Cross-correlation matrix

Template:Other uses2

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 matrixis 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

$\operatorname {R} _{\mathbf {X} \mathbf {Y} }{\stackrel {\mathrm {def} }{=}}\ \operatorname {E} [\mathbf {X} \mathbf {Y} ^{\rm {T}}]$

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} }{\stackrel {\mathrm {def} }{=}}\ \operatorname {E} [\mathbf {Z} \mathbf {W} ^{\rm {H}}]

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

Uncorrelatedness

Two random vectors $\mathbf {X} =(X_{1},...,X_{m})^{\rm {T}}$ and $\mathbf {Y} =(Y_{1},...,Y_{n})^{\rm {T}}$ 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 covariance $\operatorname {K} _{\mathbf {X} \mathbf {Y} }$ 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

The cross-covariance matrix is related to the cross-correlation 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

Hayes, Monson H., Statistical Digital Signal Processing and Modeling, John Wiley & Sons, Inc., 1996. ISBN 0-471-59431-8.
Solomon W. Golomb, and Guang Gong. Signal design for good correlation: for wireless communication, cryptography, and radar. Cambridge University Press, 2005.
M. Soltanalian. Signal Design for Active Sensing and Communications. Uppsala Dissertations from the Faculty of Science and Technology (printed by Elanders Sverige AB), 2014.