statistics of regenerative digital transmission

42
Statistics of Regenerative Digital Transmission By W. R. BENNETT (Munuscript received April 16, 1958) Statistics of a synchronous binary message pulse train applied to a re- generative repeater are related to those of the original binary message, which is assumed ergodic. It is shown that the ensemble of possible message pulse trains is a nonstationary random process having a periodically varying mean and ouiocooariance. A spectral density is calculated which shows line spectral components at harmonics of the pulse rate and a continuous density function, both with intensity proportional to the square of the absolute value of the Fourier transform of the standard pulse at the frequency considered. The continuous component has many properties similar to thermal noise but differs in that, under certain conditions described, it can exhibit regu- larly spaced axis crossings, can be exactly predicted overfinite intervals and is capable of producing discrete components when nonlinear operations are performed on it, even though no line spectral terms are originally present. The analytical results are applied to the problem of deriving a timing wave from the message pulse train by shock-exciting a tuned circuit with impulses occurring at the axis crossings. I. INTRODUCTION An ideal regenerative digital repeater is defined here as a nonlinear time-varying device with response expressed as the product of a stair- case output vs. input function such as shown in Fig. l(a) and a time- sampling function such as shown in Fig. l(b). The first function (a) converts a continuous range of possible input values to a discrete or quan- tized set of output values. The second function (b) is a time-varying response function which ideally makes the repeater sensitive only during infinitesimally narrow time intervals with regular spacing T = lifT' Taken together, the two functions quantize and sample the input wave at uniformly spaced instants of time. The input wave consists of a dis- crete-valued message component plus noise and distortion accumulated 1501

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  • Statistics of RegenerativeDigital Transmission

    By W. R. BENNETT(Munuscript received April 16, 1958)

    Statistics of a synchronous binary message pulse train applied to a re-generative repeater are related to those of the original binary message, whichis assumed ergodic. It is shown that the ensemble of possible message pulsetrains is a nonstationary random process having a periodically varyingmean and ouiocooariance. A spectral density is calculated which shows linespectral components at harmonics of the pulse rate and a continuous densityfunction, both with intensity proportional to the square of the absolute valueof the Fourier transform of the standard pulse at the frequency considered.The continuous component has many properties similar to thermal noisebut differs in that, under certain conditions described, it can exhibit regu-larly spaced axis crossings, can be exactly predicted overfinite intervals andis capable of producing discrete components when nonlinear operations areperformed on it, even though no line spectral terms are originally present.The analytical results are applied to the problem of deriving a timingwave from the message pulse train by shock-exciting a tuned circuit withimpulses occurring at the axis crossings.

    I. INTRODUCTION

    An ideal regenerative digital repeater is defined here as a nonlineartime-varying device with response expressed as the product of a stair-case output vs. input function such as shown in Fig. l(a) and a time-sampling function such as shown in Fig. l(b). The first function (a)converts a continuous range of possible input values to a discrete or quan-tized set of output values. The second function (b) is a time-varyingresponse function which ideally makes the repeater sensitive only duringinfinitesimally narrow time intervals with regular spacing T = lifT'Taken together, the two functions quantize and sample the input waveat uniformly spaced instants of time. The input wave consists of a dis-crete-valued message component plus noise and distortion accumulated

    1501

  • 1502 THE BELT, SYSTEM TECHNICAL JOURNAL, NOVEMBER 1958

    in transmission from the previous repeater. If the noise and distortionare sufficiently small to confine the departure from the proper discretemessage value within one quantization interval at the sampling instants,the regenerated wave matches the message and errorless transmissionis achieved for any number of repeater spans.

    To economize bandwidth, the actual pulses transmitted are not in-finitesimally sharp, nor even rectangular as indicated in Fig. l(b), butare multiples of a standard finite width pulse get) which may be consid-ered to be generated by a linear network from the sharp samples. A typi-cal outgoing wave from the repeater is therefore expressible by:

    00

    x(i) = L angel - n T),n=-oo

    (1)

    where the sequence an , n going from - 00 to 00, represents the messagevalues. In much of our work we shall specialize our attention to the binarycase, in which there are only two possible values of an , which we shallusually take as zero and unity. The corresponding output vs, input func-tion of Fig. 1(a) then takes the form shown in Fig. 2. A typical outgoingwave train is shown in Fig. 3 for the message sequence

    .. 10010110 ...

    The ideal repeater requires an absolute dock to supply the timingcontrol exemplified by Fig. l(b). In practical transmission systems ofconsiderable length, it may not be feasible to supply such absolute timinginformation. An alternate procedure, for example, derives the timingwave from the signal pulse train itself by applying a portion of the wave

    Wl/J

    ~ twa:

    1---

    0-2T -T 0 T 2T

    TIME-

    (b)

    o +INPUT-

    t Al~ Aof------r-+...-------j0-f-

    6-A,

    Fig. 1 - Specification of ideal regenerative repeater characteristic.

  • STATISTlrs OF ltEGEXERATIVt: DIGITAL TltA!'\SMISSION 1503

    train to a circuit tuned to the pulse repetition frequency. If the circuithas a sufficiently high Q it continues to oscillate at the desired frequencyduring the gaps when no message pulses are present. Design of the tunedcircuit requires a statistical analysis of the pulse train. This is one ofmany important statistical problems associated with practical, as dis-tinguished from ideal, regenerative repeaters.

    In this paper, we consider the following specific problems:1. Properties of a digital message pulse train as a random noise source.2. Derivation of the pulse repetition frequency from a pulse train by

    shock excitation of a tuned circuit. Under this topic, the following ef-fects are studied:

    (a) message statistics(b) message pulse shape(c) Q(d) mistuning(e) noise.

    a. Effect of time jitter in received pulse train on recovered analog sig-nal.

    The important problem of analyzing accumulated effects in a chain oflionideal regenerative repeaters is discussed in a companion paper byII. E. Rowe.' Both this paper and that of Rowe are intended to pro-

    tI-::>a.I-::>o

    , --

    1/2INPUT-

    IIo

    Fig. 2 ~ Binary output-vs.vinput response.

    T1ME-

    Io

    Fig. 3 - Typical binary wave train.

  • 150-1 THE mct.r, SYSTEM TECHNICAL JOURNAL, NOVEMBER 1958

    vide analytical background for the two preceding papers in this issue." 3Reference is also made to a paper by E. D. Sunde" which treats manyof the same problems in a somewhat different way and gives results forspecific repeater embodiments.

    II. A DIGITAL MESSAGE AS A RANDOM NOISE SOURCE

    The digital message may be regarded as a random time series of dis-erete numbers. In the notation of stochastic processes, consider the en-semble of possible messages 1I1(t), where a typical M(t) consists of theinfinite discrete time sequence: ... a-2 , a-l , ao , al , a2 .... The valuesof the a's belong to a specified set of discrete numbers and each possiblesequence has a probability of occurrence associated with it. We assumethat the message ensemble is ergodic; that is, averages over the ensembleat fixed time are identical with averages over time in any member of theensemble, except for a set of probability zero. It follows that the en-semble is also stationary; that is, averages over the ensemble do not de-pend on position in the sequence. For our purposes we do not go beyondsecond-order statistics and shall base our calculations on two ensembleaverages, the ordinary mean and the autocovariance.

    The mean ml is defined by the ensemble average for any n:

    (2)

    The value of ml is a constant for the message ensemble. For example,in the binary case in which zero and one occur with equal probabilityml is equal to one half. The autocovariance R(n) is defined by the en-semble average

    (3)

    The value of R(n) depends on n but not on k, It is determined by thedependency of successive message values. If the message consists of in-dependent numbers, the value of R(O) is the average squared messagevalue and R(n) is equal to the square of the mean for n not equal to zero,since the average of the product of independent quantities is equal to theproduct of their averages. From the ergodic property, these quantitiesmay also be derived from almost all members of the ensemble indi-vidually, thus:

    r 1 Nml = un?N L an, (4)N~"" - + 1 n--Nr I NR(n) = ;:;~ 2N + L akak+n. (5)1 k=-N

  • STATISTICS OF HEGE:'\ER.-\TIVE DIGIT,\L TRA:\'Sl\HSSIO:'I1 li){);')

    The actual wave sent over the line from one repeater to the next is thefunction ;r(i) given by (1). We may regard x(t) as a typical member ofthe pulse ensemble Ix(t) 1. Unlike the message ensemble 1M(t) I, thepulse ensemble {;r(t)I consists of continuous functions of time rather thandiscrete time series. We shall now relate the statistics of the pulse en-semble to those of the discrete message ensemble.

    We first evaluate the ensemble average of {x(t) by holding t fixedand averaging over all members. Then, since the average of a sum isequal to the sum of individual averages, and since the standard pulseget) is the same for all members of the ensemble, we find

    00

    av (;r(O) = nv L ang(t - n'I')11=-00..

    = L av (an)g(t - nT)n=-oo..

