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Zbl 0986.42016
Meyer, Yves
Signal processing and mathematical analysis. (Le traitement du signal et l'analyse mathématique.) (French)
[J] Ann. Inst. Fourier 50, No.2, 593-632 (2000). ISSN 0373-0956; ISSN 1777-5310/e

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 colleagues. The bibliography is not exhaustive and is intended to be a guide for further reading.
[Richard A.Zalik (Auburn University)]

MSC 2000:: *42C40 Wavelets
94A12 Signal theory

Keywords: wavelet; time-frequency atom; Fourier analysis; chirp; Riesz basis

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