Fourier analysis, named after Joseph Fourier's introduction of the Fourier series, is the decomposition of a function in terms of a sum of sinusoidal basis functions (vs. their frequencies) that can be recombined to obtain the original function. That process of recombining the sinusoidal basis functions is also called Fourier synthesis (in which case Fourier analysis refers specifically to the decomposition process). In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ... In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ... The Fourier series is a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ... In mathematics, the discrete Fourier transform (DFT), occasionally called the finite Fourier transform, is a transform for Fourier analysis of finite-domain discrete-time signals. ... Given a discrete set of real or complex numbers: (integers), the discrete-time Fourier transform (or DTFT) of is: // Its name implies that the {x[n]} sequence represents the values (aka samples) of a continuous-time function, , at discrete moments in time: , where is the sampling interval (in seconds), and... This is a list of linear transformations of functions related to the Fourier transform. ... Jean Baptiste Joseph Fourier (March 21, 1768 - May 16, 1830) was a French mathematician and physicist who is best known for initiating the investigation of Fourier series and their application to problems of heat flow. ... The Fourier series is a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ... In mathematics, the trigonometric functions (also called circular functions) are functions of an angle. ... In functional analysis and its applications, a function space can be viewed as a vector space of infinite dimension whose basis vectors are functions not vectors. ... FreQuency is a music video game developed by Harmonix and published by SCEI. It was released in November 2001. ...

The linear operation that transforms a function into the coefficients of the sinusoidal basis functions is called a Fourier transform in general. However, the transform is usually given a more specific name depending upon the domain and other properties of the function being transformed, as described below. Moreover, the original concept of Fourier analysis has been extended over time to apply to more and more abstract and general situations, and the general field is often known as harmonic analysis. See also: List of Fourier-related transforms. In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ... Harmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. ... This is a list of linear transformations of functions related to the Fourier transform. ...

Applications

Fourier analysis has many scientific applications — in physics, number theory, combinatorics, signal processing, probability theory, statistics, option pricing, cryptography, acoustics, oceanography, optics and diffraction, geometry, and other areas. (In signal processing and related fields, Fourier analysis is typically thought of as decomposing a signal into its component frequencies and their amplitudes.) This wide applicability stems from many useful properties of the transforms: A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... Combinatorics is a branch of pure mathematics concerning the study of discrete (and usually finite) objects. ... Signal processing is the processing, amplification and interpretation of signals, and deals with the analysis and manipulation of signals. ... Probability theory is the branch of mathematics concerned with analysis of random phenomena. ... This article is about the field of statistics. ... It has been suggested that this article or section be merged into option. ... The German Lorenz cipher machine, used in World War II for encryption of very high-level general staff messages Cryptography (or cryptology; derived from Greek κρυπτός kryptós hidden, and the verb γράφω gráfo write or λεγειν legein to speak) is the study of message secrecy. ... Acoustics is a branch of physics and is the study of sound (mechanical waves in gases, liquids, and solids). ... Thermohaline circulation Oceanography (from Ocean + Greek γράφειν = write), also called oceanology or marine science, is the branch of Earth Sciences that studies the Earths oceans and seas. ... For the book by Sir Isaac Newton, see Opticks. ... The intensity pattern formed on a screen by diffraction from a square aperture Diffraction refers to various phenomena associated with wave propagation, such as the bending, spreading and interference of waves passing by an object or aperture that disrupts the wave. ... Calabi-Yau manifold Geometry (Greek γεωμετρία; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. ... FreQuency is a music video game developed by Harmonix and published by SCEI. It was released in November 2001. ... It has been suggested that pulse amplitude be merged into this article or section. ...

• The transforms are linear operators and, with proper normalization, are unitary as well (a property known as Parseval's theorem or, more generally, as the Plancherel theorem, and most generally via Pontryagin duality).
• The transforms are invertible, and in fact the inverse transform has almost the same form as the forward transform.
• The exponential basis functions are eigenfunctions of differentiation, which means that this representation transforms linear differential equations with constant coefficients into ordinary algebraic ones. (For example, in a linear time-invariant physical system, frequency is a conserved quantity, so the behavior at each frequency can be solved independently.)
• By the convolution theorem, Fourier transforms turn the complicated convolution operation into simple multiplication, which means that they provide an efficient way to compute convolution-based operations such as polynomial multiplication and multiplying large numbers.
• The discrete version of the Fourier transform (see below) can be evaluated quickly on computers using fast Fourier transform (FFT) algorithms.

