater than 2/. For the sake of convenience of
description, we write

If there is error in one block of the encoder subsystem, ¢ will have one
nonzero element among its coordinates. Suppose the /th coordinate of
is a nonzero element A of A. Then e (Biss «2 Xe.* 2). 2 OS H,
Let us denote the /th row vector of the matrix VW by
R; Uy Cl2,y °° *, Cir).
Then the following equations hold:
6 4 AR,
(21)
NC) » ]
Conversely, if for a given output vector y the vector 6 computed by
20) satisfies (21) and there is one block of the encoder subsystem in
error, then the error vector ¢ will have \ as its /th coordinate and zero
as the other coordinates.
rom the above discussion, it is clear that a corrector subsystem which

observes the rules given below will accomplish the job of correcting
errors in one block of the encoder subsystem and produce the k Boolean
p-function fi( X ),fo( X),+++.fe( X) as its output. The rules are

il. If 6 is not the null vector, find out the integer / for which the
vector AR, for some A K has maximum number of common coordi-
nates with 6. If / > k, the k outputs are 8B; = y;,7 = 1,2,---,k. Ifl < k,
the k outputs are 8B; = y;, Be Y2,°°', Bi = yr tA,-°+, Be Vk.

In this section we shall give an example to show how the theory de-
veloped in this paper can be applied.

Suppose m = 3, p = 2,k = 3 and ¢t = 1. From Section V, we can
see that for the minimally redundant system r 2. Suppose the three
Boolean two-functions to be synthesized are

fi X) [ful X), fie( X)]
(X,-X.,X1 @ X2),
1,(X) = [for(X), foo( X)]
a ee ee
Ss(X) = [fa X), foo X)]
(X,-X;,X1 © X;)
where the symbols ® and - are respectively used to denote the Boolean

operations of additions (or) and multiplication (AND) between two
binary variables. Let k denote the field containing four elements. Let ¢

minimum function and every element of the field satisfies the equation
x 1. The four elements are shown below in terms of the primitive
element ¢, and their correspondence with binary 2-vectors is also pointed

In view of the correspondence between the binary 2-vectors and the
elements of K, any particular value of a Boolean 2-function will be con-
sidered as an element of A. For example, if

Addition and multiplication between the elements of K are shown in
the tables given below:

3 2 Q@ a3, 3
The sum of two elements a; and a; is obtained by adding the cor-
responding polynomials in ¢ modulo 2 (7, 7 = 0, 1, 2, 3). The product
of two elements is obtained by multiplying the corresponding poly-
nomials modulo 2 and modulo (¢ + ¢ + 1). From Section V we have

M1 = Mo = M3 = a, and my = ay, Me = ae and Mo; a;. There-
fore the five encoder Boolean 2-functions are given by

tion (AND) between two binary variables.
The truth table for the two Boolean functions ¢5; and ¢5» is given

The computation of this table will be illustrated by one example. Sup-
pose X, = 0, X2 = 1, X3; = 1. Then fi(X) = (0,1) = ae, fo(X)
(1,1) = az and f3(X) = (0,1) = ae. So gs(X) = ae + avaz + azar =
a. = (0,1). And so ¢3(X) = O and ¢g5.(X) = 1. The corrector sub-
system uses the outputs y; = (ya,v2), 7 = 1, 2, 3, 4, 5, of the encoder
subsystem as inputs. The final outputs 8; = (8a, Bi), 7 = 1, 2, 3 of
the corrector subsystem will be built up in several stages. From Section
Vib) = va + vn t+ v2 + ¥3 and db: = ys + ¥1 + avy + ayy;. At the
first stage the corrector subsystem synthesizes n; = avy,,7 = 2, 3. The
truth tables for (na, 72), 7 = 2,3 are given below:

The addition between the binary variables in (22) is modulo 2 addi-
tion. At the third stage, the three binary 2-tuples, « , «& and e,, which
are the first three coordinates of the error vector ¢ are synthesized as
Boolean functions of the 6’s. The part of the truth table in which at
least one of the e’s takes the value | is given below:

The truth table given below is computed from the rules given in
Section V. In the case of our example,

Suppose 6); 0, db: 1, do; 1 and 602 1. Then 6, a») and bo 3
Since 6s aoa, and 6, a: , the vector 6 is a scalar multiple of the
second row vector of 1/7. Therefore it follows that « ao, € a. and
6; = ay. Hence, ey QO, €12 O, €2) QO, €20 1, €3) Q and eg» l.

small and the Boolean functions required to be synthesized were chosen
to be very simple. Therefore the corrector subsystem would probably
require more equipment than the encoder subsystem. However, it
should be noted that the design of the corrector subsystem is independent
of the number of binary input variables and the nature of the original
Boolean functions. This design depends only on p and k. Therefore
when the number of input variables is large and the Boolean functions
required to be synthesized are complicated, the amount of equipment
required for the corrector subsystem may be small in comparison to
that required for the whole system. This is very desirable, since we
assume that the corrector subsystem is highly reliable. The example
shows how we can build up the logical design of the corrector sub-
system in any general case. However, the author believes that it. is
possible to build up much more economical corrector subsystems using

longer time to correct the errors. Such economical corrector subsystems

liable systems which correct faults of more than one block are also given.
Vil. ACKNOWLEDGMENTS

The author wishes to thank T. H. Crowley, D. B. Armstrong, J. P.
Runyon and B. A. Tague for many stimulating and useful discussions.