    = nil L get - nT).n=-oo

    (6)

    It thus appears that the ensemble average varies with time. This is notreally unexpected since it seems reasonable that we should find values inthe middle of a pulse interval to be distributed quite differently fromthose at the beginning or end. In fact, we should expect the ensembleaverage to vary periodically at the pulsing rate, and we readily demon-strate this by noting that

    00

    nv [x(t + 7') = ml L get + T - nT)00

    = nil L g[t - (n - 1)1'] = av x(t)n=--oo

    (7)

    by substitution of the summation index n' = n - 1.The periodicity of the ensemble average suggests a formal Fourier

    series expansion, thus00

    av xCi) = L c.; exp (2m'll" jfTi), (8)m--CIO

    c; = IT[' av :r(t) exp (-2m'll"jITO di

    .. iT= mdT L get - nT) exp (-2m'll"jfTt) di.n-ao 0

    (9)

    We now substitute t - nT = 11 in the integral. The limits then become-nT to - (n - 1)7' while the exponential becomes exp (-2m'll"jfTu),

  • I50G THR BELL SYSTEM TECHXIC_-\.L JOUHXAL. XOVEMBER lUa8

    since i,T = 1. The summation of finite integrals with adjoining limitsmay then be replaced by a single infinite integral, giving

    em = nu]; i: g(u) exp (-2m1rjjru) duo (10)If we introduce the Fourier transform of the standard unit pulse by

    G(j) = i: g(l) exp (-21rjjt) dt (ll)we observe that

    (12)

    find, hence,

    ""uv {.r(t)j = me]; L GCmi,) exp (2m1rjj,t). (13)rn=-oo

    The ensemble average is thus expressed as a Fourier series in time withthe amplitude of the mth harmonic of the signaling frequency propor-tional to the amplitude of the spectral representation of the unit pulseat that same harmonic frequency. We note that an ensemble consistingof precisely this set of harmonics and no other terms would exhibit justthis same result for its ensemble average. It is convenient, therefore, toresolve our pulse ensemble into a set of such discrete nonrandom har-monic components plus a remainder which has random properties andzero mean. \Ve consider therefore the ensemble

    ee

    {y(t) = (;r(t) - av {x(t)} = L (an - ml)g(t - nT). (14)Then,

    av {y(t) = O. (15)

    (16)

    We explore farther by evaluating the autocovnrinnce function Ru( T, t)for the ensemble {y(t)}. By definition,

    Ru(T, t)

    = av {y(t)y(t + T) I

    = av {rJ~oo (am - ml)g(t - mT) n too (an - ml)g(t - nT + T)}oo ""

    = L L av[(am - ml)(an - ml)]g(t - rnT)g(t - nT + T)m=-oo n--co

    "" ""= L L [R(m - n) - mI2]g(t - mT)g(1 - nT + T).m=-co n--ClO

  • STATISTICS OF REGEXt~R,\TIVE DIGIT.\L TRAXS~IISSIOX 150i

    By replacing t by t + T and increasing both summation indices by unity,we see that Ry ( T, t) is periodic in t with period T. Hence, as in the caseof the mean,

    '"Ry(T, t) = L dk exp (2k1r if,t) ,

    co '"= I, L L [R(m - II) - 1Il12]

    m=-(lCl n=-oc

    iT get - mT)g(t - nT + T) exp (-2hjI,t) dt.

    (17)

    (18)

    'Vc substitute m - n = m'; t - nT = r, and find after some renrrunge-merit analogous to that used in obtaining (to):

    where

    ee

    (h = I, L [R(m) - m/]F(T + mT, I,J,), (I !.I)

    (20)

    By the convolution theorem.

    F(T, I) = i: G(X)G*(X + .n exp [-J21rT(X + j)] dX. (21)In the specinlcnse of independent message values,

    R(O) = av (a/) = m2; R(n) = av 2 (n,,) = m}, n ~ 0 (22)and

    rl,. = I, val' (n,,) F( T, 1,/,),when' val' (n,,) is the variance of the n'" defined by

    (24)

    There is no reason, in general, for F( T, 1./,) to vanish for all nonzerovalues of k, and hence the autocovarinnce of the !I-ensemble docs notreduce to an expression independent of time. It appears, therefore, thatseparating out the periodic components which account for the fluotuat-ing mean does not make the remainder a stationary process. We there-fore cannot invoke the Fourier transform relationship between auto-covariance and power spectrum to find a power spectrum for our pulseensemble, even after the periodic components have been removed.

  • 1508 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1958

    A procedure which has often been used in similar situations is to makethe process stationary by assuming a random phase relationship betweenthe members of the ensemble. We do not wish to do this here becausesynchronous phase is an important property to be preserved in digitalregeneration. However, there is a meaningful definition of a powerspectrum which can be used without reference to the fluctuating auto-covariance. 'We shall calculate the spectral density in this way, but, asmight be expected, the resulting function is the same as would be ob-tained by randomizing the phases of the ensemble.

    Consider first the ensemble YN(t) which includes only the pulses fromn= -Nton=N:

    N

    YN(t) = I: (an - ml)g(t - nT).n=-N

    (25)

    Then, for unit pulses which possess a Fourier transform, the Fouriertransform 8N(f) of YN(t) exists and is given by

    N

    SN(J) = I: (an - ml)g(J) exp (- 2n1r1fT). (26)n=-N

    By Parseval 's theorem,

    l: y/(t) dt = l: I SN(J) 12 df, (27)

    (28)

    Let the average of the ensemble y/(t) over the interval -NT to N'I' berepresented by tWNy/(t). Then,

    1 [NT 2aVN y/(t) = (2N + 1)'1' lNT av YN (t) dt

    1 100 2= (2N + 1)'1' -00 av ISN(n I df

    1 [1 00 I N TJ 2= (2N + 1)'1' NT + 00 av YN (t) dt.Also,

    av., Y002(t) = lim aVN y/(t)N-oo

    where

    = lim t" av I SN(J) 12 dfN-oo L; (2N + 1)'1'1: w(J) df,

    (f) I , av I SN(J) 12

    w = In1 .N-oo (2N + 1)'1'

    (29)

    (30)

  • ST,\TISTICS OF REfTE:\ER.\TIVE nWITAL TRA,",SMISSIOX 1509

    From (29), lO(J) as defined by (:30) satisfies the requirement for a spec-tral density. From (2G),

    nv 1 SN{f) 12

    = :LV [SN{f)SN*(f)]N N

    = L L avf(a n - 1nl)(am - ml)]G(f)G*(j) exp [21TjfT(m - n)] (31)".=-l'/ n=-N

    N N

    = L L [R(m - n) - m/I I GU) 12 exp [21TjfT(m - n)].m=-N 11=--N

    Let m - n = k, givingN m+N

    uv I SN{f) 12 = L L [R(k) - 1/1/11 GU) 12 exp (21TjkfT). (32)m~-N k~m-N

    The order of summation can be interchanged by the formula:N m+N 0 k+N 2N NL L = L L +L L .

    m=-N k=m-N 1.=-2N m=-N k=l m=k-N(33)

    (34)

    Since the summand does not depend on m, we perform the summationon m by counting the terms. The result is:

    nv I SNU) 12 = (2N + 1) 1GU) 12 {R(O) _m12

    + 2I: (1 - ')N k ) [R(k) - m/] cos 21TkfTJl.k-l _ + 1

    From (30), then,

    wU) == fr 1 GU) 12 {R(O) - l/!J2 + 2~ [R(k) - m/] cos 21TkfT}. (35)In the special case of independent message values, R(O) = m2 = ml andR(k) = rn/ for k rf. 0, giving a result in agreement with the well-knownsolution" based on a random phase ensemble average, namely

    This spectral density function is defined for both positive and negativefrequencies. If the densities at negative frequencies are added to those ofthe corresponding positive frequencies, the above result is multiplied bytwo.

    The spectrum of Iy(t) I is thus continuous and is proportional to thesquared absolute value of the Fourier transform of the unit pulse. Thespectrum of the actual pulse ensemble has added to the above con-

  • 1510 THE BELL SYSTEM TECHXlCAL JOURNAL, ~OVEMBER 1958

    tinuous spectrum a set of line spectral components given by (13). Theline spectral terms occur at harmonics of the signaling frequency andhave amplitudes proportional to the unit pulse spectrum evaluated atthe harmonic frequencies. If the pulse spectrum vanishes at any harmonicthat harmonic does not appear in the l.r(t) I ensemble spectrum. Equa-tion (1:i) can also he written in the form

    [IV l.r :t) I = mdTG(O)~ (37)

    + 2mdT L I G(m.!,) 1 cos [mw,t + ph G(m],)].m=l

    From this expression it is clear that the de component has mean squarem/ ]/ G2(O) , while the mean square value of the mth harmonic is

    2mI2 ]/ [G(mfr) 1 2

    Fig. 4 gives curves of the continuous and line spectral density com-ponents for independent binary on-off signaling with various standardpulse shapes. For this case, if PI is the probability of the value unity, itfollows that

    m2 = PI;

    The continuous part of the pulse train spectrum has much ill com-mon with thermal noise. An audible band selected between adjacentharmonics sounds to the ear like thermal noise. If the signaling rate isplaced well above the noise band a random on-off pulse train servesadequately as a hiss source in a vocoder synthesizer. Oseillograms andspectrograms of narrow-band telegraph noise not including a harmonicwould appear to the eye to he indistinguishable from thermal noise. Thecurves of Fig. 4 provide useful quantitative data relating the amount ofnoise produced in this way for specific cases.

    It must not be concluded, however, that random telegraph noise is inall respects equivalent to thermal noise with the same spectral density.The telegraph noise remains a nonstationary process and has a phasestructure among its components which thermal noise does not possess.The nonstntionary properties of the telegraph wave ensemble are Hotthe most general but are of a special kind which have sometimes beendescribed as "periodically stationary." We have suggested the name"cyclostationary" for this type of process, i.e., one in which the ensemblestatistics vary periodically with time. Analogously, the name "cyelo-ergodic" will be used for cyclostationary processes ill which statisticsover the ensemble for fixed time are the same as statistics over the in-

  • STATISTICS OF REGEXERATIVE DIGIT,\L TRAXSMISSIOX 1511

    0.6,---,------------, 0.6,----.------------,

    2.0

    f k-T~ TIME,t_

    .L -Hl + COS 7Tt)zX-' TI -,

    , I.-T..j TIME,t_CURVE GIVEN BY 1- (SIN 7Tf)"

    2 7Tf0.21----\1-----.--- ,III

    O~I-----;::7.....::!1'":":::-

    ~ 0.08-c::lo I

  • 1512 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1958

    is made smaller, and the overlapping of adjacent pulses in time becomescorrespondingly greater. From the standpoint of digital transmission,reading the message is, of course, a paramount objective and the band-width of the transmission medium must be made large enough and thenoise and distortion small enough to enable a correct reading to be made.

