Alias-Free Sampling of Random Noise

We study the question of when the power spectrum of a stationary- random process is uniquely determined by the values of the process measured in various ways at discrete instants of time. As is well known, in the case of equi-spaced sampling, the power spectrum is not determined by the sampled data, i. e. aliasing is present. After a review of the equi-spaced case, we examine the case where the time markers are not equi-spaced, but are subject to jitter, and find that in general aliasing persists. We then consider the case where the time markers are generated by an additive scheme, whereby each sampling time is derived from the previous one by the addition of an independent random variable, with characteristic function 9^ (3). It is found that additive random sampling is alias-free if 0{s) is one-to-one on the real axis, but not alias-free if ^{s) takes the same value at two distinct points of the open upper half-plane. Various alias-free sampling methods are exhibited, notably Poisson sampling.