Generalized sampling procedures on multiply-generated shift-invariant spaces

Abstract

Sampling procedures represent input signals with time-continuous variables by discrete values derived from the input signals. The Shannon`s sampling theorem was the beginning of this field. However, his assumption that the input signals are band-limited was not realistic, so by enlarging the admissibility of the input signals, generalized sampling theories have been developed. In this paper, generalized sampling procedures on multiply-generated shift-invariant spaces are developed. The results are similar to ones on the principle shift-invariant spaces. For the orthogonal projections which are the ideal sampling procedures, the dual of the given Riesz bases and orthonormal bases for the multiply-generated shift-invariant spaces are shown. The methods how to obtain orthonormal bases are introduced: the Cholesky factorization and the Gram-Schmidt process. Finally, we discuss about the errors of the ideal and the nonideal sampling procedures.