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{{Fourier transforms}}
 
{{Fourier transforms}}
'''Fourier analysis''', named after [[Joseph Fourier]]'s introduction of the [[Fourier series]], is the decomposition of a function in terms of a sum of [[trigonometric function|sinusoidal]] [[basis function]]s (vs. their [[frequency|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).
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In mathematics, '''Fourier analysis''' is the study of the way general [[function (mathematics)|functions]] may be represented or approximated by sums of simpler [[trigonometric functions]]. Fourier analysis grew from the study of [[Fourier series]], and is named after [[Joseph Fourier]].
   
The [[linear operator|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]].
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Today, the subject of Fourier analysis encompasses a vast spectrum of mathematics. In the sciences the process of decomposing a function into simpler pieces is often called Fourier analysis, while the operation of rebuilding the function from these pieces is known as '''Fourier synthesis'''. In mathematics, the term ''Fourier analysis'' often refers to the study of both operations.
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The decomposition process itself is called a [[Fourier transform]]. The transform is often given a more specific name which depends upon the domain and other properties of the function being transformed. 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]]. Each [[Transform (mathematics)|transform]] used for analysis (see [[list of Fourier-related transforms]]) has a corresponding [[inverse function|inverse]] transform that can be used for synthesis.
   
 
==Applications==
 
==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 [[frequency|frequencies]] and their [[amplitude]]s.) This wide applicability stems from many useful properties of the transforms:
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Fourier analysis has many applications of interest to psychologists – in [[signal processing]], [[probability theory]], [[statistics]], [[acoustics]], [[optics]], [[diffraction]], structure and other areas.
   
* The transforms are [[linear operator]]s and, with proper normalization, are [[unitary operator|unitary]] as well (a property known as [[Parseval's theorem]] or, more generally, as the [[Plancherel theorem]], and most generally via [[Pontryagin duality]]).
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So, for example, breaking up a [[time series]] or a periodic phenomena such as a sound wave into its sinusoidal components of different frequencies and amplitudes and fitting them to a [[Fourier transform]] makes them amenable to calculation
* 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 [[derivative|differentiation]], which means that this representation transforms linear [[differential equation]]s 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 [[multiplication algorithm|multiplying large numbers]].
 
* The [[Discrete Fourier transform|discrete]] version of the Fourier transform (see below) can be evaluated quickly on computers using [[fast Fourier transform]] (FFT) algorithms.
 
   
==Variants of Fourier analysis==
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This wide applicability stems from many useful properties of the transforms''':'''
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:
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* The transforms are [[linear operator]]s and, with proper normalization, are [[unitary operator|unitary]] as well (a property known as [[Parseval's theorem]] or, more generally, as the [[Plancherel theorem]], and most generally via [[Pontryagin duality]]) {{harv|Rudin|1990}}.
{| class="wikitable"
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* The transforms are usually invertible.
|-
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* The exponential functions are [[eigenfunctions]] of [[derivative|differentiation]], which means that this representation transforms linear [[differential equation]]s with [[constant coefficients]] into ordinary algebraic ones {{harv|Evans|1998}}. Therefore, the behavior of a [[LTI system|linear time-invariant system]] can be analyzed at each frequency independently.
! Name || Time domain || Frequency domain || Formula
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* 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 [[Multiplication algorithm#Fourier transform methods|multiplying large numbers]] {{harv|Knuth|1997}}.
|-
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* The [[Discrete Fourier transform|discrete]] version of the Fourier transform (see below) can be evaluated quickly on computers using [[Fast Fourier Transform]] (FFT) algorithms. {{harv|Conte|de Boor|1980}}
|(Continuous) [[Fourier transform]] || Continuous, Aperiodic || Continuous, Aperiodic || <math>s \left( t \right) = \int\limits_{-\infty}^\infty S\left( \omega\right) e^{i\omega t}\,d\omega</math>
 
|-
 
|[[Fourier series]] || Continuous, Periodic || Discrete, Aperiodic || <math>s(t) = \sum_{k=-\infty}^{\infty} S_k \cdot e^{i \omega_k t}</math>
 
|-
 
|[[Discrete-time Fourier transform]] || Discrete, Aperiodic || Continuous, Periodic || <math>X(\omega) = \sum_{n=-\infty}^{\infty} x[n] \,e^{-i \omega n}</math>
 
|-
 
|[[Discrete Fourier transform]] || Discrete, Periodic || Discrete, Periodic || <math>X[k] = \sum_{n=0}^{N-1} x[n] \,e^{-i 2 \pi \frac{k}{N} n}</math>
 
|-
 
|Generalization || arbitrary [[locally compact]] [[abelian]] [[topological group]] || arbitrary locally compact abelian topological group || See [[harmonic analysis]]
 
|}
 
The table shows that
 
* 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 factor]]s; For simplicity, this table uses the definitions without them. Also, this table does not contain multi-dimensional versions.
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Fourier transformation is also useful as a compact representation of a signal. For example, [[JPEG]] compression uses a variant of the Fourier transformation ([[discrete cosine transform]]) of small square pieces of a digital image. The Fourier components of each square are rounded to lower [[precision (arithmetic)|arithmetic precision]], and weak components are eliminated entirely, so that the remaining components can be stored very compactly. In image reconstruction, each image square is reassembled from the preserved approximate Fourier-transformed components, which are then inverse-transformed to produce an approximation of the original image.
   
===Fourier transform===
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===Applications in signal processing===
: ''Main article: [[Fourier transform]]''
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{{Unreferenced section|date=September 2008}}
Most often, the unqualified term [[Fourier transform]] refers to the transform of functions of a continuous [[real number|real]] argument, representing any [[Integrable function#Square-integrable|square-integrable]] function <math>s \left( t \right)</math> as a linear combination of [[complex exponential]]s with frequencies <math>\omega\,</math>''':'''
 
   
:<math>
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When processing signals, such as [[Sound|audio]], [[radio wave]]s, light waves, [[seismic waves]], and even images, Fourier analysis can isolate individual components of a compound waveform, concentrating them for easier detection and/or removal. A large family of signal processing techniques consist of Fourier-transforming a signal, manipulating the Fourier-transformed data in a simple way, and reversing the transformation.
s \left( t \right) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} S\left( \omega\right) e^{i\omega t}\,d\omega.</math>
 
   
The quantity, <math>S(\omega)\,</math>, provides both the amplitude and initial phase (as a complex number) of basis function''':''' <math>e^{i\omega t}\,</math>.
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Some examples include:
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* [[Equalization]] of audio recordings with a series of [[bandpass filter]]s;
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* Digital radio reception with no [[superheterodyne]] circuit, as in a modern cell phone or [[radio scanner]];
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* [[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;
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* [[Cross correlation]] of similar images for [[co-alignment]];
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* [[X-ray crystallography]] to reconstruct a crystal structure from its diffraction pattern;
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* [[Fourier transform ion cyclotron resonance]] mass spectrometry to determine the mass of ions from the frequency of cyclotron motion in a magnetic field.
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* Many other forms of spectroscopy also rely upon Fourier Transforms to determine the three-dimensional structure and/or identity of the sample being analyzed, including Infrared and Nuclear Magnetic Resonance spectroscopies.
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* Generation of sound [[spectrogram]]s used to analyze sounds.
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* Passive [[sonar]] used to classify targets based on machinery noise.
   
The '''function''', <math>S(\omega)\,</math>, is the '''Fourier transform''' of <math>s(t)\,</math>, denoted by the operator <math>\mathcal{F}\,</math>''':'''
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==Variants of Fourier analysis==
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[[File:Variations of the Fourier transform.tif|thumb|400px|Illustration of using [[Dirac comb]] functions and the [[convolution theorem]] to model the effects of sampling and/or [[periodic summation]]. At lower left is a [[discrete-time Fourier transform|DTFT]], the spectral result of sampling s(t) at intervals of T. The spectral sequences at (a) upper right and (b) lower right are respectively computed from (a) one cycle of the periodic summation of s(t) and (b) one cycle of the periodic summation of the s(nT) sequence. The respective formulas are (a) the [[Fourier series]] <u>integral</u> and (b) the [[Discrete Fourier transform|DFT]] <u>summation</u>. The relative computational ease of the DFT sequence and the insight it provides into S(f) make it a popular analysis tool.]]
   
