# List of Fourier-related transforms

This is a list of

linear transformation s of functions related toFourier analysis . Such transformations map a function to a set ofcoefficient s ofbasis function s, where thebasis function s are sinusoidal and are therefore strongly localized in thefrequency spectrum . (These transforms are generally designed to be invertible.) In the case of the Fourier transform, each basis function corresponds to a singlefrequency component.Applied to functions of continuous arguments, Fourier-related transforms include:

*Two-sided Laplace transform

*Mellin transform , another closely related integral transform

*Laplace transform

*Fourier transform , with special cases**:**

**Fourier series

*** When the input function/waveform is periodic, the Fourier transform output is aDirac comb function, modulated by a discrete sequence of finite-valued coefficients that are complex-valued in general. These are called**Fourier series coefficients**. The term**Fourier series**actually refers to the inverse Fourier transform, which is a sum of sinusoids at discrete frequencies, weighted by the Fourier series coefficients.

*** When the non-zero portion of the input function has finite duration, the Fourier transform is continuous and finite-valued. But a discrete subset of its values is sufficient to reconstruct/represent the portion that was analyzed. The same discrete set is obtained by treating the duration of the segment as one period of a periodic function and computing the Fourier series coefficients.

**Sine and cosine transforms **:**When the input function has odd or even symmetry around the origin, the Fourier transform reduces to a sine or cosine transform.

*Hartley transform

*Short-time Fourier transform (or short-term Fourier transform) (STFT)

*Chirplet transform

*Fractional Fourier transform (FRFT)

*Hankel transform : related to the Fourier Transform of radial functions.For usage on

computer s, 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)**:**Equivalent to the Fourier transform of a "continuous" function that is constructed from the discrete input function by using the sample values to modulate aDirac comb . The DTFT output is always a**periodic**function. An alternative viewpoint is that the DTFT is a transform to a frequency domain that is bounded (or "finite"), the length of one period.

**Fourier series , orDiscrete Fourier transform (DFT)**:**

*** When the input sequence is periodic, the (periodic) DTFT output is also aDirac comb function, modulated by the coefficients of a**Fourier series**. The coefficients can also be computed directly from the sample values (without actually doing the DTFT), in which case it is more commonly known as**DFT**. The number of discrete values in one period of the DFT is the same as in one period of the input sequence.

*** When the non-zero portion of the input sequence has finite duration, the DTFT is continuous and finite-valued. But a discrete subset of its values is sufficient to reconstruct/represent the portion that was analyzed. The same discrete set is obtained by treating the duration of the segment as one period of a periodic function and computing the Fourier series coefficients / DFT"'.

** Discretesine and cosine transforms **:**When the input sequence has odd or even symmetry around the origin, the DTFT reduces to aDiscrete sine transform (DST) orDiscrete cosine transform (DCT).

*Z-transform , a generalization of the DTFT.

*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). TheNyquist-Shannon sampling theorem is critical for understanding the output of such discrete transforms.**ee also***

Integral transform

*Wavelet transform

*Fourier transform spectroscopy

*Harmonic analysis

*List of transforms

*List of operators

*Bispectrum **References*** A. D. Polyanin and A. V. Manzhirov, "Handbook of Integral Equations", CRC Press, Boca Raton, 1998. ISBN 0-8493-2876-4

* [*http://eqworld.ipmnet.ru/en/auxiliary/aux-inttrans.htm Tables of Integral Transforms*] at EqWorld: The World of Mathematical Equations.

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