In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... In functional analysis, a unitary operator is a bounded linear operator U on a Hilbert space satisfying U*U=UU*=I where I is the identity operator. ... In mathematics, Parsevals theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform. ... In mathematics, the Plancherel theorem is a result in harmonic analysis, first proved by Michel Plancherel. ... In mathematics, in particular in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform. ... In mathematics, an eigenfunction f of a linear operator A on a function space is an eigenvector of the linear operator; it satisfies for some scalar λ, the corresponding eigenvalue. ... For a non-technical overview of the subject, see Calculus. ... A simulation of airflow into a duct using the Navier-Stokes equations A differential equation is a mathematical equation for an unknown function of one or several variables which relates the values of the function itself and of its derivatives of various orders. ... In mathematics, constant coefficients is a term applied to differential operators, and also some difference operators, to signify that they contain no functions of the independent variables, other than constant functions. ... FreQuency is a music video game developed by Harmonix and published by SCEI. It was released in November 2001. ... In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution is the point-wise product of Fourier transforms. ... In mathematics and, in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version of g. ... In mathematics, a polynomial is an expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ... A multiplication algorithm is an algorithm (or method) to multiply two numbers. ... In mathematics, the discrete Fourier transform (DFT), occasionally called the finite Fourier transform, is a transform for Fourier analysis of finite-domain discrete-time signals. ... The Fast Fourier Transform (FFT) is an efficient algorithm to compute the discrete Fourier transform (DFT) and its inverse. ...

Variants of Fourier analysis

There exists a number of transforms that can be seen as special cases or generalizations of each other. The most popular are summarized in the following table:

Name	Time domain	Frequency domain	Formula
(Continuous) Fourier transform	Continuous, Aperiodic	Continuous, Aperiodic
Fourier series	Continuous, Periodic	Discrete, Aperiodic
Discrete-time Fourier transform	Discrete, Aperiodic	Continuous, Periodic
Discrete Fourier transform	Discrete, Periodic	Discrete, Periodic
Generalization	arbitrary locally compact abelian topological group	arbitrary locally compact abelian topological group	See harmonic analysis

The table shows that In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ... The Fourier series is a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ... Given a discrete set of real or complex numbers: (integers), the discrete-time Fourier transform (or DTFT) of is: // Its name implies that the {x[n]} sequence represents the values (aka samples) of a continuous-time function, , at discrete moments in time: , where is the sampling interval (in seconds), and... In mathematics, the discrete Fourier transform (DFT), occasionally called the finite Fourier transform, is a transform for Fourier analysis of finite-domain discrete-time signals. ... In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. ... In mathematics, an abelian group, also called a commutative group, is a group (G, * ) such that a * b = b * a for all a and b in G. In other words, the order in which the binary operation is performed doesnt matter. ... In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. ... Harmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. ...

• discreteness in one domain implies periodicity in the opposite transformed domain and the converse is true.
• continuity in one domain implies aperiodicity in the transformed domain and the converse is true.
• pure reality in one domain (i.e. imaginary part is zero everywhere) implies conjugate symmetry in the transformed domain and the converse is true.

There are various ways to include normalization factors; For simplicity, this table uses the definitions without them. Also, this table does not contain multi-dimensional versions. The concept of a normalizing constant arises in probability theory and a variety of other areas of mathematics. ...

Fourier transform

Main article: Fourier transform

Most often, the unqualified term Fourier transform refers to the transform of functions of a continuous real argument, representing any square-integrable function as a linear combination of complex exponentials with frequencies : In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ... In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, the term integrable function refers to a function whose integral may be calculated. ... Eulers formula, named after Leonhard Euler (pronounced oiler), is a mathematical formula in the subfield of complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. ...