: <math>S(\omega) = \left(\mathcal{F}s\right)(\omega)= \mathcal{F}\{s\}(\omega)\,</math>
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=== (Continuous) Fourier transform ===
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{{main|Fourier Transform}}
   
And the inverse transform (shown above) is written''':'''
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Most often, the unqualified term '''Fourier transform''' refers to the transform of functions of a continuous [[real number|real]] argument, and it produces a continuous function of frequency, known as a ''frequency distribution''. One function is transformed into another, and the operation is reversible. When the domain of the input (initial) function is time (''t''), and the domain of the output (final) function is [[frequency|ordinary frequency]], the transform of function ''s''(''t'') at frequency ''ƒ'' is given by the complex number''':'''
   
: <math>s(t) = \left(\mathcal{F}^{-1}S\right)(t)= \mathcal{F}^{-1}\{S\}(t)\,</math>
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:<math>S(f) = \int_{-\infty}^{\infty} s(t) \cdot e^{- i 2\pi f t} dt.</math>
   
Together the two functions are referred to as a ''transform pair''. See [[continuous Fourier transform]] for more information, including''':'''
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Evaluating this quantity for all values of ''ƒ'' produces the ''frequency-domain'' function. Then ''s''(''t'') can be represented as a recombination of [[complex exponentials]] of all possible frequencies''':'''
* formula for the forward transform
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:<math>s(t) = \int_{-\infty}^{\infty} S(f) \cdot e^{i 2\pi f t} df,</math>
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which is the inverse transform formula. The complex number, ''S''(''ƒ''), conveys both amplitude and phase of frequency ''ƒ''.
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  +
See [[Fourier transform]] for much more information, including''':'''
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* conventions for amplitude normalization and frequency scaling/units
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* transform properties
 
* tabulated transforms of specific functions
 
* tabulated transforms of specific functions
* discussion of the transform properties
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* an extension/generalization for functions of multiple dimensions, such as images.
* various conventions for amplitude normalization and frequency scaling/units
 
<!-- There are a number of different conventions that are used for the FT: different normalizations, different signs in the exponent...this is discussed in the [[continuous Fourier transform]] article. Please do not change this formula to match your favorite convention...we need to keep the different FT articles consistent.
 
-->
 
<!-- moved to "See Also"
 
A generalization of this transform is the [[fractional Fourier transform]], by which the transform can be raised to any real "power".
 
-->
 
<!-- probably too much information. More appropriate for the [[continuous Fourier transform]] article.
 
When <math>f \left( t \right)</math> is an [[even and odd functions|even or odd function]], the sine or cosine terms disappear and one is left with the [[Sine and Cosine transforms|cosine transform]] or [[Sine and Cosine transforms|sine transform]], respectively. Another important case is where <math>f \left( t \right)</math> is purely real, where it follows that <math>F\left( - \omega \right) = F\left( \omega \right)^*</math> (where the <math>*</math> denotes [[complex conjugation]]). Similar special cases appear for all other variants of the Fourier transform as well.
 
-->
 
   
====Multi-dimensional version====
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=== Fourier series ===
The formulation for the Fourier transform given above applies in one dimension. The Fourier transform, however, can be expanded to arbitrary dimension <math>n</math>. The more generalised version of this transform in dimension <math>n</math>, notated by <math>\mathcal{F}_n</math> is:
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{{main|Fourier series}}
:<math>s \left( \mathbf{x} \right) = \left( \mathcal{F}^{-1}_n S \right) \left( \mathbf{x} \right) = \frac{1}{(2\pi)^{n/2}} \int S \left( \boldsymbol{\omega} \right) e^{i \left\langle \boldsymbol{\omega} ,\mathbf{x} \right\rangle} \, d \boldsymbol{\omega},</math>
 
where <math>\mathbf{x}</math> and <math>\boldsymbol{\omega}</math> are <math>n</math>-dimensional [[vector (mathematics)|vector]]s, <math>\left\langle \boldsymbol{\omega} ,\mathbf{x} \right\rangle</math> is the [[inner product]] of these two vectors, and the integration is performed over all <math>n</math> dimensions.
 
   
===Fourier series===
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The Fourier transform of a periodic function, ''s''<sub>''P''</sub>(''t''), with period ''P'', becomes a [[Dirac comb]] function, modulated by a sequence of complex [[coefficients]]''':'''
: ''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 <math>s \left( t \right)</math> (with period <math>\tau \,</math>), and represents these functions as a [[series (mathematics)|series]] of sinusoids''':'''
 
   
:<math>s(t) = \sum_{k=-\infty}^{\infty} S_k \cdot e^{i \omega_k t}, \,</math>
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:<math>S[k] = \frac{1}{P}\int_{P} s_P(t)\cdot e^{-i 2\pi \frac{k}{P} t}\, dt</math> &nbsp; &nbsp; for all integer values of ''k'',
   
where <math>\omega_k = 2\pi k / \tau \,</math>, and <math>S_k \,</math> is a (complex) amplitude.
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and where <math>\scriptstyle \int_P</math>&nbsp; is the integral over any interval of length ''P''.
   
For [[real number|real]]-valued <math>s(t)\,</math>, an equivalent variation is''':'''
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The inverse transform, known as '''Fourier series''', is a representation of ''s''<sub>''P''</sub>(''t'') in terms of a summation of a potentially infinite number of harmonically related sinusoids or [[complex exponentials|complex exponential]] functions, each with an amplitude and phase specified by one of the coefficients''':'''
   
:<math>s(t) = \frac{1}{2}a_0 + \sum_{k=1}^\infty\left[a_k\cdot \cos(\omega_k t)+b_k\cdot \sin(\omega_k t)\right],</math>
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:<math>s_P(t)=\sum_{k=-\infty}^\infty S[k]\cdot e^{i 2\pi \frac{k}{P} t} \quad\stackrel{\mathcal{F}}{\Longleftrightarrow}\quad \sum_{k=-\infty}^{+\infty} S[k]\ \delta \left(f-\frac{k}{P}\right).</math>
   
where <math>a_k = 2\cdot \operatorname{Re}\{S_k\} \,</math> &nbsp; and &nbsp; <math>b_k = -2\cdot \operatorname{Im}\{S_k\} \,</math>.
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When ''s''<sub>''P''</sub>(''t''), is expressed as a [[periodic summation]] of another function, ''s''(''t'')''':''' &nbsp;&nbsp;<math>s_P(t)\ \stackrel{\text{def}}{=}\ \sum_{k=-\infty}^{\infty} s(t-kP),</math>
   
===Discrete-time Fourier transform===
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the coefficients are proportional to samples of ''S''(''ƒ'') at discrete intervals of '''1/P:''' &nbsp;&nbsp;<math>S[k] =\frac{1}{P}\cdot S\left(\frac{k}{P}\right).\,</math><ref group="note">
: ''Main article: [[Discrete-time Fourier transform]]''
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:<math>\int_{P} \left[\sum_{k=-\infty}^{\infty} s(t-kP)\right] \cdot e^{-i 2\pi \frac{k}{P} t} dt = \underbrace{\int_{-\infty}^{\infty} s(t) \cdot e^{-i 2\pi \frac{k}{P} t} dt}_{\stackrel{\mathrm{def}}{=}\ S(k/P)}</math>
For use on computers, both for scientific computation and [[digital signal processing]], one must have functions, x[n], that are defined for [[discrete time|discrete]] instead of [[continuous time|continuous]] domains, again finite or periodic. A useful "discrete-time" function can be obtained by [[Sampling (signal processing)|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''':'''
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</ref>
   
:<math>x[n] = \frac{1}{2 \pi}\int_{-\pi}^{\pi} X(\omega)\cdot e^{i \omega n} \, d \omega.</math>
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A sufficient condition for recovering ''s''(''t'') (and therefore ''S''(''ƒ'')) from just these samples is that the non-zero portion of ''s''(''t'') be confined to a known interval of duration ''P'', which is the frequency domain dual of the [[Nyquist–Shannon sampling theorem]].
   
But in this case, the limits of integration need only span one period of the periodic function, <math>X(\omega ) \,</math>, which is derived from the samples by the [[discrete-time Fourier transform]] (DTFT)''':'''
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See [[Fourier series]] for more information, including the historical development.
   