1. Armstrong, D. B., this issue, p. 577
Moore, kK. F. andShannon, C. E., Reliable Circuits Using Less Reliable Relays,
J. Frank. Inst., 262, 1956, pp. 191; 281
3. Tryon, J. G., Redundant Logic Circuitry, U.S. Patent No. 2,942,193
Von Neumann, J., Probabilistic Logics and the Synthesis of Reliable Organisms
from Unreliable Components, Automata Studies, Annals of Math. Studies
No. 34, Princeton Univ. Press, Princeton, N. J., 1956, pp. 44-98
Lofgren, L., Automata of High Complexity and Methods of Increasing Their
teliabilitv by Redundancy, Inf. & Cont., 1, 1958, p. 127
Hamming, R. W., Error Detecting and Error Correeting Codes, B.S.T.J., 29,
1950, p. 147
Bose, R. C., Mathematical Theory of the Symmetrical Factorial Design, Sank
hva, 8, 1947, p. 155
Zierler, N., A Class of Cyelic, Linear, Krror-Correcting Codes in p™ Symbols
Group Report 55-19, Lincoln Laboratory

Helix waveguide consisting of closely wound insulated copper wire
covered with an electrically absorptive or reactive jacket (Fig. 1) is a
good transmission medium for circular electric waves.’ In long distance
communication with waveguides it is useful as a mode filter, for nego-
tiating bends or particularly as the transmission line proper instead of a
metallic waveguide.”

The loss of circular electric waves decreases steadily with frequency
only in a perfect metallic waveguide.”" A similar situation prevails for
helix waveguide. Any deviations from a round and straight guide will
add to the transmission loss. At such imperfections power is converted
from the circular electric wave into other propagating modes and re-
converted. The mode conversion-reconversion effects increase the loss
and degrade the transmission characteristics.

* Parts of this paper were presented at the I.R.E. Professional Group on Mi
crowave Theory and Techniques Symposium, San Diego, California, 1960

where A, are the amplitudes of the TE,, or TM,, modes normalized
with respect to power and x, are their propagation constants. z is the ax-
ial coordinate. A square matrix with only the diagonal terms ky, = jha,

The perfect helix waveguide may be considered a perturbed metallic
waveguide. The boundary conditions for the tangential electric field

It is now expedient to transform from the normal modes of metallic
waveguide A, to the normal modes of helix waveguide L,

Its transpose is equal to its inverse. The diagonal terms y,,, of T are the
propagation constants of normal modes in helix waveguide. The circular
electric wave remains unaffected by this transformation :

It is now possible to consider an imperfect helix waveguide, perturbed
by curvature or cross-sectional deformations in the same terms. The
matrix AK’ then has off-diagonal elements also in the row and column
associated with the circular electric wave:

These new off-diagonal elements represent coupling between the circular
electric wave and other modes in an imperfect helix waveguide. They are
the coupling coefficients of generalized telegraphist’s equations. When
Maxwell’s equations for the imperfect helix waveguide are converted
into generalized telegraphist’s equations, it is found that the coupling
coefficients xo, in K’ for curvature and cross-sectional deformation are
independent of the wall impedance and the same as in metallic wave-
guide. Generalized telegraphist’s equations for deformed helix waveguide
are found in the Appendix.

The independence of the xo,’s from the wall impedance has an impor-
tant bearing on the coupling coefficients between the normal modes in
helix waveguide. If the new K’ for the imperfect helix waveguide is trans-
formed by the previous modal matrix L, there results

The new matrix I” has off-diagonal elements cp, which are caused by the
imperfection. They are the coefficients of coupling between the normal
modes of the perfect helix waveguide. In terms of the elements of K’
and L they are:

m=1
If this expression is squared and summed over all n, then, since the l’s
are elements of an orthonormal matrix, the resulting expression does not
contain any l’s:

It is recalled that the «xo, are coupling coefficients between normal
modes in an imperfect metallic waveguide. Consequently, the sum of the
squares of the coupling coefficients co, in an imperfect helix waveguide is
independent of the wall impedance and the same as in metallic wave-

The following statement will be proved: The average mode conver-
sion loss of certain kinds of random imperfections in helix waveguide
depends only on the sum of the square of all coupling coefficients.

