List of Fourier-related transforms

# List of Fourier-related transforms

Discussion
 Ask a question about 'List of Fourier-related transforms' Start a new discussion about 'List of Fourier-related transforms' Answer questions from other users Full Discussion Forum

Encyclopedia
This is a list of linear transformation
Linear transformation
In mathematics, a linear map, linear mapping, linear transformation, or linear operator is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. As a result, it always maps straight lines to straight lines or 0...

s of function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

s related to Fourier analysis. Such transformations map
Map (mathematics)
In most of mathematics and in some related technical fields, the term mapping, usually shortened to map, is either a synonym for function, or denotes a particular kind of function which is important in that branch, or denotes something conceptually similar to a function.In graph theory, a map is a...

a function to a set of coefficient
Coefficient
In mathematics, a coefficient is a multiplicative factor in some term of an expression ; it is usually a number, but in any case does not involve any variables of the expression...

s of basis function
Basis function
In mathematics, a basis function is an element of a particular basis for a function space. Every continuous function in the function space can be represented as a linear combination of basis functions, just as every vector in a vector space can be represented as a linear combination of basis...

s, where the basis functions are sinusoidal
Trigonometric function
In mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle...

and are therefore strongly localized in the frequency spectrum
Frequency spectrum
The frequency spectrum of a time-domain signal is a representation of that signal in the frequency domain. The frequency spectrum can be generated via a Fourier transform of the signal, and the resulting values are usually presented as amplitude and phase, both plotted versus frequency.Any signal...

. (These transforms are generally designed to be invertible.) In the case of the Fourier transform, each basis function corresponds to a single frequency
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...

component.

## Continuous transforms

Applied to functions of continuous arguments, Fourier-related transforms include:
• Two-sided Laplace transform
Two-sided Laplace transform
In mathematics, the two-sided Laplace transform or bilateral Laplace transform is an integral transform closely related to the Fourier transform, the Mellin transform, and the ordinary or one-sided Laplace transform...

• Mellin transform
Mellin transform
In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform...

, another closely related integral transform
• Laplace transform
• Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...

, with special cases:
• Fourier series
Fourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...

• When the input function/waveform is periodic, the Fourier transform output is a Dirac comb
Dirac comb
In mathematics, a Dirac comb is a periodic Schwartz distribution constructed from Dirac delta functions...

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
Sine and cosine transforms
In mathematics, the Fourier sine and cosine transforms are special cases of thecontinuous Fourier transform, arising naturally when attempting to transform odd and even functions, respectively.The general Fourier transform is defined as:...

: When the input function has odd or even symmetry around the origin, the Fourier transform reduces to a sine or cosine transform.
• Hartley transform
Hartley transform
In mathematics, the Hartley transform is an integral transform closely related to the Fourier transform, but which transforms real-valued functions to real-valued functions. It was proposed as an alternative to the Fourier transform by R. V. L. Hartley in 1942, and is one of many known...

• Short-time Fourier transform
Short-time Fourier transform
The short-time Fourier transform , or alternatively short-term Fourier transform, is a Fourier-related transform used to determine the sinusoidal frequency and phase content of local sections of a signal as it changes over time....

(or short-term Fourier transform) (STFT)
• Chirplet transform
Chirplet transform
In signal processing, the chirplet transform is an inner product of an input signal with a family of analysis primitives called chirplets.-Similarity to other transforms:...

• Fractional Fourier transform
Fractional Fourier transform
In mathematics, in the area of harmonic analysis, the fractional Fourier transform is a linear transformation generalizing the Fourier transform. It can be thought of as the Fourier transform to the n-th power where n need not be an integer — thus, it can transform a function to an...

(FRFT)
• Hankel transform
Hankel transform
In mathematics, the Hankel transform expresses any given function f as the weighted sum of an infinite number of Bessel functions of the first kind Jν. The Bessel functions in the sum are all of the same order ν, but differ in a scaling factor k along the r-axis...

: related to the Fourier Transform of radial functions.

## Discrete transforms

For usage on computer
Computer
A computer is a programmable machine designed to sequentially and automatically carry out a sequence of arithmetic or logical operations. The particular sequence of operations can be changed readily, allowing the computer to solve more than one kind of problem...

