Editing Fourier transform (section)

== Computation methods ==
The appropriate computation method largely depends how the original mathematical function is represented and the desired form of the output function.  In this section we consider both functions of a continuous variable, <math>f(x),</math> and functions of a discrete variable (i.e. ordered pairs of <math>x</math> and <math>f</math> values).  For discrete-valued <math>x,</math> the transform integral becomes a summation of sinusoids, which is still a continuous function of frequency (<math>\xi</math> or <math>\omega</math>).  When the sinusoids are harmonically related (i.e. when the <math>x</math>-values are spaced at integer multiples of an interval), the transform is called [[discrete-time Fourier transform]] (DTFT).

=== Discrete Fourier transforms and fast Fourier transforms ===
Sampling the DTFT at equally-spaced values of frequency is the most common modern method of computation.  Efficient procedures, depending on the frequency resolution needed, are described at {{slink|Discrete-time Fourier transform|Sampling the DTFT|nopage=n}}.  The [[discrete Fourier transform]] (DFT), used there, is usually computed by a [[fast Fourier transform]] (FFT) algorithm.

=== Analytic integration of closed-form functions ===
Tables of [[closed-form expression|closed-form]] Fourier transforms, such as {{slink||Square-integrable functions, one-dimensional}} and {{slink|Discrete-time Fourier transform|Table of discrete-time Fourier transforms|nopage=y}}, are created by mathematically evaluating the Fourier analysis integral (or summation) into another closed-form function of frequency (<math>\xi</math> or <math>\omega</math>).<ref name="Zwillinger-2014">{{harvnb|Gradshteyn|Ryzhik|Geronimus|Tseytlin|2015}}</ref>  When mathematically possible, this provides a transform for a continuum of frequency values.

Many computer algebra systems such as [[Matlab]] and [[Mathematica]] that are capable of [[symbolic integration]] are capable of computing Fourier transforms analytically. For example, to compute the Fourier transform of {{math|1=cos(6π''t'') ''e''<sup>−π''t''<sup>2</sup></sup>}} one might enter the command {{code|integrate cos(6*pi*t) exp(−pi*t^2) exp(-i*2*pi*f*t) from -inf to inf}} into [[Wolfram Alpha]].<ref group=note>The direct command {{code|fourier transform of cos(6*pi*t) exp(−pi*t^2)}} would also work for Wolfram Alpha, although the options for the convention (see {{Section link|2=Other_conventions}}) must be changed away from the default option, which is actually equivalent to {{code|integrate cos(6*pi*t) exp(−pi*t^2) exp(i*omega*t) /sqrt(2*pi) from -inf to inf}}.</ref>

=== Numerical integration of closed-form continuous functions ===
Discrete sampling of the Fourier transform can also be done by [[numerical integration]] of the definition at each value of frequency for which transform is desired.<ref>{{harvnb|Press|Flannery|Teukolsky|Vetterling|1992}}</ref><ref>{{harvnb|Bailey|Swarztrauber|1994}}</ref><ref>{{harvnb|Lado|1971}}</ref>  The numerical integration approach works on a much broader class of functions than the analytic approach.

=== Numerical integration of a series of ordered pairs ===
If the input function is a series of ordered pairs, numerical integration reduces to just a summation over the set of data pairs.<ref>{{harvnb|Simonen|Olkkonen|1985}}</ref>  The DTFT is a common subcase of this more general situation.