A SIGNAL ACCOMPLISHMENT
Frequency Analysis by R. B. Randall.
Brüel & Kjaer, 344 pp., hardback, $50.
It was certainly a treat to get to review the third edition of this book published by Brüel & Kjaer under the title Frequency Analysis. The second edition was released in 1977, and in the intervening 10+ years, a great deal has happened in the field of signal analysis. I have followed these developments closely in a variety of textbooks and especially have been impressed by the many articles that have appeared in the Brüel & Kjaer Technical Review. Much of the content of this new edition comes out of those articles. For those not familiar with the name Brüel & Kjaer, it is appropriate to mention that this company is one of the most renowned, worldwide, as a manufacturer of acoustic measuring instruments. We used to say, "If you haven't measured it with a B & K microphone, you haven't measured it." To a great extent, this is still true.
Frequency Analysis is much more than a simple revision of a 12-year-old book, and the title may be a bit misleading since this work includes discussion of a wide variety of signal analysis techniques. At least half of it is new material, and the remainder has been substantially rewritten. The book is accurate and authoritative, and includes coverage of most of the significant aspects of analysis of audio frequency and vibrational signals. It is a book which requires some mathematical preparation for a complete understanding of the analysis processes described. However, it also contains a broad variety of practical application examples and a thorough discussion of the application of both time- and frequency-domain instruments to the analysis of signals.
One test of a good book is, of course, its accuracy, and this book is tops in that respect. But a more subtle test is to see if it lends new insight into issues that are already well understood. This book does just that. Without being verbose, the author describes mathematical processes with a clarity rarely found in textbooks on matters as technical as signal analysis. It is clear that this author is trying, and I would say successfully, to make the reader understand the processes of analysis as well as presenting the equations and techniques. As is apparent, my enthusiasm for this book is quite high.
Even the foreword and introduction are well written and informative.
In the past decade, there has been greatly increased emphasis on digital, rather than analog, analysis of signals.
The use of digital filters to replace analog filters and the ascendancy of the fast Fourier transform, which is of course implemented digitally, has led to greatly improved accuracy, stability, and precision of the analysis process.
Nowhere has the impact of digital processing been more greatly felt than in the area of precision audio frequency measurements.
However, this book recognizes that it is not so much digital techniques which distinguish the two principal analysis schemes--time domain and frequency domain--but the fact that in the time domain, data is analyzed continuously in real-time sequence, while in the frequency domain, data is analyzed by means of the Fourier transform in data blocks. Recognizing the breakdown in this manner leads to a more thorough and understandable treatment of the whole body of signal analysis methods.
In addition to the standard and necessary topics, Frequency Analysis presents a fine treatment of the Hilbert transform and cepstral techniques.
Stationary, or so-called steady-state, signals are treated, as are non-stationary, or transient, signals. While the discussion is mathematically correct, the author does not bury the reader in equations. In fact, equations are used rather sparingly, and a very large number of graphs, unique figures, and examples are used to clarify the text at many points in the discussion.
A review of this book, chapter by chapter, demonstrates its logic and content. The first chapter, only four pages long, is an introduction. Chapter 2 has the ominous title, "Theoretical Frequency Analysis." In it are discussed time-domain signals, complex representation of time-domain signals, the Fourier transform, convolution, and the Hilbert transform. These are indeed theoretical topics, but they must be understood since they are the necessary basis of signal analysis. They are treated, in about 60 pages, with a clarity that is as good as I have seen. I believe that both the mathematically inclined novice and the expert will learn much from this treatment. The numerous figures in Chapter 2 are especially helpful in understanding the equations which describe time-function signals.
The third chapter, 68 pages, gets down to one of the two main topics of the book: The filter analysis of stationary signals. This chapter gives broad and thorough coverage of the properties and use of both analog and digital filters to do what is generally considered traditional frequency analysis of time-dependent but stationary signals.
This material is very much the same as in the older, 1977 edition, but it has been rewritten for greater clarity and modernized to include the newest techniques. The chapter is amazingly thorough in discussing octave, fractional octave (such as one-third octave), and narrow-band frequency analysis. The discussion of the choice of the correct filter bandwidth and speed of analysis for typical applications is excellent. A bit of commercialization is found in the specific description and use of the B & K Digital Frequency Analyzer Type 2131 as the target instrument for the examples, but the principles apply for any similarly equipped instrument. The section on calibration of the measurements is an important one, since it is all too easy to record a lot of data with modern instruments and then overlook the meaning of all the "nice graphs and charts" that are so readily produced. This thorough chapter also includes many details on the "how to" of selecting correct filter response, averaging time, dwell time, speed of analysis, and the like.
The fourth chapter, "Fast Fourier Transform," is a bit brief at 39 pages, but really is quite adequate in that it lays the groundwork for the following four chapters. The fast Fourier transform is the modern foundation of all digital time- and frequency-domain signal processing. The treatment given is only as mathematical as necessary and is augmented by the attractive use of diagrams which help to explain the fast transform method. It may not satisfy the more mathematically inclined but does lead nicely into the following discussions of abasing, picket-fencing, windowing, and other artifacts of the sampling process. The treatment of zoom techniques and the section on practical analysis of stationary signals are very good.
Two short chapters follow. The chapter on transient analysis is rather brief and places considerable emphasis on analysis of long transients. While this is a good introduction to the subsequent chapter on non-stationary signals, both chapters are treated in such a way as to be of principal interest for analysis of machinery, vibration, or speech rather than for most audio signal applications. The treatments are not adequate for the rather more elaborate analysis of musical signals, which are the ultimate non-stationary signals of main interest to the audio community.
Chapter 7 is devoted to two-channel spectral analysis. While only 42 pages long, it gives a nice introduction to the use of two-channel methods for obtaining the transfer function of systems, cross spectra, and coherence functions. Some discussion of noise and leakage in the measurement procedure is included. Auto- and cross-correlation functions are treated briefly, and the impulse response of systems is reviewed in some detail. The applications and discussion are of primary value to readers interested in vibration of structures. There is limited discussion of two-channel measurements applied to acoustical problems. B & K's Analyzer Type 2032 is used extensively as the measurement instrument for the examples, as might be expected. The chapter is a brief summation of the material that has appeared in the B & K Technical Review over the past 12 years. Unfortunately, many applications to acoustic measurements, such as sound intensity and acoustic room response, are omitted-except for a reference or two. Of course, the extensive applications of two-channel spectral analysis to electronic equipment, audio signals, and acoustics would more than till a book by themselves. The material here is only the starting point of what could and should be an entire book.
The final chapter, 33 pages, is on a topic called "Cepstrum Analysis." This is a subject which is very much for specialists in the field of machinery and vibration analysis. One can, if he wishes, learn about such terms as "quefrency," "rahmonics," "littering," and the like. Very specialized terms, indeed. It is an adequate treatment for the specialist but not of much immediate use to those interested in audio systems and signals.
The book is rounded out with three short appendices on Fourier analysis, narrow-band noise statistics, and the transfer functions of physical systems.
All in all, I like this book. It is a good summary for the specialist and still a book from which those with a more casual interest in frequency analysis will learn a great deal. It does not, however, cover applications to acoustics and audio signal analysis, which would make it a complete book for the audio practitioner. Incidentally, in Wisconsin's dry winter climate, the covers on my copy curled up rather severely.
This book is more cardboard-bound than cloth-bound. R. A. Greiner
(adapted from Audio magazine, Mar. 1989)
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