Applications first, then the maths

In this book we are presented with a wealth of information on frequency-domain analysis that can be approached by students from many different backgrounds. The material is at a level suitable for undergraduates from a wide range of disciplines including applied mathematics, engineering of all types, physics, chemistry and any experimental sciences where frequency analysis is applied. The coverage is sufficiently complete that the book will also serve as a useful reference for graduate students and other researchers.

A characteristic of many other texts on this topic is that they are aimed either solely at the mathematician or at the practitioner, being either too applied for the mathematician, without the necessary rigour, or too theoretical for the engineer and applied scientist. The ambitious task taken on here by David Kammler is to present a near-complete coverage of the basic theoretical and applied material on this topic that will be accessible to all with a knowledge of calculus and linear algebra. I recommend that students should also have a basic working knowledge of Fourier transforms and Fourier series to benefit from this text.

In the early chapters, a fairly informal approach is taken and the main results are presented, but we are alerted to the missing mathematical details that must be developed later. The framework is appealing in that it unifies the analysis of discrete/continuous time and periodic/non-periodic functions. In chapter seven, the required mathematics are developed in an approachable form that relates back to the earlier material. The approach taken is the reverse of that usually adopted in a mathematics text, in which the necessary detailed mathematics are developed first, and only then (if at all) can the reader start to gain the practical insights and overall appreciation of the topic that are so essential for practical use. I think this is an important divergence from the norm, since it allows the reader first to become thoroughly acquainted with the topic before being weighed down with theoretical detail. It is an approach that will allow this rich topic to be understood more fully by students from relevant disciplines.

The coverage in chapter six of the fast Fourier transform algorithms is particularly illuminating and certainly introduced me to some new derivations and view points on a well-established topic. One important element that could have been included here or earlier is the study of windowing or tapered analysis, which is at the heart of many applications of spectral analysis. Chapter eight, on sampling, is also very fine, although I would have appreciated an introduction of the main results on a more informal footing in an earlier chapter.

Chapters 9 to 12 are devoted to further applications of the theory to partial differential equations, wavelets, music and probability. The substantial coverage given to these topics emphasises the very broad range of fields in which Fourier analysis is routinely or less routinely applied, and the commitment of the author to the application as well as the theory of Fourier methods.

The book includes a vast number of exercises, which are presented with comments and discussion in a way that can easily be worked through by an individual. There are also extensive appendices that summarise all the main results in a readable way that will be a useful quick reference. If I have a criticism of this book, it is that it is over-detailed and long at 710 pages plus appendices. However, this is not a serious criticism when the price is very reasonable and the style is such that one can achieve a good appreciation of the main results without losing the flow in a superficial first read, and then go back and cover the details at leisure.

Simon J. Godsill is lecturer in information engineering, University of Cambridge.