Truncation Error of Sampling-Theorem Expansions

Abstract

Upper bounds are obtained for the error, termed truncation error, which arises in reconstituting a band-limited function by summing over only a finite number (instead of the requisite infinite number) of samples of this function in an appropriate sampling-theorem expansion. Upper bounds are given for the truncation errors of the Cardinal and Fogel sampling expansions and for "self-truncating" versions of these two sampling expansions; these latter sampling expansions are "self-truncating" in the sense that the upper bounds on their truncation errors are almost always much lower than the upper bounds on the truncation errors of their prototype sampling expansions. All of the upper bounds are given as functions of three parameters: M, the maximum magnitude of the band-limited function; q, the per unit guard band (assuming that the band-limited function is sampled at a rate greater than the Nyquist rate); and N, the measure of the number of samples in the finite summation of terms of the sampling expansion.