A Generalized Sampling Theorem for Frequency Localized Signals

TL;DR

This paper introduces a generalized sampling theorem for frequency localized signals, allowing for soft bandlimitation and approximate reconstruction, broadening the applicability of classical sampling theory.

Contribution

It extends traditional sampling theorems to include prefilters with soft bandlimitation and provides an approximate reconstruction method with error estimates.

Findings

Reconstruction formula for frequency localized signals.

Error estimates for approximate reconstruction.

Examples using sinc, Gaussian, and B-spline functions.

Abstract

A generalized sampling theorem for frequency localized signals is presented. The generalization in the proposed model of sampling is twofold: (1) It applies to various prefilters effecting a "soft" bandlimitation, (2) an approximate reconstruction from sample values rather than a perfect one is obtained (though the former might be "practically perfect" in many cases). For an arbitrary finite-energy signal the frequency localization is performed by a prefilter realizing a crosscorrelation with a function of prescribed properties. The range of the filter, the so-called localization space, is described in some detail. Regular sampling is applied and a reconstruction formula is given. For the reconstruction error a general error estimate is derived and connections between a critical sampling interval and notions of "soft bandwidth" for the prefilter are indicated. Examples based on the sinc-function, Gaussian functions and B-splines are discussed.