Power spectra of random spike fields and related processes

Abstract

In this article, we review known results and present new ones concerning the power spectra of large classes of signals and random fields driven by an underlying point process, such as spatial shot noises (with random impulse response and arbitrary basic stationary point processes described by their Bartlett spectra) and signals or fields sampled at random times or points (where the sampling point process is again quite general). We also obtain the Bartlett spectrum for the general linear Hawkes spatial branching point process (with random fertility rate and general immigrant process described by its Bartlett spectrum). We then obtain the Bochner spectra of general spatial linear birth and death processes. Finally, we address the issues of random sampling and linear reconstruction of a signal from its random samples, reviewing and extending former results.