Sampling-50 years after Shannon

TLDR

Regular sampling, with a uniform grid, has seen a strong revival in recent years due to its mathematical links to wavelet theory, and irregular sampling and radial basis functions are also briefly discussed. The paper reviews the state of sampling fifty years after Shannon, reinterpreting it as a Hilbert-space orthogonal projection and extending it to shift‑in‑variant spaces such as splines and wavelets, while summarizing results on approximation error and sampling rates for arbitrary inputs. The authors reinterpret Shannon sampling as an orthogonal projection in Hilbert space, extend it to shift‑in‑variant spaces such as splines and wavelets, and review related sampling variations—including wavelets, multiwavelets, Papoulis generalized sampling, finite elements, and frames—under a unified perspective. These developments enable simpler, more realistic interpolation models compatible with a broader range of anti‑aliasing prefilters, and provide insights into approximation error and sampling rates for non‑bandlimited inputs.

Abstract

This paper presents an account of the current state of sampling, 50 years after Shannon's formulation of the sampling theorem. The emphasis is on regular sampling, where the grid is uniform. This topic has benefitted from a strong research revival during the past few years, thanks in part to the mathematical connections that were made with wavelet theory. To introduce the reader to the modern, Hilbert-space formulation, we reinterpret Shannon's sampling procedure as an orthogonal projection onto the subspace of band-limited functions. We then extend the standard sampling paradigm for a presentation of functions in the more general class of "shift-in-variant" function spaces, including splines and wavelets. Practically, this allows for simpler-and possibly more realistic-interpolation models, which can be used in conjunction with a much wider class of (anti-aliasing) prefilters that are not necessarily ideal low-pass. We summarize and discuss the results available for the determination of the approximation error and of the sampling rate when the input of the system is essentially arbitrary; e.g., nonbandlimited. We also review variations of sampling that can be understood from the same unifying perspective. These include wavelets, multiwavelets, Papoulis generalized sampling, finite elements, and frames. Irregular sampling and radial basis functions are briefly mentioned.