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{{ sidebar
| name     = Random variables
| class    = hlist
| image    = [[File:CorrelationIcon.svg|100px]]
| title    = [[Random variable|Random variables]]
| pretitle = Part of a series on [[Statistics]]

| content1 = '''Correlation and covariance of random vectors'''
* [[Expected value]]

* [[Variance]]

* [[Covariance]]

* [[Central moment]]

* [[Standardized moment]]

| content2 = '''Correlation and covariance of stochastic processes'''

* [[Conditional independence]]

* [[Independence (probability theory)]]

* [[Uncorrelatedness (probability theory)]]

| content3 = '''Distribution functions'''
* [[Cumulative distribution function]]

* [[Characteristic function (probability theory)]]

* [[Moment-generating function]]

* [[Probability density function]]

}}<noinclude>{{doc|content=}}

[[:Category:Statistics templates]]

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{{ sidebar
| name     = Stochastic processes
| class    = hlist
| image    = [[File:CorrelationIcon.svg|100px]]
| title    = [[Stochastic process|Stochastic processes]]
| pretitle = Part of a series on [[Statistics]]

| content1 = '''Correlation and covariance of random vectors'''
* [[Markov chain]]

* [[Wide-sense stationary]]

* [[Cross-covariance function]]

* [[Autocovariance function]]

* [[White noise]]

| content2 = '''Correlation and covariance of stochastic processes'''

* [[Conditional independence]]

* [[Independence (probability theory)]]

* [[Uncorrelatedness (probability theory)]]

| content3 = '''Distribution functions'''
* [[Cumulative distribution function]]

* [[Characteristic function (probability theory)]]

* [[Moment-generating function]]

* [[Probability density function]]

}}<noinclude>{{doc|content=}}

[[:Category:Statistics templates]]

[[:Category:Sidebar templates]]
[[:Category:Statistics templates]]
[[:Category:Sidebar templates by topic]]


=== Symmetry properties ===
[[File:SymmetryPropertiesFourierSeries.svg|thumb|right|400px|Symmetry poperties of the Fourier series.]]

* If <math>f</math> is a real function, then <math>c_{n} = c_{-n}^*</math> ([[Hermitian function|Hermitian symmetric]]) which implies:
** <math>\Re{(c_{n})} = \Re{(c_{-n})}</math> (real part is [[even function|even symmetric]])
** <math>\Im{(c_{n})} = -\Im{(c_{-n})}</math> (imaginary part is [[odd function|odd symmetric]])
** <math>|c_{n}| = |c_{-n}|</math> (absolut value is even symmetric)
** <math>\arg{(c_{n})} = \arg{(-c_{-n})}</math> (argument is odd symmetric)
* If <math>f</math> is a real and even function (<math>f(x)=f(-x) \forall x \in \mathbb{R}</math>), then all coefficients <math>c_n</math> are real and <math>c_{n} = c_{-n}</math> (even symmetric) which implies:
** <math>b_n = 0</math> for all <math>n \ge 1</math>
* If <math>f</math> is a real and odd function (<math>f(x)=-f(-x) \forall x \in \mathbb{R}</math>), then all coefficients <math>c_n</math> are purely imaginary and <math>c_{n} = -c_{-n}</math> (odd symmetric) which implies:
** <math>a_n = 0</math> for all <math>n \ge 0</math>
* If <math>f</math> is a purely imaginary function, then <math>c_{n} = -c_{-n}^*</math> which implies:
** <math>\Re{(c_{n})} = -\Re{(c_{-n})}</math> (real part is odd symmetric)
** <math>\Im{(c_{n})} = \Im{(c_{-n})}</math> (imaginary part is even symmetric)
** <math>|c_{n}| = |c_{-n}|</math> (absolut value is even symmetric)
** <math>\arg{(c_{n})} = \arg{(-c_{-n})}</math> (argument is odd symmetric)
* If <math>f</math> is a purely imaginary and even function (<math>f(x)=f(-x) \forall x \in \mathbb{R}</math>), then all coefficients <math>c_n</math> are purely imaginary and <math>c_{n} = c_{-n}</math> (even symmetric).
* If <math>f</math> is a purely imaginary and odd function (<math>f(x)=-f(-x) \forall x \in \mathbb{R}</math>), then all coefficients <math>c_n</math> are real and <math>c_{n} = -c_{-n}</math> (odd symmetric).

