BSTJ 41: 2. March 1962: Probability Distribution for the Phase Jitter in Self-Timed Reconstructive Repeaters for PCM. (Aaron, M.R.; Gray, J.R.)

Probability Distributions for the Phase

Jitter in Self-Timed Reconstructive

Repeaters for PCM

By M. R. AARON and J. R. GRAY

(Manuscript received August 25, 1961)

Probability distributions for the timing jitter in the output of an idealized
self -timed repeater for reconstructing a PCM signal are approximated.
Primary emphasis is focused on self-timed repeaters employing complete
retiming. In this case the probability distribution for the timing jitter reduces
to the computation of the phase error in the zero crossings at the output of
the tuned circuit excited by a jitter-free binary pulse train. It is assumed
that the tuned circuit is mistuned from the pulse repetition frequency, and
the individual pulses are either impulses or raised cosine pulses. Both
random pulse trains and random plus periodic trains are considered. In
general, the probability dislribidions are skewed in the direction of increasing
phase error. The approach to the normal law in the neighborhood of the
mean when the circuit Q becomes arbitrarily large is demonstrated. Results
obtained from the analytical approach are compared with two computer
methods for the case of random impulse excitation of a tuned circuit char-
acterized, by a Q of 125 and mistuning of 0.1 per cent. Excellent agreement
between the three techniques is displayed. For no mistuning and raised
cosine excitation two methods for computing the phase error are given and
numerical results obtained from both techniques agree closely.

Some attention is given to an idealized version of a reconstructive repeater
employing partial retiming and it is shown that the timing performance of
such a repeater for random signals is very much inferior to the completely
retimed repeater.

I. INTRODUCTION

Over the past several years the problem of maintaining pulse spacing
within very close bounds in PCM transmission has received considerable
attention both theoretically and experimentally. The effects of timing
jitter in degrading repeater performance, in introducing distortion in

503

504 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1962

the decoded analog signal, and in enhancing the difficulty of dropping
or adding several pulse trains in time have been documented. 1 " Sources
of mistiming in a self-timed reconstructive repeater are well catalogued
and include: noise, crosstalk, mistiming, finite pulse width effects, and
amplitude to phase conversion in nonlinear devices. The first four of
these sources have been considered in various analyses of timing jitter
in self -timed and separately-timed PCM repeaters. Amplitude to phase
conversion in nonlinear circuits has received attention primarily from
the experimental viewpoint.

The majority of the theoretical work to date has been concerned with
timing errors in self-timed repeaters when the timing-wave extractor is
a simple tuned circuit. For a random pulse train exciting the tuned circuit
in the presence of noise and mistiming, results have been obtained for
the mean displacement and the standard deviation of the zero crossings
from their ideal location. This analysis is appropriate to repeaters em-
ploying complete retiming. These time displacements can also be
considered as phase errors and we will use this terminology in what
follows. If the probability density function for the phase error is normal,
the mean and standard deviation are sufficient for a complete statistical
description. In this paper we will show that in general the probability
density function is not normal, and is inherently unsymmetrical about
the mean.

An approximation to the probability density and the cumulative
distribution for the phase error at the output of a mistimed resonant
circuit will be derived for both random and random plus periodic pulse
trains. A completely random pulse train is defined to be one in which
pulses and spaces are equally likely. The individual pulses of the binary
pulse train are assumed to be jitter free and are either impulses or raised
cosine pulses. The approach to the normal law when the circuit Q is
large is demonstrated. For a value of Q of 125, and a mistiming of 0.1
per cent from the pulse repetition frequency a comparison of numerical
results obtained from the analytical approach and two computer methods
is made. Agreement among the three approaches is excellent.

Our plan of attack is to place all of the manipulations required to
specify the tuned circuit response to the most general pulse trains in
the Appendix and concentrate on most of the probabilistic notions in
the main body of the paper. Appendix A covers the response of the
tuned circuit to a random or random plus periodic binary pulse train of
arbitrary pulse shape, and Appendix B is concerned with the specializa-
tion to raised cosine pulses. Section II of the text deals with the terminol-
ogy required, covers the tuned circuit response to impulses, and briefly

PHASE JITTER IN PCM REPEATERS 505

summarizes the results of Appendices A and B. In .Section III, the
probability density function for the phase error is derived. Section IV
is devoted to the cumulative distribution function and Section V alludes
to the semi-invariants that are required in the evaluation of the density
and cumulative distribution functions. These semi-invariants are de-
rived in Appendix C. The approach of the probability density function
for the phase error to the normal law as the circuit Q becomes arbitrarily
large is displayed in Section VI with the algebraic support relegated to
Appendix D. The comparison of numerical results mentioned previously
with other computer approaches is made in Section VII. For zero mis-
tuning, but finite pulse width excitation, it can be shown that the proba-
bility distributions for the phase error can be related directly to the
probability distribution for the timing wave amplitude. This is demon-
strated in Section VIII. A discussion of further numerical results is given
in Section IX. We consider an idealized model of a partially retimed
repeater in Section X for purposes of comparison with the results of
Section IX. A wrap-up of the procedures, results, and future work
concludes the paper.

II. RESPONSE OF THE TIMING CIRCUIT

Before we go on to the general equation for the phase error due to
finite pulse width and mistiming, we will specialize to impulse excitation
of a simple tuned circuit characterized by its Q and mistiming from the
pulse repetition frequency. This should provide the casual reader with
some feel for how the more general equation for the phase error arises
without going through the detailed manipulations of Appendices A and
B. The procedure adopted in the analysis to follow is equivalent to that
of II. E. Rowe. 2

Assuming the input to the timing circuit to be a train of jitter-free
unit impulses occurring at random with spacing T, the excitation may
be represented as

7i==e

f(t) = Ha n 8(t - nT), (1)

where a„ is a random variable taking the values or 1 with probability
5,* 8(1 — nT) is a unit impulse whose time of arrival is nT, and the
spacing T is the reciprocal of the pulse repetition frequency f r . For a
parallel resonant circuit the impulse response is given by

* Unless otherwise specified, the ease of equal likelihood will he considered in
all calculations.

506 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1962

h(t) = A e- {r,Q)f ° l cos (2tU + <p), (2)

where

Q = 2irf„RC, and <p = tan -1 ^.

Here / is the natural resonant frequency as disting uished from the
steady-state resonant frequency/. = (l/2r)y/V LC. Combining (1)
and (2), the total response to all impulses occurring in time slots up to
and including the one at t = may be written as

F{t) = A n f,a n e- {vlQ)f ° {t -" T) cos [2n/.(« - nT) + *]. (3)

n =— oo

This expression gives the output of the timing circuit for values of t
in the interval between t = and the arrival time of the next impulse.
Rewriting (3) in the form of a carrier with both amplitude and phase
modulation we get

F(t) = AVx* + y* e7 {TlQ)f ° l cos [2irf t + tp + 0), (4)

where

= tan"^

x = f; a H e - { " Q)/ "" r cos 2irf nT, and

y = jra lie - WQ)/u " r sm2Tf nT.

In the above x and y represent the in-phase and quadrature components
of the response. If the tank could be tuned exactly to the pulse repetition
frequency (/„ = f r = l/T), then the phase modulation would disappear
and the amplitude modulation would be dependent on x alone. In prac-
tical applications this is not possible and the phase shift does occur.
If we denote the fractional mistiming Af/f r by k, we may write f in
terms of f r as follows

/ =/r(l +fc).

In this case (4) becomes, neglecting k with respect to unity in the
exponential term

PHASE JITTER IX PCM REPEATERS 507

F(t) = AV^Ty - <T (W0)/ '' cos [2x/ r (l + k)t +v + 0], (5)
with

00

a„ fi cos 2irkn,

n=0

CO

// = E«»e" (T/0)n sin27rA-n,

n=0

and

= tan -1 y/x.

To illustrate the relationship between the timing deviation td and
the phase error 0, it is assumed that repeater delays have been adjusted
so that the timing wave supplied to the regenerator in the absence of
mistiming is properly aligned with the signal impulses in the information-
bearing channel. In this case, the negative-going zero crossing occurring
ideally at t„ = T/4 determines the instant of regeneration. When mis-
timing is present this zero crossing is displaced such that it occurs at
the instant tj = T{\ - 6/2*). The difference t - U will then give the
timing deviation which, expressed as a fractional part of the pulse
spacing, is

r~2V (6)

From (G) and the definition of 0, the phase error corresponding to
the timing deviation is related to the random variables x and y by

6 = tan" 1 ^. (7)

In deriving (7) it should be recalled that only the incidental approxima-
tion k « 1 has been made. When we consider a binary pulse train in
which the pulses representing the binary "one" are of arbitrary pulse
shape, it is necessary to make other approximations to arrive at a tract-
able expression for the phase error. Furthermore, the excitation en-
compasses the infinite past as well as the tails of succeeding pulses to
accommodate driving pulses that may overlap or are not time limited.
The most general result given by (59) is an extension along two lines
of Howe's relationship for the timing jitter in the output of the tuned
circuit due to mistiming and finite pulse width. First, the results are
applicable to arbitrary pulse shape. Secondly, our relationship for the

508 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1962

phase error is based on a different approximation in the case of finite

width pulses.

