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

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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.