==Table of Fourier Series coefficients==
Some common pairsof periodic functions and their Fourier Series coefficients are shown in the table below.  The following notation applies:

*<math>f(x)</math> designates a periodic function defined on <math>0 \le x < T </math>.
*<math>a_k, b_k, a_0</math> designates a ...
*<math>c_k</math> designates a ...

{| class="wikitable"
|-
! Time domain <br /> <math>f(x)</math>
! Plot
! Frequency domain (sine-cosine form)  <br /> <math>\begin{align}& a_0 \\ & a_n \quad \text{for } n \ge 1 \\ & b_n \quad \text{for } n \ge 1\end{align}</math>
! Remarks
! Reference
|-
|<math>f(x)=A \left| \sin\left(\frac{2\pi}{T}x\right)\right| \quad \text{for } 0 \le x < T</math>
|[[File:PlotRectifiedSineSignal.svg|250px|center]]
|<math>\begin{align}
a_0 = & \frac{4A}{\pi}\\
a_n = & \begin{cases}
        \frac{-4A}{\pi}\frac{1}{1-n^2} & \quad n \text{ even} \\
        0 & \quad n \text{ odd}
        \end{cases}\\
b_n = & 0\\
\end{align}</math>
|Full-wave rectified sine
|<ref name=Papula>{{cite book | author=Papula, Lothar| title=Mathematische Formelsammlung: für Ingenieure und Naturwissenschaftler| publisher=Vieweg+Teubner Verlag | year=2009 | isbn=3834807575}}</ref>{{rp|p. 193}}
|-
|<math>f(x)=\begin{cases}
0 \quad &\text{for } 0 \le x < t_1 \\
A \left| \sin\left(\frac{2\pi}{T}x\right)\right| \quad &\text{for } t_1 \le x < T/2 \\
0 \quad &\text{for } T/2 \le x < T/2+t_1 \\
A \left| \sin\left(\frac{2\pi}{T}x\right)\right| \quad &\text{for } T/2+t_1 \le x < T 
\end{cases}</math>
|[[File:PlotRectifiedSineSignalCut.svg|250px|center]]
|<math>\begin{align}
a_0 = & \frac{2A}{\pi}\cos^2\left(\frac{\alpha}{2}\right)\\
a_n = & \begin{cases}
        \frac{2A}{\pi (1-n^2)}\left( n \sin(\alpha) \cdot \sin(n \alpha) + 1 + \cos(\alpha) \cos(n \alpha) \right) & \quad n \text{ even} \\
        0 & \quad n \text{ odd}
        \end{cases}\\
b_n = & \begin{cases}
        \frac{2A}{\pi (1-n^2)}\left( \cos(\alpha) \cdot \sin(n \alpha) - n \sin(\alpha) \cos(n \alpha) \right) & \quad n \text{ even} \\
        0 & \quad n \text{ odd}
        \end{cases}
\end{align}</math>
|Full-wave rectified sine cut by a phase-fired controller
:<math>\alpha = \frac{2\pi t_1}{T}</math>
|
|-
|<math>f(x)=\begin{cases}
A \sin\left(\frac{2\pi}{T}x\right) & \quad \text{for } 0 \le x < T/2 \\
0 & \quad \text{for } T/2 \le x < T\\
\end{cases}
</math>
|[[File:PlotHalfRectifiedSineSignal.svg|250px|center]]
