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On the Statistics of Random Pulse Processes

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INFORMATION AND CONTROL 18, 326-341 (1971)
On
the Statistics of Random Pulse Processes FR~DERmK J.
EUTL~R
Computer Information and Control Engineering Program University of Michigan Ann Arbor Michigan 48104
AND
OSCAR A. Z. LENEMAN
Statistics are obtained for pulse trains in which the pulse shapes as well as the time base are random. The general expression derived for the mean and spectral density of the pulse train require neither independence of intervals between time base points nor independence of the pulses. The spectral density appears as an infinite series that can be summed to closed form in many applica- tions (e.g., pulse duration modulation with skipped and jittered samples). If the time base is a Poisson point process and the pulse shapes are independent, stronger results become available; we are then able to calculate joint charac- teristic functions for the pulse process, thus providing a more complete statistical description. Examples are given, illustrating use of the above results for pulse duration modulation (with arbitrary pulse shapes) and telephone traffic.
INTRODUCTION
In various applications involving pulse trains, both the pulse shape and the time base are random in nature. As one example, consider pulse duration modulation (PDM) of a random signal with irregular sampling times caused by jitter and the random loss of pulses. A second example concerns disturb- ances in a receiver due to an electrical storm; the times and effects of lightning bolts are each random. There are also phenomena not ordinarily * Research sponsored by the Air Force Office of Scientific Research, AFSC, USAF, under Grant No. AFOSR-70-1920, and by Lincoln Laboratory, a center for research operated by the Massachusetts Institute for Technology with support from the U.S. Air Force. The United States Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation hereon. 326
ON THE STATISTICS OF RANDOM PULSE PROCESSES
327 regarded as having random pulse shapes and time base, but which may be interpreted as such a pulse train. An example of this type is the number of telephone lines in use when the length of calls as well as their srcination times are random. In this paper, we show how some of the statistics of pulse trains with random pulse shapes and random time base may be calculated. Two techniques are discussed. The first of these results in general expressions for the means and spectra of such pulse trains; these expressions are valid also for correlated pulses and time bases with intervals between pulses that may be neither independent nor identically distributed. The other technique yields even more information, namely the first and second order probability densities for the pulse process, but under the more restrictive condition that the time base is a Poisson point process? The second order statistics of the impulse train
s(t) = ~ %3(t -- t.)
(1.1)
were considered by the authors in an earlier paper (1968). The time base {tn} was assumed to be a stationary point process [Beutler and Leneman (1966a) and (1966b)], while {%} was taken to be a wide stationary discrete parameter process with specified covariance. From the second order properties of
s(t)
it is easy to deduce similar results for ~
c~h(t -- t~),
thus treating pulses of fixed shape with random amplitude and time base; the pulse train
~.~_~ ~h(t -- tn)
merely represents the
s(t)
of (1.1) after its passage through a linear timc-invariant filter with response function h('). However, this model is incapable of generalization to pulse trains in which the shapes of the respective pulses may vary also. For that case we must analyze the statistics of
y(t) = ~ h.(t -- t.),
(1.2)
~ o
in which {t~} is again the random time base (a stationary point process) and h~(') is the n-th pulse. A typical pulse train of this type is shown in Fig. 1. It is seen that ~o
~h(t -- t~)
becomes a special case of
y(t)
when we take
hn(t ) = ~h(t)
in (1.2) In the next section, we shall find universal formulas for the mean and spectral density of the
y(t)
of (1.2) under the following hypotheses. It is 1 For this restricted case, the spectrum (only) is calculated in Mazzetti (1964).
328
BEUTLER AND LENEMAN
tn- 2
FIG. 1.
hn(t-tn) hn+q ( I -t n-l-l)
A realization of a train of randomly shaped pulses.
supposed that {t~} is an ergodic stationary point process, and that the random functions {h~(t)} are independent of
{tn}
with means
E[h~(t)] z m(t)
(1.3) that do not depend on n. The transform covariance for {h~(t)} is F,*(o~) = E[H~+,(oJ) H~(~o)], (1.4) where the overline denotes complex conjugacy and Hk(co) is the Fourier transform of
hk(t).
The point of (1.4) is that the indicated expectation satisfies the weak stationarity condition that it does not depend on m, but only on n. The class of stationary point processes for which the spectral density expression is obtained embraces most (t~} considered to be realistic time bases. Poisson point processes are included, as well as uniformly timed sampling that has been subjected to random jitter and/or random deletion (skipping) of pulses. Other possible variations on the stationary point processes include systematic skipping of some srcinally existent points, and points at intervals of varying lengths following a planned or random sequence. Stationary point processes are analyzed in Beutler and Leneman, (1966a) and (1966b), and the statistics of point processes required for this paper are calculated therein.