To determine mode conversion, first of all, generalized telegraphist’s
equations have to be solved. With the elements of the perturbed IT
matrix the coupled line equations are

When only a circular electric wave of unit amplitude, /(0)
incident and the imperfections are small, a first-order solution at z
aL eL—u

The coupling coefficients are proportional to the geometric imperfec-
tion 6:

Con( 2) = C'o,6(z). ( 15)
Let the imperfections be a stationary random process with covariance
p(u) = (6(z)é6(z2 + u)) (16)

If, furthermore, the correlation between imperfections any appreciable
distance apart is small the covariance drops off very rapidly with in-
creasing argument. Then in the expression for (| /o(L) | ) the exponential
and the factor (lL. — uw) are constant for any weight of p(w), and one
obtains:

the spectral density S of the random imperfections 6 and the coupling
factors Co,

With (12) and (15), it may be concluded from (19) that the average loss
for circular electric waves in an imperfect helix waveguide is independent
of the wall impedance and the same as in metallic waveguide that has
the same geometrical imperfections.

The above derivation has assumed the covariance to drop off fast or
the correlation distance to be small. This is the case for any imperfection
created in the manufacturing process of the waveguide. Any manufac-
turing imperfections some reasonable distance apart are hardly corre-
lated to each other. The effects of manufacturing imperfections for helix
waveguide are therefore the same as in metallic waveguide. It is rela-
tively easy to determine tolerances for metallic waveguide.’ The above
rule lets these tolerances be valid for helix waveguide and the very in-
volved calculations for helix waveguide are not necessary.

Before accepting this rule the range of correlation distance for which
it is valid must be examined. As a typical example, the covariance has
been assumed exponential:

The average TE, loss at 55 kme has then been calculated for various
helix waveguides of 2-inch inside diameter as a function of the correla-

tion distance Ly.’ Fig. 2 shows for deformed helix waveguide the rms of
elliptical diameter differences which increase the TEo; loss by 10 per cent
of the loss in a perfect copper pipe. Up to a correlation distance of one
foot the curves almost coincide and indicate independence of the wall
impedance.

For a curved helix waveguide the range is even larger. Fig. 3 shows for
a curved helix waveguide the rms curvature under the same conditions.
Random curvature of up to a 10-foot correlation distance causes nearly
the same average TE, loss in helix waveguide and in metallic waveguide.

For a correlation distance larger than 10 feet there is an ever growing
dependence of curvature loss on wall impedance. But such curvature
distributions do not occur in the manufacturing process. They are, how-
ever, representative of laying tolerances when the waveguide is installed
with long bows to follow right of ways or the contour of the landscape.
A properly designed helix waveguide can tolerate much more laying

Fig. 3 — TE: loss in round waveguide with random curvature, 2-inch inside
diameter, at 55.5 kme.

Generalized Telegraphist’s Equation for Noncylindrical Helix Waveguide

Maxwell’s equations have been converted into generalized teleg-
raphist’s equations for curved helix waveguide elsewhere.” They have
been represented in terms of normal modes of the metallic waveguide as
well as in terms of normal modes of helix waveguide. In the former repre-
sentation the coefficients of curvature coupling between circular electric
waves and the modes of metallic waveguide were independent of the wall
impedance and the same as in metallic waveguide.

The same is true for any cross-sectional deformation in helix wave-
guide. To prove this, Maxwell’s equations will be converted into gen-
eralized telegraphist’s equations in terms of modes of metallic waveguide
for the deformed helix waveguide. This representation is different from
another analysis, where the equations are written in terms of the normal
modes of helix waveguide.”

where a is the nominal radius and the deformation 6 is for the moment
assumed to be only a function of ¢ of cylindrical coordinates (rgz), then
the boundary conditions at r = a, are

ug
The fields at r = a; can by series expansion be written in terms of the
fields at r = a. The boundary conditions then take the approximate
form:
OK
_—_ - ¥ ao
or

Maxwell’s equations in cylindrical coordinates for exponential time de-
pendence e”*' are

where » and ¢ are permeability and permittivity of the waveguide
interior.
The electromagnetic field is derived from two sets of wave functions,

| 0 oT 0 faT —
—{r—|]+. ; ea
r Lor or dg \rdg
where x is a separation constant, which takes on discrete values for the
various normal modes. The 7'-functions are normalized so that

| (flux T) (flux 77) dS = x | T’ dS = 1,
where S is the nominal cross section of the guide and the gradient and
flux vectors of 7 are defined by:
a. isc
grad, 7’ = —, grad, T = —,

To transform Maxwell’s equations into generalized telegraphist’s
equations the series expansions (32) are substituted for the field com-
ponents in (26) through (31). Certain combinations of the equations
are integrated over the nominal cross section, and advantage is taken of
the orthogonality relations (36) and (37).

