Editing Cross-correlation matrix

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{{Correlation and covariance}}

The '''cross-correlation matrix''' of two [[random vector]]s is a matrix containing as elements the cross-correlations of all pairs of elements of the random vectors. The cross-correlation matrix is used in various digital signal processing algorithms.

==Definition==
For two [[random vector]]s <math>\mathbf{X} = (X_1,\ldots,X_m)^{\rm T}</math> and <math>\mathbf{Y} = (Y_1,\ldots,Y_n)^{\rm T}</math>, each containing [[random element]]s whose [[expected value]] and [[variance]] exist, the '''cross-correlation matrix''' of <math>\mathbf{X}</math> and <math>\mathbf{Y}</math> is defined by<ref name=Gubner>{{cite book |first=John A. |last=Gubner |year=2006 |title=Probability and Random Processes for Electrical and Computer Engineers |publisher=Cambridge University Press |isbn=978-0-521-86470-1}}</ref>{{rp|p.337}}

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

and has dimensions <math>m \times n</math>. Written component-wise:

:<math>\operatorname{R}_{\mathbf{X}\mathbf{Y}} =
\begin{bmatrix}
\operatorname{E}[X_1 Y_1] & \operatorname{E}[X_1 Y_2] & \cdots & \operatorname{E}[X_1 Y_n] \\ \\
\operatorname{E}[X_2 Y_1] & \operatorname{E}[X_2 Y_2] & \cdots & \operatorname{E}[X_2 Y_n] \\ \\
 \vdots & \vdots & \ddots & \vdots \\ \\
\operatorname{E}[X_m Y_1] & \operatorname{E}[X_m Y_2] & \cdots & \operatorname{E}[X_m Y_n] \\ \\
\end{bmatrix}
</math>

The random vectors <math>\mathbf{X}</math> and <math>\mathbf{Y}</math> need not have the same dimension, and either might be a scalar value.

==Example==
For example, if <math>\mathbf{X} = \left( X_1,X_2,X_3 \right)^{\rm T}</math> and <math>\mathbf{Y} = \left( Y_1,Y_2 \right)^{\rm T}</math> are random vectors, then
<math>\operatorname{R}_{\mathbf{X}\mathbf{Y}}</math> is a <math>3 \times 2</math> matrix whose <math>(i,j)</math>-th entry is <math>\operatorname{E}[X_i Y_j]</math>.

==Complex random vectors==
If <math>\mathbf{Z} = (Z_1,\ldots,Z_m)^{\rm T}</math> and <math>\mathbf{W} = (W_1,\ldots,W_n)^{\rm T}</math> are [[complex random vector]]s, each containing random variables whose expected value and variance exist, the cross-correlation matrix of <math>\mathbf{Z}</math> and <math>\mathbf{W}</math> is defined by

:<math>\operatorname{R}_{\mathbf{Z}\mathbf{W}} \triangleq\ \operatorname{E}[\mathbf{Z} \mathbf{W}^{\rm H}]</math>

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

==Uncorrelatedness==
Two random vectors <math>\mathbf{X}=(X_1,\ldots,X_m)^{\rm T} </math> and <math>\mathbf{Y}=(Y_1,\ldots,Y_n)^{\rm T} </math> are called '''uncorrelated''' if
:<math>\operatorname{E}[\mathbf{X} \mathbf{Y}^{\rm T}] = \operatorname{E}[\mathbf{X}]\operatorname{E}[\mathbf{Y}]^{\rm T}.</math>

They are uncorrelated if and only if their cross-covariance matrix <math>\operatorname{K}_{\mathbf{X}\mathbf{Y}}</math> matrix is zero.

In the case of two [[complex random vector]]s <math>\mathbf{Z}</math> and <math>\mathbf{W}</math> they are called uncorrelated if
:<math>\operatorname{E}[\mathbf{Z} \mathbf{W}^{\rm H}] = \operatorname{E}[\mathbf{Z}]\operatorname{E}[\mathbf{W}]^{\rm H}</math>
and
:<math>\operatorname{E}[\mathbf{Z} \mathbf{W}^{\rm T}] = \operatorname{E}[\mathbf{Z}]\operatorname{E}[\mathbf{W}]^{\rm T}.</math>

==Properties==
===Relation to the cross-covariance matrix===
The cross-correlation is related to the ''cross-covariance matrix'' as follows:
:<math>\operatorname{K}_{\mathbf{X}\mathbf{Y}} = \operatorname{E}[(\mathbf{X} - \operatorname{E}[\mathbf{X}])(\mathbf{Y} - \operatorname{E}[\mathbf{Y}])^{\rm T}] =  \operatorname{R}_{\mathbf{X}\mathbf{Y}} - \operatorname{E}[\mathbf{X}] \operatorname{E}[\mathbf{Y}]^{\rm T}</math>
: Respectively for complex random vectors:
:<math>\operatorname{K}_{\mathbf{Z}\mathbf{W}} = \operatorname{E}[(\mathbf{Z} - \operatorname{E}[\mathbf{Z}])(\mathbf{W} - \operatorname{E}[\mathbf{W}])^{\rm H}] =  \operatorname{R}_{\mathbf{Z}\mathbf{W}} - \operatorname{E}[\mathbf{Z}] \operatorname{E}[\mathbf{W}]^{\rm H}</math>

==See also==
*[[Autocorrelation]]
*[[Correlation does not imply causation]]
*[[Covariance function]]
*[[Pearson product-moment correlation coefficient]]
*[[Correlation function (astronomy)]]
*[[Correlation function (statistical mechanics)]]
*[[Correlation function (quantum field theory)]]
*[[Mutual information]]
*[[Rate distortion theory#Rate–distortion functions|Rate distortion theory]]
*[[Radial distribution function]]

==References==
{{reflist}}

==Further reading==
* Hayes, Monson H., ''Statistical Digital Signal Processing and Modeling'', John Wiley & Sons, Inc., 1996. {{ISBN|0-471-59431-8}}.
* Solomon W. Golomb, and [[Guang Gong]]. [http://www.cambridge.org/us/academic/subjects/computer-science/cryptography-cryptology-and-coding/signal-design-good-correlation-wireless-communication-cryptography-and-radar Signal design for good correlation: for wireless communication, cryptography, and radar]. Cambridge University Press, 2005.
* M. Soltanalian. [http://theses.eurasip.org/theses/573/signal-design-for-active-sensing-and/download/ Signal Design for Active Sensing and Communications]. Uppsala Dissertations from the Faculty of Science and Technology (printed by Elanders Sverige AB), 2014.

{{DEFAULTSORT:Correlation Function}}
[[Category:Covariance and correlation]]
[[Category:Time series]]
[[Category:Spatial analysis]]
[[Category:Matrices (mathematics)]]
[[Category:Signal processing]]