Cross-correlation matrix: Difference between revisions

Revision as of 19:32, 29 December 2018

Latest revision as of 21:03, 14 April 2025

(11 intermediate revisions by 10 users not shown)

Line 1:

{{Otherother uses2uses|Correlation function (disambiguation)}}

{{Multiple issues|

{{Unreferenceddisputed|date=December 20092018}}

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

}}

{{Correlation and covariance}}

{{Other uses2|Correlation function}}

{{disputed}}

{{Unreferenced|date=December 2009}}

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==

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}} \stackrel{\mathrm{def}}{=}triangleq\ \operatorname{E}[\mathbf{X} \mathbf{Y}^{\rm T}]</math>

|cellpadding= 6

|border

Line 35 ⟶ 37:

<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}} \stackrel{\mathrm{def}}{=}triangleq\ \operatorname{E}[\mathbf{Z} \mathbf{W}^{\rm H}]</math>

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

Line 46 ⟶ 48:

:<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|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

Line 54 ⟶ 56:

==Properties==

* The ''cross-covariance matrix'' is related===Relation to the cross-correlationcovariance matrix as follows:===

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>

==References==

* 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.

==See also==

Line 73 ⟶ 71:

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

*[[Mutual information]]

*[[Rate distortion theory#Rate–distortion_functionsRate–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 data analysis]]

[[Category:Matrices (mathematics)]]

[[Category:Signal processing]]