On the Approximation of L2 Inner Products from Sampled Data

Hagai Kirshner and Moshe Porat,
Department of Electrical Engineering, 
Technion - Israel Institute of Technology 
Haifa 32000, Israel


Abstract

Most signal processing applications are based on discrete-time signals although the origin of many
sources of information is analog. In this work 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 $\Lebesgue$ inner product, and to determine the maximum potential
representation error induced by the sampling process. The maximum representation error can be also
determined for the Riemann-type sum approximation scheme. Examples of practical applications are
given and discussed.
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IEEE Trans. on Signal Processing,
Vol. 55, No. 5, pp. 2136-2144 (2007).

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