MODE CONVERSION IN METALLIC AND HELIX WAVEGUIDE 625
(25) is substituted for EF. . In the second term (31) is substituted for F
Maas tes ; ot IT (x) OT (im

2r “rn “rr? alr ao aber
g 07 n| OF tn’ 2 Ok, 07 m)
+I | (1 — 6) — pe dy a | 6— — de,
0 0g or a 0 or or a
where h” = wpe — x’.
Add 07;,)/dr times (26) and 07’,
over the nominal cross section:

Add —0T (»)/rdg times (29) and d7;,,,;/dr times (30) and integrate over
the nominal cross section:

dl m] . , o f ry ’
—* Jwe Vim Xim] | H, Tm) aS. (43)
dz Js
lor the right-hand side of (43) integrate 7’,,,,; times (28) over the nom-

AL im
dz
dL tm Rim) , . a r 2 2 ™ rr rry
J } ay = =z J (n)JX{m] X{[n] / 6 T' tm F ke dg.
dz Wu Wu on 0
All the integrals are taken along the nominal circumference.
Alternatively, in terms of voltages and currents the equations are
more conveniently written in terms of amplitudes A of forward- and B
of backward-traveling waves. The mode current and voltage are related
to the mode amplitudes by
V, = VK, (A, + B,),
I
(A, — B,),
where K,, is the wave impedance:
. he, ’
Kin) = : K,, (52)
we Ryn)

If the currents and voltages in the generalized telegraphist’s equations

MODE CONVERSION IN METALLIC AND HELIX WAVEGUIDE — 625
are represented in terms of the traveling wave amplitudes, the system of
coupled equations can be written in matrix notation as

represents the amplitudes of metallic waveguide modes. The square
matrix

626 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1961
Let the first element of A be the amplitude Ao, of a circular electric
wave. The propagation constant of this wave is

Morgan, 8S. P., Mode Conversion Losses in Transmission of Circular Electric
Waves through Slightly Noneylindrical Guides, J. Appl. Phys. 21, 1950, p.
329.

towe, H. EK. and Warters, W. D)., Transmission Deviations in Waveguide Due
to Mode Conversion: Theory and Experiment, Proce. I..E., 106, Pt. B,
Suppl. 18, 1959, p. 30

Schelkunoff, S. A., Conversion of Maxwell’s Equations into Generalized Teleg
raphist’s Equations, B.S.T.J., 34, 1955, p. 995

In a perfect helix waveguide the circular electric wave loss is increased
by eddy currents, finite pitch of the helix, radiation through the wire spac-
ing and effects of the wire coating. Only the contributions from eddy cur-
rents and pitch are large enough to limit wire size and spacing.

Experimental helix waveguides have tilted turns. These tilts cause cou-
pling between circular electric and unwanted modes. From the coupling
between modes in curved and in offset helix waveguide, the coupling in a
helix waveguide with tilted turns is found. For helix waveguide with slightly
irregular winding of arlitrary form, generalized telegraphist’s equations
are derived.

Tilts and other irregularities in the winding increase the circular elec-
tric wave loss. The average increase is a function of the covariance of irregu-
larities. Winding tilts with an exponential covariance and an rms value of
0.6° increase the TK, loss in 2-inch inside diameter wavequide at 55 kme
at the most by 10 per cent of the loss in a perfect copper pipe with smooth
walls. Present fabrication procedures insure a smaller wire tilt than this.

Helix waveguide consisting of closely wound insulated copper wire
covered with an electrically absorbing or reactive jacket is a good trans-
mission medium for circular electric waves.’ In long distance communica
tion with circular electric waves it is useful as a mode filter, for negotiat
ing bends or particularly as transmission line proper instead of a plain
metallic waveguide.

As in metallic waveguide, the loss of circular electric waves decreases
steadily with frequency only in a perfect helix waveguide. Any devia-
tions from a round and straight guide and from a uniform and low pitch
will add to the loss of circular electric waves.

Deviations from straightness and deformations of the cross section of
helix waveguide have been analyzed before and their effect on circular
electric wave transmission has been determined.’ When these imperfee-

tions are caused in the manufacturing process they are statistically dis-
tributed over the guide length with a small correlation distance. Then
they add nearly the same average loss to the circular electric wave

straightness and for cross-sectional deformations are therefore the same
for helix waveguide as they are for metallic waveguide.

Deviations of the winding from a low-pitch uniform spiral are imper-
fections peculiar to the helix waveguide. Their effect on circular electric
wave transmission will be analyzed here and tolerances on the winding
of the helix waveguide for low-loss transmission will be determined.

Helix waveguide is usually wound from round wire with an insulating
layer. Even when such a helix is perfectly accurate and uniform its differ-
ences from a smooth metallic waveguide add to the circular electric
wave loss. The various effects can be listed as follows:

The circumferential wall currents of circular electric waves are uni-
formly distributed in a smooth wall. In the spaced wires of the helix
waveguide their distribution is nonuniform. The heat loss is therefore
increased over the smooth wall loss. In Fig. 1 this loss increase is plotted
over the spacing for a wire size small compared to the wavelength, using
Morrison’s calculations.

