Frequency localization of wavelet packets
type: Item
instance of: scholarly article
title: Frequency localization of wavelet packets
zbMATH Open document ID: 0829.42024
DOI: 10.4171/RMI/175
published in: Revista Matemática Iberoamericana
publication date: 1996-01-21T00:00:00Z
full work available at URL: https://eudml.org/doc/39475
review text: Summary: Orthonormal bases of wavelet packets constitute a powerful tool in signal compression. It has been proved by Coifman, Meyer and Wickerhauser that ``many'' wavelet packets \(w_n\) suffer a lack of frequency localization. Using the \(L^1\)-norm of the Fourier transform \(\widehat {w}_n\) as localization criterion, they showed that the average \(2^{- j} \sum_{n=0}^{2^j -1} |\widehat {w}_n |_{L^1}\) blows up as \(j\) goes to infinity. A natural problem is then to know which values of \(n\) create this blow-up in average. The present work gives an answer to this question, thanks to sharp estimates on \(|\widehat {w}_n |_{L^1}\) which depend on the dyadic expansion of \(n\), for several types of filters. Let us point out that the value of \(|\widehat {w}_n |_{L^1}\) is a weak localization criterion, which can only lead to a lower estimate on the variance of \(\widehat {w}_n\).
Mathematics Subject Classification ID: 42C40
zbMATH DE Number: 813825
zbMATH Keywords: wavelet packets
zbMATH Keywords: signal compression
zbMATH Keywords: frequency localization
zbMATH Keywords: dyadic expansion
zbMATH Keywords: filters
author: Éric Séré
MaRDI profile type: Publication
OpenAlex ID: W2036762846
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description: scientific article; zbMATH DE number 813825