Mathematical Principles of Signal Processing

A1 Fourier Transforms of Stable Signals.- A2 Fourier Series of Locally Stable Periodic Signals.- A3 Pointwise Convergence of Fourier Series.- B1 Filtering.- B2 Sampling.- B3 Digital Signal Processing.- B4 Subband Coding.- C1 Hilbert Spaces.- C2 Complete Orthonormal Systems.- C3 Fourier Transforms of Finite-Energy Signals.- C4 Fourier Series of Finite-Power Periodic Signals.- D1 The Windowed Fourier Transform.- D2 The Wavelet Transform.- D3 Wavelet Orthonormal Expansions.- D4 Construction of an MRA.- D5 Smooth Multiresolution Analysis.- The Lebesgue Integral.- References.- Glossary of Symbols.

Fourier analysis and its modern extension of wavelet analysis provides one of the most useful tools in many applied sciences. This text bridges the gap between engineering and mathematics, providing a rigorously mathematical introduction to the subject, while emphasizing their uses in signal processing and other applications in communications engineering. An appendix provides the necessary background on Lebesgue integrals.

From the reviews:

MATHEMATICAL REVIEWS

"While many books exits, dealing either with 'theory' or 'applications', the interplay between signal processing and mathematics makes it difficult to find in a single volume the essentials of modern signal processing presented in a way which would be both rigorous for mathematicians and accessible for engineers. This challenge is precisely the purpose of Bremaud's book, and the goal is clearly attained. The presentation is clear, concise and self-contained, all the necessary concepts are put on a firm ground and proofs are worked out in great detail...In summary, the interested reader will find in Bremaud's book an invaluable reference because of its coverage, scope and style, as well as of the unified treatment it offers of (signal processing oriented) Fourier and wavelet basics."

"This book wants to bridge the gap between engineering and mathematics ... . What I liked most is the part on subband coding and the fact that bioorthogonal bases are discussed ... . A nice feature of the text is that it stimulates the reader to actively assimilate the material because the theorems (almost all the proofs are included) are interlaced with exercises challenging to immediately apply what has been explained." (Adhemar Bultheel, Belgian Mathematical Society Simon Stevin Bulletin, Vol. 11 (2), 2004)

"Brémaud has prepared this graduate-level book to bridge the gap between the mathematical theory of Fourier analysis and the application of that theory to signal processing and other areas of communications engineering. ... To be readable by an engineering audience, the book includes a self-contained appendix on Lebesgue integration. ... Given the importance of the subject matter and the thoroughness of the treatment, this book is recommended for acquisition by any library that supports graduate programs in mathematics, physics or engineering." (D. Z. Spicer, CHOICE, December, 2002)

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