This book “was developed over a five-year span as the text for an undergraduate electrical engineering course at Stanford University,” according to the author. Williams adheres quite admirably to the goals, basic ideas, and scope he describes in his preface. The book is not about hardware. It is about designing the algorithms that transform the digital or digitalized input signals into the desired digital output signals. In a readable albeit mathematically rigorous narrative, Williams explains the commonly used mathematical tools for analyzing linear digital filter signals and shows how these tools can be applied to design algorithms that satisfy commonly used design criteria.
Williams makes admirable efforts “to provide a rigorous treatment of digital filters without demanding a raft of prerequisites. The reader must have a good understanding of differential and integral calculus . . . and some exposure to complex variables.”
I was delighted to see that Williams based his approach on the idea that “digital filters are far easier than analog filters, so there is no reason to make the obvious obtuse by ‘building’ digital filters on an analog foundation.” In Chapter 6 when he treats the design of recursive filters, Williams must (reasonably) depart from this policy by admitting that “we begin the design with a good analog filter and approximate it with a digital filter.”
In the preface Williams lists some other pedagogical ideas that guided him:
“An introductory treatment should be rigorous, yet easy to understand. I’ve done this by narrowing the scope of material and making extensive use of relatively simple mathematics.”
“Most beginning digital filter designers readily understand the material, but have trouble with the applications. Therefore, motivation and examples are as important as development and explanation.” Williams does this admirably by interspersing meaningful and worked-out examples within the text. With each chapter there are exercise problems, for which a solution manual is available.
“Most people want to design as quickly as possible and not spend too much time on tool developments without motivation. I’ve organized the book so that the reader can use the tools to design as quickly as possible.” Williams has done exactly that.
“Most people enjoy learning from stories rather than dry textbook stuff. Therefore, I’ve used a storytelling approach to develop the material. Topics build on topics in a direct and natural way.” The natural and lucid way in which Williams accomplishes this is an outstanding feature of the book. He shows great sympathy for the student by not prefacing some equations with “It can easily be shown that. . . .” He shows the steps. At the same time, I must say that Williams’ storytelling style will frustrate the reader who tries to use the book as a handbook to quickly look up some topic about which he or she has become a little fuzzy. But a book cannot be all things to all people.
Here is a brief description of the contents, chapter by chapter:(1) Introduction and Motivation--Digital filters manipulate input signals (which ideally may be sampled analog signals) to produce output signals. The concept of the impulse response is developed and the properties of digital filters (linear and time-invariant) are explained.
(2) Frequency Response--“I chose to approach the frequency response rather formally and show that it is a special case of an eigenvalue. This approach introduces the reader to the very powerful concepts of eigenfunctions and eigenvalues, and it begs many of the transform issues that plague other approaches.”
(3) Design of Nonrecursive Filters--“We begin by defining and motivating the concept of a squared design error. We minimize the design error and thereby produce a technique for designing nonrecursive filters: the Fourier method.”
(4) Windowing--Since the Fourier design technique leads to infinite-length filters, this chapter introduces some commonly used windows that permit the truncation of an infinite-length filter with a minimum of degradation in the frequency response. With these tools the reader is able to design general nonrecursive filters.
(5) Recursive Filters and the z-Transform--The z-transform is introduced and its utility shown in the analysis of recursive digital filters. The recursive filter is conveniently represented in terms of its poles and zeros. Instability, unique to recursive filters, is explored.
(6) Design of Recursive Filters--The technique of developing the filter design in the analog (continuous-signal) domain is used. Beginning with a brief introduction to analog filter design, Williams explains the impulse invariant technique and the bilinear transformation technique of approximating, with a digital filter, the analog design. The reader is now able to design both recursive and nonrecursive digital filters.
(7) Polynomial Modeling of Digital Signals--Polynomial curve fitting is introduced as a technique for smoothing, interpolating, extrapolating, differentiating, and integrating digital signals.
(8) Discrete and Fast Fourier Transforms--A pedagogically superb explanation of the discrete Fourier transform (DFT) and its implementation as well as its limits and pitfalls is presented. The fast Fourier transform (FFT) is developed in a readable way as a computationally efficient way to implement the DFT. This chapter provides the practical computational tool for the Fourier design of nonrecursive filters.
The layout of the book is good. Figures are abundant and helpful, although I noticed inadequate labeling on Figures 3–11. I noticed no more typos than I would expect in any first edition. From the context the reader must occasionally read “recursive” for “nonrecursive,” or vice versa, for example in the title of Chapter 6. The style is clear and friendly, but on a few occasions Williams uses local slang or in jokes that can confuse international readers who rely heavily on a dictionary when reading English. For example: “let’s begin by massaging equation (8.2).” I am not able to compare this text with other digital signal processing texts and reference works, which Williams generously lists in Appendix D, but I would be happy to use this textbook if I ever teach a course on digital filtering. I would also recommend the book as a readable, self-contained text for self-study.
