The Whittaker–Kotel’nikov– Shannon Theorem, Spectral Translates and Plancherel’s Formula

Abstract

The classical sampling or WKS theorem on reconstructing signals from uniformly spaced samples assumes the signals to be band-limited (i.e., with spectrum in a bounded interval [-W,W]). This assumption was later weakened to a disjoint translates condition on the spectrum which led to an extension of the sampling theorem to multi-band signals with spectra in a union of finite intervals. In this article the disjoint translates condition is replaced by a more natural ‘null intersection’ condition on spectral translates. This condition is shown to be equivalent to an analogue of Plancherel’s isometric formula when the spectrum has finite measure. Thus to some extent multi-band sampling theory has a logical structure similar to classical Fourier analysis. The relationship between the null intersection condition and the isometric formula is illustrated by considering the consequences when the null intersection condition does not hold. In this case, the sampling representation cannot hold for any function, whereas the isometric formula can still hold for some functions.