On the statistics of random pulse processes

Abstract

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.