On the Approximation of $L_{2}$ Inner Products From Sampled Data

Abstract

Most signal processing applications are based on discrete-time signals although the origin of many sources of information is analog. In this paper, we consider the task of signal representation by a set of functions. Focusing on the representation coefficients of the original continuous-time signal, the question considered herein is to what extent the sampling process keeps algebraic relations, such as inner product, intact. By interpreting the sampling process as a bounded operator, a vector-like interpretation for this approximation problem has been derived, giving rise to an optimal discrete approximation scheme different from the Riemann-type sum often used. The objective of this optimal scheme is in the min-max sense and no bandlimitedness constraints are imposed. Tight upper bounds on this optimal and the Riemann-type sum approximation schemes are then derived. We further consider the case of a finite number of samples and formulate a closed-form solution for such a case. The results of this work provide a tool for finding the optimal scheme for approximating an L2 inner product, and to determine the maximum potential representation error induced by the sampling process. The maximum representation error can also be determined for the Riemann-type sum approximation scheme. Examples of practical applications are given and discussed