List of Fourier-related transforms

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.

Continuous transforms

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 a Dirac 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)
  • Rectangular mask short-time Fourier transform
• Chirplet transform
• Fractional Fourier transform (FRFT)
• Hankel transform: related to the Fourier Transform of radial functions.

Discrete transforms

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): Equivalent to the Fourier transform of a "continuous" function that is constructed from the discrete input function by using the sample values to modulate a Dirac comb. When the sample values are derived by sampling a function on the real line, ƒ(x), the DTFT is equivalent to a periodic summation of the Fourier transform of ƒ. The DTFT output is always periodic (cyclic). An alternative viewpoint is that the DTFT is a transform to a frequency domain that is bounded (or finite), the length of one cycle.
  • discrete Fourier transform (DFT):
    • When the input sequence is periodic, the DTFT output is also a Dirac comb function, modulated by the coefficients of a Fourier series^[1] which can be computed as a DFT of one cycle of the input sequence. The number of discrete values in one cycle of the DFT is the same as in one cycle 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 cycle of a periodic function and computing the DFT.
  • Discrete sine and cosine transforms: When the input sequence has odd or even symmetry around the origin, the DTFT reduces to a discrete sine transform (DST) or discrete cosine transform (DCT).
    • Regressive discrete Fourier series, in which the period is determined by the data rather than fixed in advance.
  • Discrete chebyshev transforms (on the 'roots' grid and the 'extrema' grid of the chebyshev polynomials of the first kind). This transform is of much importance in the field of spectral methods for solving differential equations because it can be used to swiftly and efficient go from grid point values to chebyshev series coefficients.
• Generalized DFT (GDFT), a generalization of the DFT and constant modulus transforms where phase functions might be of linear with integer and real valued slopes, or even non-linear phase bringing flexibilities for optimal designs of various metrics, e.g. auto- and cross-correlations.
• Discrete-space Fourier transform (DSFT) is the generalization of the DTFT from 1D signals to 2D signals. It is called "discrete-space" rather than "discrete-time" because the most prevalent application is to imaging and image processing where the input function arguments are equally spaced samples of spatial coordinates $(x,y)$. The DSFT output is is periodic in both variables.
• 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 use 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.

Notes

1. The Fourier series represents $\scriptstyle \sum_{n=-\infty}^{\infty}f(nT)\cdot \delta(t-nT),\,$ where T is the interval between samples.

See 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
• Tables of Integral Transforms at EqWorld: The World of Mathematical Equations.
• A.N. Akansu and H. Agirman-Tosun, "Generalized Discrete Fourier Transform With Nonlinear Phase", IEEE Transactions on Signal Processing, vol. 58, no. 9, pp. 4547-4556, Sept. 2010.