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Book Review
Transforms and Fast Algorithms for Signal Analysis and Representations, by Guoan Bi and Yonghong Zeng, Birkhauser 2004, ISBN 0-8176-4279-X
Reviewed by Vladimir Botchev [vladimir.botchev@analog.com]
This is perhaps the best text on transforms for signal processing since Nussbaumer’s Fast FourierTransform and Convolution Algorithms (Springer Verlag 1982) and Elliott and Rao, Fast Transforms Algorihms, Analyses, Applications (Academic Press 1982). Its nine chapters encompass almost all the knowledge needed to apply signal processing transforms successfully in practice.
It is expected that the reader has had some exposure to transforms, so the first introductory chapter is very short; it mainly provides a sort of plan of things to come. Chapter Two introduces some number theory and—more specifically—polynomial arithmetic. The authors have chosen to provide proofs only to essential theorems, like the Chinese remainder theorem, a good decision for a reference book, but perhaps not so good for a textbook. However, this book does lean more towards serving as a professional reference than does the more academic Nussbaumer text. The lack of proofs felt by some readers is largely offset by an abundance of concrete examples.
Chapter Three details various algorithms for the fast Fourier transform and provides at the end some code for transforms of unusual length (9, 5)—which can be used as building blocks for larger transforms. Chapters 4 and 5 provide perhaps more than one would like to know about 1-D and multi-dimensional Hartley transforms. While there are authors who seem to be “fans” of the Hartley transform (like Bracewell, who wrote entire books about it), signal-processing practitioners look at it more as a curiosity than a real “winner” for most tasks where a transform is needed. I have not found it to be faster than some ingenious real Fourier transforms, such as Bergland’s FFT.
Chapters Six and Seven provide a full account of the discrete-cosine transform, in all its flavors and types, including prime-factor realizations and multidimensional transforms. Simple programs are provided to illustrate some of the algorithms. Chapter 8 is unique, in the sense that—to the best of my knowledge—this material has never appeared in a book before. The chapter deals with integer transforms, which might be thought of as derived from the DCT and discrete sine transforms; these are gaining in popularity in various transform coding schemes for both video and audio. Chapter Nine is a useful introduction into time-frequency analysis. Time-frequency analysis methods are not exactly transforms, as they are understood in previous chapters, but they do use many of the transforms discussed thus far.
It might seem strange that wavelet transforms are missing, but in my opinion the authors made a good decision not to dilute a very well-written reference with a topic that perhaps suffers already a plethora of books and papers. In conclusion, this book is a highly practical and very welcome addition to the collection of texts on signal-processing techniques and applications.
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