In appendix B we specialize to the case of raised cosine pulses in order
to make use of some of Howe's results. For this case the phase error is
given by (73) and takes the form

= ^+0, (8)

X +

where a, b, and c are constants that depend upon Q, k, and the pulse
width T/s of the raised cosine pulse, x and y are correlated random vari-
ables that depend upon Q, k, and the pulse pattern. They arc defined
below (5) with the additional constraint that a„ = 1 when we consider
finite width pulse; i.e., a pulse definitely occurs at the origin. In our
notation, a positive phase error corresponds to the zero, crossing of
interest occurring prior to the reference. The largest pulse width we
consider is 1.57. This avoids the necessity of considering the effect of
the presence or absence of a following pulse on the negative-going zero
crossing of interest. Similarly, for positive-going zero crossings we do
not have to use special methods for considering the occurrence or non-
occurrence of a preceding pulse. This is not a serious analytical restric-
tion, since larger pulse widths can be handled by the machinery provided
in Section A-4. As a practical matter in the design of a self-timed recon-
structive repeater for operation in a long repeater chain, wider pulses
would introduce intolerable phase jitter. In the following, we will also
neglect the constant c in (8), since it is independent of pulse pattern
and can in principle be compensated for in either the timing path or
information-bearing path in a self-timed reconstructive repeater.

III. PROBABILITY DENSITY FOR THE PHASE ERROR

3.1 Preliminaries

From the above, the random variable of interest is

e = l+? vj. (9)

.i- + b xi

To determine the probability density p(6) or the cumulative distribu-
tion F(0), we consider the joint probability density of the correlated
random variables a;, and //i , p(.r, ,//,). F{6) = Pr (yi/xi) ^ d), which
may be written

F(e) = f dxi f dy&ixitVi) + / dxi / dyip(xi , yi) .

PHASE JITTER IN PCM REPEATERS 509

Differentiation of F(6) with respect to 6 plus rearrangement yields

p(B) = I xip(x lt 6xi) dxi + f xip(~xi,-dxi)dxi. (10)

•mi Jo

Therefore if p(.v\ , //i) is known, p(0) can be determined by integration.
As is typical of this class of problems when .i\ and iji are not correlated
normal variables, the exact determination of p(.i\ , y\) is rarely obtain-
able. Therefore, we find it essential to proceed along approximate lines.
We can write the characteristic function <p(u,v) for p(x x , iji) as

pfrvO = [j^[jii^ {UI ^ yi) vU-uyv). (ID

If we take the partial derivative of (11) with respect to u, evaluate it
at u = —6r, divide both sides by 2iri, and integrate over r from — x to
* , we get

L f t . * - h [ '"• f " J ' L" W-'-* W. - ») ■

llCl J—x Oil ii=— 9b -7T J-x J— qo ■/— eo

When we interchange the order of integration to integrate over v first,

i r aj^o ^ _ r r/ri r dmMyj _ dXi) (xm)

l-Kl J-oc Oil ii= —Bv J-tc J— co

where 8(i/i — d.Vi) is the Dirac delta function. Integration over iji then
results in

i r«f0v)i (/ ,, = r,,„ ( ,,, ftl , )rf ,,

27T?. J- oo dll I H =-9r J- oo

= / xip(xi,0xi) dxi — / Xip(—xi,—6xi)dxi.

(12)

A comparison of (10) with (12) reveals that they are equal provided
that X\ is always positive, in which case p(—.v, , — ftci) is zero. Under
this condition''

d<p(u.v)

Sirl J-x an !,= _«,.

In the following we will use (13) to approximate p(6); before doing
so we make a few remarks about the range of the random variables Xj
and 6.

The result in (13) is given as an exercise for the reader on p. 317 of Ref. ft.

510 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1962

3.2 Minimum Values of xi and yi

Our comments in this section will largely be confined to the case of
impulse excitation in which case x x = x and yi = y, where x and y are
denned following (5). From the definition of x it can be seen that it
attains its minimum value for the set of a n = 1 in which the argument
of cos 2irkn is in the second and third quadrants (modulo 2tt). With this
pulse pattern it is easily shown that

ft sin 2xke ( 1 + e ) __ _ 2 e

Zmin -= ^ _ e - WkQ )^i _ 2 p cos 2tt/c + 2 ) " " V (1 - e-(*/2*o>)

where = e -(x/e> and # = average value of y (from Appendix D).
For the values of k and Q that we consider, namely kQ less than about
0.1 and Q =■ 100, an excellent approximation for x m i n is

When /cQ is fixed at 0.1,

^ 4/vQ 2 2.5,
T

and for Q = 100, x min = -0.005. The ratio x mia /x, where x = average
value of x, can be shown to be

-(T/4A.-Q)

x

which for kQ = 0.1 is -0.00016, or very close to zero. Based on un-
published work of one of the authors, the probability of x/x of even going
negative is so remote as to be completely unimportant and decreases
with increasing Q for kQ fixed.

Another interesting way of looking at the probability of x becoming
negative is to consider the probability of pulses occurring in the first
quadrant of the argument of cos 2-irkn to constrain the minimum value
of x to zero. This can occur in any of several ways. One possibility is to
choose a single pulse (a single a„ = 1) in the sector of the first quadrant
bounded by n = and the largest integral value of n that satisfies

/3" cos 2-irkn > \ x m i n |.

For Q = 100 and kQ = 0.1, the above is satisfied for a value of n that
is less than about 148. The probability of at least one pulse in this range
of n is 1 - (1 - p) 148 which is about 1 - 10~ 18 for equally likely pulses
and spaces. Therefore, x is positive with probability very close to unity.

PHASE JITTER IN PCM REPEATERS 511

For increasing values of Q, with kQ fixed at 0.1, the probability that
x is > approaches unity even more closely.

By an argument that parallels the above, the probability that y <
for k > and impulse excitation is very small. Similarly, probability
y > for A* < is extremely small.

For raised cosine excitation, .r rain is increased by 1 + b, which for
the pulse widths considered herein is always >0.25, thereby making
x min positive for the Q's of interest to us. We also note that long strings
of zeros as required in attaining x min cannot be tolerated in a PCM
repeater with a simple tuned circuit timing extractor, since the timing
wave amplitude would fall well below the point at which it would be
useful in the repeater. A higher minimum on the timing wave amplitude
can be assured by constraining the transmitted pulse train to avoid such
long strings of spaces. 7 In this paper we simulate this constraint by the
introduction of a forced periodic pattern of pulses in the otherwise
random train. This serves to increase .r m in and decrease the range of 6
as we shall see below and in Sections VII and VIII.

3.3 Range of 6

For random impulse excitation, it is apparent from (5) that is un-
bounded when wc choose a single a n = 1 for n large and all the rest zero.
However, with a = 1 and the values of Q we consider, x is always
positive, and from the results of Section 3.2 6 is essentially confined to
(0, tt/2) for k > and [0, - (tt/2)] for k < 0. In the following we seek
tighter bounds under the practically important case a„ = 1. Experi-
mentally, a = 1 means that we examine only those time slots containing
pulses.

For the general form of 0, D. Slepian and E. N. Gilbert of Bell Tele-
phone Laboratories* have developed an algorithm for determining the
pattern that yields the maximum value of 0. Their result is particularly
simple when kQ «. 1; then we can approximate x by

i + 2>„

i

and y by

2irk £ a n

i

e -(x/Q)n

IIC

-(»/«)"

Under this condition Gilbert and Slepian have shown that the pulse

* Private communication.

ol2

THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1962

1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
ft

Fig. 1 — n c vs /3 for random impulse excitation.

pattern giving the largest value of is specified by all pulses present for
n ^ n c and pulses absent for n < n c . The value of n c is obtained from*

n+l

= n c (l + h) - ^r

a
2irV

(14)

(i - py

where |9 = e~ {vlQ) . For random impulse excitation a = = 6. For this
case, n e versus /3 obtained from (14) is shown in Fig. 1. For /3 < i, all
pulses present (n c = 1) yields the maximum value for 0. In the range
\ < < 0.639 the pulse immediately adjacent to the origin is dropped
out to obtain m „ x and so on.

The maximum value attained in a specified interval is achieved for
the largest /3 in the interval and the maximum value is given simply by
2tt/i- times the n c defined by the /3 interval. The /3 intervals corresponding
to constant n c get smaller and smaller as approaches one. This is
illustrated in Fig. 2, where we have plotted n c against Q rather than /3,
showing a continuous approximation to the actual staircase character-
istic. We note that for Q = 100, n c = 80 and inilx = 2xfcn c = IOOtt^.
With k = 10~ 3 , ma x = O.KW radians.

* See Appendix E for the proof.

PHASE JITTER IN PCM REPEATERS

513

n c

ISO

140

100

60

60

120
Q

180

Fie. 2 — n, vs Q.

For finite width pulses, a and 6 are non-zero. With raised cosine pulses
of pulse width less than 1.5 time slots a < 0.05 and b > -0.75 with the
largest negative value of b corresponding to the consideration of positive
going time slots. When the mistiming, A-, is positive, the effect of finite
pulse width then is to raise the maximum value of //,• over the impulse
case and consequently to raise 0„ mx . On the other hand, when k < 0,
I1111X can be reduced over the impulse case. We will demonstrate this
effect in connection with the cumulative distribution in Section IX of

the paper.

As noted previously, the long string of spaces implied by large n,
make the timing wave amplitude so small as to be useless in a real re-
peater. The timing wave amplitude can be increased by forcing a periodic
pulse pattern. With the constraint that every A/th pulse must occur,
the pattern that yields the maximum value for is as before where n c
is now given by

n '

(1 - 0) 2

a
2*k

+

,[

L + 6 +

1

+

(i -

2ttA-(1 - /3") 2 27rfc(l - 0»]

rMP

(r+l)M

(15)

wherer is the largest integer less than n c /M. It can be seen that (15)

514 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1962

reduces to (14) as M — > °o as expected. Furthermore, since the difference
in the last two terms of (15) is positive and the term added to 1 •+ b is
also positive, it is apparent that the effect of the periodic pattern is to
reduce n c and consequently max as expected.

