Editing Autocorrelation (section)

==Autocorrelation of random vectors{{anchor|Matrix}}==

The (potentially time-dependent) '''autocorrelation matrix''' (also called second moment) of a (potentially time-dependent) [[random vector]] <math>\mathbf{X} = (X_1,\ldots,X_n)^{\rm T}</math> is an <math>n \times n</math> matrix containing as elements the autocorrelations of all pairs of elements of the random vector <math>\mathbf{X}</math>. The autocorrelation matrix is used in various [[digital signal processing]] algorithms.

For a [[random vector]] <math>\mathbf{X} = (X_1,\ldots,X_n)^{\rm T}</math> containing [[random element]]s whose [[expected value]] and [[variance]] exist, the '''autocorrelation matrix''' is defined by<ref name=Papoulis>Papoulis, Athanasius, ''Probability, Random variables and Stochastic processes'', McGraw-Hill, 1991</ref>{{rp|p.190}}<ref name=Gubner/>{{rp|p.334}}

{{Equation box 1
|indent = :
|title=
|equation = <math>\operatorname{R}_{\mathbf{X}\mathbf{X}} \triangleq\ \operatorname{E} \left[ \mathbf{X} \mathbf{X}^{\rm T} \right] </math>
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F5FFFA}}

where <math>{}^{\rm T}</math> denotes the [[transpose]]d matrix of dimensions <math>n \times n</math>.

Written component-wise:

<math display=block>\operatorname{R}_{\mathbf{X}\mathbf{X}} =
\begin{bmatrix}
\operatorname{E}[X_1 X_1] & \operatorname{E}[X_1 X_2] & \cdots & \operatorname{E}[X_1 X_n] \\ \\
\operatorname{E}[X_2 X_1] & \operatorname{E}[X_2 X_2] & \cdots & \operatorname{E}[X_2 X_n] \\ \\
 \vdots & \vdots & \ddots & \vdots \\ \\
\operatorname{E}[X_n X_1] & \operatorname{E}[X_n X_2] & \cdots & \operatorname{E}[X_n X_n] \\ \\
\end{bmatrix}
</math>

If <math>\mathbf{Z}</math> is a [[complex random vector]], the autocorrelation matrix is instead defined by

<math display=block>\operatorname{R}_{\mathbf{Z}\mathbf{Z}} \triangleq\ \operatorname{E}[\mathbf{Z} \mathbf{Z}^{\rm H}] .</math>

Here <math>{}^{\rm H}</math> denotes [[Hermitian transpose]].

For example, if <math>\mathbf{X} = \left( X_1,X_2,X_3 \right)^{\rm T}</math> is a random vector, then <math>\operatorname{R}_{\mathbf{X}\mathbf{X}}</math> is a <math>3 \times 3</math> matrix whose <math>(i,j)</math>-th entry is <math>\operatorname{E}[X_i X_j]</math>.

===Properties of the autocorrelation matrix===
* The autocorrelation matrix is a [[Hermitian matrix]] for complex random vectors and a [[symmetric matrix]] for real random vectors.<ref name=Papoulis />{{rp|p.190}}
* The autocorrelation matrix is a [[positive semidefinite matrix]],<ref name=Papoulis />{{rp|p.190}} i.e. <math>\mathbf{a}^{\mathrm T} \operatorname{R}_{\mathbf{X}\mathbf{X}} \mathbf{a} \ge 0 \quad \text{for all } \mathbf{a} \in \mathbb{R}^n</math> for a real random vector, and respectively <math>\mathbf{a}^{\mathrm H} \operatorname{R}_{\mathbf{Z}\mathbf{Z}} \mathbf{a} \ge 0 \quad \text{for all } \mathbf{a} \in \mathbb{C}^n</math> in case of a complex random vector.
* All eigenvalues of the autocorrelation matrix are real and non-negative.
* The ''auto-covariance matrix'' is related to the autocorrelation matrix as follows:<!--

--><math display=block>\operatorname{K}_{\mathbf{X}\mathbf{X}} = \operatorname{E}[(\mathbf{X} - \operatorname{E}[\mathbf{X}])(\mathbf{X} - \operatorname{E}[\mathbf{X}])^{\rm T}] =  \operatorname{R}_{\mathbf{X}\mathbf{X}} - \operatorname{E}[\mathbf{X}] \operatorname{E}[\mathbf{X}]^{\rm T}</math><!--

-->Respectively for complex random vectors:<!--

--><math display=block>\operatorname{K}_{\mathbf{Z}\mathbf{Z}} = \operatorname{E}[(\mathbf{Z} - \operatorname{E}[\mathbf{Z}])(\mathbf{Z} - \operatorname{E}[\mathbf{Z}])^{\rm H}] =  \operatorname{R}_{\mathbf{Z}\mathbf{Z}} - \operatorname{E}[\mathbf{Z}] \operatorname{E}[\mathbf{Z}]^{\rm H}</math>