    A further salient feature of the telegraph noise ensemble as contrastedwith thermal noise is the possibility of regularly spaced axis crossings orzeros of the received wave. From (1) and (14) we see that both x(t)and Yet) will have zeros at points spaced T seconds apart if the stand-ard pulse y(t) has such zeros. That is, if for some to

    then

    g(to + nT) = 0, n = 0, 1, 2, ... , (:39)

    x(lo + nT) = y(to + nT) = 0, n = 0, 1, 2, . ". (40)

    (42)

    This does not mean that there cannot be other zeros as well, and theother zeros in general can have irregular spacings. It is possible, however,to exclude irregular zeros in specific cases. For example, if the standardpulse is the unit impulse response of a series-tuned circuit consisting ofresistance R, inductance L, and capacitance C, the function get) becomesYo(t) defined by

    Wo LyoU) = (w02 + cl//2 exp ( -at) cos (wot + fJ) j t > (41)

    a = R/2L; w02LC = 1 - R2C/4Lj tan fJ = R/(2woL).By setting Wo = 2'lr/T, we obtain nulls at to and t1 in the interval

    s t ~ T,where

    27rto/T = 7r/2 - e,27rtdT = 37r/2 - fJ.

    These represent the only nulls in the interval from zero to T and theyrepeat at to + nT and t1 + nT for all integer values of n, as shown inthe full-line waveform of Fig. 5. Furthermore, yoU) changes in sign frompositive to negative at t = to + nT and from negative to positive att = t1 + nT. No other sign changes occur. If we consider Yo(t - mT)with m any positive or negative integer, we note that the values betweento + nT and t1 + nT are the same as those of Yo(t) between to + (n - m)Tand t1 + (n - m.)T, and hence they are all negative. Likewise, the valuesof yo(t - mT) between t1 + nT and to + (n + I)T are all positive. The

  • STATISTICS OF REGENERATIVE DIGITAL TRANSMISSION 1513

    t

    i-a.tcost2~t +8J/

    -a.(t-2T) ~21T(t-2T) ~f cos T +9/

    o T 2T 3TTIME,T-

    4T 5T

    (43)

    (44)

    Fig. 5 - Response of tuned circuit to synchronous impulses.

    dashed curve of Fig. 5 shows the relation for m = 2. The conclusion isthat the summations from which x(t) and yet) are formed consist ofwaves having common nulls and the same sign between nulls. It followsthat both x(t) and y(t) for the case of g(t) - go(l) have nulls only atto + nT and t} + nT, the axis crossings are from plus to minus at to + n/I',and the axis crossings are from minus to plus at t1 + n'I',

    We have thus proved that, if go(l) is the standard pulse, the continuousnoise spectrum arising from a random choice of signaling pulses has noeffect on the nulls of the composite wave. Telegraph noise therefore isprofoundly different from thermal noise in that it does not perturb axiscrossings in a situation in which thermal noise certainly would. Produc-tion of an invariant null spacing permits the recovery of the timing orclock wave from the pulse train and is accordingly of vital importancein the self-timed regenerative repeater.

    In general, the standard pulse will not have the invariant null propertyexhibited by the special function gn(t). It may, however, be possible toconvert the actual g(t) into a new pulse which does have the requiredproperty. We note that the Fourier transform of go(t) is given hy

    ) _ j27rfCGoU - 1 + j27rfRC - 471'2 j2LC'An ensemble with standard pulse get) and transform G(f) can be con-verted into one with standard pulse go(t - To) by inserting a linear net-work with transmittance function

    y(n = ~o':~ exp (-j27rfTo),

    provided that Y(f) is physically realizable. Since Go(f) is the response of

  • 1514 THE BEI,L SYSTEM TECHNICAL JOURNAL, NOVEMBER 1958

    a tuned circuit to a constant spectrum, the problem reduces to the re-alizabilityof the reciprocal of G(f) with an arbitrary delay factor.

    Suppose, however, that the received pulse shape is such that Y(J)can not be realized. For example, (:~) for the discrete or line spectrumcomponents shows that, if the pulse transform G(f) vanishes at the pulserepetition frequency IT and all its harmonics, there are no discrete fre-quency components representing the signaling frequency in the pulsetrain. In (44) we would then have G(.f) = 0 at I = IT and, since Fo(fr)is not zero, the value of Y(.fT) would have to be infinite. It might seemthat in this case the timing information is lost. However, we shall showthat this difficulty only exists for linear time-invariant methods of deriv-ing the timing information.

    To see how we can materialize a discrete component from a con-tinuous spectrum, consider a specific ease of an uncurbed rectangularsignaling pulse

    Then, from (11),

    g(t) = 1; o < t < T. (45)

    (40)

    It is dear that G(nfT) vanishes for all integer values of n. If a pulse byitself contains zero amplitude at all harmonics of IT' no succession ofpulses cnn have nonzero amplitude at these frequencies. However, wenote that, in a binary on-off system using these uncurbed pulses, a wavesuch as that shown in Fig. 6 is obtained, in which every message digram

    Io 0

    I I

    o 0 0

    Fig. 6 - Train of uncurbed rectungulur pulses.

    onnnOVVO

    C\/\o 0 1 1 1 0 1 0~ f.T

    -

    THRESHOLOLEVEL

    1\ 1\ --A_LAAA- 0 000OOll1VOIVo 00111010 00111010

    1\ 1\1\1\00111010

    CIRCUITTUNEOTO I/T

    Fig. 7 - Recovery of timing wave from transitions of binary pulse train.

  • (47)

    i-il'ATli-iTH':-; OF HE;C;Jo;:'I:EIUTIVE DIGIT,\L TK:\:'IISMIi'lHIO:'" 1515

    01 produces an upward transition, every digram 10 produces II down-ward transition, and the wave is constant for unbroken sequences of1's or O's. If this wave is applied to the circuit of Fig. 7, which includesa differentiator, full-wave rectifier, threshold triggered pulse generatorand tuned circuit, an output wave containing pulses of form

    Yo(t - nT + To)is obtained where n represents the integer values in the message sequenceat which 01 01' 10 digrums oceur. The effective message source is trans-formed from the original sequence of zero and unity values to a newsequence in which the unity values are associated with 01 or 10 transi-tions and the ZP!'O:- with 11 or 00 digrarns. A new continuous spectrumis thereby obtained and a discrete line spectrum absent from the originalwave is created. Nonlinear operations have been used to attain this end.The system will still operate if either (but not both) the rectifier or thenonlinear triggered pulse generator is omitted. If we were dealing withthermal noise, even nonlinear operations would not suffice to constructa discrete spectrum.

    The impulse response of a tuned circuit is only one example of a stand-urd pulse shnpe which yields correct timing information from a syn-chronous tolegrnph pulse train. Another standard pulse having the samep,';Rential properties is till' response of the same tuned circuit to a rec-tangular pulse of duration equal to half the signaling period. When theexciting pulse has unit height, the response becomes Yl(t), defined by

    WOLYl(l) =

    jRin wo/; 0 < t < '1'/2exp ( -at) sin wol - exp (aT/2) sin wo(t - '1'/2); t > 'J'/2,When WilT = 211", it mar bc verified that Ol(t) has positive-going nxisero;;sinll;s at I = 0, '1',21', . ,. ; negative going axis crossings at I = 1'/2,:~T/2, f>T /2, ... ; and no other axis crossings. Hence the surnmationof any such pulses beginning only at multiples of T has the same nullsand no others,

    The problem of producing regularly spaced axis crossings from a re-ceived pulse train is studied in more detail in the next section, whichtreats HlP problems nssoeinted with derivation of timing information.W eouclude Olll' discussion here with some further remarks aboutsp('olldary pul: trnins triggered from transitions in a primary pulse(rain. HllI'h HI'('olldar,Y trains arc of major importuuce ill pulse fl'l'l[uPII('~'derivntion techniques, The example given ubove of a primary uncurbedrectnugulur pulse train is not of much intere-t in digital transmission,

  • 1516 THE BELL SYSTEM TECHKICAL JOUR"AL, KOVEMBEH 1958

    since it would require excessive bandwidth to approximate the squarewaveforms. A more practical type of pulse is the raised cosine of Fig. 8.Such a waveform can be transmitted with fair nccuracy over a band-width slightly less than the signaling frequency. Closest spacing withoutoverlapping hi obtained when one such pulse is started at the instant apreceding one reaches its peak. A flat-topped resultant wave then oc-curs between successive individual pulse peaks, as shown in Fig. 8. Atrnin of raised cosine pulses can therefore give timing information onlyon the transitions 01 and 10, and the circuit of Fig. 7 is approprinte forsignaling frequency derivation. It may be verified from Fig. 4 that theraised eosine pulse has null values in its Fourier transform at the signal-ing frequency and its harmonics.

    The message associated with the transition pulse train is, of course,different from the original message and has a different set of statistics.It is of interest to nlr-ulute the relation between the two for the case ofbinary on-off message source with independent signal values. 1"01' such a

    o 1 0 I 1 0 0 1 1 1 0 0 o

    Fig. 8 - Train of raised cosine pulses.

    source, the values of a" are selected independently, with the probabilityPI that unity Ot'CUrR and 1 - PI that zero occurs in any position. Thenthe mean !-irst and second powers are given by

    nil = nI~ = PIand the nutneovnriunce by

    (48)

    RCO) = ml, RCn) = n ~ O. (49)

    The rule for constructing the transition message is given by the fol-lowing table:

    Orlginal Message

    1 preceded b~' O .1 preceded by 1 .o preceded b~' f).o preceded by 1 ..

