Editing Fourier transform (section)

=== Basic properties ===
The Fourier transform has the following basic properties:<ref name="Pinsky-2002">{{harvnb|Pinsky|2002}}</ref>

==== Linearity ====

<math display="block">a\ f(x) + b\ h(x)\ \ \stackrel{\mathcal{F}}{\Longleftrightarrow}\ \ a\ \widehat f(\xi) + b\ \widehat h(\xi);\quad \ a,b \in \mathbb C</math>

==== Time shifting ====

<math display="block">f(x-x_0)\ \ \stackrel{\mathcal{F}}{\Longleftrightarrow}\ \ e^{-i 2\pi x_0 \xi}\ \widehat f(\xi);\quad \ x_0 \in \mathbb R</math>

==== Frequency shifting ====

<math display="block">e^{i 2\pi \xi_0 x} f(x)\ \ \stackrel{\mathcal{F}}{\Longleftrightarrow}\ \ \widehat f(\xi - \xi_0);\quad \ \xi_0 \in \mathbb R</math>

==== Time scaling ====

<math display="block">f(ax)\ \ \stackrel{\mathcal{F}}{\Longleftrightarrow}\ \ \frac{1}{|a|}\widehat{f}\left(\frac{\xi}{a}\right);\quad \ a \ne 0 </math>
The case <math>a=-1</math> leads to the ''time-reversal property'':
<math display="block">f(-x)\ \ \stackrel{\mathcal{F}}{\Longleftrightarrow}\ \ \widehat f (-\xi)</math>

<div class="skin-invert">{{Annotated image
| caption=The transform of an even-symmetric real-valued function <math>(f(t) = f_{RE})</math> is also an even-symmetric real-valued function <math>(\hat f_{RE}).</math>  The time-shift, <math>(g(t) = g_{RE} + g_{RO}),</math> creates an imaginary component, <math>i\cdot \hat g_{IO}.</math>  (see {{slink||Symmetry}}.
| image=Fourier_unit_pulse.svg
| image-width = 300
| outer-css = color: black;
| annotations =
{{Annotation|20|40|<math>\scriptstyle f(t)</math>}}
{{Annotation|170|40|<math>\scriptstyle \widehat{f}(\omega)</math>}}
{{Annotation|20|140|<math>\scriptstyle g(t)</math>}}
{{Annotation|170|140|<math>\scriptstyle \widehat{g}(\omega)</math>}}
{{Annotation|130|80|<math>\scriptstyle t</math>}}
{{Annotation|280|85|<math>\scriptstyle \omega</math>}}
{{Annotation|130|192|<math>\scriptstyle t</math>}}
{{Annotation|280|180|<math>\scriptstyle \omega</math>}}
}}</div>

==== Symmetry ====
When the real and imaginary parts of a complex function are decomposed into their [[Even and odd functions#Even–odd decomposition|even and odd parts]], there are four components, denoted below by the subscripts RE, RO, IE, and IO.  And there is a one-to-one mapping between the four components of a complex time function and the four components of its complex frequency transform:<ref name="ProakisManolakis1996">{{cite book|last1=Proakis|first1=John G. |last2=Manolakis|first2=Dimitris G.|author2-link= Dimitris Manolakis |title=Digital Signal Processing: Principles, Algorithms, and Applications|url=https://archive.org/details/digitalsignalpro00proa|url-access=registration|year=1996|publisher=Prentice Hall|isbn=978-0-13-373762-2|edition=3rd|page=[https://archive.org/details/digitalsignalpro00proa/page/291 291]}}</ref>

<math>
\begin{array}{rlcccccccc}
\mathsf{Time\ domain} & f & = & f_{_{\text{RE}}} & + & f_{_{\text{RO}}} & + & i\ f_{_{\text{IE}}} & + & \underbrace{i\ f_{_{\text{IO}}}} \\
&\Bigg\Updownarrow\mathcal{F} & &\Bigg\Updownarrow\mathcal{F} & &\ \ \Bigg\Updownarrow\mathcal{F} & &\ \ \Bigg\Updownarrow\mathcal{F} & &\ \ \Bigg\Updownarrow\mathcal{F}\\
\mathsf{Frequency\ domain} & \widehat f & = & \widehat f_{_\text{RE}} & + & \overbrace{i\ \widehat f_{_\text{IO}}\,} & + & i\ \widehat f_{_\text{IE}} & + & \widehat f_{_\text{RO}}
\end{array}
</math>

From this, various relationships are apparent, for example''':'''
* The transform of a real-valued function <math>(f_{_{RE}}+f_{_{RO}})</math> is the ''[[Even and odd functions#Complex-valued functions|conjugate symmetric]]'' function <math>\hat f_{RE}+i\ \hat f_{IO}.</math>  Conversely, a ''conjugate symmetric'' transform implies a real-valued time-domain.
* The transform of an imaginary-valued function <math>(i\ f_{_{IE}}+i\ f_{_{IO}})</math> is the ''[[Even and odd functions#Complex-valued functions|conjugate antisymmetric]]'' function <math>\hat f_{RO}+i\ \hat f_{IE},</math> and the converse is true.
* The transform of a ''[[Even and odd functions#Complex-valued functions|conjugate symmetric]]'' function <math>(f_{_{RE}}+i\ f_{_{IO}})</math> is the real-valued function <math>\hat f_{RE}+\hat f_{RO},</math> and the converse is true.
* The transform of a ''[[Even and odd functions#Complex-valued functions|conjugate antisymmetric]]'' function <math>(f_{_{RO}}+i\ f_{_{IE}})</math> is the imaginary-valued function <math>i\ \hat f_{IE}+i\hat f_{IO},</math> and the converse is true.

==== Conjugation ====
<math display="block">\bigl(f(x)\bigr)^*\ \ \stackrel{\mathcal{F}}{\Longleftrightarrow}\ \ \left(\widehat{f}(-\xi)\right)^*</math>
(Note: the ∗ denotes [[Complex conjugate|complex conjugation]].)

In particular, if <math>f</math> is '''real''', then <math>\widehat f</math> is [[Even and odd functions#Complex-valued functions|even symmetric]] (aka [[Hermitian function]]):
<math display="block">\widehat{f}(-\xi)=\bigl(\widehat f(\xi)\bigr)^*.</math>

And if <math>f</math> is purely imaginary, then <math>\widehat f</math> is [[Even and odd functions#Complex-valued functions|odd symmetric]]:
<math display="block">\widehat f(-\xi) = -(\widehat f(\xi))^*.</math>

==== Real and imaginary parts ====
<math display="block">\operatorname{Re}\{f(x)\}\ \ \stackrel{\mathcal{F}}{\Longleftrightarrow}\ \ 
\tfrac{1}{2} \left( \widehat f(\xi) + \bigl(\widehat f (-\xi) \bigr)^* \right)</math>
<math display="block">\operatorname{Im}\{f(x)\}\ \ \stackrel{\mathcal{F}}{\Longleftrightarrow}\ \ 
\tfrac{1}{2i} \left( \widehat f(\xi) - \bigl(\widehat f (-\xi) \bigr)^* \right)</math>

==== Zero frequency component ====
Substituting <math>\xi = 0</math> in the definition, we obtain:
<math display="block">\widehat{f}(0) = \int_{-\infty}^{\infty} f(x)\,dx.</math>

The integral of <math>f</math> over its domain is known as the average value or [[DC bias]] of the function.