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^[1]^: p.337

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

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 $\operatorname {E} [X_{i}Y_{j}]$ .

Cross-correlation matrix of 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}}.

Properties

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}}

References

^ Gubner, John A. (2006). Probability and Random Processes for Electrical and Computer Engineers. Cambridge University Press. ISBN 978-0-521-86470-1.