    Transition Message

    1()o1

    The value 1 occurs in the transition message from two events each hav-ing probability //lICl - m.) and hence has probability 2mlCl - 1nl). The

  • STATISTIC'S OF nEGF:'\ER,\TIVE DIGITAL TRAKSMISSION 1517

    mean first and second powers of the transition series are therefore givenby

    (50)

    We shall represent the autocovnrinuce of the transition message en-semble by p(n), The value of p(O) is simply the mean square, and hence

    p(O) = 21111(1 - ml) = 1'1 (51)To evaluate p(l) WI' note that t1!P product of adjaepnt values in the tran-sition message is zero except when the original values are 10 preceded hy0, 01' 01 preceded hy 1. The probability of obtaining the first isml(1 - 1111)2 and of obtaining the second m/(1 - 1ll1). Hence the prohn-hility of a II sequence in the transition messages is given by the sum ofthese two probabilities, and since we obtain unity for the product in thiscase and zero otherwise,

    p(l) = ml(l - 1111)2 + m/(1 - 1111) = ml(l - 1111) = rI/2. (52)For values of n greater than one the constraint imposed by the trnnsi-

    tions effectively disappears, since we may fill in all possible 0 and Ivalues between the critical end points. Thus, in the ease of n = 2 weobtain values of unity in the transition series two intervals apart for thefollowing original message sequences:

    1 0 0 1

    I 0 1 0

    0 1 0 1

    0 L 1 O.These have total probability

    p(2) = 411112(1 - 1111)2 = rt (5~)To evaluate p(~), we merely fill in all possible l's and O's in a columnbetween the second and third above and obtain the same total proba-hility because the compound event represented by the first and lasttwo columns is independent of that represented by the middle column.The probability of the first compound event is rl~' as calculated above,and that of a second is unity. Hence the complete autocovnriunce is ex-pressed by

    p(G) = 1'1,

    pel) = p(-1) = 1'd2,p(n) = p(-n) = 1'12

    for In I > 1. (54)

  • 1518 THE BELL SYSTEM TECH"ICAL JOnR"':\I~, XOVEMBI.;H 1058

    Thus, the transition series formed from the independent binary seriesbecomes a digram series: that iI', one in which adjacent values are de-pendent but nonadjacent values are independent. The discrete cornpo-nents in the spectrum me the same HI' in the independent ease exceptthat ml is replaced by 1"1 The continuous spectrum is evaluated by sub-stituting p for Rand 1'1 for ml in (:~5). The result after 1'1 has been re-placed by its value from (50) is

    w(f) =2ml(l - Inl) iT IG(r) I ~ [1 - 2m, + 2m/ + (I - 2ml)\~oH 2,,=fT]. (55)

    In thc special case in which Inl = !, this reduces to the same spectraldensity obtained for the independent message case, (:36).

    III. DEIUVA'l'ION or' TIMING Il'\FORMATION FHOM A RANDOM 'l'ELEGRAPH

    MESSAGE

    The existence of standard telegraph pulses which shock-excite a tunedcircuit to produce a sustained oscillation at the signaling rate for almostall message sequences has been demonstrated by example in Section II.

    In practical timing recovery circuits the ideal of a constant-amplitudecorrect-frequency output is not perfectly realizable. Factors which in-fluence the result include the message pulse pattern, the Q of the tunedcircuit, the presence of noise and the precision to which the naturaloscillatiou frequency of the tuned circuit can be made to match the sig-naling frequency,

    We shall analyze the problem of timing recovery under the conditionsin which the ideal performance is almost, but not quite, obtained. Theerrors in this case are relatively small in magnitude and correspondingapproximations can be made in the calculations which will show withsufficient accuracy what, the quantitative requirements must be to holdthe errors within specified small ranges,

    Assume that the tuned circuit oscillates at a frequency very near to[, the signaling frequency, and that Q is sufficiently high to make theinfluence of any one pulse on the amplitude of oscillation slight. Con-sider the ensemble of possible responses in the interval from t = 0 tot = 1', with the process having begun at 1 = - 00. Almost all the wave-forms are approximately sinusoidal and the negative-going axis crossingscluster about t = I", where I" is defined by

    av:r(t,.) = 0; av:r'(I,,) < O. (56)

    The nctual axis crossing for the typical member of the ensemble occurs

  • STATISTICS 01" HI;nF::'\EE.\TIVJ' DIGITAL TIU:,\Sl\IIH8IO" 1519

    at I,. + f/2'l1fr, where f is a small angle representing the phase error andhas a distribution of va lues over the ensemble, We assume that the wave-form through this axis erossing is that of a sine wave of frequency [,am! amplitude equal to the value of ,1'(1) one-quarter period after the axiscrossing, i.e.:

    .r(t) '" -.I'(t.. + f/Wr + 'I'll) sin [wr(1 - I,.) - fl, (57)with I f I 1. Then, rotuining only first powers of e, we haw

    :1'(1,.) == f x(te + 'I'/4),We further assume that :1'(1,. + 7'/4) can be written in the 1'01'111

    :I'(te + 7'/4) = (l + n av :1'(1,. + T/4),where I t I 1. By averaging both sides of this equation over thesemble we deduce that :1\' t = O.

    It follows from the uhovo assumptions that

    uv .I'{I,.) == uv [f(l + n av .1'(1,. + 'I'/4)1== uv f uv .I'(te + 7'/4),

    (58)

    (59)

    en-

    (liO)

    (131)

    Hence, av :1'(1,.) = 0 implies a v f == 0, The prineipnl quantity of interestis the rms phase error, which is the square root of uv f~, By the assumedcyclcergodicity of the process, an ensemble average of i is equal to anuverage of the value of f~ over all intervals of any member of the ensem-ble. From (58) and (50),

    uv .r~(t,.) - [f~(l + 2r + r~) av~ xt], + 1'/4l]- uv E':! :\,,.2 .r(te + T,/i-l).

    Hence, to the first 01'111'1' of small quantities,

    ~ av x~(lJa v f = ., ( + T/4) .av-.r Ie (62)

    The mean square phase error is thus expressed in terms calculable fromthe genera! expressions previously derived for means and r-ovnriances ofthe pulse ensemble.

    It is also of interest 10 evaluate the mean square value of t to validatethe treatment of t as a small quantity and to estimate the amount ofamplitude variation, From (;')0),

    uv .l'~(t,. + 1'/4) = :1\' (I + 2t + r~) av~ .1'(1,. + 1'/4)= (1 + av r~) :l\'~ .I'(t,. + 1'/4).

  • 1520 THE BELL SYSTEM TECHNiCAL JOURNAL, NOVEMBER 1958

    Hence,

    av y2(te+ T /4)aVLt:(te + T /-1)

    Ry(O, te + T /4)av2 .1:( te + T / -1) . (64)

    It is to be noted that the above treatment evaluates only the meansquare total phase error and does not give a frequency resolution, Thespectral composition is important, particularly in the case of a chain ofrepeaters and is discussed in the companion paper by Rowe.'

    We next calculate the average values uppeuriug in the equations formean square phase error and mean square amplitude ratio variation forspecific pulse shapes. We assume that the circuit is not tuned to the pulserepetition frequency so that a basis for estimating the error from mis-tuning can he established. For the case in which y(t) = go(t), the unitimpulse response of a single tuned r-ircuit, we write (-H) in the form

    where

    yo(t) = .-\ exp (-al) cos (wut + 6); t> 0, (65)

    J1 = wu(Hl + 1)1/2/2RQ2; a = wu/2Q;Q = wuL/R, tan 0 = 1/2Q.

    From (l), the ensemble average of .r(l) in the interval 0 ~ t ~ Tis

    '"avl;1'(1)1 =1111 L go(t-nT)o

    = 111.1 L Re exp [(j - a)wu(t - nT) + jO).n=-ao

    The series is geometric and may be summed to obtain

    (66)

    (67)

    av;1'(t) 1111.1 exp f-wu(t - T /2) /2Q]2IlQ2(cosh2 wu'1'/-1Q - cos" woT/2)

    (G8)f. wuT woT [( '1') ]. \ sinh -1Q cos 2 cos Wo t - '2 + 0- cosh ~~' sin "';'1' sin fwo(t - T /2) + OJ}.

    The value of i, in the noise-free mistuned case can be obtained readilyfrom (68) by equating the terms inclosed in the hrnces to zero, giving

    te = '1'/2 - 6/wu + wo l nrc tan [tanh (wu'1'/-1Q) cot (woT/2). (69)

  • STATISTICS OF REGECI:"EH.\TIVE DIGIT..\L TR.'CI:"I';:\IlSSIO:\" l;i21

    We note that I,. npprouehcs '1'/-1 as Q approaches infinity. If we as-slime IWI'('P('t tuning hy setting '-'0'1' = 211", (Ci8) becomes

    av .1'(1) =

    IIItWr(--1(/ + J)li~ exp [- (wrl - 1I")/2Q] cos [wr(t - '1'/2) + 81 (70)-- --- --1RQ~ sinh 1I"/2Q

    That is, the nveruge waveform of the tuned circuit response becomes adamped sine wnv of the coned frequency. When Q is large, the damp-ing is small. The limiting form as Q nppronches infinity is an undampedsine wu ve:

    I, ( ) IIIIW r ( r I'))un nv r I = -1" cos Wr I - ;_.Q_~ 11" l

    (71 )

    This result is perfect for frequency recovery, It is to be noted that it docsnot mutter whether the message values are independent or not. In thespecial ease in which pulses are generated by transitions only, 1111 wouldbe replaced by 1'1 = 21111 (1 - lilli, as given by (50).

