Editing Fourier transform (section)

== Background ==
=== History ===
{{Main|Fourier analysis#History|Fourier series#History}}
In 1822, Fourier claimed (see {{Slink|Joseph Fourier|The Analytic Theory of Heat}}) that any function, whether continuous or discontinuous, can be expanded into a series of sines.<ref>{{harvnb|Fourier|1822}}</ref> That important work was corrected and expanded upon by others to provide the foundation for the various forms of the Fourier transform used since.

=== Complex sinusoids ===
<div class="skin-invert-image">{{multiple image
| total_width       = 300
| align             = right
| image1            = Sine voltage.svg
| image2            = Phase shift.svg
| footer            =
The red [[sine wave|sinusoid]] can be described by peak amplitude (1), peak-to-peak (2), [[root mean square|RMS]] (3), and [[wavelength]] (4). The red and blue sinusoids have a phase difference of {{mvar|θ}}.
}}</div>
In general, the coefficients <math>\widehat f(\xi)</math> are complex numbers, which have two equivalent forms (see [[Euler's formula]]):
<math display="block"> \widehat f(\xi) = \underbrace{A e^{i \theta}}_{\text{polar coordinate form}}
= \underbrace{A \cos(\theta) + i A \sin(\theta)}_{\text{rectangular coordinate form}}.</math>

The product with <math>e^{i 2 \pi \xi x}</math> ({{EquationNote|Eq.2}}) has these forms:
<math display="block">\begin{aligned}\widehat f(\xi)\cdot e^{i 2 \pi \xi x}
&= A e^{i \theta} \cdot e^{i 2 \pi \xi x}\\
&= \underbrace{A e^{i (2 \pi \xi x+\theta)}}_{\text{polar coordinate form}}\\
&= \underbrace{A\cos(2\pi \xi x +\theta) + i A\sin(2\pi \xi x +\theta)}_{\text{rectangular coordinate form}}.\end{aligned}</math>
which conveys both [[amplitude]] and [[phase offset|phase]] of frequency <math>\xi.</math> Likewise, the intuitive interpretation of {{EquationNote|Eq.1}} is that multiplying <math>f(x)</math> by <math>e^{-i 2\pi \xi x}</math> has the effect of subtracting <math>\xi</math> from every frequency component of function <math>f(x).</math><ref group="note">A possible source of confusion is the [[#Frequency shifting|frequency-shifting property]]; i.e. the transform of function <math>f(x)e^{-i 2\pi \xi_0 x}</math> is <math>\widehat{f}(\xi+\xi_0).</math>&nbsp; The value of this function at &nbsp;<math>\xi=0</math>&nbsp; is <math>\widehat{f}(\xi_0),</math> meaning that a frequency <math>\xi_0</math> has been shifted to zero (also see [[Negative frequency#Simplifying the Fourier transform|Negative frequency]]).</ref> Only the component that was at frequency <math>\xi</math> can produce a non-zero value of the infinite integral, because (at least formally) all the other shifted components are oscillatory and integrate to zero. (see {{slink||Example}})

It is noteworthy how easily the product was simplified using the polar form, and how easily the rectangular form was deduced by an application of Euler's formula.

=== Negative frequency ===
{{See also|Negative frequency#Simplifying the Fourier transform|l1=Negative frequency § Simplifying the Fourier transform}}

Euler's formula introduces the possibility of negative <math>\xi.</math>&nbsp;  And {{EquationNote|Eq.1}} is defined <math>\forall \xi \in \mathbb{R}.</math>  Only certain complex-valued <math> f(x)</math> have transforms <math> \widehat f =0, \ \forall \ \xi < 0</math>  (See [[Analytic signal]].  A simple example is <math> e^{i 2 \pi \xi_0 x}\ (\xi_0 > 0).</math>)&nbsp; But negative frequency is necessary to characterize all other complex-valued <math> f(x),</math> found in [[signal processing]], [[partial differential equations]], [[radar]], [[nonlinear optics]], [[quantum mechanics]], and others.

For a real-valued <math> f(x),</math>  {{EquationNote|Eq.1}} has the symmetry property <math>\widehat f(-\xi) = \widehat {f}^* (\xi)</math>  (see {{slink||Conjugation}} below). This redundancy enables {{EquationNote|Eq.2}} to distinguish <math>f(x) = \cos(2 \pi \xi_0 x)</math> from <math>e^{i2 \pi \xi_0 x}.</math>&nbsp;  But of course it cannot tell us the actual sign of <math>\xi_0,</math> because  <math>\cos(2 \pi \xi_0 x)</math> and <math>\cos(2 \pi (-\xi_0) x)</math> are indistinguishable on just the real numbers line.

=== Fourier transform for periodic functions ===
The Fourier transform of a periodic function cannot be defined using the integral formula directly.  In order for integral in {{EquationNote|Eq.1}} to be defined the function must be [[Absolutely integrable function|absolutely integrable]].  Instead it is common to use [[Fourier series]].  It is possible to extend the definition to include periodic functions by viewing them as [[Distribution (mathematics)#Tempered distributions|tempered distributions]].

This makes it possible to see a connection between the [[Fourier series]] and the Fourier transform for periodic functions that have a [[Convergence of Fourier series|convergent Fourier series]].  If <math>f(x)</math> is a [[periodic function]], with period <math>P</math>, that has a convergent Fourier series, then:
<math display="block">
\widehat{f}(\xi) = \sum_{n=-\infty}^\infty c_n \cdot \delta \left(\xi - \tfrac{n}{P}\right),
</math>
where <math>c_n</math> are the Fourier series coefficients of <math>f</math>, and <math>\delta</math> is the [[Dirac delta function]].  In other words, the Fourier transform is a [[Dirac comb]] function whose ''teeth'' are multiplied by the Fourier series coefficients.

=== Sampling the Fourier transform ===
{{Broader|Poisson summation formula}}
The Fourier transform of an [[Absolutely integrable function|integrable]] function <math>f</math> can be sampled at regular intervals of arbitrary length <math>\tfrac{1}{P}.</math> These samples can be deduced from one cycle of a periodic function <math>f_P</math> which has [[Fourier series]] coefficients proportional to those samples by the [[Poisson summation formula]]:
<math display="block">f_P(x) \triangleq \sum_{n=-\infty}^{\infty} f(x+nP) = \frac{1}{P}\sum_{k=-\infty}^{\infty} \widehat f\left(\tfrac{k}{P}\right) e^{i2\pi \frac{k}{P} x}, \quad \forall k \in \mathbb{Z}</math>

The integrability of <math>f</math> ensures the periodic summation converges.  Therefore, the samples <math>\widehat f\left(\tfrac{k}{P}\right)</math> can be determined by Fourier series analysis:
<math display="block">\widehat f\left(\tfrac{k}{P}\right) = \int_{P} f_P(x) \cdot e^{-i2\pi \frac{k}{P} x} \,dx.</math>

When <math>f(x)</math> has [[compact support]], <math>f_P(x)</math> has a finite number of terms within the interval of integration.  When <math>f(x)</math> does not have compact support, numerical evaluation of <math>f_P(x)</math> requires an approximation, such as tapering <math>f(x)</math> or truncating the number of terms.