Customer Reviews

Fast Fourier Transform and Its Applications

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Fourier transforms used to scare me. I got brain-lock, and couldn't even use them effectively in a package like Matlab or IDL. I got lost in integrals and delta functions and found myself rapidly sinking into mathematical quicksand. Evoking qliphotic demons from within a pentagram of power would have been simpler and less arcane.

Then along came Brigham. Although his book had all the gnarly math of any other Fourier transform explanation I had ever seen, he also drew diagrams--diagrams which allowed me to "get" what the language of mathematics had so clearly expressed. All of a sudden the integrals were tamed. I wasn't in quicksand, just a damp sidewalk at Adventureland, waiting for the Jungle Cruise.

And that was just the first couple of chapters! Brigham quickly moved into transform theory, applying the Fourier integral to convolution and correlation. Then into sampled waveforms and the discrete Fourier transform and its applications.

Finally, he presented the Fast Fourier Transform. Once again, he clarifies without obfuscating. I found the FFT moving from the hyper-arcane to the land of "Well, duh!" (Beware: The actual FFT code included is not particularly efficient. Find source code for implementation *elsewhere*.) He extends the FFT to convolution and correlation, as well as to two dimensions. He doesn't skimp on applications, either. He clarifies interferometry, time-difference-of-arrival, power spectrum analysis, and beamforming.

If you're not a signal processing wonk already, read this book. You may find it a powerful cure for DSP-phobia.

I used the book to teach a Fourier Transform course for fellow engineers at Lockheed Martin. I taught straight from the book and only included some of my own personal FFT applications. The book gives a development of Fourier Transform & Series Fourier Transform, along with practical knowledge on how to apply them, and also gives numerous application at various level of difficulty. This is my favorite FFT reference. I'm still studying some of the applications in the last three chapters. I like the author's figures for showing the convolution integral using Fourier Transforms. All the concepts were shown graphical with figures. The publisher needs to print this book in hard back since I used my paper back edition so much that its coming apart.

I also would recommend "Fourier-Related Transforms, Fast Algorithms and Applications" by Norman Morrison and "Introduction to Fourier Analysis" by Okan Ersoy as good academic references but lacked the application focus I was looking for.

I'm delighted to see that Brigham's book is in print. (I bought my first copy 20 years ago!) I'm planning to use as part of a course on signal processing. I've read several books on FT's and this is the best. It is one of the few technical books whose presentation is genuinely complete and clear. For example, the relationships between the discrete and the continuous FT, the role of the convolution integral, and the Nyquist theorem are presented logically with both algebraic proofs and graphical explanations. I'm always on the look out for monographs that focus on one subject and make you feel like you've become an expert on it when you're through. This fits the bill.

This is an excellent, excellent text covering continuous and discrete-time signals/systems. Yes, it's ultimate focus is on the FFT, but about 50% of the content is basic continuous- and discrete-time signal analysis concepts, explained thoroughly, with superb graphic examples, and textual information from an obviously talented thinker, writer, communicator. It is extremely comprehensive but written 'patiently', bordering on the active voice style, which seems to be in short supply these days. If you had that well known and universally despised "S&S" textbook in college, torch it, buy this text, clean the slate, start over. There's no need to live with spectrophobia. This text has none of the usual inward, regurgitated extra-academic nonsense, the author clearly set out to write a fresh and original book. Read it slowly, the examples are vivid. You'll attain a visual feel for just what a linear transformation is, what the Fourier Transform is, and you'll develop the necessary intuition to allow you to be able to imagine both time and frequency domain relationships in a qualitative, systems oriented way (and this is where you want to be, trust me). The sections on convolution, correlation, discrete sampling, DFTs, finite-length effects, well, they're just perfect, with numerous mind-altering graphic examples. Many of these examples show side-by-side what happens in time and frequency domains simultaneously. It's clear that the author, unlike so many others, had a genuine interest in teaching and communicating. There's plenty of rigorous stuff if you want to go in deep, and I recommend it. The FFT sections are from the ground up - after a time you'll be able to design your own FT strategies and implement them with confidence. The best engineering text on the planet. Good Luck!

I am mainly a neurobiologist by training, but transform theory is very useful in visual neurobiology and visual psychophysics, and I've looked at and read dozens of books on various aspects of signal processing and transform theory. Much of this is surprisingly applicable to the brain science area, as the revolutionary work of David Marr and other scientists showed a quarter of a century ago.

But getting back to the present book, this is one of the best books I've read in the signal analysis area. Brigham's presentation of various aspects of the FT, including the continuous FT, digital FT, convolution integrals, and so on, is clear and concise, whether he's discussing theory or applications. Also, his disussion of the Nyquist sampling theorem is the best and easiest to understand I've read.

Interestingly, this theorem has quite practical applications, not just in digital sample theory, but in real life. According to the Nyquist theorem, no information is lost in converting from analog to digital form if the sampling frequency is twice that of the highest frequency in the signal. Well, have you ever used those audio headphones they have on commercial jetliners? The Nyquist theorem means they can switch the audio outputs at high frequency using well-known time-domain switching techniques rather than run copper to each passenger's seat. I've read that this saves 300 pounds of copper wire in a typical plane, the weight savings of which can of course be more profitably used for transporting other things. If you consider that 300 pounds is about the weight of your average couple, you can see how the savings would add up after even a few flights.