2.2 Pitch!
If the helix of radius a is closely wound from a single wire of insulation
diameter D, then the pitch angle y is given by
D
tany = 5—.
2a
If for a faster manufacturing process n wires are wound simultaneously,
nD

The wall currents of circular electric waves are strictly circumferential.
In the helix waveguide their path is disturbed by the finite pitch. Power
of circular electric waves is dissipated into the wall impedance Z, which
the surrounding jacket presents to the waveguide interior. The added
circular electric wave loss due to finite pitch is

where Zy = VWyu/e is the wave impedance and k = w/e the propaga-
tion constant of free space. k,, is the mth root of J;(.c) = 0, and h,,

A reactive jacket will not dissipate any power. A helix waveguide,
designed for transmitting the circular electric wave around bends, has a
quarter wave jacket with a very large wall impedance.” In this case, to
keep a, low, ¥ has to be chosen small.

Even though the helix is closely wound the wires are spaced by the
wire insulation. With the electric field of circular electric waves parallel
to the wires, the space between acts as waveguide below cutoff. Being
short, this cutoff waveguide will transmit some circular electric wave
power, which is then absorbed by the jacket. The circular electric wave
loss caused by this power absorption has been investigated for various
forms of wire cross section.’ For round helix wires this loss is so small
compared to the eddy current losses of Fig. 1 that it may be entirely
neglected for any wire spacing. Consequently the increase in eddy current
losses, rather than the power dissipation through the gaps, limits the
wire spacing.

The insulating layer of the helix wires adds to the circular electric
wave loss in two different ways. Its dielectric constant tends to concen-

trate the electric field into the layer. Thus the wall currents and the wall
current losses are increased. In addition, the finite loss factor of any
insulating material causes dielectric losses in the small but finite electric
field of the circular electric wave. Both of these effects can be calculated
with a sufficient approximation from attenuation formulas for the round
waveguide with a dielectric lining.”

Again, it is found that the effects of the insulating layer are so small
compared to the eddy current losses that they may be neglected.

Number of wires, wire size, and wire spacing through insulation are
therefore determined by the pitch effect of (2) and the eddy current loss
of Fig. 1. To speed the winding the numbers of wires should be large.
To increase the effects of a reactive or resistive jacket on unwanted
modes the wires should be widely spaced.” The increase in circular elec-
tric wave loss from Fig. | and equation (2) sets a limit, however, to num-

The preceding discussion has considered only the loss in a perfectly
wound helix waveguide. A practical helix will not have perfectly uniform
windings. One imperfection in particular has been most notable in re-
search models of helix waveguide made by Bell Telephone Laboratories.
This imperfection is tilts in the winding.

Aside from the finite pitch, a single turn of the helix is usually not in a
transverse plane, but is slightly inclined and forms a small angle @ with
the axis. Even an improved winding method with an automatic feed
control has not entirely eliminated this inclination.”

Such inclined helix turns give rise to mode conversion. There is a simple
way to analyze circular electric wave propagation in helix waveguide
with nonuniformly tilted winding, in which the results of previous cal-
culations are used. Consider a perfect helix waveguide, a section of
which between 2 0 and z L has been deflected in an arbitrary
manner by 2x(z), as shown in Fig. 2. With this deflection is associated a
change of guide direction dx/dz and for gentle deflections a curvature
1/R dx (dz.

One way to calculate propagation through this deflected section is to
use the formulas for wave propagation in the curved helix waveguide’
and evaluate them for the curvature distribution d°r/dz°. Thus, when

is curvature coupling between a circular electric mode m and the modes n

Fig. 2 — Curved helix waveguide as superposition of offset and tilted winding

of first order circumferential dependence (i.e., TE), and TM,,). The
coupling coefficient is

where k, k,, and h,, have the same meanings as in (2). k, is the radial
propagation constant of the coupled mode n and h, Vk? — (k,?/a?)
is the axial propagation constant of the coupled mode n. NV, is a normali-

A mode m of unit amplitude incident at z = 0 converts power in the
deflected section to the coupled modes n. For gentle deflection the ampli-
tude and phase of mode n at z = L is given by

Another way to calculate propagation through the deflected section
is to consider it as a continuously offset waveguide with a continuously
varying tilt 6 of the winding. Both the offset x and the tilt 6 = dxr/dz
will then cause coupling between a circular electric mode m and modes n
of first-order circumferential dependence. The coupling coefficient for
offset has been calculated® before:

Equation (10) should give the same result as (6). Integrating by parts
brings (10) into a form that can be directly compared with (6):

Equations (11) and (6) can only given identical results when
Co ; C,
i me @
(hyn — h,)? hn — h,
Hence the coupling coefficient for tilted winding is
Co
Mi, = (h, —h,)Ce+ .
h,, a hin
Substituting from (4) and (8) into (12), and from (12) into (9):

With these coupling coefficients, generalized telegraphist’s equations

The summation in (14) is to be extended not only over all the modes n
but also over their two polarizations according to the orientation of 6;
6 may not only be in the plane of Fig. 2, it can also be perpendicular to
that plane.