The quantity, , provides both the amplitude and initial phase (as a complex number) of basis function: .

The function, , is the Fourier transform of , denoted by the operator :

And the inverse transform (shown above) is written:

Together the two functions are referred to as a transform pair. See continuous Fourier transform for more information, including: In mathematics, the continuous Fourier transform is a certain linear operator that maps functions to other functions. ...

• formula for the forward transform
• tabulated transforms of specific functions
• discussion of the transform properties
• various conventions for amplitude normalization and frequency scaling/units

Multi-dimensional version

The formulation for the Fourier transform given above applies in one dimension. The Fourier transform, however, can be expanded to arbitrary dimension n. The more generalised version of this transform in dimension n, notated by is:

where and are n-dimensional vectors, is the inner product of these two vectors, and the integration is performed over all n dimensions. A vector in physics and engineering typically refers to a quantity that has close relationship to the spatial coordinates, informally described as an object with a magnitude and a direction. The word vector is also now used for more general concepts (see also vector and generalizations below), but in this... In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ...

Fourier series

Main article: Fourier series

The continuous transform is itself actually a generalization of an earlier concept, a Fourier series, which was specific to periodic (or finite-domain) functions (with period ), and represents these functions as a series of sinusoids: The Fourier series is a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ... The Fourier series is a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ... In mathematics, a series is often represented as the sum of a sequence of terms. ...

where , and is a (complex) amplitude.

For real-valued , an equivalent variation is: In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

where   and   .

Discrete-time Fourier transform

Main article: Discrete-time Fourier transform

For use on computers, both for scientific computation and digital signal processing, one must have functions, x[n], that are defined for discrete instead of continuous domains, again finite or periodic. A useful "discrete-time" function can be obtained by sampling a "continuous-time" function, x(t). And similar to the continuous Fourier transform, the function can be represented as a sum of complex sinusoids: Given a discrete set of real or complex numbers: (integers), the discrete-time Fourier transform (or DTFT) of is: // Its name implies that the {x[n]} sequence represents the values (aka samples) of a continuous-time function, , at discrete moments in time: , where is the sampling interval (in seconds), and... Digital signal processing (DSP) is the study of signals in a digital representation and the processing methods of these signals. ... Discrete time is non-continuous time. ... Continuous time occurs when time is sampled continuously. ... In signal processing, sampling is the reduction of a continuous signal to a discrete signal. ...

But in this case, the limits of integration need only span one period of the periodic function, , which is derived from the samples by the discrete-time Fourier transform (DTFT): Given a discrete set of real or complex numbers: (integers), the discrete-time Fourier transform (or DTFT) of is: // Its name implies that the {x[n]} sequence represents the values (aka samples) of a continuous-time function, , at discrete moments in time: , where is the sampling interval (in seconds), and...

Discrete Fourier transform

Main article: Discrete Fourier transform

The DTFT is defined on a continuous domain. So despite its periodicity, it still cannot be numerically evaluated for every unique frequency. But a very useful approximation can be made by evaluating it at regularly-spaced intervals, with arbitrarily small spacing. Due to periodicity, the number of unique coefficients (N) to be evaluated is always finite, leading to this simplification: In mathematics, the discrete Fourier transform (DFT), occasionally called the finite Fourier transform, is a transform for Fourier analysis of finite-domain discrete-time signals. ...

    for

When the portion of x[n] between n=0 and n=N-1 is a good (or exact) representation of the entire x[n] sequence, it is useful to compute:

which is called discrete Fourier transform (DFT). Commonly the length of the x[n] sequence is finite, and a larger value of N is chosen. Effectively, the x[n] sequence is padded with zero-valued samples, referred to as zero padding. In mathematics, the discrete Fourier transform (DFT), occasionally called the finite Fourier transform, is a transform for Fourier analysis of finite-domain discrete-time signals. ...

The inverse DFT represents x[n] as the sum of complex sinusoids:

The table below will note that this actually produces a periodic x[n]. If the original sequence was not periodic to begin with, this phenomenon is the time-domain consequence of approximating the continuous-domain DTFT function with the discrete-domain DFT function.