But getting back to the book, I first encountered this work 20 years ago, and I'm delighted to see it's still around. My neurobiology student friends and I all got turned on to this book early in grad school, and it was a great help. I still fondly recall the many enjoyable hours we spent talking about how Fourier analysis and other techniques such as the Laplace transform, temporal modulation transfer fuction, two-dimensional convolution integrals, two-dimensional point-spread functons, filter techniques such as the finite and infinite response filters, Kalman filter, Hammond filter, and so on, could be applied to our area. And it all started with this book. It's a true classic that has never been exceeded for it's clear and concise exposition of a very important mathematical tool in both engineering and neurophysiology.

The book I have is ISBN 0-13-307496. It was published in 1974.

I am very happy about this book, I first read it in 1979 when I was 19, and I found it really marvellous. I agree with the other reviewers, but I must add a note of caution - the edition I have contains some errors. They are as follows;

p155, p157 - the factors w(11) and w(10) are incorrectly placed on the butterfly diagram 10.3 and 10.4 respectively,

p166, p168, p169, equations 10-26 and most equations following to the end of the chapter - the factors R(N-n) and I(N-n) should be R(N-1-n) and I(N-1-n), respectively.

I hope I'm right about this, but the convention is that the indices are from 0 to N-1, and therefore if n=0, then N-n is N - which not an allowed index.

Apart from these sort of errors (I havn't been through the whole book with a fine toothcomb), its really very good, actually extraordinarily clear.

One of its main benefits is that it doesn't veer away from the FFT to the very complicated developments such as fractional transforms and other developments which might confuse the sort of audience it's aimed at (which is definitely the graduates).

But if you want to look deeply into FFTs for a real application you will need a lot more. I must mention,for instance, that the implementation of an FFT needs fairly careful error propagation and rounding analysis, and this isn't covered at all in the book. Neither are prime factor FFTs. In fact the chapter "FFT algorithms for arbirary factors" is only a method of factoring into powers of two, and certainly not the prime factor decomposition which was developed later by Winograd, Chuo, and others.

It must also be said that while the DCT is practically a kissing cousin of the FFT, this naturally isn't covered in this text... but neither are the finite field implementations that are now taking many peoples imaginations to faster and faster FFTS.

Also, there are jolly useful things to know about, such as the FFT when you only need a subset of the output data points. There are pruning algorithms which greatly simplify the computations.

But it's very good as a starter, I wouldn't do without my copy!

Fourier transforms elegance and its vivid applications are brought out excellently by the author. It was wonderful the way the author let the pictures speak. They are alone worth the price, but wait you ask "Is it not that Engineer's refrence look tough with lot of equations ?" . Well this is not a typical engineers book. It is a right brained approach to a decidedly left brained topic.

This is stimulating and scintillating overview of Fourier Transform. When he shows how a periodic function could be explained with a convultion with an impulse or when he draws the nyquist theorm out of equations, he makes maths an art form.

An excellent work by an artist on Fourier Transform. A must have book for all DSP engineers.

This book is not just another terse math or signal processing book. It tries to provide an alternative to standard DSP techniques that develop the FFT adequately enough, but show nothing about applications and have the student believing that "The Butterfly Element" is something he can buy an armful of at an electronics store and assemble. All developments in this book use graphical techniques and examples that insure clarity in the presentation. The book provides not only a readable introduction to the FFT but a thorough and unified reference for applying it to various fields of interest. It is great for self-study. The text is divided into five major subject areas:

1. The Fourier Transform and its properties
2. The Discrete Fourier Transform - It is developed from the continuous Fourier Transform both graphically and theoretically. Its properties are examined as are numerous waveform classes via illustrative examples. Discrete convolution and correlation are defined and compared with the continuous equivalents via examples.
3. The FFT - The FFT algorithm is developed along with an explanation of why the FFT is efficient. Computer programs are developed that can calculate the FFT.
4. Basic Applications of the FFT - Presents the application of the FFT to the computation of discrete and inverse discrete Fourier transforms. There is an emphasis on graphical examination of resolution and common FFT user mistakes such as aliasing, time domain truncation, noncausal time functions, and periodic functions. The applications examined include Laplace transform computation, discrete convolution and correlation, and two-dimensional Fourier transform convolution. Computer programs are provided.
5. Signal Processing and System FFT Applications - The design and application of digital filters using the FFT are explored. A novel application of the FFT to multichannel band-pass filtering is developed in a way that can readily be expanded by the reader.

I highly recommend this book to readers who want a complete explanation and investigation of the FFT and its applications that is clear enough for self-study.

This book is great, its concepts are in both mathmatical and intuitive forms. The graphical approach makes it easy to understand yet you still get the equations for actual implementations. The second half of the book is all about examples/applications that are applicable towards all disciplines. Excellent choice for understanding of the FFT!

I purchased the first edition of this book way back on January 10, 1975, when I was young design engineer, just breaking into the Digital Signal Processing business. I bought the book at the Stanford University Book store for a whopping $19.95. The receipt is still in the book.

The book was considered to be a DSP industry bible back then, and in my opinion, it is still the best DFT/FFT book on the market today.

Mr. Brigham seems to be very detailed oriented. He methodically progresses from one subject to the next and explains each topic in a clear and concise manner. The book is loaded with extremely detailed graphics that give the reader a very good picture of the operation, properties, and mechanics of the Discrete Fourier Transform. His mathematical derivations of various DFT/FFT properties are straight forward and can be read and understood in short order.

I consider Mr. Brigham's book to be an essential engineering resource. I have relied upon, and utilized the information provided in this book for over 30 years of successful DSP design.

If I ever misplace my time worn book, would I purchase it again? The answer is a definite yes. The money is well spent.