FIRST AND SECOND ORDER STATISTICS
The mean and spectral density for the
y(t)
of (1.2) can be derived in terms of simple closed form expressions. The spectral density appears as an infinite series in terms of/'n* and fn*, where the latter is the generating function of the distance separating n successive points, i.e., of tk+ ~ -- te. As will be seen from the examples that follow the derivation, the infinite series repre- senting the spectral density of the pulse train can be summed to an analytical expression for many models of interest in applications. The arguments used will be heuristic, but the validity of the final results can be established either by alternative methods (in the time domain) or by justifying the various types in the calculation.
ON THE STATISTICS OF RANDOM PULSE PROCESSES
329 The mean
y(t)
is computed by taking expectations of (1.2) successively on {hn(t)} and {t~}. Because of the independence of these two random processes
E[y(t)] :- E[ ~ h~(t -- tn)] = E[_~ m(t -- t~)].
(2.1)
Now if we let
s(t) = ~ 3(t -- t~)
it is possible to write
co
,~(t - t,,) = f m(t --
~-) &)
,4~-
--¢o
and hence (2.2) (2.3) From an interchange of expectation and integration in (2.3) one then obtains (2.4)
co
Ely(t)] -- ~ f re(u) du.
In the formula (2.4)
E[s(t)]
= fi represents the average number of pulses per unit time. The expectation of
s(t)
has been derived in Beutler and Leneman (1968), and values of the mean number of points per unit time/9 are available for a wide variety of stationary point processes in Beutler and Leneman (1966b). The spectral density S u can be adduced by first computing the correlation
E[y(t + r) y(t)]
and then taking the Fourier transform of the expectation. This method has been used for the
s(t)
of (1.1) in Beutler and Leneman (1968), but becomes inconvenient when an attempt is made to generalize the same technique to
y(t).
The same result may be attained more simply by utilizing the direct method [Davenport and Root (1958), p. 108], which means that we use the formula
1 I T e -~tdt 2 .
S~(w) = lim
E (-T f y(t) )
(2.5)
T~co 0
The indicated limit is best taken by letting T =
fi-lN,
with N then tending toward infinity through the integers to attain the desired limit. Such T is convenient because it allows us to suppose that for large N approximately N pulses fall into the interval (0,
fl-lN].
This is indeed true if {t,~} is an ergodic stationary point process as defined in Section 3.6 of Beutler and Leneman (1966a); i.e., if the average number of points in (0, T] tends toward fi for
330
BEUTLER AND LENEMAN
almost every realization as T-+ oo. ~ We also assume that the interval (0,/~-IN] is sufficiently large so that the contributions of pulses intruding from outside the interval and tails of pulses lost by restricting the interval are negligible compared with N. Then over (0, fi-lN]
y(t)
is approximated ~-~+N--1
h~(t- t~),
where t~ is the first point past the srcin. Insofar as the statistics of the sum are concerned, k is irrelevant because of the station- arity of {t~} and the assumption (1.4) which permits us to translate the indices of H~. Accordingly, we take k = 1 and obtain asymptotically
f~o-l Ny( t )
e -i~t dt = ~ H~(~o) e i~t .
(2.6)
1
This expression is substituted into (2.5). On multiplying it by its conjugate and taking its expectation we find that
B-1N 2 N
E ( y(t) e -i°,t dt ~- ~ F*_~(co) E{exp[--iw(t• --
t,n)]}. (2.7)
°0 m,q~=l
The right hand expectation represents (for n >/m) the characteristic function for n -- m successive intervals between points. If the probability density function for the length of k successive intervals n is called fk, we may definers* as the corresponding characteristic function and writef~* as
co
fk*(ioo) = e-i~f~(x) dx = E{exp[--ico(t~+~ --
tj)]}.
0
(2.8) For many stationary point processes of interest in applications, the fk* have been calculated in Beutler and Leneman (1966b). The negative indices on the right side of (2.8) produce expectations of the complex conjugate for each term, so that it is consistent to define f* (it0) =
f*(ioJ)
and F_*n(o~ =
I'n*(oJ ).
With this convention the left side of (2.7) becomes
-IN 2 N
E ~ ytt) e -'~*dt = • F*_~(~olf*(ioJ).
(2.9)
~0 m,n=l
Neither this assumption nor the one following are needed to effect the final result, but they greatly facilitate its derivation. The alternative approach through a time domain argument also yields the same final expression for the spectral density, and this provides an additional check on its validity. There exists stationary point processes for which different sets of lengths of n successive intervals do not all have the same distribution functions, but these do not appear to be of physical interest. See Beutler and Leneman (1966a), Section 4.2.

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