3.4 Probability Density Function, p(6)

With the above preliminaries disposed of, we will proceed to use (13)
to develop an approximate expression for p(d). To do this we assume
that the logarithm of the characteristic function possesses a power
series expansion in the neighborhood of u = = v. The general form of
this series is 10

log v ( Ul v) = Z Z hL (iuYdvy (i6)

r =o «=o rlsl

T + S 7*

where the X„ are the semi-invariants of the distribution for x\ and y t .
Since

% =v L^ v] '

we may write

vie) = J_. /"°±[iogd

2ti J- - du

exp [log <p]

dv. (17)

Using (17) and performing the differentiation indicated in the integrand,
we get

We now remove terms from the double summation for which r -f- s ^ 2.
The remaining terms we treat as u, and expand e u in a power series
retaining only the first two terms (e u ~ 1 + u). In this case p(6) be-
comes approximately

rlsl

p(.o)~ P .(e) +ZE^(-i)Wfl), (19)

r s
r+»>2

where

■, r* • i»» j 2-i

Po(e) = j d \j^ j ^ - exp -iv(\ lo - Xoi) - | (a 2O 2 - 2X U + X 02 ) ,

PHASE JITTER IN PCM REPEATERS

515

or p„(0) = (d/dd)f (d), where f (d) is defined by comparison with the

above.

Similarly,

*■« -»['££*«

r+e

•exp -iv(\ v ,e - Xoi) - - (X 2O 2 - 2Xn0 + X 02 )

or

]■

An upper limit for the double summation in (19) is set in order to make
the approximation for p(6) consistent with the number of terms used
in the power series expansion for c". The reason for 6 as an upper limit
will become apparent when we discuss the semi-invariants, X r « , in detail
in Section V. Performing the differentiations and integrations indicated
in (19) we finally arrive at

pM

where

and

1_ A,(6) [_ A (d) 2 ~\
V / 27ryl 1 (e)' eXP L 2A l (d)j

r+a=6

1 + EEf-D r Tr

rlsl L (V2A 1 (d)V +e

r+«>2

H ~w

2AM/

II

(r+

+

(_M0)_\

) - l W2A 1 (d)J A,

,(0)

(V2A 1 (e)r

(r+s)-l

6A 2 (d)_

A„(d) = \ 10 (d - e ),

.1,(0) = \., Q e- - 2\ u e + x u2 ,

A 2 (0) = X 1O [0( X 2( ,0„ - X„) - CKnOo - X 02 )],

A rH {6) = sX 2o 2 + (r - s)\ n 6 - rXo2 ,

X()l

X10

(20)

516 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1962

The H's are Hermite polynomials defined by

The result in (20) gives a general expression for p(d) as a function of
the semi-invariants of the distribution of Xi and y x . The solution ob-
tained is approximate in that it depends upon an asymptotic expansion
analogous to the Edgeworth Series. As noted by Cramer, 9 one is not
particularly interested in whether series of this type converge or not,
but whether a small number of terms suffice to give a good approximation
to the probability density function over a specified range of its argu-
ment. In our case, the statistical properties of the input pulse pattern,
and the parameters of the timing circuit are controlling in this regard.
With this in mind, the determination of the range in B over which a
valid approximation may be obtained in various cases is deferred for
the present.

IV. CUMULATIVE DISTRIBUTION FUNCTION

The cumulative distribution function F(6) may be determined using
the results derived in the preceding section. Beginning with (19) we
may write

vie) -//(*) + ifl^, (-iYU'(o). (2i)

r+k>-<

By definition*

Fid) = I p(u) du.

J— CO

Integrating (21) between the limits indicated, F(d) becomes

Fid) = /„(*) + iTs ^ i-iYUe) +i (22)

r+»>2

Referring back to (19) and performing the integration over v necessary
to determine f„(d) and/ r *(0), we get

* The significance of the lower limit of integration in the definition of F{6)
will be discuBsed in connection with the numerical results.

PHASE JITTER IN' PCM REPEATERS 517

tK 2^2 lV2A l ($)j V2*A 1

/ U9) \
.exp ["- ^1 ft ± ( 1)r H ™- l \V2Mf))

(23)

where 4 (ff), Ai{B) and ff r +»-i have been previously denned.

V. SEMI-INVARIANTS FOR THE DISTRIBUTION OP X AND y

In this section we consider the coefficients of the power series expan-
sion for the logarithm of the characteristic function <p(u,v). These are
determined as functions of the parameters of the timing circuit, and the
excitation and provide the necessary information for an explicit solution
for p(d) and F(6). A closed form for the X„ is obtainable for all excita-
tions of interest under the condition p = \ (pulses and spaces equally
likely). [The semi-invariants for any p can be obtained by appropriate
differentiations of log <p(u,v ). We have not expended the energy for this
exercise.] The semi-invariants are shown below for random impulse
excitation under the condition kQ « t and are derived for all excitations
we consider in Appendix D.*

Xl ° " 2(1 -p) Xo1 ~ (T^rpy (24)

(-l)'B r+a (2 r+t - 1) d* I 1 \

X„ !,+.>: = - - • (2xA) - (j— z^ (25)

where /3 = e _lT/0) , g = tt Q (r + s), and the B r+> are Bernoulli numbers.
Since B r+H = for r + s odd and > 1 , we note that the odd order semi-
invariants given in (24) and (25) vanish beyond order 1. Therefore
since the X™ for r + s — 3 are zero, one can extend the upper limit in
the double summation in (19) to (3, and still maintain consistency with
the fact that only 2 terms in the power series expansion for the expo-
nential, e", were used in the approximation for p(8). This conclusion is
valid for all excitations of interest.

VI. BEHAVIOR OF p(d) FOR LARGE Q

When the Q of the resonant circuit becomes large, the past history of
the input signal becomes increasingly important in determining the

* The more general semi-invariants without the restriction kQ « -k are given
in Appendix D; however, they are too long to be repeated here.

518 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1962

statistical properties of x and y. This follows from the form of the ex-
ponential term in the expressions for x and y given in (5). Invoking the
Central Limit Theorem under this condition, one would expect the
values of x and y to begin heaping up about their respective means with
the probability density function p(x,y) approaching a two dimensional
normal distribution. Analogous behavior is expected of 6 and we will
now consider p(0) as given by (20) in the neighborhood of its mean for
large Q. The discussion is restricted to the case of random impulse
excitation, but the results for other excitations parallel those of this

section.

To determine p(6) near its mean, we write, using the previous condi-
tion kQ « ir,

. V ,

2irk ^3 a„n e

(26)

2 a n

-an

where

For this to hold as Q becomes arbitrarily large, we require the kQ
product to be constant. Since

Z—an
a n e ,

n=0

6 can also be written as

e 2tt/c i- [log X] = -2irk i- [log % + log x\ , (27)

da da\_ x J

where x is the average value of x. Expanding log x/x in a power series
in the neighborhood of 1 (x near x), and keeping only the first term,
becomes

•~-**c *•«-**£ [Ml- (28)

Differentiating the above with respect to a we get for 6 in the neighbor-
hood of its mean

e~i + *v-*y t (29)

PHASE JITTER IN PCM REPEATERS 519

In determining this result we make use of the fact that

y = -2rk £- [.?]. (30)

da

Using (29) one can determine the logarithm of the characteristic func-
tion of 0, and the associated semi-invariants of the distribution. When
this is done, the mean of is

H-l-S (3D

which also can be derived directly from (29). The standard deviation
and the 4th semi-invariant are given by

/ gggjg

V (1 -/?)(l+/3) 3

-2(2Tfc)V [ _ 4/3 3 (l - g) 6/3 2 (l + /3 4 )(1
(1 -/3<) L (1 -0 4 ) (1 ~ P)

-0)

4/3(1 - /3) 3 (1 + 4/3 4 + /3 8 )

+

(1 - /3*) 8

(1 -/3) 4 (1 + ll/3 4 + 11/8' + 18") '
(1 - 04)4

(32)

with /3 = e~ a . These same results can be derived using (20) and including
only the first correction term from the double sum (i.e., only those X„
for which r + s = 4). The details of the calculation along with the \ TS
of interest are given in Appendix D. The final result for p(9) is

/ ft — a \

#4

V27rff 2cr- \ 4! 4cr 4 /

The above equation for p(0) is in the form of, -the standard Edgeworth
approximation. In the limit as Q becomes large (0 — »,1), and with kQ
constant, p(8) reduces to

with 0„ - 2fcQ and <r = k\ZtrQ. Equation (26) indicates the approach
to the normal law as Q becomes large with the first correction term going
as l/Q. The above results for O and a correspond to those derived

520 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1962

earlier by Bennett 1 by another method. If we rewrite a as kQ\/ir/Q we
notice that p(0) becomes more peaked with increasing Q, and falls off
quite rapidly as departs from the mean. In the high Q case the concen-
tration about 0„ becomes more pronounced as expected.

It is to be emphasized that the general properties of p(0) for large Q
demonstrated here will be true for the other inputs also. For example,
with random impulse excitation plus 1 out of M pulses forced, the
average value will remain the same as above but a will be a function of
M;

The effect of M is to reduce a and therefore increase the concentration
about the mean. As M becomes large (fewer pulses required to occur),
the effect of M becomes insignificant for this large Q case.

VII. NUMERICAL RESULTS FOR p(0) AND 1 - F(d) ! IMPULSE EXCITATION

7.1 p(0)

To determine the behavior of the probability density function for
finite Q, we must use the general form of the approximation to p(0)
given by (20), since most of the approximations made in the previous
section for Q arbitrarily large are no longer valid. By way of illustration
we consider the case Q = 100, A- = 10 -3 with impulse excitation and all
pulses random (p = j). For negative mistiming, A- = -10" , the curve
for p(0) will be identical with that for k positive except that is re-
placed with - 0. The result for the probability density function is shown
in Fig. 3. The calculations* upon which this curve is based include
the first and second correction terms of (20); i.e., terms for which r +
s = 4 and r 4- s = 6. Points beyond 9 = 0.18 radians on the lower end
and = 0.35 radians on the upper end are not included, since the ap-
proximation begins to fail at these extremes. More specifically, the
probability density obtained from (20) goes negative somewhere be-
tween = 0.13 radians and = 0.12 radians and = 0.35 and = 0.36
radians. However, as we shall see later, up to these points the results
for the cumulative distribution are in good agreement with computer
simulation. The cumulative distribution is also shown on Fig. 3 to point
out the fact that the median occurs slightly below the approximate mean
given by 2AQ. In addition, it is apparent from the shape of p(0) and

* Equation (20) and all subsequent calculations for p(0) and F{B) were pro-
grammed for the IBM 7090 computer by Miss E. G. Cheatham.

