Cross-correlation matrix

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The cross-correlation matrix of two random vectors 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.

DefinitionEdit

For 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>, each containing random elements 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>Template:Cite book</ref>Template:Rp

Template:Equation box 1 \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.

ExampleEdit

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 vectorsEdit

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 vectors, 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 transposition.

UncorrelatednessEdit

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 vectors <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>

PropertiesEdit

Relation to the cross-covariance matrixEdit

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>

ReferencesEdit

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