From (14) the loss which is added to the mode m by a tilted winding
may be calculated. With

The obliquely wound helix of the preceding section is just a special
case of a general irregular winding. In Fig. 3 a turn of such an oblique
helix has been drawn in more detail. Aside from a small pitch the wire
follows the curve

around the circumference. Its direction deviates by y from the trans-
verse direction, where

» SIN pe. (18)
A summation of cos pg would only add identical terms with different
polarization; it has been omitted from (18). The boundary conditions

where Z is the wall impedance which the outside jacket presents through
the helix to the waveguide interior.
Wave propagation in such an irregular structure is best analyzed by

into generalized telegraphist’s equations’ for the boundary conditions
(19) and (20).

An appropriate form of representation is in terms of normal modes of
the perfect helix waveguide. With two sets of wave functions

the normal mode fields of the perfect helix waveguide are the individual

Equation (32) is the characteristic equation for the perfect helix wave-
guide. Its roots k, = x,a determine the propagation constants
h, =k — Yn

of the normal modes of the perfect helix waveguide.
The transverse fields of the normal modes are orthonormal in that

The integral in (34) extends over the cross section of the guide; 6,,,, is
the Kronecker symbol.

The z-dependence of the voltage and current coefficients in (29) is
found by substituting the sums of (29) for the transverse field com-
ponents into Maxwell’s equations.

In the line integral along the boundary, F, from the boundary condition
(20) is substituted. In the surface integral over the cross section, (30) is
substituted for F, . Subsequently the boundary conditions of the perfect
helix waveguide may be used to simplify (38) to first order in y:

Expression (30) for H, holds only for the normal modes of the perfect
waveguide. It has been obtained by differentiating the sum (29) for EF,
in (23). The individual terms of this sum vanish at r = a, while in the
present case according to (19) EF, has a finite value there. Hence the
sum (29) for £, is nonuniformly convergent and differentiation makes it
diverge. To replace H, in (41), substitute in (23) FE, from (29), multiply

These equations represent an infinite set of coupled transmission lines.
For the present purpose it is more convenient to write these transmision-

traveling wave amplitudes, the following equations for coupled traveling

lor a tilted winding with y from (17),
ew jet.
In this case circular electric waves interact only with modes of first
circumferential order (p 1). Helix irregularities of higher order in
will cause coupling to modes of correspondingly higher circumferer tial

The design of a helix waveguide is started by selecting wire size and
spacing. A tolerable amount of added TE, loss is specified, and with
lig. 1 and equation (2) wire size and spacing are determined.

If, for example, eddy current losses in the helix should not be more
than 10 per cent of the loss in a guide with smooth walls, then from Fig. 1
the ratio of wire diameter to insulation diameter should be

To determine the actual wire size with (2) the wall impedance has to
be specified. Different applications of the helix waveguide require differ-
ent values for the wall impedance. A typical and also very critical ex-

ample is the helix waveguide for intentional bends. In this case, by sur-
rounding the helix with a quarter-wave jacket and a metallic shield the
wall impedance is made very high.

Z=j Kn tan k,, 6, (49)
We A
where ¢, isthe permittivity of the jacket, h,° = avV/ wye, — h,? the radial
propagation constant in it, and 6 the relative thickness.
For a quarter-wave jacket of a low-loss material the wall impedance is
real and approximately

Fiber glass laminated with epoxy resin has a relative permittivity at
millimeter wavelengths of ¢./e9 = 4 — j(0.04). The relative wall imped-
ance from (49) is then Z/Z» $1.4. In 2-inch inside diameter wave-
guide with smooth walls, the TE, loss at 55.5 kme is aga = 2.77 X 10°
Less than 10 per cent of this figure is added to the TE loss in the
present example when the pitch is

soe <4x 10°. (51)
a
No. 37 wire (AWG) with a heavy Formex coat has d 0.0045 and
D = 0.0054. It very nearly satisfies conditions (48) and (51) when the
helix is wound from one wire only (n = 1). Lower wall impedance values
such as are used for helix mode filters or all helix guide would not re-
quire as low a pitch as (51).

For the winding process, tolerances for irregularities must ‘be specified.
In (15) the added loss is expressed in terms of the 6 of a tilted winding.
The loss caused by higher-order irregularities can also be determined by
(15) when the corresponding coupling coefficients (47) are substituted.