:<math>X(\omega) = \sum_{n=-\infty}^{\infty} x[n] \,e^{-i \omega n}.\,</math>
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=== Discrete-time Fourier transform (DTFT) ===
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{{main|Discrete-time Fourier transform}}
   
===Discrete Fourier transform===
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The DTFT is the mathematical dual of the time-domain Fourier series. Thus, a convergent [[periodic summation]] in the frequency domain can be represented by a Fourier series, whose coefficients are samples of a related continuous time function:
: ''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''':'''
 
   
:<math>X[k] = X\left(\frac{2 \pi }{N} k\right)= \sum_{n=-\infty}^{\infty} x[n] \,e^{-i 2 \pi \frac{k}{N} n},</math> &nbsp; &nbsp; for <math>k = 0, 1, \dots, N-1. \,</math>
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:<math>S_{1/T}(f)\ \stackrel{\text{def}}{=}\ \underbrace{\sum_{k=-\infty}^{\infty} S\left(f - \frac{k}{T}\right) \equiv \overbrace{\sum_{n=-\infty}^{\infty} s[n] \cdot e^{-i 2\pi f n T}}^{\text{Fourier series (DTFT)}}}_{\text{Poisson summation formula}} = \mathcal{F} \left \{ \sum_{n=-\infty}^{\infty} s[n]\ \delta(t-nT)\right \},\,</math>
   
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''':'''
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which is known as the DTFT. Thus the '''DTFT''' of the ''s''[''n''] sequence is also the '''Fourier transform''' of the modulated [[Dirac comb]] function.<ref group="note">We may also note that: &nbsp;
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<math>\scriptstyle
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\sum_{n=-\infty}^{+\infty} T\ s(nT)\ \delta(t-nT)\ =\ \sum_{n=-\infty}^{+\infty} T\ s(t)\ \delta(t-nT)\ =\ s(t)\cdot T \sum_{n=-\infty}^{+\infty} \delta(t-nT).</math>
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<br>Consequently, a common practice is to model "sampling" as a multiplication by the [[Dirac comb]] function, which of course is only "possible" in a purely mathematical sense.</ref>
   
:<math>X[k] = \sum_{n=0}^{N-1} x[n] \,e^{-i 2 \pi \frac{k}{N} n},</math>
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The Fourier series coefficients, defined by:
   
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]].
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:<math>s[n]\ \stackrel{\mathrm{def}}{=} \ T \int_{1/T} S_{1/T}(f)\cdot e^{i 2\pi f nT} df = T \underbrace{\int_{-\infty}^{\infty} S(f)\cdot e^{i 2\pi f nT} df}_{\stackrel{\mathrm{def}}{=} \ s(nT)}\,</math>
   
The inverse DFT represents x[n] as the sum of complex sinusoids''':'''
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is the inverse transform. With '''s[n] = T•s(nT)''', this Fourier series can now be recognized as a form of the [[Poisson summation formula]]. Thus we have the important result that when a discrete data sequence, ''s''[''n''], is proportional to samples of an underlying continuous function, ''s''(''t''), one can observe a periodic summation of the continuous Fourier transform, ''S''(''ƒ''). That is a cornerstone in the foundation of [[digital signal processing]]. Furthermore, under certain idealized conditions one can theoretically recover ''S''(''ƒ'') and ''s''(''t'') exactly. A sufficient condition for perfect recovery is that the non-zero portion of ''S''(''ƒ'') be confined to a known frequency interval of width ''1/T''. When that interval is [-0.5/T, 0.5/T], the applicable reconstruction formula is the [[Whittaker–Shannon interpolation formula]].
   
:<math>x[n] = \frac{1}{N} \sum_{k=0}^{N-1} X[k] e^{i 2 \pi \frac{k}{N} n}, \quad \quad n = 0, 1, \dots, N-1. \,</math>
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Another reason to be interested in ''S''<sub>''1/T''</sub>(''ƒ'') is that it often provides insight into the amount of [[aliasing]] caused by the sampling process.
   
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.
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Applications of the DTFT are not limited to sampled functions. See [[Discrete-time Fourier transform]] for more information on this and other topics, including:
  +
* normalized frequency units
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* windowing (finite-length sequences)
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* transform properties
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* tabulated transforms of specific functions
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=== Discrete Fourier transform (DFT) ===
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{{main|Discrete Fourier transform}}
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The DTFT of a periodic sequence, ''s''<sub>''N''</sub>[''n''], with period ''N'', becomes another [[Dirac comb]] function, modulated by the coefficients of a '''Fourier series'''.&nbsp; And the integral formula for the coefficients simplifies to a summation (see [[Discrete-time Fourier transform#Periodic data|DTFT/Periodic data]])''':'''
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:<math>S_N[k] =\frac{1}{NT} \underbrace{\sum_N s_N[n]\cdot e^{-i 2\pi \frac{k}{N} n}}_{S_k}\,</math>, &nbsp; &nbsp; where <math>\scriptstyle \sum_N</math>&nbsp; is the sum over any n-sequence of length '''N'''.
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The ''S''<sub>''k''</sub> sequence is what's customarily known as the '''DFT''' of ''s''<sub>''N''</sub>. &nbsp;It is also N-periodic, so it is never necessary to compute more than N coefficients. &nbsp;In terms of ''S''<sub>''k''</sub>, the inverse transform is given by''':'''
  +
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:<math>s_N[n] = \frac{1}{N} \sum_{N} S_k\cdot e^{i 2\pi \frac{n}{N}k},\,</math> &nbsp; &nbsp; where <math>\scriptstyle \sum_N</math>&nbsp; is the sum over any k-sequence of length '''N'''.
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When ''s''<sub>''N''</sub>[''n''] is expressed as a [[periodic summation]] of another function''':''' &nbsp;<math>s_N[n]\ \stackrel{\text{def}}{=}\ \sum_{k=-\infty}^{\infty} s[n-kN],</math> &nbsp; and &nbsp; <math>s[n]\ \stackrel{\text{def}}{=}\ T\cdot s(nT),\,</math>
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the coefficients are equivalent to samples of ''S''<sub>1/''T''</sub>(''ƒ'') at discrete intervals of '''1/P = 1/NT:''' &nbsp;&nbsp;<math>S_k = S_{1/T}(k/P).\,</math> &nbsp; (see [[Discrete-time Fourier transform#Sampling the DTFT|DTFT/Sampling the DTFT]])
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Conversely, when one wants to compute an arbitrary number (N) of discrete samples of one cycle of a continuous DTFT, &nbsp;<math>S_{1/T}(f),\,</math>&nbsp; it can be done by computing the relatively simple DFT of ''s''<sub>''N''</sub>[''n''], as defined above. &nbsp;In most cases, ''N'' is chosen equal to the length of non-zero portion of ''s''[''n'']. &nbsp;Increasing ''N'', known as ''zero-padding'' or ''interpolation'', results in more closely spaced samples of one cycle of &nbsp;''S''<sub>''1/T''</sub>(''ƒ''). &nbsp;Decreasing ''N'', causes overlap (adding) in the time-domain (analogous to [[aliasing]]), which corresponds to decimation in the frequency domain. (see [[Discrete-time Fourier transform#Sampling the DTFT|Sampling the DTFT]]) In most cases of practical interest, the ''s''[''n''] sequence represents a longer sequence that was truncated by the application of a finite-length [[window function]] or [[FIR filter]] array.
   
 
The DFT can be computed using a [[fast Fourier transform]] (FFT) algorithm, which makes it a practical and important transformation on computers.
 
