Digital representations of operators on band-limited random signals

Abstract

Sampling representations of bounded linear operators (BLOs) acting on extended classes of band-limited signals are derived. These are classes of deterministic signals that are not necessarily square-integrable and random signals with covariance functions that are not necessarily square-integrable on the plane. These classes consist of signals whose Fourier transforms, defined as generalized functions (or distributions), have supports that are subsets of compact sets. The merit of these representations lies in the fact that the image of a signal under an operator may be reconstructed from the samples of the original signal rather than the samples of the image. For example, the derivative of a signal, of any order, may be expressed in terms of the samples of the signal itself instead of the samples of the derivative.