Signal processing is a subject closely related to mathematics, computer science, and electrical engineering. It is currently of great interest, and the theoretical results obtained here have led to important new algorithms and techniques, with plenty of applications. In this book, Damelin and Miller have decided to focus on the mathematical foundations of signal processing.

The first two chapters are devoted to those foundations of signal processing that are located in mathematical analysis. Chapters 3 and 4 then deal with Fourier analysis and thus provide the classical path to the core of signal processing, which is first touched on in chapter 5’s discussion of compressive sampling. A discretized form of the transform theory, culminating of course in the famous fast Fourier transform algorithm, is the topic of chapter 6. Based on this foundation, chapter 7 then deals with another core part of classical signal processing: namely, the theory of filters. Chapters 8 to 11 cover modern and very important mathematical tools like wavelets, frames, and filter banks, and their applications, in great detail. The book concludes with chapter 12, which describes techniques for representing large sets of data using small amounts of memory, without giving up too much accuracy.

In summary, Damelin and Miller provide a very detailed and thorough treatment of all the important mathematics related to signal processing. This includes the required background information found in elementary mathematics courses, so their book is really self-contained. The style of writing is suitable not only for mathematicians, but also for practitioners from other areas. Indeed, Damelin and Miller managed to write their text in a form that is accessible to nonspecialists, without giving up mathematical rigor. Therefore, it is highly suitable for advanced undergraduate-level students who are looking for self-study materials and teachers who are looking for a textbook to base their courses on.