Oscar A.Z. Leneman

The Spectral Analysis of Impulse Processes

An expression for the spectral density of the impulse process s(t) = {if236-1} is derived under the assumption that \αn\}} is a stationary process, and that \tn\}} is a stationary point process independent of \αn\}}. The spectral density appears as an infinite series in terms of the correlation of \αn\}} and the interval statistics of \tn\}}. The same result was obtained by Leneman by a different argument under considerably more restrictive conditions of validity. Various models of impulse processes are discussed relative to random sampling of random processes. Random and systematic loss of samples, separate read-in and read-out jitters, and correlated random scaling errors can all be represented by appropriate assumptions on \αn\}} and \tn\}}. Finally, closed form expressions are calculated for the spectral density of s(t) and the sampled process under combinations of the sampling errors mentioned in the preceding paragraph.

Random sampling of random processes: Impulse processes

The concept of random impulse trains is useful as it facilitates statistical studies on discrete pulse modulated control and communication systems. As a result, this paper discusses the improper random process s ( t ) = ∑ n = − ∞ ∞ α n δ ( t − t n ) where the impulse occurrence times t n constitute a stationary point process and where the intensity modulating coefficients α n constitute a stationary random process, independent of the point process t n . With some heuristics and using basic results of the theory of stationary point processes, the autocorrelation function and the spectral density function of the impulse process are evaluated. The results obtained are remarkably simple and are illustrated with various examples.

Random sampling of random processes: Stationary point processes

This is the first of a series of papers treating randomly sampled random processes. Spectral analysis of the resulting samples pre-supposes knowledge of the statistics of tn, the random point process whose variates represent the sampling times. We introduce a class of stationary point processes, whose stationarity (as characterized by any of several equivalent criteria) leads to wide-sense stationary sampling trains when applied to wide-sense stationary processes. Of greatest importance are the nth forward [backward] recurrence times (distances from t to the nth point thereafter [preceding]), whose distribution functions prove more useful to the computation of covariances than interval statistics, and which possess remarkable properties that facilitate the analysis. The moments of the number of points in an interval are evaluated by weighted sums of recurrence time distribution functions, the moments being finite if and only if the associated sum converges. If the first moment is finite, these distribution functions are absolutely continuous, and obey some convexity relations. Certain formulas relate recurrence statistics to interval length statistics, and conversely; further, the latter are also suitable for a direct evaluation of moments of points in intervals. Our point process requires neither independent nor identically distributed interval lengths. It embraces most of the common sampling schemes (e.g., periodic, Poisson, jittered), as well as some new models. Of particular interest are point processes obtained from others by a random deletion of points (skip processes), as for instance a jittered cyclically periodic process with (random or systematic) skipping. Computation of the statistics for several point processes yields new results of interest not only for their own sake, but also of use for spectral analyses appearing in other papers of this series.

Random Sampling of random processes: Optimum linear interpolation

Abstract The problem of optimum mean-square linear interpolation for random sampling is discussed in this paper. The technique of study is very simple and further extends the work done in periodic sampling and in periodic sampling with. jitter. For example, two sampling schemes are considered: nearly periodic sampling with skips and Poisson sampling. There is no restriction on the bandwidth of the signal; and, sampling errors in amplitude are also considered. Some of the results obtained are remarkably simple.

On the statistics of random pulse processes

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 applications (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 characteristic 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.

Correlation Function and Power Spectrun of Randomly Shaped Pulse Trains

This paper presents a new and simple technique for evaluating the correlation function and the power spectrum of a randomly shaped pulse train defined as y(t) = ∞Σh_n(t-t_n) n=-∞ where the {h_n(·)} are random functions that describe the shape of the pulses and where the random occurrence times {t_n} constitute a stationary point process. The obtained results are very simple, and various illustrations are given.

Random sampling of random processes: Mean-square behavior of a first order closed-loop system

This paper discusses the mean-square performance of a first order random sampled-data system with feedback, where the sampling times constitute a stationary point process, with independent and identically distributed sampling intervals. The paper presents some new results for the cases of periodic sampling, periodic sampling with skips, and Poisson sampling.

A note on the mean-square behavior of a first-order random sampling system

This correspondence presents results concerning the mean-square behavior at sampling times of a first-order sampled data system with feedback, where the sampling times constitute a stationary point process with independent and identically distributed sampling intervals.

Mean-square error of step-wise interpolation for various sampling schemes

Abstract Expressions are given for the mean-square error of a step-wise reconstruction procedure. This is accomplished for various sampling schemes: periodic sampling, periodic sampling with skips, nearly-periodic sampling (with jitter) and Poisson sampling. For illustration, the sampled signal is assumed to be either a wide-sense Markov process or band-limited white noise. The results obtained are plotted and permit simple interpretations.