The DFT can be computed using a fast Fourier transform (FFT) algorithm, which makes it a practical and important transformation on computers. The Fast Fourier Transform (FFT) is an efficient algorithm to compute the discrete Fourier transform (DFT) and its inverse. ...

Fourier transforms on arbitrary locally compact abelian topological groups

The Fourier variants can also be generalized to Fourier transforms on arbitrary locally compact abelian topological groups, which are studied in harmonic analysis; there, the Fourier transform takes functions on a group to functions on the dual group. This treatment also allows a general formulation of the convolution theorem, which relates Fourier transforms and convolutions. See also the Pontryagin duality for the generalized underpinnings of the Fourier transform. In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. ... Abelian, in mathematics, is used in many different definitions: In group theory: Abelian group, a group in which the binary operation is commutative Category of abelian groups Ab has abelian groups as objects and group homomorphisms as morphisms Metabelian group is a group where the commutator subgroup is contained in... In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. ... Harmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. ... In mathematics, in particular in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform. ... In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution is the point-wise product of Fourier transforms. ... In mathematics and, in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version of g. ... In mathematics, in particular in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform. ...

Time-frequency transforms

Time-frequency transforms such as the short-time Fourier transform, wavelet transforms, chirplet transforms, and the fractional Fourier transform try to obtain frequency information from a signal as a function of time (or whatever the independent variable is), although the ability to simultaneously resolve frequency and time is limited by an (mathematical) uncertainty principle. The short-time Fourier transform (STFT), or short-term Fourier transform, is a Fourier-related transform used to determine the sinusoidal frequency and phase content of a signal as it changes over time. ... The wavelet transform is a transformation to basis functions that are localized in scale and in time as well (where the Fourier transform is only localized in frequency, never giving any information about where in space or time the frequency happens). ... Comparison of wave, wavelet, chirp, and chirplet In signal processing, the chirplet transform is an inner product of an input signal with a family of analysis primitives called chirplets. ... The fractional Fourier transform (FRFT) is a linear transformation generalizing the continuous Fourier transform, and 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 intermediate domain between time and... In quantum physics, the outcome of even an ideal measurement of a system is not deterministic, but instead is characterized by a probability distribution, and the larger the associated standard deviation is, the more uncertain we might say that that characteristic is for the system. ...

Interpretation in terms of time and frequency

In terms of signal processing, the transform takes a time series representation of a signal function and maps it into a frequency spectrum, where ω is angular frequency. That is, it takes a function in the time domain into the frequency domain; it is a decomposition of a function into harmonics of different frequencies. In information theory, a signal is the sequence of states of a communications channel that encodes a message. ... In statistics, signal processing, and econometrics, a time series is a sequence of data points, measured typically at successive times, spaced at (often uniform) time intervals. ... Familiar concepts associated with a frequency are colors, musical notes, radio/TV channels, and even the regular rotation of the earth. ... It has been suggested that this article or section be merged into Angular velocity. ... Look up time in Wiktionary, the free dictionary. ... FreQuency is a music video game developed by Harmonix and published by SCEI. It was released in November 2001. ... In mathematics, two functions and are orthogonal if their inner product is zero. ... This article is about the components of sound. ...

When the function f is a function of time and represents a physical signal, the transform has a standard interpretation as the frequency spectrum of the signal. The magnitude of the resulting complex-valued function F at frequency ω represents the amplitude of a frequency component whose initial phase is given by: arctan (imaginary part/real part). In information theory, a signal is the sequence of states of a communications channel that encodes a message. ... The magnitude of a mathematical object is its size: a property by which it can be larger or smaller than other objects of the same kind; in technical terms, an ordering of the class of objects to which it belongs. ... It has been suggested that pulse amplitude be merged into this article or section. ... This article is about a portion of a periodic process. ...

However, it is important to realize that Fourier transforms are not limited to functions of time, and temporal frequencies. They can equally be applied to analyze spatial frequencies, and indeed for nearly any function domain.

