Cross-correlation matrix

Example

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

Complex random vectors

If ${\displaystyle \mathbf{Z}=(Z_1,\dots ,Z_m{)}^{\mathrm{T}}}$ and ${\displaystyle \mathbf{W}=(W_1,\dots ,W_n{)}^{\mathrm{T}}}$ 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

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

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

Uncorrelatedness

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

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

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

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

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

and

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

Properties

Relation to the cross-covariance matrix

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

${\displaystyle {\mathrm{K}}_{\mathbf{X}\mathbf{Y}}=\mathrm{E}[(\mathbf{X}-\mathrm{E}[\mathbf{X}])(\mathbf{Y}-\mathrm{E}[\mathbf{Y}]{)}^{\mathrm{T}}]={\mathrm{R}}_{\mathbf{X}\mathbf{Y}}-\mathrm{E}[\mathbf{X}]\mathrm{E}[\mathbf{Y}{]}^{\mathrm{T}}}$

Respectively for complex random vectors:

${\displaystyle {\mathrm{K}}_{\mathbf{Z}\mathbf{W}}=\mathrm{E}[(\mathbf{Z}-\mathrm{E}[\mathbf{Z}])(\mathbf{W}-\mathrm{E}[\mathbf{W}]{)}^{\mathrm{H}}]={\mathrm{R}}_{\mathbf{Z}\mathbf{W}}-\mathrm{E}[\mathbf{Z}]\mathrm{E}[\mathbf{W}{]}^{\mathrm{H}}}$