Editing Autocorrelation (section)

== Autocorrelation of deterministic signals ==
In [[signal processing]], the above definition is often used without the normalization, that is, without subtracting the mean and dividing by the variance. When the autocorrelation function is normalized by mean and variance, it is sometimes referred to as the '''autocorrelation coefficient'''<ref name="dunn">{{cite book |first=Patrick F. |last=Dunn |title=Measurement and Data Analysis for Engineering and Science |location=New York |publisher=McGraw–Hill |year=2005 |isbn=978-0-07-282538-1 }}</ref> or autocovariance function.

=== Autocorrelation of continuous-time signal ===
Given a [[Signal (electronics)|signal]] <math>f(t)</math>, the continuous autocorrelation <math>R_{ff}(\tau)</math> is most often defined as the continuous [[cross-correlation]] integral of <math>f(t)</math> with itself, at lag <math>\tau</math>.<ref name=Gubner/>{{rp|p.411}}

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|equation = <math>R_{ff}(\tau) = \int_{-\infty}^\infty f(t+\tau)\overline{f(t)}\, {\rm d}t = \int_{-\infty}^\infty f(t) \overline{f(t-\tau)}\, {\rm d}t</math>
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where <math>\overline{f(t)}</math> represents the [[complex conjugate]] of <math>f(t)</math>. Note that the parameter <math>t</math> in the integral is a dummy variable and is only necessary to calculate the integral. It has no specific meaning.

=== Autocorrelation of discrete-time signal ===
The discrete autocorrelation <math>R</math> at lag <math>\ell</math> for a discrete-time signal <math>y(n)</math> is

{{Equation box 1
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|equation = <math>R_{yy}(\ell) = \sum_{n \in Z} y(n)\,\overline{y(n-\ell)}</math>
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The above definitions work for signals that are square integrable, or square summable, that is, of finite energy. Signals that "last forever" are treated instead as random processes, in which case different definitions are needed, based on expected values. For [[wide-sense-stationary random process]]es, the autocorrelations are defined as

<math display=block>\begin{align}
R_{ff}(\tau) &= \operatorname{E}\left[f(t)\overline{f(t-\tau)}\right] \\
R_{yy}(\ell) &= \operatorname{E}\left[y(n)\,\overline{y(n-\ell)}\right] .
\end{align}</math>

For processes that are not [[Stationary process|stationary]], these will also be functions of <math>t</math>, or <math>n</math>.

For processes that are also [[Ergodic process|ergodic]], the expectation can be replaced by the limit of a time average. The autocorrelation of an ergodic process is sometimes defined as or equated to<ref name="dunn"/>

<math display=block>\begin{align}
R_{ff}(\tau) &= \lim_{T \rightarrow \infty} \frac 1 T \int_0^T f(t+\tau)\overline{f(t)}\, {\rm d}t \\
R_{yy}(\ell) &= \lim_{N \rightarrow \infty} \frac 1 N \sum_{n=0}^{N-1} y(n)\,\overline{y(n-\ell)} .
\end{align}</math>

These definitions have the advantage that they give sensible well-defined single-parameter results for periodic functions, even when those functions are not the output of stationary ergodic processes.

Alternatively, signals that ''last forever'' can be treated by a short-time autocorrelation function analysis, using finite time integrals. (See [[short-time Fourier transform]] for a related process.)

===Definition for periodic signals===
If <math>f</math> is a continuous periodic function of period <math>T</math>, the integration from <math>-\infty</math> to <math>\infty</math> is replaced by integration over any interval <math>[t_0,t_0+T]</math> of length <math>T</math>:

<math display=block>R_{ff}(\tau) \triangleq \int_{t_0}^{t_0+T} f(t+\tau) \overline{f(t)} \,dt</math>

which is equivalent to

<math display=block>R_{ff}(\tau) \triangleq \int_{t_0}^{t_0+T} f(t) \overline{f(t-\tau)} \,dt</math>

===Properties===
In the following, we will describe properties of one-dimensional autocorrelations only, since most properties are easily transferred from the one-dimensional case to the multi-dimensional cases. These properties hold for [[Stationary process#Weak or wide-sense stationarity|wide-sense stationary processes]].<ref>{{cite book|last1=Proakis|first1=John|title=Communication Systems Engineering (2nd Edition)|date=August 31, 2001|publisher=Pearson|isbn=978-0130617934|page=168|edition=2}}</ref>

* A fundamental property of the autocorrelation is symmetry, <math>R_{ff}(\tau) = R_{ff}(-\tau)</math>, which is easy to prove from the definition. In the continuous case,
** the autocorrelation is an [[even function]] <math>R_{ff}(-\tau) = R_{ff}(\tau)</math> when <math>f</math> is a real function, and
** the autocorrelation is a [[Hermitian function]] <math>R_{ff}(-\tau) = R_{ff}^*(\tau)</math> when <math>f</math> is a [[complex function]].
* The continuous autocorrelation function reaches its peak at the origin, where it takes a real value, i.e. for any delay <math>\tau</math>, <math>|R_{ff}(\tau)| \leq R_{ff}(0)</math>.<ref name=Gubner/>{{rp|p.410}} This is a consequence of the [[rearrangement inequality]]. The same result holds in the discrete case.
* The autocorrelation of a [[periodic function]] is, itself, periodic with the same period.
* The autocorrelation of the sum of two completely uncorrelated functions (the cross-correlation is zero for all <math>\tau</math>) is the sum of the autocorrelations of each function separately.
* Since autocorrelation is a specific type of [[cross-correlation]], it maintains all the properties of cross-correlation.
* By using the symbol <math>*</math> to represent [[convolution]] and <math>g_{-1}</math> is a function which manipulates the function <math>f</math> and is defined as <math>g_{-1}(f)(t)=f(-t)</math>, the definition for <math>R_{ff}(\tau)</math> may be written as:<!--
--><math display=block>R_{ff}(\tau) = (f * g_{-1}(\overline{f}))(\tau)</math>