Alias-free randomly timed sampling of stochastic processes
Frederick J. Beutler · IEEE Transactions on Information Theory · 1970
The notion of alias-free sampling is generalized to apply to random processesx(t)sampled at random timest_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)\}, wherer(n) = E[x(l_{m+n}) \overline{x(t_m)}]. The actual sampling timest_nneed not be known to effect recovery of the spectrum ofx(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 (fort \leq t_oort \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) \}.