Signal processing and mathematical analysis
type: Item
instance of: scholarly article
title: Signal processing and mathematical analysis
zbMATH Open document ID: 0986.42016
author: Yves Meyer
publication date: 2000-05-17T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/75430
review text: This is a review of certain mathematical aspects of signal processing from a historical perspective. Section 1 discusses the ``numerical revolution'', the early contributions of Wiener, von Neumann, Shannon, Rosenblueth, Wigner, Brillouin, Gabor, Wilson, Morlet and Mallat, and ends with a statement of Daubechies' theorem on the existence of orthonormal wavelet bases of compact support. Section 2 introduces the concepts of analysis and synthesis in signal processing. Sections 3 and 4 deal with Shannon's theorem and its generalizations. Section 5 discusses the time--frequency plane and mentions Gabor wavelets, Wilson bases and Dolby filtering, Heisenberg's uncertainty principle, the time--frequency plane, and Gibbs cells. Section 6 contains a definition of Riesz basis and a brief description of windowed Fourier analysis and Gabor wavelets, including the Balian--Low theorem and its relationship to critical sampling. Wilson bases are described in Section 7. Sections 8 and 9 are titled ``Search for the optimal basis'' and ``Chirps in mathematics and signal processing''. Section 10 deals with frequency--modulated Wilson bases. After a brief historical discussion, Section 11 gives an elementary proof of the differentiability at \(x=1\) of Riemann's series \(\sigma(x) = \sum_{n=1}^{\infty} n^{-2} \sin (\pi n^2 x)\). The purpose of this proof is to illustrate the author's position that when studying a given property, certain series expansions are more relevant than others. In the case at hand, the property is differentiability at a given point. The expansion of \(\sigma(x)\) in a Fourier series does not yield the desired proof. On the other hand, the proof of the assertion readily follows by expanding \(\sigma(x)\) in a series of chirps. Sections 12 and 13 discuss chirp functional spaces and analysis by \(r\)--regular chirp wavelets. Finally, Section 14 gives an application to the detection of gravitational waves and briefly discusses the work of B. Torrésani and his colleag…
reviewed by: Richard A. Zalik
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 94A12
zbMATH DE Number: 1448501
zbMATH Keywords: wavelet
zbMATH Keywords: time-frequency atom
zbMATH Keywords: Fourier analysis
zbMATH Keywords: chirp
zbMATH Keywords: Riesz basis
MaRDI profile type: Publication
OpenAlex ID: W2015155313
cites work: Q4865484
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DOI: 10.5802/AIF.1766
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 1448501