PHASE JITTER IX PCM REPEATERS

521

1.0

Q =

100

k = to- 3

uT °- 6

n

M

\0.5

•S-
o.

P(«)| 1

l/ FW \

0.2

IN RADIANS

Fig. 3 — p(6) and F(0) us a function of for A- = 10" J and Q = 100. Random
impulse excitation.

F(0) that the probability density is skewed in the direction of increasing
phase error. This is more easily visualized from Fig. 4 where we have
shown p(d) as in Fig. 3 plotted on log paper. The normal probability
density with the same mean and variance as our computed curve is also
shown to further illustrate the skewness.

On Fig. 5 we have plotted p(0), as defined in (20), to illustrate the
contribution of its constituent terms. From this figure we see that the
principal term (always positive) predominates over most of the range.
At the tails, the terms involving X„ for r + s = 4 pulls p(6) in and
forces the density to become negative. The last term in the approxima-
tion, for which r + s = (i, serves to extend the region over which p(9)
remains positive.

When \/M pulses are forced, the skewness is reduced, as is the vari-
ance. There are several ways of explaining this effect. First, as discussed
in Section 3, the denominator of 6 in (8 ) or (9) is raised, thereby reducing

522

THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1962

0.10

0.15

0.25
6 IN RADIANS

0.30

Fig. 4 — j)(6) for k = 10 -3 and Q = 100. The normal curve with the same mean
and variance is also shown for comparison. Random impulse excitation.

the range of variation of the timing wave amplitude and confining 6 to a
narrower range. This is expected from the physical standpoint, since
forcing a periodic pattern with the remaining pulses and spaces equally
likely is similar to increasing the probability of occurrence of a pulse in
an all-random sequence. Since the pulses, when they occur, have the
proper spacing, they will tend to correct for the departure of the zero
crossings from the mean that has occurred during the free response of
the tuned circuit in the absence of a pulse. Indeed, in the limit when
M — 1 (all pulses definitely occur), all the probability is concentrated
at the mean, 2kQ, which is identical to the steady state phase shift of

PHASE JITTER IN PCM REPEATERS

;-)2:i

the tuned circuit in response to a sine wave at the pulse repetition fre-
quency. This behavior is also predicted mathematically from (20) and
the fact that A rs goes to zero for r + s > 1 when M = 1. The same effect
occurs when Q approaches infinity with kQ constant and it can be shown
from the results of the previous section that p(0) goes to 5(0) when the
limit is taken. In this light, we can view the introduction of forced pulses
as effectively increasing the Q of the tuned circuit while maintaining kQ
fixed.

Q= 100

k = 10" 3

\

\

\

to-'

V

\\ PRINCIPAL
\\ TERM

JO" 2

\

<

\\

i

\

in -3

>

J

ALL THREE \\
TERMS \\

\

f

\

A

>

lO" 4

\

PRINCIPAL TERM J

\

1

\

iO" 5

0.12

0.16

0.24 0.28

IN RADIANS

0.32

0.36

Fig. 5 — Contributions of various terms involved in the p(0) approximation
given by (20). Random impulse excitation is assumed, with k = 10~ 3 and Q = 100.

524

THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1962

10'

10"

10"

5

Q = 10C

m.

w

w

\

\\

\

\\\

\

\v

V

\\

\

\\

\\

III

\

\\

\

\\.y

= 00

rv

~\

Y

\ \

\

\

\

'

^

A

\

\\

\

\\

, \

\

\

c

V

/

\

\

\

5

\

0.15

0.25 0.30

6 IN RADIANS

0.35

Fig. 6
100.

The effect on p(0) of requiring \/M impulses to occur, k = 10 3 , Q =

In practical applications, the effect of a pulse at the origin is of par-
ticular interest. Mathematically, this corresponds to M = » . Physically
this means we examine and record phase error only for those time slots
containing a pulse. Fig. 6 illustrates the narrowing of the density func-
tion for M = *> (pulse at the origin), and M = 16, 8, and 4. It is
interesting to note that, for these cases, the probability density function
remains positive over the range of we have used in the computations
from 0.1 to 0.4 radian. This encompasses values of p(d) < 10" ' on the
left of the mean and p(d) < 10~ 5 to the right of the mean. This is to be
expected since A r „ decrease with decreasing M for r + s ^ 2, thereby

PHASE JITTER IX PCM REPEATERS

525

reducing the importance of the terms involving the Hermite polynomials
in (20) and improving the approximation.

Fig. 7 depicts the behavior of p(0) as Q grows with kQ fixed at 0.1.
The results are consistent with the predictions of the previous section.

7.2 1 - F{6)

For a closer inspection of the behavior of the distribution at its tails,
1 — F{6) will be examined. This function as evaluated from (23) for

102

IO" 3

kQ=o.i

.

LA

\

\

\

'

> \

\

Q = 5(

\

\20

\°

= 100

\

V

1

\

\

\

>

V

\

\

1 1

1

\

\

0.10 0.15 0.20 0.25 0.30 0.35 0.40

6 IN RADIANS

Fig. 7 — The effect on p{0) of increasing Q with kQ = 0.1 and random impulse
excitation.

526

THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1962

Q = 100, k = 10~ 3 , and purely random excitation (p = 2) is shown in
Fig. 8. The plot shown gives the probability that deviates from its
mean by more than some constant C times a. In the same figure a
comparison of the calculated approximation with the normal curve of
identical mean and standard deviation indicates a substantial departure
from the normal law as the phase error increases. When periodic patterns
are interspersed with the random train, the departure from the mean is
further reduced, as can be seen from Fig. 9. Similar behavior is exhibited
in Fig. 10, where Q is increased from 100 to 500 and kQ maintained
constant at 0.1.

I

5

Q =

00

\

k= 10- 3

10"'

s

\\

\

\
\

u

\ \

\ '

\
\

Al 5

\ ALL PULSES
\ RANDOM
\ ff= 0.018

g 10- 3
K

a.

^

\

\

normal\

CURVE \
0" = 0.0I8\

\

\

\

1

I0~ 5

D

I

3

c

4

5

1

Fig. 8 — Comparison of 1 - F(6) with the normal curve in the vicinity of the
tails The normal curve is computed assuming the same mean and variance used
in determining 1 - F(9). Random impulse excitation with Q = 100 and k = 10 s
is assumed for computing 1 — Fifi).

PHASE JITTER IN PCM REPEATERS

527

^

h

\

Q = ioo

k=io" 3

\

\

\\

A

I

\

\

\

\\

[

\

\M = oo

M = 4

\

\\

\ X

\

\

\

\

\

\

0.20 0.25 0.30

O IN RADIANS

Fig. 9 — The effect on 1 — F(0) of requiring \/M impulses to occur, k = 10~ 3 ,
Q = 100.

7.3 Comparison with other approaches

Since we have made approximations in arriving at our expression for
the phase error, it is natural to ask how these approximations affect our
computed results. A comparison of our results with two other approaches

528

THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 19G2

kQ=o.i

4

10"'

■ \

\ \

V

\\

\

1

\\

lO- 2

1

\\

\

1

\

10- 3

\

1

tz

\

\Q = I00

io-«

\200 \

\

i

I

\

Q = 500l

\

\

in-*

0.10

O.IS

0.20 0.25 0.30 0.35 0.40

6 IN RADIANS

Fig. 10 — The effect on 1 — F{d) of increasing Q with kQ = 0.1 and random
impulse excitation.

will be made for the case of impulse excitation. We recall from Section 2
that the phase error under impulse excitation is given by

tan = -.
x

For kQ sufficiently small we can write

PHASE JITTER IN PCM REPEATERS 529

Z 0:6"
n =

The approximation of tan 6 by its argument is not crucial in this case,
since a straightforward transformation can be made on the probability
distribution to correct for this approximation [i.e., p(0) = sec 2 0p(tan 9)].

H. Martens* shows that (35) can be manipulated to yield a recursion
relationship for the phase error that is in a convenient form for digital
computer evaluation. T. V. Crater and S. O. Rice used this approach in
some of their work, and a probability distribution so determined is
shown by the dots in Fig. 11 for Q = 125. For the same value of Q, we
have computed the probability distribution from the series in (23), and
it is displayed as the solid curve of Fig. 11. It can be seen that the agree-
ment between the two approaches is excellent. The scattering of the
"experimental" points at the 10 -3 level and below is due to the limited
number of pulse positions considered by Crater and Rice. Specifically,
10 pulse positions were processed after an initial transient of some
5X10 pulse positions had elapsed.

In addition, S. O. Rice in unpublished work has shown that the tail
of the distribution should behave as A(%) /2 *"', where A is an unknown
constant. When we take the values of 6 at the 10 3 and 10" 4 levels and
substitute these in Rice's asymptotic form and form a ratio, the con-
stant .4 cancels out and we should obtain 10. The actual value for the
ratio is 10.9, which tends to indicate that the asymptotic behavior has
virtually been reached. This suggests that an extrapolation of the distri-
bution to larger values of 6 by merely continuing with the same slope
should be valid.

We also note that we can write

where we have made use of 6„ = 2I>Q. With hQ constant, one would
expect the cumulative probability to fall off faster for larger Q, as is
indeed the case. The slopes of the curves of Fig. 10 follow Rice's pre-
dictions quite closely.

