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Fast Transforms in Audio DSP

Since most audio signal processing applications benefit from zero padding (see §8.1), in which case we can always choose the FFT length to be a power of 2, there is almost never a need in practice for more ``exotic'' FFT algorithms than the basic ``pruned'' power-of-2 algorithms. (Here ``pruned'' means elimination of all unnecessary operations, such as when the input signal is real [77,22].)

An exception is when processing exactly periodic signals where the period is known to be an exact integer number of samples in length.A.8 In such a case, the DFT of one period of the waveform can be interpreted as a Fourier seriesB.3) of the periodic waveform, and unlike virtually all other practical spectrum analysis scenarios, spectral interpolation is not needed (or wanted). In the exactly periodic case, the spectrum is truly zero between adjacent harmonic frequencies, and the DFT of one period provides spectral samples only at the harmonic frequencies.

Adaptive FFT software (see §A.7 below) will choose the fastest algorithm available for any desired DFT length. Due to modern processor architectures, execution time is not normally minimized by minimizing arithmetic complexity [24].

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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 © 2014-10-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University