    To complete the estimate of rrns phase error ami amplitude ratio varia-tion, we must evulunte the average value of .r(t) one-quarter period awayfrom I,., and the average squares of :r(t) at t = t, and I = Ie + '1'/--1, Thevalue of av :1'(1,. + 7'/--1) is ohtnined by substituting t = t; + 1'/--1 in (58),The evaluation of u.vcrnge squares may be done in genernl from thocovarinnce Iormuln of either (Ifi) 01' (17)-(21), recalling that

    uv i(l)

    av iU)

    (72)

    Two binary ('asc:o:: of cousidernhle importance which simplify considerablyare those of completely independent message values and the messug de-rived from trnusit ions between independent values. These arc:

    l ndcpeudcnt Binant JIessayc

    H(O) 1111; N(II) = II =;t! 0,

    mT)y(l - /11'1' + T), (7--1)

    uv i/(t)00

    1II1(l - 1111) L: l(t - 111'1'),m=-oo

  • 1522 THE BELL SYSTEM TECHNICAL JOURNAl., NOVEMBER I fl58

    Transition J1Jes.~age for Independent Binar,!} Value

    R(O) = 1'1 = :!111,(1 -ml), R(I) = R( - t)

    = I'd:!, R(n) = 1"~; In I > 1,00

    Ill/(T, t) = 1', 1: get - mT)(1 - 1',)g(1 - m'I' + T)m--oo

    + (! - 1',)g[I-(m - 1)1' + T] + (j - 1',)g[t -em + 1)1' + T),00

    nv y2(t) = Rl/(O, I) = 1'1 1: [(1 - I'I)l(t - mT)+ (1 - 2r,)g(t - mT)g[t - (m + I) 1'J.

    (75)

    (76)

    (77)

    For the independent binary message ensemble with (/(1) = (/11(1),o < t < 1',

    00

    nv y2(t) = Rl/(O, I) = ml(l - 1rh) L (/02(1 - 1/1'1')m=-CIO

    4 sinh wo'l'/2Q (cosh? woT/2Q - cos2 woT) (78)

    .( cosh~ woT/2Q - COH2 wuT + sinh" (wu1'/2Q) COl' WilT-cos [wu(2t - 1') + 20J - sinh (wu1'/Q) cosh (wu'l'/2Q).sin WOT sin [wo(2t - 1') + 2011.

    If woT = 2'11'", this reduces to:

    av y2(t) =

    mil - ml)w/(4Q2 + I) exp [-wr(t - 1'/2)/0) cos2 (wrt + 0) (79)8R2Q4 sinh '11'"/Q

    In the limit :II' Q approaches infinity, then,

    2( ) lIh(l - Inl)W/ 2:lV!J I ---+ 2'11'"R2Q cos wrt. (80)

    IV. IWALUA1'ION OF HMS PHASE ERROR WHE:" INDEPENDEN'r ON-~It-OFF

    UNI'l' IMPULSES ARE APPLIED 1'0 MISTUKED CIRCUI1'

    We illustrate the use of the equations derived above by carryingthrough the evaluation of (G2) when the message pulse source consists ofindependent on-or-off unit impulses and the ratio of tuning error to the

  • STATISTICS OF HEGE:-IEHATIVE DIGITAL TRA"ISMISSIO)/ 152:3

    signaling rate is small compared with unity. We adopt the followingnotation:

    k(wo - Wr)/Wr = fractional tuning error,a = woT/4Q = (1 + k)7r/2Q,{3 = (1 + k)7I",

    WO[(te - T /2) + 0) = .

    (81)

    (82)

    (84)

    (86)

    From (69), then,

    sin = tanh a cot {3 (1 + tanh' a cae (3)-1/2,cos = (1 + tanh" a cae (3)-I/2.

    Then, from (78), we may write, after noting that av x(te) = 0 impliesav :1:2(tc) = av y2(te):av:r2(te) =

    m1(1 - m1)wo\~Q2 + 1) exp[ -wo(tc - T/2)/Q] [sinh" 20' (83)16R2Q'1 sinh 20' (cosh" 20' - cos' 2{3)

    + sin" 2{3 + sinh" 20' cos 2{3 cos 2 - sinh 20' cosh 20'sin 2{3 sin 2 I.From straightforward trigonometrical manipulations,

    2 tanh 0: cot {3sin 2 = ---~------''-,--I + tanh20: cot" {3'1 - tanh" 0: cot2 {3

    cos 2 = -c--c---=-=---~1 + tanh" 0: coP (3 .

    When the values of sin 2 and cos 2 are substituted in (83) we find,after some reduction:

    nv .r2(te ) =m\(l - J11j)wo2(4Q2 + 1) sin 2{3 (85)

    exp I_Q-1[0 + arc tan (tanh a cot (3)1lWR2Q'1 sinh 20: (cosh? 20' - cos22(3)

    In the high-Q case, the value of av xUe + T /4) will be very nearly equalto the maximum value of av :I:(t) as given by (68). Making this assump-tion and simplifying, we obtain:

    av 2 :rUr + T /4) ==m12wo2(4Q2 + 1) exp I-Q-I[Wo/4.fr + 0 + arc tan (tanh 0: cot f3)J I

    16R2Q2(cosh2 0: - C082 (3)

  • 1524 THE BELL SYSTEM 'l'ECHNICAL JOURNAL, NOVEMBER 1958

    Then, from (62),

    nv f2 == (1 - ml) sin2 2{3 exp (-wo/4frQ)4m\ sinh 2a(cosh2 a - sin" (3)

    (1 - Uti) sin2 2k1r exp [-(1 + k)1r/2Q] (87)4m\ sinh (1 + lc)1r/Q [cosh" (1 + k)1r/2Q - sin" k1rj'

    In the limit for k 1 and Q 1, which is the region of practical interest,av f2 --+ (1 - m\)1rk2 Q/m\ . (88)

    The rms phase error is the square root of this quantity. For equal proba-bility of pulses and spaces, m\ = t, and

    ( 2)1/2. 1._ r-nQEr nu = av E = It'V'7r\l. (89)

    Fig. 9 shows a set of curves of rms phase error vs. Q for fixed values oftuning errol'. It is to be noted that this performance is attained when thereference phase of negative axis-crossings is tc defined by av x(t c) = O.This value of time is displaced from the time to defined by (42), whichwould be the correct reference if there were no mistuning. The differenceis accounted for by the steady-state phase shift of the network at the

    FREQUENCY ERRORK = MIDBAND FREQUENCYQ = MIDBAND FREQUENCY

    (3 DB) BANDWIDTH

    ow

    ~ 10-4~, I~iE ~ / I/' -- ~E:~~0~~~ ~S~~~L6p~I~TUMIII , V PHASE SHIFT~ , ~.!;.-- ---- REFERENCE IS INCOMING PULSE

    10-~ I-- OO)~O:.:?-1-__-t-+-__ & ASSUMES NO CORRECTION FORY PHASE SHIFT OF TUNED CIRCUIT~ ~~---+---II----I---+~~--1-~~~---,---.~.--.----.~~-,---t

    s 10310-6 L~--J_L_--J_L_--:!L-.l-~---:!-.L-----:!--.L-~----l

    10

    Figv U - Phase error in timing wave from mistuned recovery circuit.

  • 8T,\TISTICS OF REGEXEH.\TIYE DIGlT.\L TR,\XS:\IISSIOX };j2;"j

    pulse frequency. If we were to take the axis-crossing reference time as/0, the mean square phase error would he determined by a \. ./(fo) insteadof nv .r2(fc) in (liZ). Sinep :IV .r(tlJ) does not vanish, the expression fornv x2(to) must he written as

    (90)

    (91)

    (92)

    From (42),

    /0 = 1'/4 - 1'O/Z1r,

    to - 1'/2 = - 1'/4 - O/wr = - (0 + 1r/2)/w r ,cos [wo(fo - 1'/2) + OJ cos [(1 + k)1r/2 - kO]

    - sin [k(1r /2 - 0)],

    sin [wo(t - 1'/2) + OJ = cos [k(1r/2 - 0)],cos 2[woCto - 1'/2) + OJ - cos k( 1r - 20),sin 2[woCt - 1'/2) + OJ - sin k(1r - 20).Substituting these values and taking the limit for k 1, and Q 1,we find the rrns phase error in the region of iuterest to he given approxi-mately hy

    / -'- k(l + 1/4Q) -'-'J .frill" - (1.:2 + 1/4Q2)1/2 - ~Ql.

    The dashed curves on Fig. 9 show the values of error thus obtained andiudiento the importance of a phase adjustment to optimize for the tunedcircuit in use.

    It remains to evaluate the variation in peak height of the tuned cir-euit response. This requires evaluating av t 2 from (64), which in turn re-quires calculation of uv y2(tc + T /4) from (78). As in the ease of av2!I(tl' + 1'/4) it is sufficient to take the mnximum value of (78) OWl' a periodin t. The evaluation of av ( is facilitated by immediate comparison of theresulting expression for av 2 yUc + 1'/4) with that for av2 xtt; + 1'/4)given hy (86). We find

    r, 2(t + 1'/4) = ,2 . (t + 1',/ ') B(l - nl\)(cosh6 a - cosh" (3) (93).l\ I) I' rn .1 I' , ot ,. 11/J sinh" a (cosh" 2a - cos' 2(3)

    where

    B = cosh" 2a - C082 2{3+ sinh 2a[sinh22a ('OS2 2{3 + cosh" 2a sin2 2{3]l/2. (94)

  • 1526 THE BELL SYSTEM TECHCIlICAL JOURNAL, NOVEMBER 1958

    Then, by some further manipulation, we find:

    av 1;2 =(l - 1n1)fsinh22a + sin 2 2{3 + sinh 2a (sinh2 2a + sin2 2(3)1/2J (95)

    411h sinh 2a (cosh" a - sin" (3)

    In the limit when Q 1 and k 1, we obtain

    (96)

    (97)

    With no tuning error, the ratio

    r2 (l - 1nlh-av ~ --> -'-----~"-=----21nIQ

    can also be obtained by taking the ratio of (80) to the square of (71).When lc !Q, the limiting form is

    (98):LV 1;2 == (1 - 1nIh- (l + 3k2Q2).211hQ

    If I., is fixed and not equal to zero, the limit as Q is made very large is

    2 (1 - tnIhrQ/,;2 (99)av I; =111-1

    The analysis given here is valid only when nv 1;2 1.V. EFFECT OF NOISE ON TIMING

    The question of timing errors caused by noise superimposed on thereceived pulse train can be resolved into two parts: (1) the shift in refer-ence time at the input to the tuned circuit and (2) the resultant variation

    \ARC TAN g' (tal

    ________ to -,__L __2!::I~G~~__IIII

    VnIII

    A -i~---Qn ---~

    Fig. 10 - Geometrical coustruet.ion showing relut.ion between noise und timing errol'.