While the above comparisons are comforting, they only indicate that
our final expressions for p(6) and F(6) are accurate for computing these
quantities from the initial defining equation for 6. Approximations have
been made in arriving at the starting relationship. A check on these
initial approximations may be obtained from a simulation of the problem.

* Unpublished memorandum.

530

THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 19G2

277k

Fig. 11 — Comparison of 1 - F(B) computed by (23) with the results of the
Crater-Rice simulation for Q = 125. Random impulse excitation is assumed.

One such simulation has been accomplished by Miss M. R. Branower
using a combination of analogue and digital computers. The principal
errors introduced in this process involve the stability of the analogue
computer with time and the number of pulses processed. For attuned
circuit characterized by a Q of 125 and mistiming k = +10" , the
computer simulation yields the results of Fig. 12. Results obtained using
(23), the exact semi-invariants of Appendix C, and the tan transforma-
tion mentioned previously yield the "computed curve" of Fig. 12.

PHASE JITTER IN PCM REPEATERS

531

Again the results are in very close agreement. To indicate the effect of
the approximation kQ « tt, we have repeated the computed curve of
Fig. 11 on Fig. 12.

VIII. RAISED COSINE EXCITATION

8.1 Results for 1 - F{6)

With raised cosine excitation, the computations are performed as before
and only the semi-invariants X„ for r + s = 1 are changed from the

10"

CD 10"

10- 3

10"

o ANALOG SIMULATION
DATA DUE TO

N N

V

fx

\

\
\

V

\

COMPUTED FROM (23)
(TAN 6 CORRECTION
NOT INCLUDED);
\ APPROXIMATE
l\ SEMI- INVARIANTS

\ <

\\

\

—

S. S

\ \

V

\

COMPUTED FROM (23T

(TAN CORRECTION

INCLUDED)

\

V \
\ \
\ \

^ —

cA

\ V

v

v*

vs

\ \

v

O.i

4

0.2

6

0.<
8

8

IN RA

o.;

DIANS

o.:

2

0..

14

Fig. 12 — Comparison of 1 - F(6) computed bv (23) with the results of an
analog simulation due to M. R. Branower. Random impulse excitation with
Q = 125 and k = 10 3 is assumed. The effect of the tan approximation is shown
together with results for both approximate and exact semi -variants.

532

THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1962

1

"

ST***-

N s

N

\

^

^

^

\

\

1

\\

V

\

w

\

\

\\

A

\

V

\\

\

\

\

. I.5T, POSITIVE
\ ^' GOING

2

\

\

\ I.5T, NEGATIVE
\S^ GOING

\

V

\ A

\

\

a:

\

U.

1

\

\

4

\

T, POSITIVE
GOING ^-

¥

>

v\

10- 3

T, NEGATIVE
GOING ~-- -

A

\

V

^

\

\

\,

\

31

\

\

VA

\^

4

\

\

\

2

10" 4

\

\

\

\

, \

\

V

\ \

\

V

\

V

\

\

\

\

\

v

2

10" 5

\

\

15

0.

20

0.

25

6

0.
IN R

30
ADIAK

0.

s

35

0.

40

0.'

Fig 13 — Plot of 1 - F{6) for raised cosine excitation. Pulses of width T and
1.57' are assumed in the calculation. The distribution of the phase error for both
positive and negative-going zero crossings is shown. Q = 100, k = 10 3 .

previous case. Results obtained for this excitation are shown on Fig. 13,
where it is apparent that the use of widest pulses and positive-going zero
crossings yields the largest phase error. The effect of Q and M with this
type of input is the same as with impulses.

PHASE JITTER IN PCM REPEATERS 533

8.2 Comparison with another approach when k =

In the absence of mistiming, the phase error becomes

e = ih,' (:i0)

and the probability distribution for may be obtained by methods given
previously, or by the following relationship:

Prob (0 £ X) = Prob (tjtj ^ x)

(37)

= Prob

Therefore, if the distribution for x is known, the distribution for may
be determined from it. The random variable x is the normalized timing
wave amplitude defined by Rowe. This random variable has been con-
sidered by S. O. Rice in unpublished work and he has developed a pro-
cedure for closely approximating its probability distribution. Using the
method of moments, one of the authors also computed this distribution.
The results were in excellent agreement with Rice's results and the
cumulative distribution obtained by the moment method is shown in
Fig. 14. It can be shown that the probability density for x is unimodal
and symmetric about its mean; therefore, the data on Fig. 14 suffices to
specify the complete distribution. With this data and (37) we can
determine the distribution for 0. Alternately, we can use (23) to make
this computation. A comparison of the distribution obtained by the two
approaches is shown in Fig. 15 and it can be seen that the agreement is
very close. Thus we have found another check on our series approxima-
tion for p(0). Conversely, we can use the distribution for to compute
the distribution for .v. In this regard it is interesting to note that when
the Edgeworth expansion including semi-invariants through order is
used to approximate the distribution for x, the density function begins
to turn negative in the neighborhood of 3<r from the mean indicating
failure of the approximation. On the other hand, using the same number
of semi-invariants in the expansion for p(6), where in this case is es-
sentially the reciprocal of x, we obtain a good approximation to the
cumulative distribution for .r. This is believed to be due to the narrowness
of the range of as compared with.r; i.e., x varies from 1 to 1/(1 — /3) =
Q/tt, while l/x goes from 1 — = t/Q to 1.

534

THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1962

* I0" 3

m

g io-«

Q.

S

2

io- 5

10

1.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 I

\

.1 1.2

Fig. 14 — Probability distribution of the timing wave amplitude. Q — 100.

IX. OPTIMUM TUNING — FINITE PULSE WIDTH

In the case of impulse excitation it should be apparent that zero mis-
tuning, A: = 0, is the desired objective for no phase error. On the other
hand, with finite width pulses zero mistiming does not yield zero phase
error. Mistiming can be purposely introduced in the finite pulse width
case to make the mean value of 9 zero, to minimize the variance of 6, or
to optimize some other parameter of the 6 distribution.

An approximation to making the mean of zero may be obtained by
choosing k such that the average value of the numerator of is zero.
This means that

PHASE JITTER IN PCM REPEATERS

?/i = a + y = a +

or

k = -

(1 - 0Y-

g(i -£) 2

7T/3 •

-o,

535
(38)

(39)

For example, when Q = 100 and a = 0.65, as for raised cosine pulses of
width 1.57\ then k = -2.05 X 10~ 4 to satisfy (39). In the high Q case
(39) becomes k = — (qt/Q 2 ).

10-

r

T

_$

\

?

\

\

*

4

\ C

DMPUTED
FROM
EQ. 23
/

\\

COMPUTED FROM \\»
DISTRIBUTION FOR X \\

"^^ 1

I
\

\

\

\

\

\

0.02

0.04 0.06

9 IN RADIANS

Fig. 15 — Comparison of the distribution of as computed by (23) and that
determined from the distribution of the timing wave amplitude of Fig. 14. Raised
cosine pulses of width 1.5T drive a tuned circuit with sxQ = 100 and zero mistim-
ing. Timing deviations in the neighborhood of negative-going zero crossings are
considered.

536

THE HELL SYSTEM TECHNICAL JOURNAL, MARCH 1962

When the objective is to minimize the variance of 0, we consider a
as defined in Appendix D; i.e.

{\20da~ — 2\nft» + Xoa) /^q\

X10

A plot of a versus A- is shown in Fig. 16, where it is seen that the minimum
a occurs close to the "zero mean" value of k. Probability distributions
for values of k that encompass the optimum are shown on Fig. 17. The
narrowing of the density function for the optimum value of k is evident.
The results of this section suggest that when the tuned circuit in a
self-timed repeater is adjusted, it should be excited with a random pulse
train and the tuning adjusted to minimize the jitter on the leading edge

-3XI0" 3

-1 1

k, FRACTIONAL MISTUNING

Fig. 16 — Standard deviation of phase error as a function of mistuning with
raised cosine pulses 1.5'f wide. Negative-going zero crossings are considered. Q =
100.

PHASE JITTER IX PCM REPEATERS

537

-0.4 -0.3

-0.1 0.1

6 IN RADIANS

Fig. 17 — p(0) for raised cosine excitation with various miatunings in the
neighborhood of the optimum mistiming. Negative-going zero crossings and
pulses 1.5jT wide are assumed in making the calculations. Q = 100.

of the output pulse train as viewed, for example, on an oscilloscope. This
is the method used for the adjustment of the repeater of Ref. 8.

X. PARTIAL RETIMIMi

111 Section VIII we have shown that, in the absence of mistiming, the
variable 6 can be related to the normalized timing wave amplitude x

538 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1962

and the distribution for 6 determined from the distribution for x. Here
we will also make use of the distribution for x in order to analyze an
idealized version of a forward-acting partial retiming scheme. The
scheme we consider has been described by E. D. Sunde 6 and analyzed
for periodic pulse patterns in Ref. 7. We make the same assumptions
here as in the later reference, namely

1. The pulses exciting the tuned circuit are so narrow that they can
be considered impulses. They are obtained by processing incoming
pulses to the repeater and they excite a simple tuned circuit.

2. The timing wave is so clamped that its maximum excursion is at
ground.

3. Reconstruction of the raised cosine pulse takes place when the
algebraic sum of the timing wave and the raised cosine pulse crosses
a threshold assumed to be at half the peak pulse amplitude.

For random impulse excitation of the tuned circuit prior to t = and
the definite occurrence of a pulse at / = 0, we have, according to the
above assumptions (with no pulse overlap)

5( 1 + -t)-S0— tH (41)

for | i | ^ T/2s

where

x = £ M n ,

11=1)
a„ = 1 (the pulse at the origin definitely occurs),

and

.r = average value of x.

