Alias-free randomly timed sampling of stochastic processes

TLDR: It is shown that any spectrum whatsoever can be recovered if \{t_n\} is a Poisson point process on the positive (or negative) half-axis and randomly jittered sampling at the Nyquist rate is alias free.

Abstract: The notion of alias-free sampling is generalized to apply to random processes x(t) sampled at random times t_n ; sampling is said to be alias free relative to a family of spectra if any spectrum of the family can be recovered by a linear operation on the correlation sequence \{r(n)\} , where r(n) = E[x(l_{m+n}) \overline{x(t_m)}] . The actual sampling times t_n need not be known to effect recovery of the spectrum of x(t) . Various alternative criteria for verifying alias-free sampling are developed. It is then shown that any spectrum whatsoever can be recovered if \{t_n\} is a Poisson point process on the positive (or negative) half-axis. A second example of alias-free sampling is provided for spectra on a finite interval by periodic sampling (for t \leq t_o or t \geq t_o ) in which samples are randomly independently skipped (expunged), such that the average sampling rate is an arbitrarily small fraction of the Nyquist rate. A third example shows that randomly jittered sampling at the Nyquist rate is alias free. Certain related open questions are discussed. These concern the practical problems involved in estimating a spectrum from imperfectly known \{ r(n) \} .