Power spectra of random spike fields & related processes

This paper presents general methods for obtaining power spectra of a large class of signals and random fields driven by an underlying point processes, in particular spatial shot noises with random impulse response and arbitrary basic stationary point processes described by their Bartlett spectrum, and signals or fields sampled at random times or points, where again the sampling point process is quite general. The formulas obtained clearly show the interaction between the underlying counting process, the sampled process or the impulse response. 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. Finally we obtain the Cramér spectrum of general spatial birth and death processes. Index Terms Shot noises, ultra-wide band communication, random sampling, power spectrum, point processes, filtered point processes, Cramér spectral measure, Bartlett spectral measure, pulse-interval modulation, pulse position modulation, Hawkes processes, random fields.