Beyond Bandlimited Sampling: Nonlinearities, Smoothness and Sparsity

TL;DR

This paper surveys recent extensions of Shannon sampling theory that address non-bandlimited signals, nonlinear distortions, and nonideal sampling by viewing sampling as projection onto suitable subspaces.

Contribution

It introduces a unified framework for sampling that encompasses nonlinearities, smoothness, and sparsity, expanding the applicability of classical sampling theorems.

Findings

Uniform sampling of non-bandlimited signals

Perfect compensation for nonlinear effects

Extension of Shannon theorem to broader classes of signals

Abstract

Sampling theory has benefited from a surge of research in recent years, due in part to the intense research in wavelet theory and the connections made between the two fields. In this survey we present several extensions of the Shannon theorem, that have been developed primarily in the past two decades, which treat a wide class of input signals as well as nonideal sampling and nonlinear distortions. This framework is based on viewing sampling in a broader sense of projection onto appropriate subspaces, and then choosing the subspaces to yield interesting new possibilities. For example, our results can be used to uniformly sample non-bandlimited signals, and to perfectly compensate for nonlinear effects.