Alias-free randomly timed sampling of stochastic processes

The notion of alias-free sampling is generalized to apply to random processes sampled at random times ; 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 , where . The actual sampling times need not be known to effect recovery of the spectrum of . Various alternative criteria for verifying alias-free sampling are developed. It is then shown that any spectrum whatsoever can be recovered if 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 or ) 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 .