Alias-free randomly timed sampling of stochastic processes

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) \}
.