Fast Transforms in Audio Signal Processing

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 [59,18].)

An exception is when processing exactly periodic signals where the period is known to be an exact integer number of samples in length. (This can be arranged using pitch detection and resampling of the periodic signal. A time-varying pitch requires time-varying resampling [53].) In such a case, the DFT of one period of the waveform can be interpreted as a Fourier series of the periodic waveform, and unlike virtually all other practical spectrum analysis scenarios, spectral interpolation is not needed (or wanted).

Adaptive FFT software (see §A.4 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 [20].