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Fast Fourier Transform (FFT) Algorithms

The term fast Fourier transform (FFT) refers to an efficient implementation of the discrete Fourier transform (DFT) for highly compositeA.1 transform lengths $ N$ . When computing the DFT as a set of $ N$ inner products of length $ N$ each, the computational complexity is $ {\cal O}(N^2)$ . When $ N$ is an integer power of 2, a Cooley-Tukey FFT algorithm delivers complexity $ {\cal O}(N\lg N)$ , where $ \lg N$ denotes the log-base-2 of $ N$ , and $ {\cal O}(x)$ means ``on the order of $ x$ ''. Such FFT algorithms were evidently first used by Gauss in 1805 [31] and rediscovered in the 1960s by Cooley and Tukey [17].

In this appendix, a brief introduction is given for various FFT algorithms. A tutorial review (1990) is given in [23]. Additionally, there are some excellent FFT ``home pages'':

Pointers to FFT software are given in §A.7.

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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8.
Copyright © 2017-09-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University