  • STATIST!('S OF REGEXERATIVE DIGIT_-\.L 'I'RA~SI\IISSIOl\' 152i

    in the phase of the tuned circuit response. The first question can be con-veniently answered by a geometrical argument based on Fig. 10. Herean enlarged section of the message pulse wa ve front is approximated bya straight line AB of slope equal to that of the pulse at the unperturbederit.icnl time tn The instantaneous noise voltage I'" advances the criticaltime by 5" , where 1'"/5,, = slope, and hence

    5" = I'"jg'(to). (l00)

    Htatisties of lJ" can be evaluated from the statistics of the noise and thespecificution of the pulse waveform. In particular, if the instantaneousnoise values at corresponding instants of signaling intervals 11 intervalsapart are not independent hut have an autocorrelation or autocovnriancefunction Rv(n), the resulting time shifts have the eorresponding functionRa(n), where

    (101)

    The average is taken over all time for one member of the ensemble togive the autocorrelation or over the ensemble at fixed time to give theuutocovnriance, vVe assume here that the process is ergodic nnd hencethat the two averages are equal. Vve have also assumed that the slopeof the signal pulse is constant in the neighhorhood of the slicing level.

    In the second part of the problem, we assume perfect tuning of the re-covery circuit and write for its response

    ee

    z(t) = L a"go(t - n'I' + 5,,)."=-00

    (102)

    Here gn(t) represents any response function which reproduces the axiscrossings correctly and, in particular eases evaluated, it will be assumedas HlP impulse response of a tuned circuit. We assume that the individualvalues of 5" an' sufficiently small to enable an accurate representation byfirst-order terms in a power series, thus:

    ec

    z(t) == L a,,[goCt - nT) + lJ"go'(t - n'1')]n=-oo

    ""= :r(l) + L a"lJ"go'Ct - 11'1').

    "=-00

    Without 10:-;'''; of generality, we may take a\'lJ" = 0, since a constant timeshift may he r-ompensuted hy a hnnge in the referenee time. Then, if thenoise und mcssuge ensembles are independent,

    nv z(t) = av .r(t). (l0-l)

  • 1~j28 THE BELL SYSTEM TECHXICAL JOUR:>lAL, :>lOVEMBER 1958

    We next cnlculate the autocovariance of the perturbed tuned circuitresponse as

    nv [z(t)z(t + r)] = nv [.r(t).r(t + T))ao ao (l05)

    + L L nv (a/"a,,) uv (O",On)go'(1 -m1')go'(t - nT + r).

    The first. term on the right represents the unperturbed nutocovnriauceand has already been evaluated. The effect of the noise is given hy thesecond term and, by use of the same technique as in obtaining (16), weobtain its uutocovariance as

    Rz-r(T, I) =ao ao (106)L L R(m - n)R,,(m - n)go'(1 - mT)go'(l- n'I' + T).

    III particular for the case of independent message values,eo

    Hz_x(T, t) = mI L Rlm)Oo'(t -m'l')gu'(t - m'I' + T), (107)alit!

    cc

    nv ri(t) - ;r2(t)] = Hz-x(O, t) = l1h L R6(m )[go' (t - mT)f (108)

    We may treat the phase error caused by noise in the same way as wedid the mistuning error in deriving (62). Since, with no tuning error,av x\tc) = 0 and t, = 7'/4 - O/21r'fr,

    2 av l(t.,) ( )av t = -------~ lOU" av2x(tc + 1'/4)'

    If the input noise values are assumed independent,

    Then

    (110)

    If

    av i(tJec

    m102 L foo/(le -m1')f111=-00

    (111)

    it follows that

    go(t) = lie-at ros(wrl + fJ),

    go'(t) = Be-at cos(wrl + fJ -

  • STATISTICS OF REGE:\'ER.\TIVE DIGIT.\L TR.\:\'Sl\IISSIO:\' ];">2B

    where

    tan cI> = wr/'a. (114)

    (11;j)

    Hence, the previous evaluation of (79) ran be used for nv i(t,.) , replnciug(J by (J - cI> and multiplying the result by (ci + W r2)02. Then

    . 2( ) _ lI1]w/o\.ui + 1) exp (0 + 7r)/2Q.1\ Z t, - R"ll' . I 'Q- (' Sill 1 7r/

    The limit as Qupproaches infinity is~ .,

    2() III\W r s: )av z t; = 27r H2Q . (1 1G

    The expression for :1\' ~1I2 is found from division by the square of theamplitude factor of (70), giving

    .,w r"1r .)

    --> '~Q 0- as Q --> oo_1111

    (I17)

    The effect of noise on the recovered timing is seen to vary inversely withQand hence may be made nrbitrarily small by using a sufficiently high-Qeircuit. The effect is opposite to that of tuning error which becomes worseai'i Q is increased. The possibility of an optimum Q taking both effer-tsinto account thus exists.

    VI. lWFECT OF TIMING VARIATIONS 0:-1 A DECODED ANALOG RWNAL

    The effects of timing errors are of two main kinds. First, they increasethe difficulty of reading the message correctly because the decisions memade at a less favorable point on the pulse wave. This is railed the align-ment error and is particularly important in a long chain of regenerative re-peaters. It. hal' been discussed by E. D. Sunde" and is also analyzed in theeompanion paper by H. E. Rowe.' It will not be treated here. The secondeffect is to impair the usefulness of the received digital message eventhough the correct sequence of values is delivered. The seriousness ofthis effect varies widely with the type of message. At one extreme, therecovery of printed text from a telegraph message would hardly heaffected. An analog signal transmitted by pulse rode modulation, on theother hand, is in some danger because the decoded signal is not correctlyreproduced by irregularly spared samples. A single-channel system wouldundergo phase modulation from this cause. In the rase of a time-divisionmultiplex system, some interchannel crosstalk may result if the jittervuries with the signals in the channels. The most severe requirement oc-

  • 11);30 'rIm BELL SYSTEM TECHKICAL JOUR:\TAL, :\TOVEMBEH 1958

    curs when PGM is applied to transmit a group of frequency-multiplexedchannels, since any waveform distortion can mean interchannel crosstalkafter the channels are separated.

    We should point out at the heginning that we are not forced to accepttime jitter in the received digital wave as finally used. A master clockcould he inserted at any stage to force the pulse back into a proper time-reference framework. By means of resampling combined with pulsestretching and delay line distribution techniques, such corrections ('1\11he made even when the extent of the tuning variations exceeds the signal-ing interval. It would, of course, be desirable to avoid these operationsand hence it is important to determine the permissible extent of timejitter in specific situations. We shall consider here the case of multiplexspeech channels in frequency division transmitted by pulse code modula-tion, the FDM-PCM system.

    Our procedure will be to represent the multiplex signal by an ensembleof band-limited functions (s(t) I having a spectral density w.(f) whichvanishes for If I ;;;; [, Any member of the ensemble is completely de-fined hy its samples taken 1/2/0 apart; in fact, by the sampling theorem,we have the identity:

    (t) ~. ( /2 r ) sin 211" fo(t + n/2fo).~ = n~oo S n r 211"fo(t + n/2fo) . (118)

    The right-hand member is the response of an ideal low-pass filter to im-pulses of weight proportional to the samples. The effect of time jitter is tointroduce an irregular spacing of the recovered samples which, whenapplied to an ideal filter, produce the distorted wave

    u(t) = i: s(n/2fo) sin 211"fo(t + Un + n/2fo) n=-oo 211" fo(t + Un + n/2fo) (119)

    The timing enol's are represented by Un = u(n/2fo), where lu(t) I repre-sents a jitter ensemble with specified statistical properties.

    The evaluation of interchannel crosstalk is conveniently aceomplishedhy assuming signals present in all but one of the FDl\I chunnels. Thespectral density w.(f) is set equal to zero throughout the frequencies occu-pied by the idle channel. Values in this frequency range of w.(f), the spec-tral density of the recovered signals, then give the interchannel inter-ference caused by the time jitter. This method has the advantage ofseparating the interchannel interference sought from in-band distortionwhich is of no consequence. 'Ve refer to a previous paper" for a descrip-tion of Rome of the necessary mathematical techniques.

  • (121)

    (120)

    (122)

    (123)

    av [s(111/2/0) s(n/2/0 ) 1

    STATISTICS (JIo' HE(;J;;:-\F,IL\.TIVE DIGITAL TRAJI'SMISSIO:-1 1531

    Wp ussumo that s(t) and 1/(1} have uutor-ovnriancc fuurtions expressedin terms of the spectrnl dpn:-;itipR hy :

    i l ll w.(.n exp (j'27f'T j) elf,-IllrR,,(T) = w,,(j) exp (j27f'T.f) d].-/0It is no loss of generality to assume that lI(t) is band-limited to the range-Io to fo , since higher frequency eomponents would produce effeets in-distinguishahle from those in the hand of half the sampling rate, Thespectral densities are in turn expressible in terms of the nutocovurinncesby

    w.(J) = i: R.(T) exp (- j27f'jT) dr,w,,(J) = i: R,b) exp ( - j27f' jT) dr.

