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Why a DFT is usually called an FFT in practice

Practical implementations of the DFT are usually based on one of the Cooley-Tukey ``Fast Fourier Transform'' (FFT) algorithms [17].8.1 For this reason, the matlab DFT function is called `fft', and the actual algorithm used depends primarily on the transform length $ N$ .8.2 The fastest FFT algorithms generally occur when $ N$ is a power of 2. In practical audio signal processing, we routinely zero-pad our FFT input buffers to the next power of 2 in length (thereby interpolating our spectra somewhat) in order to enjoy the power-of-2 speed advantage. Finer spectral sampling is a typically welcome side benefit of increasing $ N$ to the next power of 2. Appendix A provides a short overview of some of the better known FFT algorithms, and some pointers to literature and online resources.


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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 © 2014-04-06 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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