Cross-correlation matrix

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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 ${\displaystyle \mathbf {X} =(X_{1},\ldots ,X_{m})^{\rm {T}}}$  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

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

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

cross-correlation matrix of 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

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

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

Uncorrelatedness

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

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

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

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.

See also

• Autocorrelation
• Correlation does not imply causation
• Covariance function
• Pearson product-moment correlation coefficient
• Correlation function (astronomy)
• Correlation function (statistical mechanics)
• Correlation function (quantum field theory)
• Mutual information
• Rate distortion theory
• Radial distribution function