List of Fourier-related transforms

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This is a list of linear transformations of functions related to Fourier analysis. 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:

• Two-sided Laplace transform, a generalization of the continuous Fourier transform
• Mellin transform, another closely related integral transform
• Laplace transform
• Fourier transform (or FT), with special cases:
  • Sine and cosine transforms (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, number theory and algebra, discrete arguments (e.g. functions of a series of discrete samples) are often more appropriate, and are handled by the transforms (analogous to the continuous cases above):

• Discrete-time Fourier transform (DTFT), the Fourier transform of a discrete sequence or, equivalently, the continuous Fourier Transform of continuous-time function consisting of a sequence of impulses.
• Z-transform, a generalization of the DTFT, analogous to the Laplace Transform as a generalization to the continuous Fourier Transform.
• Discrete Fourier transform (DFT), the Fourier transform of a discrete periodic sequence (yielding discrete periodic frequencies). (It can also be thought of as the DTFT of a finite-length sequence evaluated at discrete frequencies.)
  • Discrete cosine transform (DCT)
  • Discrete sine transform (DST)
• Modified discrete cosine transform (MDCT)
• Discrete Hartley transform (DHT)
• Also the discretized STFT (see above).
• Hadamard transform (Walsh function).

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.

[edit] See also

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

[edit] References

• A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equations, CRC Press, Boca Raton, 1998. ISBN 0-8493-2876-4
• Tables of Integral Transforms at EqWorld: The World of Mathematical Equations.