Equation (41) is based on the assumption that the average timing wave
has a peak-to-peak amplitude equal to the peak pulse height (i.e., when
x = x, the timing wave amplitude varies between —1 and 0). If we
define l p as the time at which regeneration takes place and d p = 2irt p /T
as the corresponding phase angle, then it can be seen from (41) that
this phase is a random variable dependent upon the random variable x.
We will solve for d p under the condition s = 1 , which means that the
information-bearing pulses are resolved.* Under this condition — (t/2)
< 6 P < 0. Consistent with our previous definition of phase error, we will
consider the negative of 6 P , since this makes the phase error positive

* Other pulse widths and different ratios of average timing wave amplitude to
pulse peak can be handled, but we will not consider them here.

PHASE JITTER IN PCM REPEATERS 539

when we take our reference as the phase corresponding to the time at
which the pulse peak occurs (at t = 0). In this way a positive phase
error corresponds to regeneration prior to the pulse peak and permits
direct comparison with the results of section 8 for the complete retiming
approach. Solving (41 ) for cos B p gives

x

cos d p = -^— (42)

1+?
x

and

Prob (cos0 p ^ A) = Prob (d„ ^ cos -1 A)

x

= H^ ax r Prob ( a -(r^A))

(43)

l+ I

It is apparent from the above that we can use the distribution for x to
determine the distribution for 6 P . For Q — 100, the distribution for x
is shown in Fig. 14 and with (43) enables us to obtain the distribution
for P as shown in Fig. 18. When we compare this result with that of Fig.
15, which shows 1 — F(d) for the case of complete retiming, it is ap-
parent that partial retiming results in a considerably larger variation of
phase error. This supports the contention made in Ref. 7.

XI. CONCLUSIONS AND FUTURE WORK

We have derived an approximate relationship for the probability
density and cumulative distribution for the phase error at the output of a
tuned circuit when it is excited by a random or random plus periodic
pulse train. The effects of mistiming of the tuned circuit and the finite
widths of the driving pulses have been considered. Three independent
checks of our results indicate that the expressions given are excellent
approximations to the true state of affairs for kQ < 0.1 and Q > 100.
Regions defined by these limits encompass values of k and Q of interest
in PCM systems under consideration.

More specifically, we have shown that the distributions are not normal
and are skewed in the direction of increasing phase error. When we
consider pulse positions in which a pulse definitely occurs, it has been
shown that the maximum phase error is bounded. In addition with
raised cosine excitation we have demonstrated that the mistiming can
be adjusted to minimize the mean or variance of the distribution for the

540

THE BELL SYSTEM TECHNICAL JOUKXAL, MAKCH 196S

O 10"

<

A

a.
Jh 10"

o
cr
a.

10"

1.00

1.04

1.12 1.16 1.20

6 IN RADIANS

1.24

Fig. 18 — Distribution of the phase error with partial retiming. Q = 100 and
Ic = 0. Raised cosine excitation pulse width = T.

phase error. The performance of an idealized version of a forward-acting
partial retiming scheme has been analyzed and shown to be considerably
inferior to a completely retimed repeater.

There are several desirable directions to proceed from our present
position. First, it appears to be possible, in the case where we examine
pulses only, to start from the maximum value of and work back toward
the mean to better approximate the distribution near the tails. S. O. Rice
has used this approach in related problems with success. Second, it is of
interest to determine the pattern to give the maximum phase error at
the output of a string of repeaters. This is not necessarily the pattern
that creates 1I111X in a single repeater. In this regard, we have concen-
trated on only a single repeater. Obviously it is of interest to extend our
results to a repeater string. This extension remains elusive.

XII. ACKNOWLEDGMENTS

The authors are indebted to several people for contributions to this
effort. We would like to acknowledge the work of Miss E. G. Cheatham

PHASE JITTER IX PCM REPEATERS 541

in computer programming. T. V. Crater kindly made the results of his
digital computations available prior to publication. Miss M. R. Branower
was most cooperative in providing us with distributions obtained from
an analogue simulation of the problem. Our thanks go to E. N. Gilbert
and D. Slepian for deriving the conditions for the maximum value of 6.
We are grateful to 8. 0. Rice for helpful discussions, for results on the
asymptotic behavior of the probability distribution, and for prior work
on a related problem which provided us with helpful clues on how to
proceed in our problem.

APPENDIX A. DERIVATION OF EQUATION FOR NORMALIZED TIMING ERROR

A-l. Response of tuned circuit to random pulse train

The impulse response of a parallel resonant circuit is well known to be

h(t) = Real part of [l (l + ^) e' ( ' IQ)/ "' c + *"-'] . (44)

Following Rowe," we will imply the real part in all subsequent calcula-
tions involving complex quantities. The pulse train applied to the tuned
circuit is given by

r(t) = J2a n g(t ~ nT), (45)

—00

where :

a„ = 1 with probability p,

a„ = with probability 1 — p, and
(j{t) = pulse shape representing the binary 1.
The response of the tuned circuit to r(t) is

z{t) = J r( T )h(t - t) dr. (46)

In view of (45), this can be written

z (l) = T^aMt - nT)

/(.tlT)-n r /, m \ "I

g{xT)exp \*±± - firf.T J x\ dx.

(47)

542 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1962

Define

/.-i±*=/r(l + «, (48)

with A; = fractional mistiming from the pulse repetition frequency.
Equation (47) can be manipulated to yield

z(t) = \A(t)\e iar,rt+ * lt) \ (49)

where

$(0 - ten" 1 -^ + 2*/ r fc*

£ a n

rlQ(l+k)n

_/ i(y ~ " I "

+ tan"

I sin 2irkn

+ h (Jr ~ n ) cos 2irkn ] (50)

Ea nC ' /e(,+i) "[/ 1 ^-n)cos2,k

+ 7 2 ( y, - wj sin 2irfcw

and

;, U -

4-)

= Re I'' " f(*T) exp |~fe/.T - i2«/.r) x\ efcc, and

J - M LVQ J J (51)

- Im f^ T " g(xT) exp IfafoT - frirfoT} s] dx.

In (49), |il(t) | represents the amplitude modulation on the carrier,
while *(0 represents the phase modulation, the quantity of primary
interest here.

A-2. Equation for normalized timing error

There is no loss in generality and it is convenient if the timing error
is evaluated in the neighborhood of the pulse that occurs for n = 0.

PHASE JITTER IN PCM REPEATERS 543

In this neighborhood, negative-going zero crossings occur where

2irj r t + *(0 - \

or

Similarly, positive-going zero crossings occur for

In the absence of tuning error, and with impulse excitation, $ =
and the negative and positive-going zero crossings occur close to ± 7'/4
respectively.* Using these zero crossings as a reference, it is easily seen
that the equations for normalized timing error become

•S + ?)

ei ^ \4 _ 77 (54)

for negative-going zero crossings and

for positive-going zero crossings.

With the exception of the minor generalization to arbitrary pulse
shape, the method employed thus far is identical with that used by
Rowe. 2 At this point in the evaluation of the timing error, we depart
from his approximate solutions of (54) and (55) and attempt other
approaches. Before proceeding in this direction, an indication of the
approximation used by Rowe will be given. For the high Q case, <£> will
be small and will change only a small amount for small changes in 2irf r t.
Based on this assumption,

r

=

m)

2t '

02

*(-*)

cting tan

'*-

(50)

T

2x

* Neglei

(56)

544 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1962

It should be pointed out that these initial approximations are good for
Rowe's purposes (steady-state error for \/M patterns). However, for
our purposes they need to be improved.

A-3. Approximate solution of equation for normalized timing error

One method for improving the accuracy of the initial approximation
is to expand <£> in a power series about T/4 for negative-going zero cross-
ings and retain two terms in the expansion to get

!i= *<*> . (57)

T 2r + *'(!)

The form of $ makes this approach messy and makes the determination
of the probability distribution more difficult.

Another approach that is more tractable involves the separate Taylor
expansion of h and h (51) in <S> about the reference time. If we retain
only the first two terms in the Taylor expansion, replace the arctangent
by its argument, and neglect k with respect to unity, we obtain for
negative-going zero crossings

ci _ 1 fe

T 4irQ 4

58;

E a n e MQ)n [-sin 2wkn (h(i - n) + eJi(l - n)

_ 1 + cos 2irkn{h{\ - n) + eJ*(\ - n))]

2T Z a n e MQ) "[cos2rkn(I l (l - n) + eji'd - n))

—00

-I- sin 2irkn(I 2 (l - n) + e x h'{\ - n))\

If terms in (ei/T) 2 are neglected, multiplication of both sides of (58)
by the long denominator on the right results in a linear equation for
ei/T. This equation is applicable to arbitrary pulse shape, time-limited
or not, and has been applied by one of the authors to periodic patterns
of both Gaussian and raised cosine pulses in unpublished work. The
results were compared with digital computer simulation and were in
excellent agreement, thereby giving us confidence in using this approach
for random pulse patterns. In this paper, we will concentrate on raised
cosine pulses. This enables us to make use of some of the results given
by H. E. Rowe in Section 2.5 of his paper. 2 For these time-limited pulses,
the limits of integration on the I's of (51) are modified in an obvious
way, and the upper limit on the sum over n is limited to the pulse im-

PHASE JITTER IX PCM REPEATERS 545

mediately succeeding the time slot of interest at n = for negative-
going zero crossings. The evaluation of the various I's required is dis-
cussed in Appendix B.

