Cross-correlation matrix: Difference between revisions

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{{other uses|Correlation andfunction covariance(disambiguation)}}

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{{Other uses2|Correlation function}}

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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 matrixismatrix is used in various digital signal processing algorithms.

==Definition==

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<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>.

==Cross-correlation matrix of complexComplex 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]]

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==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.

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[[Category:Covariance and correlation]]

[[Category:Time series]]

[[Category:Spatial data analysis]]

[[Category:Matrices (mathematics)]]

[[Category:Signal processing]]