The DFT can be computed using a [[fast Fourier transform]] (FFT) algorithm, which makes it a practical and important transformation on computers.
  +
  +
See [[Discrete Fourier transform]] for much more information, including''':'''
  +
* transform properties
  +
* applications
  +
* tabulated transforms of specific functions
  +
  +
===Summary===
  +
For periodic functions, both the Fourier transform and the DTFT comprise only a discrete set of frequency components (Fourier series), and the transforms diverge at those frequencies. One common practice (not discussed above) is to handle that divergence via [[Dirac delta]] and [[Dirac comb]] functions. But the same spectral information can be discerned from just one cycle of the periodic function, since all the other cycles are identical. Similarly, finite-duration functions can be represented as a Fourier series, with no actual loss of information except that the periodicity of the inverse transform is a mere artifact. We also note that none of the formulas here require the duration of <math>s\,</math> to be limited to the period, '''P''' or '''N'''. &nbsp;But that is a common situation, in practice.
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{| class="wikitable" style="text-align:left"
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|+ <math>s(t)\,</math> &nbsp;transforms (continuous-time)
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|-
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! !! Continuous frequency !! Discrete frequencies
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|-
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! Transform
  +
| <math>S(f)\ \stackrel{\text{def}}{=}\ \int_{-\infty}^{\infty} s(t)\ e^{-i 2\pi f t}\,dt</math>
  +
|| <math>\underbrace{\frac{1}{P}\cdot S\left(\frac{k}{P}\right)}_{S[k]}\ \stackrel{\text{def}}{=}\ \frac{1}{P} \int_{-\infty}^{\infty} s(t)\ e^{-i 2\pi \frac{k}{P} t}\,dt \equiv \frac{1}{P} \int_P s_P(t)\ e^{-i 2\pi \frac{k}{P} t}\,dt </math>
  +
|-
  +
! Inverse
  +
| <math>s(t) = \int_{-\infty}^{\infty} S(f)\ e^{ i 2 \pi f t}\,df </math>
  +
||<math>\underbrace{s_P(t) = \sum_{k=-\infty}^{\infty} S[k] \cdot e^{i 2\pi \frac{k}{P} t}}_{\text{Poisson summation formula (Fourier series)}}</math>
  +
|}
  +
  +
{| class="wikitable" style="text-align:left"
  +
|+ <math>s[n]\,</math> &nbsp;transforms (discrete-time)
  +
|-
  +
! !! Continuous frequency !! Discrete frequencies
  +
|-
  +
! Transform
  +
| <math>\underbrace{S_{1/T}(f) = \sum_{n=-\infty}^{\infty} \overbrace{T\cdot s(nT)}^{s[n]}\cdot e^{-i 2\pi f nT}}_{\text{Poisson summation formula (DTFT)}}</math>
  +
|| <math>\underbrace{\overbrace{S_{1/T}\left(\frac{k}{NT}\right)}^{S_k} = \sum_{n=-\infty}^{\infty} s[n]\cdot e^{-i 2\pi \frac{kn}{N}}}_{\text{Poisson summation formula}} \equiv \underbrace{\sum_{N} s_N[n]\cdot e^{-i 2\pi \frac{kn}{N}}}_{\text{DFT}}</math>
  +
|-
  +
! Inverse
  +
| <math>s[n] = T \int_{1/T} S_{1/T}(f)\cdot e^{i 2\pi f nT} \,df </math>
  +
<math>\sum_{n=-\infty}^{\infty} s[n]\cdot \delta(t-nT) = \underbrace{\int_{-\infty}^{\infty} S_{1/T}(f)\cdot e^{i 2\pi f t}\,df}_{\text{inverse Fourier transform}} </math>
  +
|| <math>s_N[n] = \underbrace{\frac{1}{N} \sum_{N} S_k\cdot e^{i 2\pi \frac{kn}{N}}}_{\text{inverse DFT}}</math>
  +
<math>s_P(nT) = \frac{1}{T}\cdot s_N[n] = \sum_{N} \underbrace{\frac{1}{P}\cdot S_{1/T}\left(\frac{k}{P}\right)}_{S_{N}[k]} \cdot e^{i 2\pi \frac{kn}{N}}</math>
  +
|}
   
 
===Fourier transforms on arbitrary locally compact abelian topological groups===
 
===Fourier transforms on arbitrary locally compact abelian topological groups===
  +
The Fourier variants can also be generalized to Fourier transforms on arbitrary [[locally compact]] [[abelian group|abelian]] [[topological group]]s, 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 [[convolution]]s. See also the [[Pontryagin duality]] for the generalized underpinnings of the Fourier transform.
   
The Fourier variants can also be generalized to Fourier transforms on arbitrary [[locally compact]] [[abelian]] [[topological group]]s, 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 [[convolution]]s. See also the [[Pontryagin duality]] for the generalized underpinnings of the Fourier transform.
+
===Time–frequency transforms===
  +
{{details|Time–frequency analysis}}
   
===Time-frequency transforms===
+
In [[signal processing]] terms, a function (of time) is a representation of a signal with perfect ''time resolution,'' but no frequency information, while the Fourier transform has perfect ''frequency resolution,'' but no time information.
[[Time-frequency transform]]s such as the [[short-time Fourier transform]], [[wavelet transform]]s, [[chirplet transform]]s, 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]].
 
   
==Interpretation in terms of time and frequency==
+
As alternatives to the Fourier transform, in [[time–frequency analysis]], one uses time–frequency transforms to represent signals in a form that has some time information and some frequency information – by the [[uncertainty principle]], there is a trade-off between these. These can be generalizations of the Fourier transform, such as the [[short-time Fourier transform]], the [[Gabor transform]] or [[fractional Fourier transform]] (FRFT), or can use different functions to represent signals, as in [[wavelet transforms]] and [[chirplet transform]]s, with the wavelet analog of the (continuous) Fourier transform being the [[continuous wavelet transform]].
   
In terms of [[Signal (information theory)|signal]] processing, the transform takes a [[time series]] representation of a signal function and maps it into a [[frequency spectrum]], where &omega; is [[angular frequency]]. That is, it takes a function in the [[time]] domain into the [[frequency]] domain; it is a [[orthogonal system|decomposition]] of a function into [[harmonic]]s of different frequencies.
+
== History ==
  +
{{see also|Fourier series#Historical development}}
   
When the function ''f'' is a function of time and represents a physical [[Signal (information theory)|signal]], the transform has a standard interpretation as the frequency spectrum of the signal. The [[magnitude (mathematics)|magnitude]] of the resulting complex-valued function ''F'' at frequency &omega; represents the [[amplitude]] of a frequency component whose [[phase (waves)|initial phase]] is given by: ''arctan (imaginary part/real part)''.
+
A primitive form of harmonic series dates back to ancient [[Babylonian mathematics]], where they were used to compute [[ephemerides]] (tables of astronomical positions).<ref>{{citation
  +
|title=The evolution of applied harmonic analysis: models of the real world
  +
|first=Elena
  +
|last=Prestini
  +
|url=http://books.google.com/?id=fye--TBu4T0C
  +
|publisher=Birkhäuser
  +
|year=2004
  +
|isbn=978-0-8176-4125-2
  +
}}, [http://books.google.com/books?id=fye--TBu4T0C&pg=PA62 p. 62]<br />{{citation
  +
|url=http://books.google.com/?id=H5smrEExNFUC
  +
|title=Indiscrete thoughts
  +
|first1=Gian-Carlo
  +
|last1=Rota
  +
|first2=Fabrizio
  +
|last2=Palombi
  +
|authorlink=Gian-Carlo Rota
  +
|publisher=Birkhäuser
  +
|year=1997
  +
|isbn=978-0-8176-3866-5
  +
}}, [http://books.google.com/books?id=H5smrEExNFUC&pg=PA11 p. 11]<br />{{Citation | edition = 2 | publisher=[[Dover Publications]] | last = Neugebauer | first = Otto | author-link = Otto E. Neugebauer | title = The Exact Sciences in Antiquity | origyear = 1957 | year = 1969 | isbn = 978-0-486-22332-2 | url = http://books.google.com/?id=JVhTtVA2zr8C}}<br />{{citation
  +
|arxiv=physics/0310126
  +
|title=Analyzing shell structure from Babylonian and modern times
  +
|first1=Lis
  +
|last1=Brack-Bernsen
  +
|first2=Matthias
  +
|last2=Brack
  +
}}</ref>
  +
The classical Greek concepts of [[deferent and epicycle]] in the [[Ptolemaic system]] of astronomy were related to Fourier series (see [[Deferent and epicycle#Mathematical formalism|Deferent and epicycle: Mathematical formalism]]).
   
