# Cross-correlation matrix

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.

## Definition

For two random vectors $\displaystyle{ \mathbf{X} = (X_1,\ldots,X_m)^{\rm T} }$ and $\displaystyle{ \mathbf{Y} = (Y_1,\ldots,Y_n)^{\rm T} }$, each containing random elements whose expected value and variance exist, the cross-correlation matrix of $\displaystyle{ \mathbf{X} }$ and $\displaystyle{ \mathbf{Y} }$ is defined by[1]:p.337

$\displaystyle{ \operatorname{R}_{\mathbf{X}\mathbf{Y}} \triangleq\ \operatorname{E}[\mathbf{X} \mathbf{Y}^{\rm T}] }$

and has dimensions $\displaystyle{ m \times n }$. Written component-wise:

$\displaystyle{ \operatorname{R}_{\mathbf{X}\mathbf{Y}} = \begin{bmatrix} \operatorname{E}[X_1 Y_1] & \operatorname{E}[X_1 Y_2] & \cdots & \operatorname{E}[X_1 Y_n] \\ \\ \operatorname{E}[X_2 Y_1] & \operatorname{E}[X_2 Y_2] & \cdots & \operatorname{E}[X_2 Y_n] \\ \\ \vdots & \vdots & \ddots & \vdots \\ \\ \operatorname{E}[X_m Y_1] & \operatorname{E}[X_m Y_2] & \cdots & \operatorname{E}[X_m Y_n] \\ \\ \end{bmatrix} }$

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

## Example

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

## Complex random vectors

If $\displaystyle{ \mathbf{Z} = (Z_1,\ldots,Z_m)^{\rm T} }$ and $\displaystyle{ \mathbf{W} = (W_1,\ldots,W_n)^{\rm T} }$ are complex random vectors, each containing random variables whose expected value and variance exist, the cross-correlation matrix of $\displaystyle{ \mathbf{Z} }$ and $\displaystyle{ \mathbf{W} }$ is defined by

$\displaystyle{ \operatorname{R}_{\mathbf{Z}\mathbf{W}} \triangleq\ \operatorname{E}[\mathbf{Z} \mathbf{W}^{\rm H}] }$

where $\displaystyle{ {}^{\rm H} }$ denotes Hermitian transposition.

## Uncorrelatedness

Two random vectors $\displaystyle{ \mathbf{X}=(X_1,\ldots,X_m)^{\rm T} }$ and $\displaystyle{ \mathbf{Y}=(Y_1,\ldots,Y_n)^{\rm T} }$ are called uncorrelated if

$\displaystyle{ \operatorname{E}[\mathbf{X} \mathbf{Y}^{\rm T}] = \operatorname{E}[\mathbf{X}]\operatorname{E}[\mathbf{Y}]^{\rm T}. }$

They are uncorrelated if and only if their cross-covariance matrix $\displaystyle{ \operatorname{K}_{\mathbf{X}\mathbf{Y}} }$ matrix is zero.

In the case of two complex random vectors $\displaystyle{ \mathbf{Z} }$ and $\displaystyle{ \mathbf{W} }$ they are called uncorrelated if

$\displaystyle{ \operatorname{E}[\mathbf{Z} \mathbf{W}^{\rm H}] = \operatorname{E}[\mathbf{Z}]\operatorname{E}[\mathbf{W}]^{\rm H} }$

and

$\displaystyle{ \operatorname{E}[\mathbf{Z} \mathbf{W}^{\rm T}] = \operatorname{E}[\mathbf{Z}]\operatorname{E}[\mathbf{W}]^{\rm T}. }$

## Properties

### Relation to the cross-covariance matrix

The cross-correlation is related to the cross-covariance matrix as follows:

$\displaystyle{ \operatorname{K}_{\mathbf{X}\mathbf{Y}} = \operatorname{E}[(\mathbf{X} - \operatorname{E}[\mathbf{X}])(\mathbf{Y} - \operatorname{E}[\mathbf{Y}])^{\rm T}] = \operatorname{R}_{\mathbf{X}\mathbf{Y}} - \operatorname{E}[\mathbf{X}] \operatorname{E}[\mathbf{Y}]^{\rm T} }$
Respectively for complex random vectors:
$\displaystyle{ \operatorname{K}_{\mathbf{Z}\mathbf{W}} = \operatorname{E}[(\mathbf{Z} - \operatorname{E}[\mathbf{Z}])(\mathbf{W} - \operatorname{E}[\mathbf{W}])^{\rm H}] = \operatorname{R}_{\mathbf{Z}\mathbf{W}} - \operatorname{E}[\mathbf{Z}] \operatorname{E}[\mathbf{W}]^{\rm H} }$