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 [74,21].)

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 series (§B.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 [23].

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.