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.
+
In modern times, variants of the discrete Fourier transform were used by [[Alexis Clairaut]] in 1754 to compute an orbit,<ref>{{citation
  +
|title=Fourier analysis on finite groups and applications
  +
|first=Audrey
  +
|last=Terras
  +
|authorlink=Audrey Terras
  +
|publisher=[[Cambridge University Press]]
  +
|year=1999
  +
|isbn=978-0-521-45718-7
  +
|url=http://books.google.com/?id=-B2TA669dJMC
  +
}}, [http://books.google.com/books?id=-B2TA669dJMC&pg=PA30#PPA30,M1 p. 30]</ref>
  +
which has been described as the first formula for the DFT,<ref name="thedft4">{{citation
  +
|first1=William L.
  +
|last1=Briggs
  +
|first2=Van Emden
  +
|last2=Henson
  +
|publisher=SIAM
  +
|year=1995
  +
|isbn=978-0-89871-342-8
  +
|url=http://books.google.com/?id=coq49_LRURUC
  +
|title=The DFT : an owner's manual for the discrete Fourier transform
  +
}}, [http://books.google.com/books?id=coq49_LRURUC&pg=PA2#PPA4,M1 p. 4]</ref>
  +
and in 1759 by [[Joseph Louis Lagrange]], in computing the coefficients of a trigonometric series for a vibrating string.<ref name="thedft">{{citation
  +
|title=The DFT: an owner's manual for the discrete Fourier transform
  +
|first1=William L.
  +
|last1=Briggs
  +
|first2=Van Emden
  +
|last2=Henson
  +
|publisher=SIAM
  +
|year=1995
  +
|isbn=978-0-89871-342-8
  +
|url=http://books.google.com/?id=coq49_LRURUC
  +
}}, [http://books.google.com/books?id=coq49_LRURUC&pg=PA2#PPA2,M1 p. 2]</ref> Technically, Clairaut's work was a cosine-only series (a form of [[discrete cosine transform]]), while Lagrange's work was a sine-only series (a form of [[discrete sine transform]]); a true cosine+sine DFT was used by [[Carl Friedrich Gauss|Gauss]] in 1805 for [[trigonometric interpolation]] of [[asteroid]] orbits.<ref name=Heideman84>Heideman, M. T., D. H. Johnson, and C. S. Burrus, "[http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1162257 Gauss and the history of the fast Fourier transform]," IEEE ASSP Magazine, 1, (4), 14–21 (1984)</ref>
  +
Euler and Lagrange both discretized the vibrating string problem, using what would today be called samples.<ref name="thedft" />
   
==Applications in signal processing==
+
An early modern development toward Fourier analysis was the 1770 paper ''[[Réflexions sur la résolution algébrique des équations]]'' by Lagrange, which in the method of [[Lagrange resolvents]] used a complex Fourier decomposition to study the solution of a cubic:<ref>
  +
{{citation
  +
|url=http://books.google.com/?id=KVeXG163BggC
  +
|title = Basic algebra
  +
|first=Anthony W.
  +
|last=Knapp
  +
|publisher=Springer
  +
|year=2006
  +
|isbn=978-0-8176-3248-9
  +
}}, [http://books.google.com/books?id=KVeXG163BggC&pg=PA501 p. 501]</ref>
  +
Lagrange transformed the roots <math>x_1,x_2,x_3</math> into the resolvents:
  +
<!-- equation to clarify connection; instantly recognizable if familiar with DFT matrix -->
  +
:<math>\begin{align}
  +
r_1 &= x_1 + x_2 + x_3\\
  +
r_2 &= x_1 + \zeta x_2 + \zeta^2 x_3\\
  +
r_3 &= x_1 + \zeta^2 x_2 + \zeta x_3
  +
\end{align}</math>
  +
where ''ζ'' is a cubic root of unity, which is the DFT of order 3.
   
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 [[Sound|audio]] or an [[image]]), manipulating the Fourier-transformed data in a simple way, and reversing the transformation. Some examples include:
+
A number of authors, notably [[Jean le Rond d'Alembert]], and [[Carl Friedrich Gauss]] used [[trigonometric series]] to study the [[heat equation]],{{Citation needed|date=April 2009}} but the breakthrough development was the 1807 paper
  +
''[[Mémoire sur la propagation de la chaleur dans les corps solides]]'' by [[Joseph Fourier]], whose crucial insight was to model ''all'' functions by trigonometric series, introducing the Fourier series.
   
* 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);
+
Historians are divided as to how much to credit Lagrange and others for the development of Fourier theory: [[Daniel Bernoulli]] and [[Leonhard Euler]] had introduced trigonometric representations of functions,<ref name="thedft4" /> and Lagrange had given the Fourier series solution to the wave equation,<ref name="thedft4" /> so Fourier's contribution was mainly the bold claim that an arbitrary function could be represented by a Fourier series.<ref name="thedft4" />
* [[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 filter]]s;
 
* 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 [[precision (arithmetic)|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.
+
The subsequent development of the field is known as [[harmonic analysis]], and is also an early instance of [[representation theory]].
   
== About notation ==
+
The first fast Fourier transform (FFT) algorithm for the DFT was discovered around 1805 by Carl Friedrich Gauss when interpolating measurements of the orbit of the asteroids Juno and Pallas, although that particular FFT algorithm is more often attributed to its modern rediscoverers [[Cooley–Tukey FFT algorithm|Cooley and Tukey]].<ref name=Heideman84/><ref>{{citation
  +
|title=Fourier analysis on finite groups and applications
  +
|first=Audrey
  +
|last=Terras
  +
|authorlink=Audrey Terras
  +
|publisher=Cambridge University Press
  +
|year=1999
  +
|isbn=978-0-521-45718-7
  +
|url=http://books.google.com/?id=-B2TA669dJMC
  +
}}, [http://books.google.com/books?id=-B2TA669dJMC&pg=PA30#PPA31,M1 p. 31]</ref>
   
The Fourier transform is a mapping on a function space. This mapping is here denoted <math>\mathcal{F}</math> and <math>\mathcal{F}\{s\}</math> is used to denote the Fourier transform of the function ''s''. This mapping is linear, which means that <math>\mathcal{F}</math> 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 <math>\mathcal{F} s</math> instead of <math>\mathcal{F}\{s\}</math>. 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 <math>\omega</math> for its variable, and this is denoted either as <math>\mathcal{F}\{s\}(\omega)</math> or as <math>(\mathcal{F} s)(\omega)</math>. Notice that in the former case, it is implicitly understood that <math>\mathcal{F}</math> is applied first to ''s'' and then the resulting function is evaluated at <math>\omega</math>, not the other way around.
+
==Interpretation in terms of time and frequency==
  +
In [[signal processing]], the Fourier transform often takes a [[time series]] or a function of [[continuous time]], and maps it into a [[frequency spectrum]]. That is, it takes a function from the time domain into the [[frequency]] domain; it is a [[orthogonal system|decomposition]] of a function into [[Sine wave|sinusoids]] of different frequencies; in the case of a [[Fourier series]] or [[discrete Fourier transform]], the sinusoids are [[harmonic]]s of the fundamental frequency of the function being analyzed.
   
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 <math>\mathcal{F}\{s(t)\}</math> 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, <math>\mathcal{F}\{ \mathrm{rect}(t) \} = \mathrm{sinc}(\omega)</math> is sometimes used to express that the Fourier transform of a rectangular function is a sinc function, or <math>\mathcal{F}\{s(t+t_{0})\} = \mathcal{F}\{s(t)\} e^{i \omega t_{0}}</math> 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 <math>t_{0}</math>. If possible, this informal usage of the <math>\mathcal{F}</math> operator should be avoided, in particular when it is not perfectly clear which variable the function to be transform depends on.
+
When the function ''ƒ'' is a function of time and represents a physical [[Signal (information theory)|signal]], the transform has a standard interpretation as the frequency spectrum of the signal. The [[magnitude (mathematics)|magnitude]] of the resulting complex-valued function ''F'' at frequency ω represents the [[amplitude]] of a frequency component whose [[phase (waves)|initial phase]] is given by the phase of&nbsp;''F''.
   