Subject to the above conditions, the normalized timing error, as de-
rived in Appendix B, can be written in the following form:

c, _ Ay + Bx + C , Q)

T~ Dy + Ex + F' { j

where

y - Z a n e- (TlQ)n sin 2-wkn,

(60)
x - E a»f- ( "° )n

n=0

and o — 1 (a pulse definitely occurs for n = 0). ^4 through F are de-
fined in Appendix B and are functions of the pulse width and Q and
mistiming of the tuned circuit. In addition, C and F are functions of the
presence or absence of a pulse in the succeeding time slot for negative-
going zero crossings if sufficient pulse overlap exists. For positive-going
zero crossings the form of the equation for the normalized timing error
is the same and the new C and F are dependent upon the presence or
absence of a pulse in the preceding time slot. This assumes that the
pulse width is less than 2.57'.

A-4. Modification of probability distributions for pulse overlaps

With the dependence on the occurrence of a succeeding pulse, as is
the case for negative-going zero crossings with sufficient pulse overlap,
we must modify the determination of the probability distribution as
given in the main body of the paper. If we denote e n /T and C = C\ ,
F = Fi for Oi = 1 (a succeeding pulse definitely occurs), and denote
e \i/T and C = (\ , F = F 2 for a x = 0, then the average probability dis-
tribution for the timing deviation will be given by

Prob fe £ \\ = V Prob (^ g XJ + (1 - p) Prob fe ^ \j . (61)

When the pulse width is less than 1.57 1 , C, = C 2 , F x = F 2 , and there-
fore e u = c 12 and the above modification is not required. A similar pro-
cedure is applicable for positive-going zero crossings.

546 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1962

APPENDIX B. RAISED COSINE PULSES

B-l. Determination of Ps

For a raised cosine pulse centered at the origin and of width T/s, I
of equation (51) becomes

I(x) =

*<-*

I(x) = r (1 + cos 2wsx 1 )e l{ * IQ) - j2r]Kxi dx l

1-1*5 (62)

'« - <i)

•>i

where

K ■ (1 + AO
The integral in (62) is readily evaluated to give

-#-

i

2ttA'

r-„[(ir/«)-i2)r]iCx -[{rlQ)-jir]Kl2a-.

— e

( 1+ ib)

+ i

+ i

I- [(WO)— j2ir]Ka A-jirtx -[(t/Q)-j2w\ A72«-,

— e

I- [ (ir ,'<?)- j"2t]Ki -ft*** -[(a-/C)-i2ir]K/28-,

(63)

— e

ci + .)+y|

The derivatives required in the evaluation of (58) may be obtained
from

dl __ (rlQ)Kx t -j2,Kx , i -j2r(K-s)x , i e -j2x(K+»)ari ,q^\

dx

In the evaluation of / and dl/dx, mistuning makes very little differ-
ence for the allowable values in practical systems. Therefore, with K = 1

A-i/4 = -./^[l + cosf,]
/!=-!/< =je* /4c [l + cos | S ]

(65)
(66)

PHASE JITTEH IN PCM REPEATERS 547

/'L=3/4 = j e w40 [l + co S ?^] (67)

/'U-3 /4 = -je W40 [l + cos^]. (68)

Equations (65) and (68) above are required for negative-going zero
crossings, while (66) and (67) are needed for positive-going zero cross-
ings.

B-2. Equation for Normalized Timing Error with Raised Cosine Pulses
From (58) we can write the equation for normalized timing error as

e l = -J- - k - 1*L (CQ)

T 4ttQ 4 2wP' K }

where N and P are denned by comparison with (58). Cross multiplica-
tion by P, neglecting terms in e x and collecting terms, yields

e ± Ay + Bx + C , .

T Dy+Ex + F' K '

where x and y are defined by (60), and A through F are as follows:

<- -k ['■©-'■©]- [« +S I*©-*®]

-[^ + y[ si " 2 ^(-'i) +c ° s2rf/ 'H)]

548 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1962

+ «, .««> {cos **[£ /,'(-*) + /,(-§)

+Gi + 8 ft '(-S)] + *"[^'(-8

For positive-going zero crossings, only the constants (7 and F are
changed.

B-3. Numerical Evaluation of Constants

In order to make use of some of Rowe's results, we will choose the
same two cases for pulse width that he used.

Case 1. s = 1 , Pulses Resolved

a. Negative-Going Zero Crossings. Since mistuning has a small effect
on the evaluation of the Z's, we neglect it in this regard. Neglecting
terms in 1/Q 2 and k/Q, after some arithmetic one arrives at

Ly - (1 - f\ X + ^ +2^16 + 0Q085fc

r 3 , 1 , t x , ftQ7 , , 0.06

isa y + g (*-i) + 0876 + - 5 -

Q > 50 and /cQ < 0.2 encompass values of practical interest. In this
region the term in y in the denominator of (71) can be neglected and
the numerator term 0.0085fc is also negligible. It is also convenient to
deal with phase error rather than timing error. Therefore, we rewrite
(71) as

0.397

aa "-gl"i-H* + 0159+ o

(72)

The multiplication by —2ir is used to avoid any questions later on as
to which way certain inequalities are to be taken. This means that 6 is
the negative of the phase error as previously defined. A positive value
of signifies that the zero crossing occurs prior to ±T/4 for negative
going and positive going zero crossings respectively. The general form
of 6 for all the cases to be considered herein then can be written as

PHASE JITTER IN POM REPEATERS 549

'- V + a + c. (73)

x + 6
For the situation under consideration in this section,

r» 1 ri\ i 0.334 , IT ,

« = 0.159 + -g- + - A-
I = -0,5 + f

h[l-H

b. Positive-Going Zero Crossings. Proceeding in the same way as in
Sections B-2 and B-3 above, the phase error for positive-going zero
crossings is as in (73) with

b = -0.75 + x

--IG-H-

In this case it should be noted that with zero mistiming (y = 0) and
with a pulse for n = and nowhere else, a positive-going zero crossing
does not occur in the neighborhood of —T/4. Under this special con-
dition, x = 1 and (73) with the constants of this section would predict
an incorrect error in the positive-going zero crossing. Of course such a
sparse pattern occurs with probability zero. Fortunately, for all other
more reasonable periodic; patterns, results obtained from (73) are in
good agreement with computer simulation.

Case 2. s = § , Pulses Overlapping, Base Width = 1.5T

a. Negative-Going Zero Crossings. In this section we will dispense
with all of the algebra and arithmetic and simply write down the final
results. For the case at hand

now i i

-^— y - -= (0.073 - 0.064A-Q).x- + 0.0264 4- =r

e ± ^ ■ (0.034 - 0.02kQ)

T 0.255.C - 0.062 + 0.048/Q

550 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1962

When this is converted to the form of (73), we have
a = 0.65 + ^ ~ 0.21k

i = -0.243 + °^

c = - i [1.8 - 1.58&Q] .
b. Positive-Going Zero Crossings

a = 0.65 - ~ + 0.94fc

b = -0.753 + !^

c= -\: [1.8 - 1.58A-Q] .

The remarks made in connection with positive-going zero crossings
for Case 1 are equally applicable here.

APPENDIX C. SEMI-INVARIANTS FOR THE JOINT DENSITY FUNCTION OF
X\ AND y\

C-l. One out of M pulses definitely occur; the remaining pulses are in-
dependent and occur with probability %; raised cosine pulses.

The characteristic function is defined as

<p(u,v) = E exp i(uxi + vyi), (75)

where E is the expectation operator, and from Appendix B

00 00

xi = £ e~ aMm cos 2TkMm + b + £ a„e' an cos 2irfcn,

00 00 x

Z/i = S e" aMm sin 2ir/cMm + a + £ a n e _an sin 2vkn,

with a = tt/Q. Substituting (76) in (75) and performing the expecta-
tion operation gives

PHASE JITTER IN PCM REPEATERS 551

<p(u,v) = exp i ^2 e~ ( " IQ)Mm (u cos 2irkMm + v sin 2-wkMm)

•exp i(ub + va) X II ex P^o e <T/<?)n (wcos2T/ai + vsm.2irkn) ) (77)

n^mM [2

00 f -(r/«)n

X II cos< — - — (u cos 2-irkn + *> sin 27r/cn)

n^mM \ 2

which may be rearranged to

tp(u.v) = exp - /\ le~ r Q (u cos 2irkMn -f- y sin 2wkMn)
|_2 „=o I,

.]

H cos < — - — (u cos 2irkn + sin 2Tkn)

_l_ g iir/ejn ^ w cog ^fcfi -{- y sin 2irkn) } | exp i(«ft + va)

-MQ)n

n cos{-

A

(78)

e
cos {

n=0

r -(ir/Q)Mn V

1 1 cos i o ( u cos 2irfcMn + y sin 2irkMn) >

n=0 [ 2 J

When we take the logarithm of (78), we obtain

log <p(u,v) = l - Y. [p Mn (ucos 2wkMn + v sin 2wkMn)
2 „=o

+ /3"(m cos 2irkn + y sin 2irkn)]
+ i(ua -\- vb) + ^2 log cos — (u cos 2irfcn + y sin 2irkn) (79)

n=0 L 2 J

- J2 log cos M-q- (u cos 2irkMn + y sin 2-irkMn) ,

where = e~ MQ) .

The first sum in (79) may be carried out, and when combined with
i(ua + vb) yields the semi-invariants X 10 and X i which are of course
the mean values for Xi and iji respectively. Since the last two terms of
(79) are similar in form, we will confine our manipulations to the next
to the last term. We denote this term by

F(u,v) = ^ log cos — (u cos 2irkn + v sin 2irkn) .
„=o \_2 J

(80)

552 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1962

Using the infinite product expansion for the cosine and the power series
expansion for the log; i.e.,

cos 2 = n

m=0

1 -'■■■■■ VJ] "<->

2z_
(2m +

and

log(l-*) = -Z- (x < 1).

