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=−∞∞αnδ(t−tn) where the impulse occurrence times tn constitute a stationary point process and where the intensity modulating coefficients αn constitute a stationary random process, independent of the point process tn. 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