==References==
+
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. This justifies their use in branches such diverse as [[image processing]], [[heat conduction]], and [[automatic control]].
* 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
+
==Notes==
* A. D. Polyanin and A. V. Manzhirov, ''Handbook of Integral Equations'', CRC Press, Boca Raton, 1998. ISBN 0-8493-2876-4
+
{{reflist|group=note}}
* 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: [http://www.dspguide.com/pdfbook.htm])''
 
   
 
==See also==
 
==See also==
* [[fractional Fourier transform]]
+
{{colbegin}}
  +
* [[Generalized Fourier series]]
  +
* [[Fourier-Bessel series]]
  +
* [[Fourier-related transforms]]
  +
* [[Laplace transform]] (LT)
  +
* [[Two-sided Laplace transform]]
  +
* [[Mellin transform]]
  +
* [[Non-uniform discrete Fourier transform]] (NDFT)
  +
* [[Quantum Fourier transform]] (QFT)
  +
* [[Number-theoretic transform]]
  +
* [[Least-squares spectral analysis]]
  +
* [[Basis vector]]s
 
* [[Bispectrum]]
 
* [[Bispectrum]]
 
* [[Characteristic function (probability theory)]]
 
* [[Characteristic function (probability theory)]]
* [[Chirplet]]
 
* [[Number-theoretic transform]]
 
* [[Laplace transform]]
 
* [[Mellin transform]]
 
 
* [[Orthogonal functions]]
 
* [[Orthogonal functions]]
* [[Pontryagin duality]]
 
 
* [[Schwartz space]]
 
* [[Schwartz space]]
* [[Two-sided Laplace transform]]
+
* [[Spectral density]]
  +
* [[Spectral density estimation]]
  +
* [[Spectral music]]
 
* [[Wavelet]]
 
* [[Wavelet]]
  +
{{colend}}
  +
  +
==Citations==
  +
{{reflist}}
  +
  +
==References==
  +
* {{citation
  +
| last1 = Conte | first1 = S. D.
  +
| last2 = de Boor | first2 = Carl
  +
| title = Elementary Numerical Analysis
  +
| edition = Third
  +
| location = New York
  +
| publisher=McGraw Hill, Inc.
  +
| isbn = 0-07-066228-2
  +
| year=1980
  +
}}
  +
* {{citation|first=L.|last=Evans|title=Partial Differential Equations|publisher=American Mathematical Society|year=1998|isbn=3-540-76124-1}}
  +
* Howell, Kenneth B. (2001). ''Principles of Fourier Analysis'', CRC Press. ISBN 978-0-8493-8275-8
  +
* Kamen, E.W., and B.S. Heck. "Fundamentals of Signals and Systems Using the Web and Matlab". ISBN 0-13-017293-6
  +
* {{citation|first=Donald E.|last=Knuth|title=The Art of Computer Programming Volume 2: Seminumerical Algorithms|edition=3rd|year=1997|publisher=Addison-Wesley Professional|isbn=0-201-89684-2|location=Section 4.3.3.C: Discrete Fourier transforms, pg.305}}
  +
* Polyanin, A.D., and A.V. Manzhirov (1998). ''Handbook of Integral Equations'', CRC Press, Boca Raton. ISBN 0-8493-2876-4
  +
* {{citation|first=Walter|last=Rudin|title=Fourier Analysis on Groups|publisher=Wiley-Interscience|year=1990|isbn=0-471-52364-X}}
  +
* {{Citation
  +
| last = Smith | first = Steven W.
  +
| url = http://www.dspguide.com/pdfbook.htm
  +
| title = The Scientist and Engineer's Guide to Digital Signal Processing
  +
| edition = Second
  +
| location = San Diego, Calif.
  +
| publisher=California Technical Publishing
  +
| year=1999
  +
| isbn=0-9660176-3-3
  +
}}
  +
* Stein, E.M., and G. Weiss (1971). ''Introduction to Fourier Analysis on Euclidean Spaces''. Princeton University Press. ISBN 0-691-08078-X
   
 
==External links==
 
==External links==
*[http://eqworld.ipmnet.ru/en/auxiliary/aux-inttrans.htm Tables of Integral Transforms] at EqWorld: The World of Mathematical Equations.
+
* [http://eqworld.ipmnet.ru/en/auxiliary/aux-inttrans.htm Tables of Integral Transforms] at EqWorld: The World of Mathematical Equations.
*[http://cns-alumni.bu.edu/~slehar/fourier/fourier.html An Intuitive Explanation of Fourier Theory] by Steven Lehar.
+
* [http://cns-alumni.bu.edu/~slehar/fourier/fourier.html An Intuitive Explanation of Fourier Theory] by Steven Lehar.
  +
* [http://www.archive.org/details/Lectures_on_Image_Processing 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
  +
* {{cite web|last=Moriarty|first=Philip|title=∑ Summation (and Fourier Analysis)|url=http://www.sixtysymbols.com/videos/summation.htm|work=Sixty Symbols|publisher=[[Brady Haran]] for the [[University of Nottingham]]|coauthors=Bowley, Roger|year=2009}}
   
[[Category:Fourier analysis]]
+
{{DEFAULTSORT:Fourier Analysis}}
[[Category:Integral transforms]]
+
[[Category:Fourier analysis|*]]
 
[[Category:Digital signal processing]]
 
[[Category:Digital signal processing]]
  +
[[Category:Mathematics]]
  +
[[Category:Time series analysis]]
   
[[ar:تحويل فوريي]]
+
{{enWP|Fourier analysis}}
[[be:Пераўтварэнне Фур'е]]
 
[[cs:Fourierova transformace]]
 
[[de:Fourier-Transformation]]
 
[[es:Transformada de Fourier]]
 
[[eu:Fourierren transformaketa]]
 
[[fa:تبدیل فوریه]]
 
[[fr:Transformée de Fourier]]
 
[[gl:Transformada de Fourier]]
 
[[id:Transformasi Fourier]]
 
[[is:Fourier–vörpun]]
 
[[it:Trasformata di Fourier]]
 
[[nl:Fouriertransformatie]]
 
[[ja:フーリエ変換]]
 
[[pl:Transformacja Fouriera]]
 
[[pt:Transformada de Fourier]]
 
[[ru:Преобразование Фурье]]
 
[[sr:Фуријеова трансформација]]
 
[[fi:Fourier'n muunnos]]
 
[[sv:Fourier-transform]]
 
[[th:การแปลงฟูริเยร์]]
 
[[vi:Biến đổi Fourier]]
 
[[tr:Fourier dönüşümü]]
 
[[zh:傅里叶变换]]
 
{{enWP| Fourier analysis}}
 

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In mathematics, Fourier analysis is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph Fourier.

Today, the subject of Fourier analysis encompasses a vast spectrum of mathematics. In the sciences the process of decomposing a function into simpler pieces is often called Fourier analysis, while the operation of rebuilding the function from these pieces is known as Fourier synthesis. In mathematics, the term Fourier analysis often refers to the study of both operations.

The decomposition process itself is called a Fourier transform. The transform is often given a more specific name which depends upon the domain and other properties of the function being transformed. 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. Each transform used for analysis (see list of Fourier-related transforms) has a corresponding inverse transform that can be used for synthesis.

ApplicationsEdit

Fourier analysis has many applications of interest to psychologists – in signal processing, probability theory, statistics, acoustics, optics, diffraction, structure and other areas.

So, for example, breaking up a time series or a periodic phenomena such as a sound wave into its sinusoidal components of different frequencies and amplitudes and fitting them to a Fourier transform makes them amenable to calculation

This wide applicability stems from many useful properties of the transforms:

Fourier transformation is also useful as a compact representation of a signal. For example, JPEG compression uses a variant of the Fourier transformation (discrete cosine transform) 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 image square is reassembled from the preserved approximate Fourier-transformed components, which are then inverse-transformed to produce an approximation of the original image.

Applications in signal processingEdit

When processing signals, such as audio, radio waves, light waves, seismic waves, and even images, Fourier analysis can isolate individual components of a compound waveform, concentrating them for easier detection and/or removal. A large family of signal processing techniques consist of Fourier-transforming a signal, manipulating the Fourier-transformed data in a simple way, and reversing the transformation.

Some examples include:

Variants of Fourier analysisEdit

File:Variations of the Fourier transform.tif

(Continuous) Fourier transform Edit

Main article: Fourier Transform

Most often, the unqualified term Fourier transform refers to the transform of functions of a continuous real argument, and it produces a continuous function of frequency, known as a frequency distribution. One function is transformed into another, and the operation is reversible. When the domain of the input (initial) function is time (t), and the domain of the output (final) function is ordinary frequency, the transform of function s(t) at frequency ƒ is given by the complex number:

S(f) = \int_{-\infty}^{\infty} s(t) \cdot e^{- i 2\pi f t} dt.

Evaluating this quantity for all values of ƒ produces the frequency-domain function. Then s(t) can be represented as a recombination of complex exponentials of all possible frequencies:

s(t) = \int_{-\infty}^{\infty} S(f) \cdot e^{i 2\pi f t} df,

which is the inverse transform formula. The complex number, S(ƒ), conveys both amplitude and phase of frequency ƒ.