Applications in signal processing

In signal processing, Fourier transformation can isolate individual components of a complex signal, concentrating them for easier detection and/or removal. A large family of signal processing techniques consist of Fourier-transforming a signal (such as a clip of audio or an image), manipulating the Fourier-transformed data in a simple way, and reversing the transformation. Some examples include: Sound is a disturbance of mechanical energy that propagates through matter as a wave. ... It has been suggested that this article or section be merged into image (disambiguation). ...

• Removal of unwanted frequencies from an audio recording (used to eliminate hum from leakage of AC power into the signal, to eliminate the stereo subcarrier from FM radio recordings, or to create karaoke tracks with the vocals removed);
• Noise gating of audio recordings to remove quiet background noise by eliminating Fourier components that do not exceed a preset amplitude;
• Equalization of audio recordings with a series of bandpass filters;
• Digital radio reception with no superheterodyne circuit, as in a modern cell phone or radio scanner;
• Image processing to remove periodic or anisotropic artifacts such as jaggies from interlaced video, stripe artifacts from strip aerial photography, or wave patterns from radio frequency interference in a digital camera;
• Cross correlation of similar images for co-alignment;
• X-ray crystallography to reconstruct a protein's structure from its diffraction pattern;
• Fourier transform ion cyclotron resonance mass spectrometry to determine the mass of ions from the frequency of cyclotron motion in a magnetic field.

Fourier transformation is also useful as a compact representation of a signal. For example, JPEG compression uses Fourier transformation of small square pieces of a digital image. The Fourier components of each square are rounded to lower arithmetic precision, and weak components are eliminated entirely, so that the remaining components can be stored very compactly. In image reconstruction, each Fourier-transformed image square is reassembled from the preserved approximate components, and then inverse-transformed to produce an approximation of the original image. A hum is a sound with a particular timbre (or sound quality), usually a monotone or with slightly varying tones, often produced by machinery in operation or by insects in flight. ... Usually hidden to the unaided eye, the blinking of (non-incandescent) lighting powered by AC mains is revealed in this motion-blurred long exposure of city lights. ... FM radio is a broadcast technology invented by Edwin Howard Armstrong that uses frequency modulation to provide high-fidelity sound over broadcast radio. ... It has been suggested that Karaoke clubs in Sri Lanka be merged into this article or section. ... A noise gate is an electronic device or software logic that is used to control the volume of an audio signal. ... For information about computer bandwidth management, see Equalization (computing). ... The frequency axis of this symbolic diagram would be logarithmically scaled. ... The Super Heterodyne receiver (or to give it its full name, The Supersonic Heterodyne Receiver) was invented by Edwin Armstrong in 1918. ... Motorola T2288 mobile phone A mobile phone is a portable electronic device which behaves as a normal telephone whilst being able to move over a wide area (compare cordless phone which acts as a telephone only within a limited range). ... A Scanner is a radio receiver generally capable of picking up AM and FM (and sometimes SSB) radio signals anywhere from 100kHz to 2. ... UPIICSA IPN - Binary image Image processing is any form of information processing for which the input is an image, such as photographs or frames of video; the output is not necessarily an image, but can be for instance a set of features of the image. ... This article is being considered for deletion in accordance with Wikipedias deletion policy. ... jaggies are those sharp edges that you see in all the wii games Jaggies is the informal name for aliasing artifacts in raster images, often caused by non-linear mixing effects producing high-frequency components and/or missing or poor anti-aliasing filtering prior to sampling. ... Strip aerial photography (or aerial strip photograhy) is a method of aerial photography that uses a high-speed, low-altitude aircraft to take a continuous picture, rather than using overlapping high-altitude photographs, as in conventional aerial photography. ... Radio Frequency Interference (RFI) is electromagnetic radiation which is emitted by electrical circuits carrying rapidly changing signals, as a by-product of their normal operation, and which causes unwanted signals (interference or noise) to be induced in other circuits. ... In statistics, the term cross-correlation is sometimes used to refer to the covariance cov(X, Y) between two random vectors X and Y, in order to distinguish that concept from the covariance of a random vector X, which is understood to be the matrix of covariances between the scalar... X-ray crystallography, also known as single-crystal X-ray diffraction, is the oldest and most common crystallographic method for determining the structure of molecules. ... Fourier Transform Ion Cyclotron Resonance, also known as Fourier Transform Mass Spectrometry, is a type of mass analyzer (or mass spectrometer) for determining the mass to charge ratio (m/z) of ions based on the cyclotron frequency of the ions in a magnetic field. ... JPG redirects here. ... The precision of a value describes the number of digits that are used to express that value. ...

