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 $$\mathbf{X} = (X_1,\ldots,X_m)^{\rm T}$$ and $$\mathbf{Y} = (Y_1,\ldots,Y_n)^{\rm T}$$, each containing random elements whose expected value and variance exist, the cross-correlation matrix of $$\mathbf{X}$$ and $$\mathbf{Y}$$ is defined by

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


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

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

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


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

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

Uncorrelatedness
Two random vectors $$\mathbf{X}=(X_1,\ldots,X_m)^{\rm T} $$ and $$\mathbf{Y}=(Y_1,\ldots,Y_n)^{\rm T} $$ are called uncorrelated if
 * $$\operatorname{E}[\mathbf{X} \mathbf{Y}^{\rm T}] = \operatorname{E}[\mathbf{X}]\operatorname{E}[\mathbf{Y}]^{\rm T}.$$

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

In the case of two complex random vectors $$\mathbf{Z}$$ and $$\mathbf{W}$$ they are called uncorrelated if
 * $$\operatorname{E}[\mathbf{Z} \mathbf{W}^{\rm H}] = \operatorname{E}[\mathbf{Z}]\operatorname{E}[\mathbf{W}]^{\rm H}$$

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

Relation to the cross-covariance matrix
The cross-correlation is related to the cross-covariance matrix as follows:
 * $$\operatorname{K}_{\mathbf{X}\mathbf{Y}} = \operatorname{E}[(\mathbf{X} - \operatorname{E}[\mathbf{X}])(\mathbf{Y} - \operatorname{E}[\mathbf{Y}])^{\rm T}] = \operatorname{R}_{\mathbf{X}\mathbf{Y}} - \operatorname{E}[\mathbf{X}] \operatorname{E}[\mathbf{Y}]^{\rm T}$$
 * Respectively for complex random vectors:
 * $$\operatorname{K}_{\mathbf{Z}\mathbf{W}} = \operatorname{E}[(\mathbf{Z} - \operatorname{E}[\mathbf{Z}])(\mathbf{W} - \operatorname{E}[\mathbf{W}])^{\rm H}] = \operatorname{R}_{\mathbf{Z}\mathbf{W}} - \operatorname{E}[\mathbf{Z}] \operatorname{E}[\mathbf{W}]^{\rm H}$$