See Fourier transform for much more information, including:

  • conventions for amplitude normalization and frequency scaling/units
  • transform properties
  • tabulated transforms of specific functions
  • an extension/generalization for functions of multiple dimensions, such as images.

Fourier series Edit

Main article: Fourier series

The Fourier transform of a periodic function, sP(t), with period P, becomes a Dirac comb function, modulated by a sequence of complex coefficients:

S[k] = \frac{1}{P}\int_{P} s_P(t)\cdot e^{-i 2\pi \frac{k}{P} t}\, dt     for all integer values of k,

and where \scriptstyle \int_P  is the integral over any interval of length P.

The inverse transform, known as Fourier series, is a representation of sP(t) in terms of a summation of a potentially infinite number of harmonically related sinusoids or complex exponential functions, each with an amplitude and phase specified by one of the coefficients:

s_P(t)=\sum_{k=-\infty}^\infty S[k]\cdot e^{i 2\pi \frac{k}{P} t} \quad\stackrel{\mathcal{F}}{\Longleftrightarrow}\quad \sum_{k=-\infty}^{+\infty} S[k]\ \delta \left(f-\frac{k}{P}\right).

When sP(t), is expressed as a periodic summation of another function, s(t):   s_P(t)\ \stackrel{\text{def}}{=}\ \sum_{k=-\infty}^{\infty} s(t-kP),

the coefficients are proportional to samples of S(ƒ) at discrete intervals of 1/P:   S[k] =\frac{1}{P}\cdot S\left(\frac{k}{P}\right).\,[note 1]

A sufficient condition for recovering s(t) (and therefore S(ƒ)) from just these samples is that the non-zero portion of s(t) be confined to a known interval of duration P, which is the frequency domain dual of the Nyquist–Shannon sampling theorem.

See Fourier series for more information, including the historical development.

Discrete-time Fourier transform (DTFT) Edit

Main article: Discrete-time Fourier transform

The DTFT is the mathematical dual of the time-domain Fourier series. Thus, a convergent periodic summation in the frequency domain can be represented by a Fourier series, whose coefficients are samples of a related continuous time function:

S_{1/T}(f)\ \stackrel{\text{def}}{=}\ \underbrace{\sum_{k=-\infty}^{\infty} S\left(f - \frac{k}{T}\right) \equiv \overbrace{\sum_{n=-\infty}^{\infty} s[n] \cdot e^{-i 2\pi f n T}}^{\text{Fourier series (DTFT)}}}_{\text{Poisson summation formula}} = \mathcal{F} \left \{ \sum_{n=-\infty}^{\infty} s[n]\ \delta(t-nT)\right \},\,

which is known as the DTFT. Thus the DTFT of the s[n] sequence is also the Fourier transform of the modulated Dirac comb function.[note 2]

The Fourier series coefficients, defined by:

s[n]\ \stackrel{\mathrm{def}}{=} \ T \int_{1/T} S_{1/T}(f)\cdot e^{i 2\pi f nT} df = T \underbrace{\int_{-\infty}^{\infty} S(f)\cdot e^{i 2\pi f nT} df}_{\stackrel{\mathrm{def}}{=} \ s(nT)}\,

is the inverse transform. With s[n] = T•s(nT), this Fourier series can now be recognized as a form of the Poisson summation formula. Thus we have the important result that when a discrete data sequence, s[n], is proportional to samples of an underlying continuous function, s(t), one can observe a periodic summation of the continuous Fourier transform, S(ƒ). That is a cornerstone in the foundation of digital signal processing. Furthermore, under certain idealized conditions one can theoretically recover S(ƒ) and s(t) exactly. A sufficient condition for perfect recovery is that the non-zero portion of S(ƒ) be confined to a known frequency interval of width 1/T. When that interval is [-0.5/T, 0.5/T], the applicable reconstruction formula is the Whittaker–Shannon interpolation formula.

Another reason to be interested in S1/T(ƒ) is that it often provides insight into the amount of aliasing caused by the sampling process.

Applications of the DTFT are not limited to sampled functions. See Discrete-time Fourier transform for more information on this and other topics, including:

  • normalized frequency units
  • windowing (finite-length sequences)
  • transform properties
  • tabulated transforms of specific functions

Discrete Fourier transform (DFT) Edit

Main article: Discrete Fourier transform

The DTFT of a periodic sequence, sN[n], with period N, becomes another Dirac comb function, modulated by the coefficients of a Fourier series.  And the integral formula for the coefficients simplifies to a summation (see DTFT/Periodic data):

S_N[k] =\frac{1}{NT} \underbrace{\sum_N s_N[n]\cdot e^{-i 2\pi \frac{k}{N} n}}_{S_k}\,,     where \scriptstyle \sum_N  is the sum over any n-sequence of length N.

The Sk sequence is what's customarily known as the DFT of sN.  It is also N-periodic, so it is never necessary to compute more than N coefficients.  In terms of Sk, the inverse transform is given by:

s_N[n] = \frac{1}{N} \sum_{N} S_k\cdot e^{i 2\pi \frac{n}{N}k},\,     where \scriptstyle \sum_N  is the sum over any k-sequence of length N.

When sN[n] is expressed as a periodic summation of another function:  s_N[n]\ \stackrel{\text{def}}{=}\ \sum_{k=-\infty}^{\infty} s[n-kN],   and   s[n]\ \stackrel{\text{def}}{=}\ T\cdot s(nT),\,

the coefficients are equivalent to samples of S1/T(ƒ) at discrete intervals of 1/P = 1/NT:   S_k = S_{1/T}(k/P).\,   (see DTFT/Sampling the DTFT)

Conversely, when one wants to compute an arbitrary number (N) of discrete samples of one cycle of a continuous DTFT,  S_{1/T}(f),\,  it can be done by computing the relatively simple DFT of sN[n], as defined above.  In most cases, N is chosen equal to the length of non-zero portion of s[n].  Increasing N, known as zero-padding or interpolation, results in more closely spaced samples of one cycle of  S1/T(ƒ).  Decreasing N, causes overlap (adding) in the time-domain (analogous to aliasing), which corresponds to decimation in the frequency domain. (see Sampling the DTFT) In most cases of practical interest, the s[n] sequence represents a longer sequence that was truncated by the application of a finite-length window function or FIR filter array.

The DFT can be computed using a fast Fourier transform (FFT) algorithm, which makes it a practical and important transformation on computers.

See Discrete Fourier transform for much more information, including:

  • transform properties
  • applications
  • tabulated transforms of specific functions

SummaryEdit

For periodic functions, both the Fourier transform and the DTFT comprise only a discrete set of frequency components (Fourier series), and the transforms diverge at those frequencies. One common practice (not discussed above) is to handle that divergence via Dirac delta and Dirac comb functions. But the same spectral information can be discerned from just one cycle of the periodic function, since all the other cycles are identical. Similarly, finite-duration functions can be represented as a Fourier series, with no actual loss of information except that the periodicity of the inverse transform is a mere artifact. We also note that none of the formulas here require the duration of s\, to be limited to the period, P or N.  But that is a common situation, in practice.

s(t)\,  transforms (continuous-time)
Continuous frequency Discrete frequencies
Transform S(f)\ \stackrel{\text{def}}{=}\ \int_{-\infty}^{\infty} s(t)\ e^{-i 2\pi f t}\,dt \underbrace{\frac{1}{P}\cdot S\left(\frac{k}{P}\right)}_{S[k]}\ \stackrel{\text{def}}{=}\ \frac{1}{P} \int_{-\infty}^{\infty} s(t)\ e^{-i 2\pi \frac{k}{P} t}\,dt \equiv \frac{1}{P} \int_P s_P(t)\ e^{-i 2\pi \frac{k}{P} t}\,dt
Inverse s(t) = \int_{-\infty}^{\infty} S(f)\ e^{ i 2 \pi f t}\,df \underbrace{s_P(t) = \sum_{k=-\infty}^{\infty} S[k] \cdot e^{i 2\pi \frac{k}{P} t}}_{\text{Poisson summation formula (Fourier series)}}
s[n]\,  transforms (discrete-time)
Continuous frequency Discrete frequencies
Transform \underbrace{S_{1/T}(f) = \sum_{n=-\infty}^{\infty} \overbrace{T\cdot s(nT)}^{s[n]}\cdot e^{-i 2\pi f nT}}_{\text{Poisson summation formula (DTFT)}} \underbrace{\overbrace{S_{1/T}\left(\frac{k}{NT}\right)}^{S_k} = \sum_{n=-\infty}^{\infty} s[n]\cdot e^{-i 2\pi \frac{kn}{N}}}_{\text{Poisson summation formula}} \equiv \underbrace{\sum_{N} s_N[n]\cdot e^{-i 2\pi \frac{kn}{N}}}_{\text{DFT}}
Inverse s[n] = T \int_{1/T} S_{1/T}(f)\cdot e^{i 2\pi f nT} \,df

