List of Fourier-related transforms

List of Fourier-related transforms

This is a list of linear transformations of functions related to the Fourier transform. Such transformations map a function to a set of coefficients of basis functions, where the basis functions are sinusoidal and are therefore strongly localized in the frequency spectrum. (These transforms are generally designed to be invertible.) In the case of the Fourier transform, each basis function corresponds to a single frequency component.

Applied to functions of continuous arguments, Fourier-related transforms include:

Fourier transform (FT), with special cases:
- Cosine transform and sine transform (for functions of even/odd symmetry)
- Fourier series (for periodic functions)
Hartley transform

For usage on computers, discrete arguments (e.g. functions of a series of discrete samples) are more appropriate, and are handled by the transforms (analogous to the continuous cases above):

Discrete Fourier transform (DFT), with special cases:
- Discrete cosine transform (DCT)
- Discrete sine transform (DST)
Modified discrete cosine transform (MDCT)
Discrete Hartley transform (DHT)
Short term Fourier transform (or short-time Fourier transform) (STFT)

The usage of all of these transforms is greatly facilitated by the existence of fast algorithms based on the fast Fourier transform (FFT). The Nyquist-Shannon sampling theorem is critical for understanding the output of such discrete transforms.