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Digital Fourier analysis : fundamentals
Kido K., Springer Publishing Company, Incorporated, New York, NY, 2014. 203 pp. Type: Book (978-1-461492-59-7)
Date Reviewed: Jan 9 2015

I think it’s fair to say that underlying any established branch of mathematics--algebra, analysis, number theory, set theory, logic--there is a theory of limitless depth. Fourier analysis--an infinite subset of an infinite set, if you’ll forgive the abuse of set theory and the questionable analogy--certainly fits that bill. Just as Fourier series were born of utility in the mathematics of heat transfer, Fourier analysis has continued to explode into ultra-usefulness within many quantitative fields. I think that most of us would agree with the book’s assertion that the “development of the fast Fourier transform (FFT) algorithm in 1965 and the subsequent inventions of microchips for signal processing accelerated . . . Fourier analysis based signal processing.” (For what it’s worth, it clarified, at least for me, the presence of “digital” in the book’s title.)

One of my vivid memories--or was it an equally vivid bad dream?--is having been in an undergraduate class where the professor initially threatened to give an F to anyone who used the = sign between a function and its Fourier expansion. Yes, that was a math class (and the threat was withdrawn as the semester progressed, the initial point having been made). That same day, I was, within the next three hours, in a physics class wherein the use of Dirac delta (spike) “functions” and Heaviside (step) “functions” and their “derivatives” could be characterized as devil-may-care uninhibited, as was the use of Fourier(-like) expansions in what physicists misnamed as Hilbert space. (Have you ever dealt with the square of a delta function?) Laurent Schwartz’s 1950 Fields Medal gave rigor its due recognition in restoring the balance that Dirac’s (1933) Nobel-winning formalism had tipped toward “physical intuition.” I am perhaps fighting the proverbial last war, but today’s computing (science) at large has, to some extent, restored the dignity and (I dare say) necessity of proof and rigor, as the societal cost of software incorrectness has risen in magnitude and publicity. The patronizing smiles in reaction to expressions of mathematical scruples are far less frequent today, as the damaging results of un-thought-out specifications, non-rigorous algorithms, and loss of numerical precision take their toll.

The books of Churchill, Lighthill, and Friedman [1, 2, 3] are among the many works that have for decades ameliorated the Fourier analysis and generalized function theory-practice dichotomy--though the “practice” part during (what the present book calls) the “analog age” was certainly hobbled by the requirement of “too much analog processing.” These forever-valid, analog-age books are “heavier,” less broad, and are directed differently than is the excellent and pedagogically focused book under review. The latter, furthermore, takes maximal cognizance and advantage of modern computers and computing. The publisher is entirely justified in characterizing this textbook as “a thorough, accessible, introduction to digital Fourier analysis for undergraduate students in the sciences.” The principles in this book are expounded so as to “reach the cornerstone of signal processing: the fast Fourier transform.”

This short and to-the-point book comprises the following seven chapters, and is replete with marvelously clear and effective graphs and illustrations: “Sine and Cosine Waves”; “Fourier Series Expansion[s]”; “Digitized Waveforms”; “Discrete Fourier Transform (DFT)”; “Fast Fourier Transform”; “DFT and [Its] Spectrum”; and “Time Window[s].” At the risk of over-indulging in reminiscence by adding to those above, I was pleasantly surprised some decades ago to hear about the late, great mathematician Kunihiko Kodaira’s [4] endorsement of Schaum’s Outline Series [5] and their genre for teaching such subjects as Fourier analysis, to which Kodaira himself had made profound contributions.

The place for the book under review is between the Schaum’s solved-problem collection and the theoretical works cited above; its flow places it closer to the latter, but with significant, stepwise, hands-on experience of each theoretical concept. There are, of course, end-of-chapter exercises. As for formulae, these are derived wherever derivations are brief and instructive, but the author occasionally sees fit to ask the student to consult other books for more detailed derivations. This pedagogic philosophy is very beneficial with respect to this book’s flow. (I had a little trouble, in the second chapter, with the book’s passing from Fourier series to Fourier integral, but regard this as a minor issue that can be handled by Wikipedia or other-book consultation.) A positive attribute that I do not regard as minor is exemplified by the choice of f (frequency), versus ω (angular frequency), as the more intuitive to students. Such pedagogical choices pervade the book, and show the author’s cognizance of his intended undergraduate audience. Another minor flaw is in the otherwise adequate index: Within, for example, the H entries, “Hanning window” precedes “Hamming window.” For R, “Rotating vector” precedes “Riesz window.” The index is not sufficiently dense for this to matter.

I recommend this excellent undergraduate textbook highly.

Reviewer:  George Hacken Review #: CR143070 (1504-0264)
1) Churchill, R. V. Fourier series and boundary value problems. McGraw Hill, New York, NY, 1941.
2) Lighthill, M. J. Introduction to Fourier series and generalised functions. Cambridge University Press, Cambridge, UK, 1958.
3) Friedman, B. Principles and techniques of applied mathematics. John Wiley, New York, NY, 1956.
4) Hirzebruch, F. Kunihiko Kodaira: mathematician, friend, and teacher. Notices of the American Mathematical Society 45, 11(1998), 1456–1462.
5) Spiegel, M. Schaum's outline of Fourier analysis. McGraw Hill, New York, NY, 1974.
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