|<math>\begin{align}
a_0 = & \frac{2A}{\pi}\\
a_n = & \begin{cases}
        \frac{-2A}{\pi}\frac{1}{1-n^2} & \quad n \text{ even} \\
        0 & \quad n \text{ odd}
        \end{cases}\\
b_n = & \begin{cases}
        \frac{A}{2} & \quad n=1 \\
        0 & \quad n > 1
        \end{cases}\\
\end{align}</math>
|Half-wave rectified sine
|<ref name=Papula/>{{rp|p. 193}}
|-
|<math>f(x)=\begin{cases}
A & \quad \text{for } 0 \le x < D \cdot T \\
0 & \quad \text{for } D \cdot T \le x < T\\
\end{cases}
</math>
|[[File:PlotRectangleSignal.svg|250px|center]]
|<math>\begin{align}
a_0 = & 2AD\\
a_n = & \frac{A}{n \pi} \sin \left( 2 \pi n D \right)\\
b_n = & \frac{2A}{n \pi} \left( \sin \left( \pi n D \right) \right) ^2\\
\end{align}</math>
| <math>0 \le D \le 1</math>
|
|-
|<math>f(x)=\frac{Ax}{T} \quad \text{for } 0 \le x < T</math>
|[[File:PlotSawtooth1Signal.svg|250px|center]]
|<math>\begin{align}
a_0 = & A\\
a_n = &  \frac{-A}{n \pi}\\
b_n = & 0\\
\end{align}</math>
| 
|<ref name=Papula/>{{rp|p. 192}}
|-
|<math>f(x)=A-\frac{Ax}{T} \quad \text{for } 0 \le x < T</math>
|[[File:PlotSawtooth2Signal.svg|250px|center]]
|<math>\begin{align}
a_0 = & A\\
a_n = &  \frac{A}{n \pi}\\
b_n = & 0\\
\end{align}</math>
| 
|<ref name=Papula/>{{rp|p. 192}}
|-
|<math>f(x)=\frac{4A}{T^2}\left( x-\frac{T}{2} \right)^2 \quad \text{for } 0 \le x < T</math>
|[[File:PlotParabolaSignal.svg|250px|center]]
|<math>\begin{align}
a_0 = & \frac{2A}{3}\\
a_n = & \frac{4A}{\pi^2 n^2}\\
b_n = & 0\\
\end{align}</math>
| 
|<ref name=Papula/>{{rp|p. 193}}
|-
|<math>f(x)=\begin{cases}
\frac{A}{DT}x & \quad \text{for } 0 \le x < DT\\
A\frac{T-x}{T(1-D)} & \quad \text{for } DT \le x < T
\end{cases}</math>
|[[File:PlotSawtooth3Signal.svg|250px|center]]
|<math>\begin{align}
a_0 = & A\\
a_n = & \\
b_n = & \\
\end{align}</math>
| <math>0 \le D \le 1</math>
|
|-
|<math>f(x)=\begin{cases}
\frac{A}{DT}x & \quad \text{for } 0 \le x < DT \\
(x-T/2)\frac{-2A}{T(1-2D)} & \quad \text{for } DT \le x < (1-D)T \\
(x-T)\frac{A}{DT} & \quad \text{for } (1-D)T \le x < T
\end{cases}</math>
|[[File:PlotSawtooth4Signal.svg|250px|center]]
|<math>\begin{align}
a_0 = & 0\\
a_n = & 0\\
b_n = & \frac{A}{D(1-2D)n^2\pi^2} \sin (2\pi n D)\\
\end{align}</math>
| <math>0 \le D \le 1/2</math>
| 
|-
| 
|[[File:PlotPeriodicDeltaImpulsesSignal.svg|250px|center]]
| 
| 
| <math>\delta</math> denotes the [[Dirac delta function]].
|}

