On Nonuniform Sampling of Bandwidth-Limited Signals

TLDR

The study investigates four special nonuniform sampling processes and extends Cauchy’s sampling theorem to a broader class of nonuniform sample point distributions, aiming to uncover properties of bandwidth‑limited signals. The authors generalize Cauchy’s sampling statement to a wider class of nonuniform sample point distributions and modify it to provide more precise information on signal determination. They derive four generalized sampling theorems that characterize unique, over‑ and under‑specification, provide explicit reconstruction formulas, estimate complexity and accuracy, and introduce a minimum‑energy signal with advantages over time‑limited signals for finite samples.

Abstract

The purpose of this investigation is to examine four special nonuniform sampling processes in detail, and to deduce some interesting properties of bandwidth-limited signals. The main results are contained in four generalized sampling theorems. These theorems not only contain the nature of determination (unique-specification, over-specification, and underspecification) of signals but also include explicit reconstruction formulas. From the reconstruction formulas, the complexity and accuracy of the nonuniform sampling processes discussed can be estimated. In addition, these theorems lead to observations regarding the allowable shapes, the "prediction," and the "energy" of bandwidth-limited signals in general. A "minimum-energy" signal is introduced which has certain advantages as compared to the ordinary "time-limited" signals when a finite number of sample values are given. Finally, a statement due to Cauchy on the sampling of bandwidth-limited signals is generalized to include a wider class of nonuniform sample point distributions and modified to give more exact information regarding the nature of determination of signals.