Sampling procedures in function spaces and asymptotic equivalence with shannon's sampling theory

Autor:	Akram Aldroubi, Michael Unser
Rok vydání:	1994
Předmět:	Discrete mathematics Sequence Control and Optimization Sinc function Function space Generating function Sampling (statistics) Information theory Orthogonal basis Computer Science Applications Convolution Combinatorics Signal Processing Analysis Mathematics
Zdroj:	Numerical Functional Analysis and Optimization. 15:1-21
ISSN:	1532-2467 0163-0563
DOI:	10.1080/01630569408816545
Popis:	We view Shannon's sampling procedure as a problem of approximation in the space S = {s: s (x) = (c * sinc)(x)c e l 2}. We show that under suitable conditions on a generating function λ e L 2, the approximation problem onto the space V = {v:v(x) = (c * λ)(x)c e l 2} produces a sampling procedure similar to the classical one. It consists of an optimal prefiltering, a pure jitter-stable sampling, and a postfiltering for the reconstruction. We describe equivalent signal representations using generic, dual, cardinal, and orthogonal basis functions and give the expression of the corresponding filters. We then consider sequences λn, where λn denotes the n-fold convolution of λ. They provide a sequence of increasingly regular sampling schemes as the value of n increases. We show that the cardinal and orthogonal pre- and postfilters associated with these sequences asymptotically converge to the ideal lowpass filter of Shannon. The theory is illustrated using several eamples.
Databáze:	OpenAIRE
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