==Properties==
This table shows some mathematical operations in the time domain and the corresponding effects in the frequency domain.
*<math>x[n]^{*}</math> is the [[complex conjugate]] of <math>x[n]</math>.
*<math>f(x),g(x)</math> designate a <math>T</math>-periodic functions defined on <math>0 < x \le T </math>.
*<math>\hat{f}[n], \hat{g}[n]</math> designates the Fourier series coefficients (exponential form) of <math>f</math> and <math>g</math> as defined in equation TODO!!!

{| class="wikitable"
|-
! Property
! Time domain
! Frequency domain (exponential form)
! Remarks
! Reference
|-
| Linearity
| <math>a\cdot f(x) + b\cdot g(x)</math>
| <math>a\cdot \hat{f}[n] + b\cdot \hat{g}[n]</math>
| complex numbers <math>a,b</math>
|
|-
| Time reversal / Frequency reversal
| <math>f(-x)</math>
| <math>\hat{f}[-n]</math>
| 
| <ref name=Shmaliy/>{{rp|p. 610}}
|-
| Time conjugation
| <math>f(x)^*</math>
| <math>\hat{f}[-n]^*</math>
| 
| <ref name=Shmaliy/>{{rp|p. 610}}
|-
| Time reversal & conjugation
| <math>f(-x)^*</math>
| <math>\hat{f}[n]^*</math>
|
|
|-
| Real part in time
| <math>\Re{(f(x))}</math>
| <math>\frac{1}{2}(\hat{f}[n] + \hat{f}[-n]^*)</math>
|
|
|-
| Imaginary part in time
| <math>\Im{(f(x))}</math>
| <math>\frac{1}{2i}(\hat{f}[n] - \hat{f}[-n]^*)</math>
|
|
|-
| Real part in frequency
| <math>\frac{1}{2}(f(x)+f(-x)^*)</math>
| <math>\Re{(\hat{f}[n])}</math>
|
|
|-
| Imaginary part in frequency
| <math>\frac{1}{2i}(f(x)-f(-x)^*)</math>
| <math>\Im{(\hat{f}[n])}</math>
|
|
|-
| Shift in time / Modulation in frequency
| <math>f(x-x_0)</math>
| <math>\hat{f}[n] \cdot e^{-i\frac{2\pi x_0}{T}n}</math>
| real number <math>x_0</math>
| <ref name=Shmaliy>{{cite book | author=Shmaliy, Y.S.| title=Continuous-Time Signals| publisher=Springer | year=2007 | isbn=1402062710}}</ref>{{rp|p. 610}}
|-
| Shift in frequency / Modulation in time
| <math>f(x) \cdot e^{i\frac{2\pi n_0}{T}x}</math>
| <math>\hat{f}[n-n_0] \!</math>
| integer <math>n_0</math>
| <ref name=Shmaliy/>{{rp|p. 610}}
|-
| Differencing in frequency
|
|
|
|
|-
| Summation in frequency
|
|
|
|
|-
| Derivative in time
| <math> \frac{d f(x)}{d x} </math> 
| <math> j \frac{2 \pi n}{T} \hat{f}[n]</math>
|
|
|-
| Derivative in time (<math>N</math> times)
| <math>  </math> 
| <math>  </math>
|
|
|-
| Integration in time
| <math>  </math> 
| <math>  </math>
|
|
|-
| Convolution in time / Multiplication in frequency
| <math>(f * g)(x) = \int_{0}^{T}f(\tau)g(x-\tau) d\tau</math>
| <math>\frac{1}{T} \hat{f}[n] \cdot \hat{g}[n]</math>
| <math>*\!</math> denotes [[Circular convolution|continuous circular convolution]].
|
|-
| Multiplication in time / Convolution in frequency
| <math>f(x) \cdot g(x)</math>
| <math>(\hat{f} * \hat{g})[n] = \sum_{k=-\infty}^{\infty}\hat{f}[k] \hat{g}[n-k]</math>
| <math>*\!</math> denotes [[Periodic convolution|Discrete convolution]].
|
|-
| [[Cross correlation]]
| <math>\rho_{xy} [n] = x[-n]^* * y[n] \!</math>
| <math>R_{xy} (\omega) = X_{2\pi}(\omega)^* \cdot Y_{2\pi}(\omega) \!</math> 
|
|
|-
| [[Parseval's theorem]]  
| <math>E_{xy} = \sum_{n=-\infty}^{\infty} {x[n] \cdot y[n]^*} \!</math>
| <math>E_{xy} = \frac{1}{2\pi}\int_{-\pi}^{\pi}{X_{2\pi}(\omega) \cdot Y_{2\pi}(\omega)^* d\omega} \!</math>
|
| <ref name=ProakisManolakis/>{{rp|p. 236}}
|}