This is a list of
linear transformationIn mathematics, a linear map is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. The expression "linear operator" is commonly used for linear maps from a vector space to itself...
s of
functionIn mathematics, a function is a relation between a given set of elements and another set of elements , which associates each element in the domain with exactly one element in the codomain...
s related to Fourier analysis. Such transformations
mapIn mathematics and related technical fields, the term map or mapping is often a synonym for function. Thus, for example, a partial map is a partial function, and a total map is a total function. Related terms like domain, codomain, injective, continuous, etc...
a function to a set of
coefficientIn mathematics, a coefficient is a constant multiplicative factor of a specific object. For example, in the expression 9x2, the coefficient of x2 is 9.The object can be such things as a variable, a vector, a function, etc...
s of
basis functionIn mathematics, a basis function is an element of a particular basis for a function space. Every 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 vectors.In...
s, where the
basis functionIn mathematics, a basis function is an element of a particular basis for a function space. Every 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 vectors.In...
s are
sinusoidalIn 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 spectrumFamiliar concepts associated with a frequency are colors, musical notes, radio/TV channels, and even the regular rotation of the earth. A source of light can have many colors mixed together and in different amounts . A rainbow, or prism, sends the different frequencies in different directions,...
. (These transforms are generally designed to be invertible.) In the case of the Fourier transform, each basis function corresponds to a single
frequencyFrequency 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.
Applied to functions of continuous arguments, Fourier-related transforms include:
- 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
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
In mathematics, the Laplace transform is a widely used integral transform. It has many important applications in mathematics, physics, optics, electrical engineering, control engineering, signal processing, and probability theory....
- Fourier transform
In mathematics, Fourier analysis is a subject area which grew out of the study of Fourier series. The subject began with trying to understand when it was possible to represent general functions by sums of simpler trigonometric functions...
, with special cases:
- Fourier series
In mathematics, a Fourier series decomposes a periodic function or periodic signal into a sum of simple oscillating functions, namely sines and cosines . The study of Fourier series is a branch of Fourier analysis...
- When the input function/waveform is periodic, the Fourier transform output is a Dirac comb
In mathematics, a Dirac comb is a periodic Schwartz distribution constructed from Dirac delta functionsfor some given period T...
function, modulated by a discrete sequence of finite-valued coefficients that are complex-valued in general.
This is a list of
linear transformationIn mathematics, a linear map is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. The expression "linear operator" is commonly used for linear maps from a vector space to itself...
s of
functionIn mathematics, a function is a relation between a given set of elements and another set of elements , which associates each element in the domain with exactly one element in the codomain...
s related to Fourier analysis. Such transformations
mapIn mathematics and related technical fields, the term map or mapping is often a synonym for function. Thus, for example, a partial map is a partial function, and a total map is a total function. Related terms like domain, codomain, injective, continuous, etc...
a function to a set of
coefficientIn mathematics, a coefficient is a constant multiplicative factor of a specific object. For example, in the expression 9x2, the coefficient of x2 is 9.The object can be such things as a variable, a vector, a function, etc...
s of
basis functionIn mathematics, a basis function is an element of a particular basis for a function space. Every 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 vectors.In...
s, where the
basis functionIn mathematics, a basis function is an element of a particular basis for a function space. Every 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 vectors.In...
s are
sinusoidalIn 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 spectrumFamiliar concepts associated with a frequency are colors, musical notes, radio/TV channels, and even the regular rotation of the earth. A source of light can have many colors mixed together and in different amounts . A rainbow, or prism, sends the different frequencies in different directions,...
. (These transforms are generally designed to be invertible.) In the case of the Fourier transform, each basis function corresponds to a single
frequencyFrequency 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
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
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
In mathematics, the Laplace transform is a widely used integral transform. It has many important applications in mathematics, physics, optics, electrical engineering, control engineering, signal processing, and probability theory....
- Fourier transform
In mathematics, Fourier analysis is a subject area which grew out of the study of Fourier series. The subject began with trying to understand when it was possible to represent general functions by sums of simpler trigonometric functions...
, with special cases:
- Fourier series
In mathematics, a Fourier series decomposes a periodic function or periodic signal into a sum of simple oscillating functions, namely sines and cosines . The study of Fourier series is a branch of Fourier analysis...
- When the input function/waveform is periodic, the Fourier transform output is a Dirac comb
In mathematics, a Dirac comb is a periodic Schwartz distribution constructed from Dirac delta functionsfor some given period T...
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
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
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
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
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
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
In mathematics, the Hankel transform of order ν of a function f is given by:where Jν is the Bessel function of the first kind of order ν with ν ≥ −1/2...
: related to the Fourier Transform of radial functions.
Discrete transforms
For usage on
computerA computer is a machine that manipulates data according to a set of instructions.Although mechanical examples of computers have existed through much of recorded human history, the first electronic computers were developed in the mid-20th century . These were the size of a large room, consuming as...
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
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 combIn mathematics, a Dirac comb is a periodic Schwartz distribution constructed from Dirac delta functionsfor some given period T...
. 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
In mathematics, a Fourier series decomposes a periodic function or periodic signal into a sum of simple oscillating functions, namely sines and cosines . The study of Fourier series is a branch of Fourier analysis...
, or Discrete Fourier transformIn mathematics, the discrete Fourier transform is a specific kind of Fourier 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 (periodic) DTFT output is also a Dirac comb
In mathematics, a Dirac comb is a periodic Schwartz distribution constructed from Dirac delta functionsfor some given period T...
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.
- Discrete 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 transformIn 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 transformA discrete cosine transform expresses a sequence of finitely many data points in terms of a sum of cosine functions oscillating at different frequencies...
(DCT).
- 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.It can be considered as a discrete equivalent of the Laplace transform...
, a generalization of the DTFT.
- 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
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).
- Hadamard transform
The Hadamard transform is an example of a generalized class of Fourier transforms...
(Walsh functionIn 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 usage of all of these transforms is greatly facilitated by the existence of efficient algorithms based on a
fast Fourier transformA fast Fourier transform is an efficient algorithm to compute the discrete Fourier transform and its inverse. There are many distinct FFT algorithms involving a wide range of mathematics, from simple complex-number arithmetic to group theory and number theory; this article gives an overview of...
(FFT). The Nyquist-Shannon sampling theorem is critical for understanding the output of such discrete transforms.
See also
- Integral transform
In mathematics, an integral transform is any transform T of the following form:The input of this transform is a function f, and the output is another function Tf. An integral transform is a particular kind of mathematical operator....
- Wavelet transform
- 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 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
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...