    The s- and 11- ensembles nrc assumed to be independent and ergodic.~ote also that the signal density spectrum is defined in terms of meansquared voltage or current values, while the jitter spectrum is expressedin terms of mean square values of time,

    Our procedure is to culculute first the autocorrelnt.ion function of theu(t)- ensemble by the definition

    ao ao

    R.(T) = av u(t)u(t + T) = av L L s(m/2jo)R(,,/2jo)(124)

    sin 27f' joU + u'" + III /'2/0 ) sin 27f'.fo(t + T + UTI + n/2jo)47f'2f02(t + U". + 1II/2fo)(t + T + lin + ,,/2jo)

    The spectral density of the distorted signal ensemble is then given by

    w.(J) = i: R.(T) exp (- j27f'jT) dr. (125)We assume ensemble and time averages to be interchangeable hy theergodic ussumption. The a verugiug 0\'('1' the s- and li- ensembles may hedone separately bemuse of their independence. Furthermore, the double:-;(']'ils may be averaged term by term since, in any case, the average of thesum is equal to the sum of the individual averages. 'Ve first note that, forthe s- ensemhle averaging,

    uv [s(m/2.f,,)s(m + n - 1II)/2.fo](126)

    R.[(n - 1II)/2.fo] = R.[(m - n)/2foJ.The avornging oyer the 1/- ensemble is more troublesome.

  • 1532 'J'Im BEJ,L SYS'J'ElII TECHNICAL JOUR",AI" ",OVEMBEH 1958

    It has been found possible to obtain a reasonably simple answer onlyin the case of a gaussian jitter ensemble. By an involved calculation,which is explained in the Appendix, we obtain for this case, settingTo = 1/2Jo,

    r 00 1"11'0RaCT) = ~ L: 1(,(111'0) exp (-iU,,~) cos (1' - nTo)z dz, (127)

    1r ,t=-OQ 0

    where

    U} = R,,(O) - RI/(uTo) .

    III terms of the complex error function,

    (128)

    (130)

    R ( ) = ~o= ~ R.(n1'o) l~ .f ('!!.U" _ .l' + nTo) (129).1' .}_/ L..J [T .\eel 'I' Z 2U ..... v 1r n=-CC1 "n .I. 0 nIn the ('ase of PC'l\f transmission of an FDl\I group, the allowable

    jitter is small. This means that the integral of the spectral density w..(.f)must be small and, hence, that RI/(O), which is equal to this integral,must be small; R .. (T) assumes its largest values at r = 0 and we con-clude a for/ion: that UII~ must be small. Hence, we approximate by re-placing exp (-i[JII~) by zero and first-power term I' in its powerseries expansion and obtain an approximate value of the integral in(127) :

    f. IUnTo) sil~ 271".(0(1' + :,1'0) (t _ 471"2 (02 U,/ )11--00 ~7I"JO(T + /11'0) .._ '}U 2[co:o; 271"Jo(T + nTo) _ 1'0 Hin 271"/0(1' + n1'o)]

    - -II (1' + nTo}2 11'(1' + uTo) .By a further muuipulation also given in the Appendix, we find that thisapproximation yields for the spectral density

    w.c,n ,: (1 - 41r2lu02 )w.( f)

    + 41r2/ [w" I/ c,n + w"1/(2/0 + f) + w"1/(2/0 - j')]; I f I < 10,where

    110 = V Ry(O) is the rms time jitter and

    (131 )

    convolution of signal and jitter spectrum.

    Wo note that the first term consists of the original power spectrum ofthe signal depressed by the "crowding" cfTed factor (271"fuo)2. The crowd-

  • STATISTIC'S OF REGEXEH.\TlYE DIGITAL TRAXS~lISSIOX 1;j33

    ing effect docs not introduce new frequencies not present in the originalsignal and therefore does not contribute to interchannel interference. Theinterference is given entirely hy the second term, which consists of dif-Ierentinted convolution spectra of signal and jitter. The differentiationis expressed by the multiplying far-tor (21rf/, which /l;iws a triangularvoltage spectrum like that of thermal noise in frequency modulation.The term wy(f) represents the direr-t r-ouvolutiou of signal and jitterspectra, while 10 ,,(2/0 .f) represents sidebands of the convolution spec-trum on the sampling frequency.

    In testing of multichannel systems by noise-band loading, as pre-viously mentioned, the signal spectrum applied is flat except for the nar-row hand assigned to the channel under test. The power spectrum ismade equal to zero throughout the test-channel frequencies (see Figs. 11and 12). The expression for w.(J) is then:

    {

    K '1O.(f) = '

    o,I f I < f and I

    It < If I < ft + Ie ,

    ft +fe < If I

  • 1534 THE BELL SYSTEM TECHXICAL JOUR:"lAL, NOVEMBER 1958

    co }..-

    Fig. 12 - Spectral relations for noise band as test signal with I. < 10/2.

    functions, we actually need to calculate for only one of these intervals.There are two contributions from (1:U) to the positive interval - onefor w,,(f) and the other from w,,(2fo - n. The term w,,(2fo + f)vanishes for f > 0 since w,,(JI) is zero for JI > 2fo.

    The contribution from w,,(2/0 - .f) to the positive interval it to it + feis defined by the interval 2fo - i = it + ic to 2/0 - f = it or f = 2fo -it - fe to f = 2.fo - ft. ,rye must therefore evaluate w,,(.f) in theranges It toft + [; and 2fo - i. - fe to 2fo - It .

    Figs. ] 1 and 12 show the limits of integration which apply for theeases of i, greater than and less than fo/2, respectively. From these dia-grams, we deduce that the distortion spectrum in the range it < I .~-It 2'

    and

  • HTATI:-;TICS OF HEGE:\EHATIVE DWIT.\L TR.\:\SMIS8IO:\ V53;j

    A special rase of interest is that of a flat jitter band over a low-fre-quency portion of the signal band, Rowe's previously riled work' onregenerative repeater r-hains indicates that the jitter spectrum peaks atthe low-frequency edge of the signal band as the numher of repeatersbecomes large. Suppose the jitter spectrum is uniform from - F to F,f < fo ami consider the rase of it > fo/2 and F < ill - ft . This corre-spends to a test channel in the upper half of the signal band but belowthe top by at least the width of the jitter band. The distortion spectrumis given by

    Waden = 47r2'f;r~u02 (1- 1 ,- l e + 1-I+F) dA~} -I-F -I,

    = 47r2.~;(U02 (-f, - I,. + .f+ F - f + F + it) (138)

    = 47r2fI~u~ (2F _ ~)2F' r c

    if fe 2F.*The mean square total interference voltage in the idle channel is.:uc2 = 2 Waif) df == 87r2f/Ku02fc.

    I,

    If the mean square signal is 80\ we have811

    2 = 2](fo,and

    (140)

    (141)

    The dependence on the square of test-channel frequency shows that wehave a triangular voltage spectrum like that obtained in FM.

    A quantity of interest defining signal quality in the typical channel ofthe frequency-division multiplex system is the ratio ill of mean squarevalue of the sine wave test tone which fully loads the channel to the rneansquare interference voltage when the channel is idle and the other chan-nels are being operated in a normal manner. The mean square full loadtest tone voltage on one channel of an FDl\I telephone system is related

    * The implications of this approximut ion should not be overlooked. If the band-width of the jitter spert rum is small compared to a channel width, interchannelinterference would result only from high-order modulation neglected in theabove approximation.

  • ]5:36 THE BELL SYSTE~I TECHXIC.\L JOFHXAL, XOVEMBER 1958

    to the mean square full loud test tone fur the entire system by the Hol-brook-Dixon' curves, which we shall express in the form

    Jl" = n Pr/P" , (1-12)

    where

    P1 full load test tune mean square value necessary to transmit onevoice channel,

    P n full load test tone mean square value necessary to transmit nsuperimposed voice channels.

    When the multiplex system is operating normally the active channelsarc loaded with voice signals and not by sine waves. The value of s02,the mean square total signal voltage with speech loading, may be ex-pressed as

    (143)

    where H is the ratio of the peak factor of a sine wave to that of a compos-ite multichannel speech wave. This is based on the assumption that thenormal speech loading reaches system overload occasionally. The peakfactor of a sine wa ve is V2, corresponding to :1 db, while the peak factorof superimposed speech chnnnels approaches that of thermal noise as thenumber of chnunels iH m...ndc large. The peak factor of thermal noise de-ponds on the value of the probability of excess chosen. A value oftenused is 4, corresponding to 12 db, for which the probability of observinggreater peaks is one in 10,000. This choice fixes H2 at eight, correspondingto 9 dh.

    We write, in nocordnnce with the above,

    ill = l'r/u/,and assume Jll specified. It follows that

    (144)

    U = .i.11,'p,..fol 4 "! 'J 0 j 0-s: nlon cSo

    H2Mn fo-11r2 ft 2u 02nfc . (145)

    Solving for 1l02, we obtain the minimum requirement for mean squaretime jitter:

    (146)"110 H2foM n

    41r2N fcnM'

    Because of the triangular distortion voltage spectrum, the requirementis most severe at the highest channel carrier frequency.

    It is convenient to express this requirement in terms of rrns phase

  • STATISTICS OF REGEXERATIVE DIGITAL TRA~SMISSIO~ 1537

    jitter allowed at the digital pulse frequency, which is 2Nfo for an N -digitPCM system. Since = 211'(2i'~fo)1I0,

    2 = 4N2H2f 03ilf" (147)nNfcM .

    Consider, for example, an 8-digit binary PC~J system transmitting a2000-channel single-sideband FDM telephone signal with 4-kc carrierspacing. The top channel is [It 8 mc and imposes the most severe require-ment. We ask for a 50-db ratio of Iull load test tone power to idle channelinterference. Assume the sampling frequency is 20 me. We evaluate M"by extrapolating from Table I of Ref. 8: H 2 = 8, N = 8, fo = 107,M; = 2000/80, n = 2000, ft = 8 X 106 , fc = .. X to', ill = 106 , andwe calculate = 0.:3 radian. This result is dose to estimates made byO. E. DeLange and E. D. Sunde by different methods.

