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This article reviews known results and contains 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 spectrum), and signals or fields sampled at random times or points (where again the sampling point process is 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 spectrum of general spatial linear birth and death processes. Finally we adress the issue of random sampling, and of the linear reconstruction of a signal from its random samples, reviewing and extending former results.