Perturbation stability of coherent Riesz systems under convolution operators
type: Item
instance of: scholarly article
title: Perturbation stability of coherent Riesz systems under convolution operators
zbMATH Open document ID: 1020.42021
author: Werner Kozek
author: Georg Zimmermann
author: Götz E. Pfander
published in: Applied and Computational Harmonic Analysis
publication date: 2002-11-28T00:00:00Z
review text: Consider transmission signals \[ f(x)= \sum^\infty_{k=-\infty} \sum^{N-1}_{l=0} c_{k,l}g_l(x- ak) \] represented with respect to some Riesz basis \(\{g_{k,l}\}\). After transmission over a communication channel, the received signal \(r(x)\) can be split into a linearly transformed version and statistically independent additive noise \[ r(x)= Kf(x)+ n(x), \] where the authors assume that the channel distortion corresponds to a translation invariant system \(Kf(x)= (h* f)(x)\), where \(h\) is a function with some practical relevant properties. The orthogonal perturbation of \(g\) by \(K= K_h\) is defined by \(d_{g,h}= \|K_hg-\langle K_hg, g\rangle/\langle g,g\rangle g\|_{L_2}\). The authors prove upper and lower bounds on the orthogonal perturbation formulated in terms of spectral spread and temporal support of \(g\), respectively. Further, they compare the three types of coherent families \[ \bullet\text{ Gabor systems, }\bullet\text{ Wilson bases, }\bullet\text{ wavelet bases} \] with respect to their performance under orthogonal perturbation. Finally, they illustrate the performance of these Riesz bases in a realistic setup when perturbed by a realization of a twisted copper cable impulse response. It follows that Gabor bases are best matched to a set of convolution operators which satisfy typical practical constraints.
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 94A11
Mathematics Subject Classification ID: 94A14
zbMATH DE Number: 1836815
zbMATH Keywords: coherent Riesz systems
zbMATH Keywords: Gabor basis
zbMATH Keywords: Wilson basis
zbMATH Keywords: wavelet basis
zbMATH Keywords: modulation
zbMATH Keywords: orthogonal perturbation
zbMATH Keywords: convolution operators
reviewed by: Gabriele Steidl
MaRDI profile type: Publication
full work available at URL: acha.2001.0375
OpenAlex ID: W2023955532
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DOI: 10.1006/ACHA.2001.0375
description: scientific article; zbMATH DE number 1836815