    VII. ACKKOVnEDGME",T

    The work described here had its inception in the work on PCM doneat Bell Laboratories in the early 1940's and its beginnings were in-fluenced by discussions with C. B. Feldman, R. L. Dietzold and L. A.lVlacColl. Later benefits were received hy comparison with results oh-tained on various features of the problem by .T. R. Pierce, H. E. Roweand E. D. Sunde. The present form has profited from suggestions byD. W. Tufts.

    APPENDIX

    By the definition of a time average,. INN

    Ru(T) = ,~I~~ (2N + 1)'1'0 m~N "~N R,[(m - n)To]

    1N1'osin ~ (t. :mTo + 11m ) sin ~ (I + T + nTo + u,,) dt-N1'" (T) (t + 1/1'1'0 + um)(t + T + n'I' + u,.)

    (148)

    The integrand is an analytic function of t in the finite plane and hencethe path of integration may be deformed without changing the value ofthe integral. We replace the part of the path passing through the zerosof the denominator by downward indentations. Sum and differenceformulas may then be applied to the sines to resolve the integral into thesum of separate components without introducing singularities ill the inte-

  • 1538 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1958

    grands. Designating the resulting path of integration by C, we then have:

    R ( ) - I' ~ ~ ToR.[(m - n)ToJxT - lin L..J L..J ?N=oo m=-oo n=-QO 4N '7["-

    1 dtc (t + m'l'o + Urn) (t + T + nTo+ Un){cos ;0 [T + (n - m)To + Un - uml- cos ;0 [2t + T + (m + n)T + tlm+ unl}.

    To evaluate the integral, write:

    (149)

    mTo + u.; =T + nTo + Un =

    (150)

    Note that a and b are real numbers. We must calculate:

    cos ;0 (b - a) cos '1~o (2t + a + b)L(t + a)(t + b) dt - L (t + a)(t + b) dt.The first integral vanishes as we let N approach infinity, as may be seenby closing the contour in an infinite semicircle below the real t-axis. Theabsolute value of the integral around the semicircle of radius 1" cannotexceed the product of length of path 1r1" and the maximum absolute valueof the integrand, which is less than l/r2 The integral around the semi-circle is therefore less in absolute value than 1r/1" and approaches zero asI" goes to infinity. The integral around the closed contour including Cand the semicircle must vanish because there are no singularities inside.Hence the integral over C also vanishes and the first integral is zero.

    The second integral may be written as the sum of two terms:

    ei.. (a+b)/To 1 ei21rt/To dt + e-i.. (a+b)/To 1 e- i 2.. t/ To dt(lSI)

    2 c(t+a)(t+d) 2 c(t+a)(t+b)'

    In the first term we dose the contour in an infinite semicircle above thereal t-axi:; and obtain an integrand which vanishes exponentially as theradius is made large. In the limit the integral along C is then equal to theintegral around the closed contour, which in turn is equal to 211/.7 timesthe sum of the residues at the poles t = -a and t = -b. In the second

  • H1'.\TIRTIf'R OF HEGEXEIUTIVE DIGlTc\L TRAI\SMISSION 1539

    term we close the contour bolow the rcnl [-axis and show that the integralvanishes. Therefore t.he complete result is given by:

    .) " (.-i!TUIT," '-,i!TI'I'I'U)_ -:-7r./ (')T(u+I,)/1'" 1 + _1_-----;,---'2 b - a a - b

    1rj [iT(a-b) To---ca - b

    2'1l' sin 'Il'(a - /1)/1'0a - b

    Hence,

    (154)ToR.[(m - n)1'II] (os [em - 1/)1'0 + 1/", - Un - T]Z

    '2N'Il'

    N N

    L L

    N N ToR.[(m-n)Tolsin; [(m-n)To+um-un-T] (153)lim L L .) , , 0N=", n--N b--N .N'Il'l(m - n)To + '11m - Un - T]

    For purposes of cnlculation it. is convenient to introduce the dummyvariable z and write the equivalent form:

    1,1'/ '1'0R.(T) = dzom=-N 11=-N

    Next we change the order of summation by letting m. - ndropping the prime after eliminating m to obtain

    11

  • Hi-lO THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1958

    for 2N + 1 values of n with m fixed and then dividing by 2N approachesthe procedure for oulculating the average of G(nT) over the tz-ensemble.

    To evaluate the ensemble average of G(nTo) , we invoke a theorem ex-pressed by equation (1.1-l) of Ref. 6, which states that, if u(t) is a gauss-ian ensemble,

    av Icos [au(t) + bu(t + T) + .all

    [a

    2 + b2 ] (157)= exp ~ Ru(O) - abRu(T) cos.a.We shall assume from this point on that 1t(t) is a gaussian ensemble.Then the above result fits our ease if we set t = n/I', II = -Z, b = z,.a = (mTo - T)Z, T = mTo . We then find

    av G(nTo) = ToR.(mTo) e-,2[Ru(Ol-Ru(mTol] cos (mTo - T)Z, (158)'!r

    andT 00-s L: R.(mTo)1r m=-oo

    i:o e- z2 [R u (0 )- R ,, (m T ,1] cos [(T - m'1'o)z] dz. (159)The integral is now evaluated by writing the cosine in exponential

    form, completing the square in the exponent, and substituting variablesto obtain the error function. The result is given as (129) of Section VI.We have replaced m by -n and made use of the fact that the autocorre-lation function is even.

    We next calculate the power spectrum waCf) by taking the Fouriertransform of the approximate autocorrelation (1.'30). The complete in-tegrand has no singularities in T and, by indenting the path of integra-tion below each point T = -nTo , we can separate the integral into com-ponent parts without singularities on the path. Denoting the depressedpath by C, we evaluate

    fe-i2lf/T sin i (T + nTo) dr

    C '!r(T + nTo)f [-j(2r/-(r/Tol),+inr _ ,-i[2lf/+

  • STATISTICS OF REGE:'>lERATIVE DIGITAL TRA:'>lSMISSIOiII 15-11

    (WI)

    r0; If I

    ~ \'(2)1 - ;),"'"1'" 111-j21r/T' 1T ( + '1' )

    Je S1l1 'i'n T n 0 dr

    C (T + nTo)3

    I> ')'1'.

    ~ 0

    f [-j[21r/-(1r/l'ol)r+jlllr -j[2.-/+(1r/To))T-j"'-j tlr= e - ec 2j(T + nTorJ (162)r0;

    = \ 1T (2 f 1T)2 j211.-/To- - 1T - - e .2 To '

    If I > 2;1l1

    If 1< ')'1'~ Il

    Then, for If I < 1/2'1'0 ; that is, If I < fo :

    ) ~ () [ 1T2.,

    2 2 ( 1 )W.(f == LJ ToRs nTo 1 - -'1'2 Un- - 2Un 1T 2f - 7Pn=-oc 0 10

    _ ') 'I'll~ 2 (')'! _~)2 [T 2J j2n.-/T~ 1T 2 1T ~ To "e

    ec

    = To L RJnToHl - -11T2/U,,2)ej211lrfT.n=-CO

    (163)

    From (120), we note that ws(t) vanishes outside the range -fo to fo,

    and hom the nnulogous relation for R,,(t),

    (164)

    (165)

  • 1542 'rIlE BELL SYS'l'EM TECHNICAL JOURNAL, NOVEMBER 1958

    Application of the convolution theorem shows that

    R.(nTo)RII(nTo) = L: Ws.y(lI)ei21rTlTUP dll,where

    L: W,,(A)W u (>\ + v) dAL: W.(A + II )WIl(A) dA.

    (166)

    (167)

    We substitute the above expressions for R.(nT II) and R,(nTu)R,,(nTo)in (163) and then apply Poisson's summation formula,

    Tl~OO L: \O(z)e i'lZ dz = 211" Tl~OO \O(2n 1l" ),to obtain, for II I < Io,

    00 ( )'J ? n

    VVu(f) == [1 - hrRIl(O)lll~ w. To - I

    ~ ~ ~ (n )+ 411" I ,,~oo W " To - I .

    (168)

    (169)

    Since both wsW and wll(f) have been assumed to vanish for II I > In =1/21'0, the only term of the first Rum which falls in the signal band -Io toIo is that for n = 0, while in the second sum the signal-band contributorsare n = 0, -1 and I. The latter conclusion follows from the faet that theconvolution of two functions limited to the same low-pass band is limitedto a low-pass band twice as great. AIRo, noting that power spectra areeven functions, we obtain O;H).

    REFERENCES

    1. Rowe, H. E., this issue, pp. 1543-1598.2. DeLange, O. E., this issue, pp. 1455-1486.3. DeLange, O. E. and Pust.elnyk , M., this issue, pp. 1487-1500.4. Sunde, E. D., Self-Timing Regenerative Repeaters, B.S.T.J., 36. July 1957, pp.

    891-937. .5. Macfarlane, G. G., On the Energy Spectrum of an Almost Periodic Succession

    of Pulses, Proc. I.R.E., 37. October 1949, pp. 1139-1143.6. Bennett, W. R., Curtis, H. E. and Rice, S. 0., Interchannel Interference in

    FM and PM Systems Under Noise Loading Conditions, B.S.T.J., 34. May1955, pp. 601-636.

    7. Holbrook, D. B. and Dixon, J. T., Load Rating Theory for MultichannelAmplifiers, B.S.T.J., 18. October 1939, pp. 624-644.

    8. Feldman, C. B. and Bennett, W. H., Band Width and Transmission Perfor-mance. B.B.T ..T.. 28. .lulv 1940. pp. ,190-595.