About notation

The Fourier transform is a mapping on a function space. This mapping is here denoted and is used to denote the Fourier transform of the function s. This mapping is linear, which means that can also be seen as a linear transformation on the function space and implies that the standard notation in linear algebra of applying a linear transformation to a vector (here the signal s) can be used to write instead of . Since the result of applying the Fourier transform is again a function, we can be interested in the value of this function evaluated at the value ω for its variable, and this is denoted either as or as . Notice that in the former case, it is implicitly understood that is applied first to s and then the resulting function is evaluated at ω, not the other way around.

In mathematics and various applied sciences it is often necessary distinguish between a function s and the value of s when its variable equals t, denoted s(t). This means that a notation like formally can be interpreted as the Fourier transform of the values of s at t, which must be considered as an ill-formed expression since it describes the Fourier transform of a function value rather than of a function. Despite this flaw, the previous notation appears frequently, often when a particular function or a function of a particular variable is to be transformed. For example, is sometimes used to express that the Fourier transform of a rectangular function is a sinc function, or is used to express the shift property of the Fourier transform. Notice, that the last example is only correct under the assumption that the transformed function is a function of t, not of t₀. If possible, this informal usage of the operator should be avoided, in particular when it is not perfectly clear which variable the function to be transform depends on.

References

• Edward W. Kamen, Bonnie S. Heck, "Fundamentals of Signals and Systems Using the Web and Matlab", ISBN 0-13-017293-6
• E. M. Stein, G. Weiss, "Introduction to Fourier Analysis on Euclidean Spaces", Princeton University Press, 1971. ISBN 0-691-08078-X
• A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equations, CRC Press, Boca Raton, 1998. ISBN 0-8493-2876-4
• Smith, Steven W. The Scientist and Engineer's Guide to Digital Signal Processing, 2nd edition. San Diego: California Technical Publishing, 1999. ISBN 0-9660176-3-3. (also available online: [1])

See also

• fractional Fourier transform
• Bispectrum
• Characteristic function (probability theory)
• Chirplet
• Number-theoretic transform
• Laplace transform
• Mellin transform
• Orthogonal functions
• Pontryagin duality
• Schwartz space
• Two-sided Laplace transform
• Wavelet

The fractional Fourier transform (FRFT) is a linear transformation generalizing the continuous Fourier transform, and 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 intermediate domain between time and... The bispectrum is a statistic used to search for nonlinear interactions. ... In probability theory, the characteristic function of any random variable completely defines its probability distribution. ... Comparison of wave, wavelet, chirp, and chirplet In signal processing, the chirplet transform is an inner product of an input signal with a family of analysis primitives called chirplets. ... The number-theoretic transform is similar to the discrete Fourier transform, but operates with modular arithmetic instead of complex numbers. ... In mathematics, the Laplace transform is a technique for analyzing linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems. ... In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform. ... In mathematics, two functions and are orthogonal if their inner product is zero. ... In mathematics, in particular in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform. ... In mathematics, Schwartz space is the function space of rapidly decreasing functions. ... 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. ... A wavelet is a kind of mathematical function used to divide a given function into different frequency components and study each component with a resolution that matches its scale. ...

External links

• Tables of Integral Transforms at EqWorld: The World of Mathematical Equations.
• An Intuitive Explanation of Fourier Theory by Steven Lehar.
• Lectures on Image Processing: A collection of 18 lectures in pdf format from Vanderbilt University. Lecture 6 is on the 1- and 2-D Fourier Transform. Lectures 7-15 make use of it., by Alan Peters