Discrete Fourier Analysis and Wavelets

Discrete Fourier Analysis and Wavelets, by S. Allen Broughton and K. Bryan, New Jersey, Wiley, 2009, 338 pp., £55.95 or US$99.95 (hardback), ISBN 978 0 470 29466 6 \end{bkreview}

Signal and image processing is an area which continues to develop and expand. It is a subject matter which has obvious practical data applications which are underlaid by, at times, complex mathematical methods. This book is authored by two professors of Mathematics at the Rose-Hulman Institute of Technology and provides a good, mathematically focused, introductory text to the methods, both classical and modern, currently utilised in signal and image processing.

The text aims to provide entry level material on a range of signal processing methods to those at upper-undergraduate or beginning postgraduate levels working within mathematics, engineering or other related areas. The fact that the text is aimed at a wide audience of practitioners perhaps explains why the text is less technically detailed than other books which cover a similar information base. Though, where any technical details or definitions are omitted, full references to more extensive works are provided. Therefore, whilst the book acts as a good introductory text, those readers looking for an in-depth description of the mathematics behind these techniques are advised to look elsewhere.

The reader is first introduced to traditional and well-established signal processing methods, such as the discrete Fourier and discrete cosine transforms. Throughout these sections, the JPEG compression algorithm is the principal example used to promote understanding and this is an example which is thoroughly developed throughout the book. The text is then concerned with detailing other processing techniques such as convolution, filtering and windowing. More modern advancements in the field, such as filter banks and wavelet methods are then considered. Many of these ideas are clearly stated and formed in such a way that the reader is provided with at least some explanation of the derivation of them. Where clarity is lacking, the reader is helpfully directed to others works or given worked examples to help further explain concepts.

One of the strengths of the text lies in the large number of illustrative examples given. Many of these examples are enhanced by pictorial demonstrations of the theory. Also, the book does provide the reader with an extensive range of exercises which are usefully supplemented with hints and some sample answers. In addition to these exercises the book also details projects for completion in MATLAB where the aim is to allow the reader to apply the mathematical theory to realistic field problems.

Much of the text is given over to general introductions of mathematical theory and notation which are fundamental to the understanding of topics later introduced. A comparatively minimal portion of the text covers the rather extensive topic of wavelet methodology though a good proportion of the text is spent explaining filter banks and how they give rise to wavelets. For ease of understanding the majority of the examples on wavelet theory are based around the simplest basis (the Haar basis). In general, the book gives a rather self-contained and in-depth guide to the discrete Fourier analysis portion of the text, whilst only giving, in the main, introductory material on wavelets. Therefore, the book can be recommended to those readers wishing to explore the Fourier aspect of the material, but those looking for a more comprehensive guide to wavelets are better directed to Vidakovic 1.