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. 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:
  • Fourier series
    • When the input function/waveform is periodic, the FT output is a Dirac comb function, modulated by a discrete sequence of finite-valued coefficients that are complex-valued in general. Fourier series refers only to the finite coefficients. The actual FT is infinite-valued at a discrete set of frequencies.
    • When the non-zero portion of the input function has finite duration, the FT 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 if it is the period of a periodic function and computing the Fourier series.
  • Sine and cosine transforms: When the input function has odd or even symmetry around the origin, the FT 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 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 (FT) of a "continuous" function that is constructed from the discrete input function by using the sample values to modulate a Dirac 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, or Discrete Fourier transform (DFT):
    • When the input sequence is periodic, the (periodic) DTFT output is also a Dirac comb function, modulated by a Fourier series. The series 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 if it is the period of a periodic function and computing the Fourier series / 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).
• Z-transform, a generalization of the DTFT, analogous to the Laplace Transform as a generalization to the continuous Fourier Transform.
• 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.

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.