System approximations and generalized measurements in modern sampling theory

Holger Boche, Volker Pohl

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review

5 Scopus citations

Abstract

This chapter studies several aspects of signal reconstruction of sampled data in spaces of bandlimited functions. In the first part, signal spaces are characterized in which the classical sampling series uniformly converge, and we investigate whether adaptive recovery algorithms can yield uniform convergence in spaces where non-adaptive sampling series does not. In particular, it is shown that the investigation of adaptive signal recovery algorithms needs completely new analytic tools since the methods used for nonadaptive reconstruction procedures, which are based on the celebrated Banach–Steinhaus theorem, are not applicable in the adaptive case.The second part analyzes the approximation of the output of stable linear time-invariant (LTI) systems based on samples of the input signal, and where the input is assumed to belong to the Paley–Wiener space of bandlimited functions with absolute integrable Fourier transform. If the samples are acquired by point evaluations of the input signal f, then there exist stable LTI systems H such that the approximation process does not converge to the desired output Hf even if the oversampling factor is arbitrarily large. If one allows generalized measurements of the input signal, then the output of every stable LTI system can be uniformly approximated in terms of generalized measurements of the input signal.The last section studies the situation where only the amplitudes of the signal samples are known. It is shown that one can find specific measurement functionals such that signal recovery of bandlimited signals from amplitude measurement is possible, with an overall sampling rate of four times the Nyquist rate.

Original languageEnglish
Title of host publicationApplied and Numerical Harmonic Analysis
PublisherSpringer International Publishing
Pages269-305
Number of pages37
Edition9783319197487
DOIs
StatePublished - 2015

Publication series

NameApplied and Numerical Harmonic Analysis
Number9783319197487
ISSN (Print)2296-5009
ISSN (Electronic)2296-5017

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