In the present problem, however, the irregularities are not known, but
at best some of their statistical properties are known. Equation (15)
can then be used to express the statistics of the loss in terms of the sta-

For an oblique winding @ is assumed to be a stationary random process
with covariance R(u) and spectral distribution S(£):

In (52), <2> is the expected value of xr.
Taking the expected value on both sides of (15) the average added
loss is obtained in terms of the covariance R(u):

lor a mere estimate the covariance is assumed to be exponential to
simplify the calculation:
TS

S(£) So (56)
AS : aaa? 8)
l + Let
with S(£) nearly flat with spectral density So for mechanical frequencies
smaller than & = 1/L,. Lo may be regarded as the cutoff mechanical
wavelength according to (56) or as a correlation distance according to
(55).
The average added loss is for L >> Lo
’ P,(2r — Aa,Lo) — Q, AB Lo =
<a,> TiN > 7 93 = ° (4)
n AB,? Le + (47 = Aa, lL)?

To evaluate (57) the characteristic equation (32) of the perfect helix
waveguide has to be solved for all the coupled modes n, and propagation
constants and coupling coefficients have to be calculated.

The helix waveguide for intentional bends is again a typical and critical
example. In this case Z x at the design frequency and the characteris-
tic equation (32) simplifies to

The roots kyo of JpyiCc) = Oand J,4(2) = 0 are good approximations
for the roots of (58). With &, Knog & w, where

the approximations can be sufficiently improved. The coupling coeffi-
cients in this case are given by

With these relations, (57) has been evaluated for a helix waveguide
with a nonuniformly tilted winding (p = 1).

Fig. 4 shows the rms value of @ as a function of the correlation distance
Ly for an added average TE ; loss of 10 per cent of the loss in a copper
pipe with smooth walls. The waveguide diameter is 2 inches and the
frequency 55.5 kme. The tolerance is most critical for a correlation dis-
tance of 1 inch. But even then an rms tilt of 0.6° can be tolerated. In
experimental models of helix waveguide the maximum tilt has, with
some care, been kept below 0.3°.

Ina perfectly wound helix waveguide the circular electric wave loss is
significantly increased only by the eddy current losses in spaced wires.
The finite pitch contributes to the circular electric wave loss only when
the wall impedance is very high or when the helix is wound from more
than one wire.

Fig 4. — TE», loss in helix waveguide with random tilt of winding, 2-inch inside
diameter, at 55.5 kme. Design for intentional bends with infinite wall impedance.

increase in circular electric wave loss can be calculated. In a 2-ineh
inside diameter helix waveguide at 55.5 kme an rms tilt angle of 0.6
adds at the most 10 per cent of the loss in a perfect copper pipe to the
average TH loss. In experimental models of helix waveguide the maxi-
mum tilt deviation has been kept below 0.3°.

. Morgan, 8. P., and Young, J. A., Helix Waveguide, B.S.T.J., 35, 1956, p. 1347
2. Unger, H. G., Helix Waveguide Theory and Application, B.S.T.J., 37, 1958,
p. 1599.
3. Unger, H. G., Noneylindrical Helix Waveguide, B.S.T.J., 40, 1961, p. 233.

}. Morrison, J. A., Heat Loss of Circular Electric Waves in Helix Waveguides,
I.R.E., Trans., MTT-6, 1958, p. 173

. Katsenelenbaum, B. Z., Attenuation of Ho. Modes in a Helical Waveguide,
Radiotekhnika i Elektronika 4, 1959, p. 428.

3. Unger, H. G., Circular Electric Wave Transmission in a Dielectric-Coated
Waveguide, B.S.T.J., 36, 1957, p. 1253

. Beck, A. C. and Rose, C. F. P., Waveguide for Circular Electric Mode Trans
mission, Proc. I.f.E., 106, Pt. B, Suppl. 13, 1959, p. 159.

. Schelkunoff, S. A., Conversion of Maxwell’s Equations into Generalized
Telegraphist’s Equations, B.S.T.J., 34, 1955, p. 995.

. Rowe, H. E. and Warters, W. D)., Transmission Deviations in Waveguide Due
to Mode Conversion: Theory and Experiment, Proc. I.E.E., 106, Pt. B,
Suppl. 13, 1959, p. 3.

D. B. Armstrrona, B.A., 1940, University of Toronto; M.S., 1951, and
Se.D., 1955, Massachusetts Institute of Technology; Bell Telephone
Laboratories, 1954—. He has been engaged in research in switching prob-
lems, which have included simulation and economic studies of telephone
systems, studies of central office systems and reliability studies of digital
systems. Member Sigma Xi.

D. W. Bonus, B.S. in E.E., 1937, New York University; Bell Tele-
phone Laboratories, 1935—. He has been concerned with problems of
electric shock and the protection of communication facilities from for-
eign potentials. This has included field investigations of lightning be-
havior, design of surge measuring devices and laboratory surge testing of
apparatus. Member A.I.E.E.

G. D. Boyp, B.S., 1954, M.S., 1955, and Ph.D., 1959, California
Institute of Technology; Bell Telephone Laboratories, 1959—. He is en-
gaged in optical maser research.