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
Discrete-time Fourier transform
In mathematics, the discrete-time Fourier transform is one of the specific forms of Fourier analysis. As such, it transforms one function into another, which is called the frequency domain representation, or simply the "DTFT", of the original function . But the DTFT requires an input function...

(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
Dirac comb
In mathematics, a Dirac comb is a periodic Schwartz distribution constructed from Dirac delta functions...

. 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
Discrete Fourier transform
In mathematics, the discrete Fourier transform is a specific kind of discrete transform, used in Fourier analysis. It transforms one function into another, which is called the frequency domain representation, or simply the DFT, of the original function...

(DFT):
• When the input sequence is periodic, the DTFT output is also a Dirac comb
Dirac comb
In mathematics, a Dirac comb is a periodic Schwartz distribution constructed from Dirac delta functions...

function, modulated by the coefficients of a Fourier series 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
Sine and cosine transforms
In mathematics, the Fourier sine and cosine transforms are special cases of thecontinuous Fourier transform, arising naturally when attempting to transform odd and even functions, respectively.The general Fourier transform is defined as:...

: When the input sequence has odd or even symmetry around the origin, the DTFT reduces to a discrete sine transform
Discrete sine transform
In mathematics, the discrete sine transform is a Fourier-related transform similar to the discrete Fourier transform , but using a purely real matrix...

(DST) or discrete cosine transform
Discrete cosine transform
A discrete cosine transform expresses a sequence of finitely many data points in terms of a sum of cosine functions oscillating at different frequencies. DCTs are important to numerous applications in science and engineering, from lossy compression of audio and images A discrete cosine transform...

(DCT).
• Regressive discrete Fourier series
Regressive discrete fourier series
In applied mathematics, the regressive discrete Fourier series is a generalization of the discrete Fourier transform where the Fourier series coefficients are computed in a least squares sense and the period is arbitrary, i.e., not necessarily equal to the length of the data. It was first...

, in which the period is determined by the data rather than fixed in advance.
• Z-transform
Z-transform
In mathematics and signal processing, the Z-transform converts a discrete time-domain signal, which is a sequence of real or complex numbers, into a complex frequency-domain representation....

, a generalization of the DTFT.
• Modified discrete cosine transform
Modified discrete cosine transform
The modified discrete cosine transform is a Fourier-related transform based on the type-IV discrete cosine transform , with the additional property of being lapped: it is designed to be performed on consecutive blocks of a larger dataset,...

(MDCT)
• Discrete Hartley transform
Discrete Hartley transform
A discrete Hartley transform is a Fourier-related transform of discrete, periodic data similar to the discrete Fourier transform , with analogous applications in signal processing and related fields. Its main distinction from the DFT is that it transforms real inputs to real outputs, with no...

(DHT)
• Also the discretized STFT (see above).
The Hadamard transform is an example of a generalized class of Fourier transforms...

(Walsh function
Walsh function
In mathematical analysis, the set of Walsh functions form an orthogonal basis of the square-integrable functions on the unit interval. The functions take the values -1 and +1 only, on sub-intervals defined by dyadic fractions...

).

The use of all of these transforms is greatly facilitated by the existence of efficient algorithms based on a fast Fourier transform
Fast Fourier transform
A fast Fourier transform is an efficient algorithm to compute the discrete Fourier transform and its inverse. "The FFT has been called the most important numerical algorithm of our lifetime ." There are many distinct FFT algorithms involving a wide range of mathematics, from simple...

(FFT). The Nyquist–Shannon sampling theorem is critical for understanding the output of such discrete transforms.

• Integral transform
• Wavelet transform
• Fourier transform spectroscopy
Fourier transform spectroscopy
Fourier transform spectroscopy is a measurement technique whereby spectra are collected based on measurements of the coherence of a radiative source, using time-domain or space-domain measurements of the electromagnetic radiation or other type of radiation....

• Harmonic analysis
Harmonic analysis
Harmonic analysis is the branch of mathematics that studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms...

• List of transforms
• List of operators
• Bispectrum
Bispectrum
In mathematics, in the area of statistical analysis, the bispectrum is a statistic used to search for nonlinear interactions. The Fourier transform of the second-order cumulant, i.e., the autocorrelation function, is the traditional power spectrum...