Cross-correlation matrix: Difference between revisions

Revision as of 12:50, 13 April 2019

Revision as of 01:11, 27 September 2019

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{{Correlation and covariance}}

{{Other uses2|Correlation function}}

{{Multiple issues|

{{disputed|date=December 2018}}

{{More citations needed|date=December 2009}}

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{{Correlation and covariance}}

The '''cross-correlation matrix''' of two [[random vector|random vectors]]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==

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==Cross-correlation matrix of 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|complex random vectors]]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>

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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|complex random vectors]]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

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*[[Correlation function (quantum field theory)]]

*[[Mutual information]]

*[[Rate distortion theory#Rate–distortion_functionsRate–distortion functions|Rate distortion theory]]

*[[Radial distribution function]]