DonaLp B. Currriss, B.S. in E.E., 1959, Newark College of Engineer-
ing; Bell Telephone Laboratories, 1951—-. Until 1959 Mr. Cuttriss was
a member of the component development department working on design
and development of voice-frequency laminated-core inductors and sémi-
conductor field-effect devices. In 1959 he transferred to transistor de-
velopment and he has been concerned with development of diffused-base
germanium transistors and with methods of producing high-quality gal-
lium arsenide. Member Tau Beta Pi.

A. GARDNER Fox, 8.B., 1934, and 8.M., 1935, Massachusetts Institute
of Technology; Bell Telephone Laboratories, 1936—. His early work was
in development of mobile radio transmitters, early radar development
and general waveguide research. Since 1944 he has been engaged in de-
sign of radio frequency amplifiers and in research on millimeter waves.
He heads a group specializing in microwave physics. Fellow I.R.E.

James P. Gorvon, B.S., 1949, Massachusetts Institute of Technology,
M.A., 1951, and Ph.D., 1955, Columbia University; Bell Telephone Lab-
oratories, 1955—. His research in quantum electronics has involved work
on molecular beam masers, paramagnetic resonance and solid state ma-
sers. Member American Association for the Advancemens of Science,
American Physical Society, Sigma Xi.

Puitip A. Gres, B.S.E.E., 1956, Carnegie Institute of Technology;
Bell Telephone Laboratories, 1956—-. His work in Systems Engineering
has included statistical and economic optimization studies of outside
plant facilities. Member Eta Kappa Nu, Phi Kappa Phi, Tau Beta Pi.

TinGYE Lt, B.Sc., 1953, University of Witwatersrand (South Africa) ;
M.S., 1955, and Ph.D., 1958, Northwestern University; Bell Telephone
Laboratories, 1957—. He has been engaged in studies of microwave an-
tennas and microwave propagation. Recently he has been primarily con-
cerned with work on optical masers. Member I.R.E., Eta Kappa Nu,
Sigma Ni.

H. C. Marre, B.S., 1949, and Ph.D., 1956, California Institute of
Technology; M.S., 1950, Massachusetts Institute of Technology; Bell
Telephone Laboratories, 1959-60; associate professor of electrical en-
gineering, California Institute of Technology, 1953—-. While at Bell Lab-
oratories during a year’s leave of absence from Cal Tech, Mr. Martel was
engaged in visual and acoustics research related to signal coding and
detection. Member I.R.E., Sigma Ni, Tau Beta Pi.

Max V. Maruews, B.S., 1950, California Institute of Technology;
M.S., 1952, and Se.D., 1954, Massachusetts Institute of Technology;
Bell Telephone Laboratories, 1955-—-. He has specialized in acoustics re-
search in speech transmission and has been especially concerned with
simulating speech experiments on a digital computer. Member Acoustical
Society of America, I.R.E., Sigma Xi.

R. L. Peek, Jr., A.B., 1921, Columbia College; Met.Ié., 1923, Colum-
bia School of Mines; Bell Telephone Laboratories, 1924——. His early
work was in materials testing. In 1936 he turned to apparatus develop-
ment involving coin collectors, electromagnets, relays and switches. Dur-

spring relay development. Since 1951 he has been in charge of a group
studying new electromagnetic devices. Member A.1.E.E.

D. K. Ray-Cuavupuurt, B.Se., 1953, Presidency College, University of
Caleutta (India); M.Se., 1955, University of Caleutta; Ph.D., 1959, Uni-
versity of North Carolina; instructor in statistics, Case Institute of

Technology, 1959-60; Bell Telephone Laboratories, summer, 1960; visit-
ing assistant professor of statistics, University of North Carolina, 1960
He has been engaged in research on problems of constructing experi-

mental designs in statistics, sampling theory and error-correcting codes.
Member Institute of Mathematical Statistics.

ERLING D. Sunpe, Dipl. Ing., 1926, Technische Hochschule, Darmstadt,
Germany; American Telephone & Telegraph Co., 1927-34; Bell Tele-
phone Laboratories, 1934—. He has made theoretical and experimental
studies of inductive interference from railway and power systems, light-
ning protection of the telephone plant, and fundamental transmission
studies in connection with the use of pulse modulation systems. Author
of Karth Conduction Effects in Transmission Systems, a Bell Laboratories
Series book. Senior member I.R..; member American Association for
the Advancement of Science, A.I.E.E., American Mathematical Society.

Hans-GrorG UnGer, Dipl. Ing., 1951 and Dr. Ing., 1954, Technische
Hochschule, Braunschweig (Germany); Siemens and Halske (Germany),
1951-55; Bell Telephone Laboratories, 1956—. His work at Bell Labora-
tories has been in research in waveguides, especially circular electric wave
transmission. He is now on leave of absence from Bell Laboratories as
professor of electrical engineering at the Technische Hochschule in
Braunschweig. Senior member I.R.I.; member German Communication
engineering Society.

The Bell System Technical Journal  1961-03: Vol 40 Iss 2