The authors concisely and accurately describe their book as “a book about mathematics, pitched at the advanced undergraduate/beginning graduate level, where ideas from signal processing are used to motivate much of the material, and applications of the theory to signal processing are featured.” The idea is that mathematics students see applications of theory (predominately real analysis) to signal processing, and students from other fields get a better appreciation for the mathematical foundations.
This works to a limited extent. Mathematics students get a high level view of signal processing but only a very limited sense of how or why it is done in practice. Students from other fields see some splendid analysis, but they may have trouble making the connection to the signal processing that they actually do. It would have been useful, perhaps at the very beginning of the book, to establish some common ground with a few basic examples.
The first four chapters are all mathematics: normed vector spaces, analysis tools (special functions, infinite products), Fourier series and Fourier transforms. The fifth chapter introduces the relatively new notion of compressive sampling (sometimes called compressed sensing) in which mathematicians at statisticians were at the forefront of the development. This chapter includes a brief section on practical implementation. The chapters that follow address individual topics of interest in signal processing. These include discrete transforms, linear filters, wavelets and multiresolution analysis. One of the most interesting features of the book is at the very end — a chapter on the parsimonious representation of data. This is a brief self-contained introduction to the subject that illustrates applications of some of the tools described and developed in previous chapters.
The book’s title is a bit of an overreach. Better might be “some mathematics related to signal processing”, since there’s also a good deal of relevant mathematics that’s not here. Despite that, this is a solid book. The exposition is well organized and complete, the writing crisp and clear.
Bill Satzer (firstname.lastname@example.org) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.
2. Normed vector spaces
3. Analytic tools
4. Fourier series
5. Fourier transforms
6. Compressive sensing
7. Discrete transforms
8. Linear filters
9. Windowed Fourier transforms, continuous wavelets, frames
10. Multiresolution analysis
11. Discrete wavelet theory
12. Biorthogonal filters and wavelets
13. Parsimonious representation of data