List of Fourier-related transforms - Definition and Overview

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:

• Laplace transform, a related continuous-time transform
• Continuous Fourier transform (often, just Fourier transform or FT), with special cases:
  • Cosine transform and sine transform (for functions of even/odd symmetry)
  • Fourier series (for periodic functions)
• Hartley transform
• Short-time Fourier transform (or short-term Fourier transform) (STFT)
• Chirplet transform
• Fractional Fourier transform (FRFT)

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):

• Z-transform, a more general transform of which the DFT is a special case
• Discrete Fourier transform (DFT), with special cases:
  • Discrete cosine transform (DCT)
  • Discrete sine transform (DST)
• Modified discrete cosine transform (MDCT)
• Discrete Hartley transform (DHT)
• Also the discretized STFT (see above).
• Discrete-time Fourier transform (DTFT), as the DFT but for functions with an infinite domain (and thus a continuous, periodic spectrum).

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

See also

• Integral transform
• Wavelet transform
• Fourier transform spectroscopy
• Harmonic analysis
• List of transforms