F(u,v) becomes

F(u,v) = -EEE .ff'tffiC (si)

£To £3> h z (2m + 1) 2; tt- ;

where C„ == e~°" cos 2*7m and S n = e~ an sin 2xfcw. The sum over j may
be obtained by virtue of

A 1 _ (2 2j - lK-p^x)"^

^o(2m + 1)* 2^(2 j) I

where the 2? 2 y are the Bernoulli numbers. With the above sum over m
and the expansion of (uC n + vS n ) 21 in a binomial series, we arrive at

'<«"> - S (-1) g(S" " § ( 2 /) s c " v(s - )! '" (82)

Proceeding in the same manner that took us from (80) to (82), it can
be verified that the last term of (79) takes the same form as the right-
hand side of (82) with n replaced by nM. These results and comparison
with the definition of the semi-invariants for a two dimensional dis-
tribution 10 lead to the following for the semi-invariants for the process
under consideration :

_ 1 I" 1 - g cos 2wk ^ 1 - M cos 2tUM 1 ,

Xl ° ~ 2 Ll - 2/3 cos 2irk + /3 2 + 1 - 2/3" cos 2*kM + P 2M \ '

__ 1 [" g sin 2irb /3 M sin 2irkM 1

01 ~ 2 Ll - 2/3 cos 2tt/c + /3 2 + 1 - 2/3" cos 2irkM + /3 2M J °'

and

Xrs ] r+s>1 = B r+ '(2 T+S - 1) £ [CnX . _ CnAfUVL (g3)

The sum over n can be shown to be a geometric series multiplied by
two finite series if the sines and cosines in S and C respectively are rep-
resented in exponential form and use is made of the binomial expansion.
After some algebra, an alternate form for (83) can be shown to be

PHASE JITTER IN PCM REPEATERS 553

X "'->' = V+OT-)-" «'■***>• (84)

where G(r,s,fi,k,M) is (shortened to (?)

g = j\j\ (—i) q

P =o 9 =og!(r - p)lq\(s - q)\

f 1

.1 - /?<'+«> exp [t2rt(r + s - 2p - 2g)]

1

(85)

,]■

1 - 0«Hf> exp [iSMfcflf (r + s - 2p - 2g)].

For m and u in the neighborhood of zero, the contributions to the series
in (79) become smaller as n becomes larger. The importance of suc-
cessive terms is judged by the exponential decay factor e~ ( " T,Q) . If we
consider all terms up to some ?i mnx where n ran x » Q/tt and kn max « 1,
then we arrive at the following inequality

^ « 1 . (86)

IT

Under the above condition cos 2irkn can be replaced by unity and sin
2wkn by 2irkn for all terms of importance in the series and (79) becomes
approximately

+ (ub + va)\ + £ log cos 1^- (u + 2rknv)\ (87)

- J2 log cos <^- (u -t- 2TrkMnr) .

Paralleling the operations performed on (80) to obtain (82) it can
be shown that the semi-invariants obtained from (87) under the con-
dition (86) are

^ = -K(I^ + (T^) + a ' Md (88)

X »U>i - ( 1) - (|i + a) - (2xA ) — \ — — I — — - J ,
with # = (r + s)ir/Q.

554 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1962

C-2. Same as I Above Except That Pulses are Impulses

For this case the semi-invariants are as above with a = = b.

C-3. Impulse Excitation, All Pulses Random

With this type of excitation, we have

_ 1 I" 1 - j3 cos 2irk "I
10 ~ 2U - 2/3cos2tt/o + /3 2 J'

1 f ]8 sin 2vk 1

01 " 2 Ll - 2/3cos27rA: + 0"J'

and

_ B r+ .(2 r+ ' - l) rM [ + Y- ("I)'

ArsJr+8>1 ' (r + •)»(«)• Lfe & PKr " P) lgl(« - 9) !

! 1.

I - /3 r+8 exp [i2irfc(r + s - 2p - 2g)] J

It is readily shown in this case that the approximate semi-invariants
[subject to (8(>)] are

Aio — ~

2 \1 - j8/ '

(l - &Y-

X(11 = ,.«** (90)

g r+ ,(2 r+8 -l) ( r rf 4 / 1 \
xa +s >i - (-D - r + s - <*« — ^— ^J,

with = (r + «)r/Q.

APPENDIX D

High Q Behavior of p(0)

To illustrate the behavior of the probability density function when
the Q of the resonator becomes large, we consider p(6) in the neighbor-
hood of the mean, 6 . We include terms of the double summation in
(19) for which r + s = 4. Since the Bernoulli numbers B r+8 = for
r + s odd and > 1, the terms X„ for r + s = 3 are zero. For 6 ~ 6 ,
therefore, p{6) becomes

PHASE JITTER IX PCM REPEATERS

555

p(6) =

An

V2tt (Xm0o 2 - 2\u0 o + Xo 2 ) J

Xio
•exp — — -

(e - e o y

H t

2 (\20S0 — 2\nd + X02)
X 1O (0 - ».) \

l) r+ S =4

(91)

, 1 \V 2(X20^o — 2Xll0 o ~{~ X02) / yy /_i\r Ah 6 J

•\/2(X 2 o0 o 2 - 2Xiift, + Xo2) J r . r!s! X10 4

x„

r+«>2

The semi-invariants of interest in the above equation are given below
and were determined using the results of the previous section for the
case "all impulses random," subject to kQ « 71- .

XlO —

1

2(1 -/3)

X 20 = i x

4(1 -0 2 )

X11 =

X40 — — =

1 1

1 - w

X01 _
Xio

irk

2ttA-/3
1 -/8

/3 2

2 (1
wk

-0 2 ) 2

/3 4

4

(1 - 4 ) 2

X02 —

Uk) 2 (3 2 (l + 2 )

X22 — —

(i - py

{TckffiW + /3 4 )
2 (1 - /3 3 ) 3

Xl3= -^ A "> 3 ( i -^4)4 (1 + 4/3 4 + /3 8 )

Xoi = -2{irkY

1

1 - |9»)»

(1 + ll/3 4 + ll/3 8 + /3 12 ).

Using the above expressions for the X's, the following quantities in (91 )
may be reduced to

An

1

(Xaoftr - 2Xi !0„ + X02) 1 y/2(2vk)fi

(1 -0)*(1 - fi)i

r+s=4

EE(-D

r X r8 do

rYslX^

r+«>2

1 2(27rA-)V 4
41 1 - /? 4

r _ 4/3 3 (l - g) 0/3 2 (l + /3 4 )(1 -ff) 2
I (1 -0 4 ) (1 -0 4 ) 2

4/3(1 - /3) 3 (1 + 4/3 4 +_^) _i_ (1 ~ g) 4 + 110* + 11/3 8 + 1

(1 - P 4 ) 8

(1 - /3 4 ) 4

']■

or

556 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1962

r+s=4

EE(-i)'

vr x r8 e: x

4

r!s! Xio 4 4l

r+s>2

The probability density therefore takes the form

1 + 2

X^VV^/

(92)

This result is in the form of the standard Edgeworth approximation
with 6 , a, and X 4 the mean, the standard deviation and the 4th semi-
invariant of the distribution, respectively. In the limit as Q becomes
large (0 — * 1) we approximate 1 — jfl by t/Q and

<r -> ky/rQ d -> 2kQ

The coefficient of the 4th Hermite polynomial approaches — (5tt/128Q).
Equation (92) then indicates the approach to the normal law with the
first correction term going as 1/Q. The results for d and <r correspond
to those derived earlier by Bennett, Rice and others.

APPENDIX E

Determination of raax

For kQ <g.TT,& good approximation for is (from Appendix B)

DO

a + 2rk E a n n(3 n
= J=? . (93)

b + E a«/3"

n=0

When a„ = 1, we have

fl k\ + E a^/3

/t = — ==% • (94)

14- 6 + E M

n=l

It is of interest to determine the pulse pattern that yields the maximum
value of 9/2-irk. This is equivalent to the determination of a one-zero
sequence of a„'s such that (94) is a maximum.

PHASE JITTER IN PCM REPEATERS 557

Assume that an initial pattern has been chosen such that d/2irk =
A /B o . If a single a n is changed from zero to one (pulse added), then
0/2** is changed to (A + np n )/(B + 0"). Clearly, we should effect
this conversion if

A + wj8" > Ao
B + B n = B

or

n ^ £ • ( 95 )

On the other hand if a one is changed to a zero (pulse removed), then
6/2irk will be increased if

A - n0 n Ao
Bo - 0" B

or

n < 4-°. (96)

■Do

The process is continued in this manner until all a n = 1 for n ^ n c and
all a H = for w < n c (except a a , which is constrained to be unity).
n c may be determined from the above process, since

n c = ?£ = i== =^ , (97)

2ttA-

i + fc + E r

which can be rearranged to

(3 n < +1 a

1 - /3) 2 2rk

+ m(l + 6). (98)

When a periodic pulse pattern of 1 out of every M pulses is forced,
0mnx is found in the same manner as above and the relationship between
the various parameters to achieve this maximum is given by (15) of
the main body of the paper.

REFERENCES

1. Bennett, W. R., B.S.T.J., 37, Nov., 1958, p. 1501.

2. Rowe, H. E., B.S.T.J., 37, Nov., 1958, p. 1543.

558 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1962

3. DeLange, 0. E., B.S.T.J., 37, Nov., 1958, p. 1455.

4. DeLange, 0. E., and Pustelnyk, M., B.S.T.J., 37, Nov., 1958, p. 1487.

5. Sunde, E. D., B.S.T.J., 36, July, 1957, p. 891.

6. DeLange, B.b.T.J., 35, Jan., 1956, p. 67.

7. Aaron, M. R., B.S.T.J., 41, Jan., 1962, p. 99.

8. Mayo, J. S., B.S.T.J., 41, Jan., 1962, p. 25.

9. Cramer, H., Mathematical Methods of Statistics, Princeton University Press,

Princeton, N. J., 1946, p. 317 and p. 224.
10. Laning and Battin, Random Processes in Automatic Control, McGraw-Hill
Book Co., New York, 1956, p. 61.