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Zbl 0986.42016
Meyer, Yves
Signal processing and mathematical analysis. (Le traitement du signal et l'analyse mathÃ©matique.)
(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. TorrÃ©sani 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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