\sum_{n=-\infty}^{\infty} s[n]\cdot \delta(t-nT) = \underbrace{\int_{-\infty}^{\infty} S_{1/T}(f)\cdot e^{i 2\pi f t}\,df}_{\text{inverse Fourier transform}}

s_N[n] = \underbrace{\frac{1}{N} \sum_{N} S_k\cdot e^{i 2\pi \frac{kn}{N}}}_{\text{inverse DFT}}

s_P(nT) = \frac{1}{T}\cdot s_N[n] = \sum_{N} \underbrace{\frac{1}{P}\cdot S_{1/T}\left(\frac{k}{P}\right)}_{S_{N}[k]} \cdot e^{i 2\pi \frac{kn}{N}}

Fourier transforms on arbitrary locally compact abelian topological groupsEdit

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.

Time–frequency transformsEdit

For more details on this topic, see Time–frequency analysis.

In signal processing terms, a function (of time) is a representation of a signal with perfect time resolution, but no frequency information, while the Fourier transform has perfect frequency resolution, but no time information.

As alternatives to the Fourier transform, in time–frequency analysis, one uses time–frequency transforms to represent signals in a form that has some time information and some frequency information – by the uncertainty principle, there is a trade-off between these. These can be generalizations of the Fourier transform, such as the short-time Fourier transform, the Gabor transform or fractional Fourier transform (FRFT), or can use different functions to represent signals, as in wavelet transforms and chirplet transforms, with the wavelet analog of the (continuous) Fourier transform being the continuous wavelet transform.

History Edit

A primitive form of harmonic series dates back to ancient Babylonian mathematics, where they were used to compute ephemerides (tables of astronomical positions).[1] The classical Greek concepts of deferent and epicycle in the Ptolemaic system of astronomy were related to Fourier series (see Deferent and epicycle: Mathematical formalism).

In modern times, variants of the discrete Fourier transform were used by Alexis Clairaut in 1754 to compute an orbit,[2] which has been described as the first formula for the DFT,[3] and in 1759 by Joseph Louis Lagrange, in computing the coefficients of a trigonometric series for a vibrating string.[4] Technically, Clairaut's work was a cosine-only series (a form of discrete cosine transform), while Lagrange's work was a sine-only series (a form of discrete sine transform); a true cosine+sine DFT was used by Gauss in 1805 for trigonometric interpolation of asteroid orbits.[5] Euler and Lagrange both discretized the vibrating string problem, using what would today be called samples.[4]

An early modern development toward Fourier analysis was the 1770 paper Réflexions sur la résolution algébrique des équations by Lagrange, which in the method of Lagrange resolvents used a complex Fourier decomposition to study the solution of a cubic:[6] Lagrange transformed the roots x_1,x_2,x_3 into the resolvents:

\begin{align}
r_1 &= x_1 + x_2 + x_3\\
r_2 &= x_1 + \zeta x_2 + \zeta^2 x_3\\
r_3 &= x_1 + \zeta^2 x_2 + \zeta x_3
\end{align}

where ζ is a cubic root of unity, which is the DFT of order 3.

A number of authors, notably Jean le Rond d'Alembert, and Carl Friedrich Gauss used trigonometric series to study the heat equation,[citation needed] but the breakthrough development was the 1807 paper Mémoire sur la propagation de la chaleur dans les corps solides by Joseph Fourier, whose crucial insight was to model all functions by trigonometric series, introducing the Fourier series.

Historians are divided as to how much to credit Lagrange and others for the development of Fourier theory: Daniel Bernoulli and Leonhard Euler had introduced trigonometric representations of functions,[3] and Lagrange had given the Fourier series solution to the wave equation,[3] so Fourier's contribution was mainly the bold claim that an arbitrary function could be represented by a Fourier series.[3]

The subsequent development of the field is known as harmonic analysis, and is also an early instance of representation theory.

The first fast Fourier transform (FFT) algorithm for the DFT was discovered around 1805 by Carl Friedrich Gauss when interpolating measurements of the orbit of the asteroids Juno and Pallas, although that particular FFT algorithm is more often attributed to its modern rediscoverers Cooley and Tukey.[5][7]

Interpretation in terms of time and frequencyEdit

In signal processing, the Fourier transform often takes a time series or a function of continuous time, and maps it into a frequency spectrum. That is, it takes a function from the time domain into the frequency domain; it is a decomposition of a function into sinusoids of different frequencies; in the case of a Fourier series or discrete Fourier transform, the sinusoids are harmonics of the fundamental frequency of the function being analyzed.

When the function ƒ 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 the phase of F.

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. This justifies their use in branches such diverse as image processing, heat conduction, and automatic control.

NotesEdit

  1. \int_{P} \left[\sum_{k=-\infty}^{\infty} s(t-kP)\right] \cdot e^{-i 2\pi \frac{k}{P} t} dt = \underbrace{\int_{-\infty}^{\infty} s(t) \cdot e^{-i 2\pi \frac{k}{P} t} dt}_{\stackrel{\mathrm{def}}{=}\ S(k/P)}
  2. We may also note that:   \scriptstyle
\sum_{n=-\infty}^{+\infty} T\ s(nT)\ \delta(t-nT)\ =\ \sum_{n=-\infty}^{+\infty} T\ s(t)\ \delta(t-nT)\ =\ s(t)\cdot T \sum_{n=-\infty}^{+\infty} \delta(t-nT).
    Consequently, a common practice is to model "sampling" as a multiplication by the Dirac comb function, which of course is only "possible" in a purely mathematical sense.

See alsoEdit

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CitationsEdit

  1. Prestini, Elena (2004), The evolution of applied harmonic analysis: models of the real world, Birkhäuser, ISBN 978-0-8176-4125-2, http://books.google.com/?id=fye--TBu4T0C , p. 62
    Rota, Gian-Carlo; Palombi, Fabrizio (1997), Indiscrete thoughts, Birkhäuser, ISBN 978-0-8176-3866-5, http://books.google.com/?id=H5smrEExNFUC , p. 11
    Neugebauer, Otto (1969) [1957], The Exact Sciences in Antiquity (2 ed.), Dover Publications, ISBN 978-0-486-22332-2, http://books.google.com/?id=JVhTtVA2zr8C 
    Brack-Bernsen, Lis; Brack, Matthias, Analyzing shell structure from Babylonian and modern times 
  2. Terras, Audrey (1999), Fourier analysis on finite groups and applications, Cambridge University Press, ISBN 978-0-521-45718-7, http://books.google.com/?id=-B2TA669dJMC , p. 30
  3. 3.0 3.1 3.2 3.3 Briggs, William L.; Henson, Van Emden (1995), The DFT : an owner's manual for the discrete Fourier transform, SIAM, ISBN 978-0-89871-342-8, http://books.google.com/?id=coq49_LRURUC , p. 4
  4. 4.0 4.1 Briggs, William L.; Henson, Van Emden (1995), The DFT: an owner's manual for the discrete Fourier transform, SIAM, ISBN 978-0-89871-342-8, http://books.google.com/?id=coq49_LRURUC , p. 2
  5. 5.0 5.1 Heideman, M. T., D. H. Johnson, and C. S. Burrus, "Gauss and the history of the fast Fourier transform," IEEE ASSP Magazine, 1, (4), 14–21 (1984)
  6. Knapp, Anthony W. (2006), Basic algebra, Springer, ISBN 978-0-8176-3248-9, http://books.google.com/?id=KVeXG163BggC , p. 501
  7. Terras, Audrey (1999), Fourier analysis on finite groups and applications, Cambridge University Press, ISBN 978-0-521-45718-7, http://books.google.com/?id=-B2TA669dJMC , p. 31

ReferencesEdit

External linksEdit

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