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.

For two random vectors ${\displaystyle \mathbf {X} =(X_{1},\ldots ,X_{m})^{\rm {T}}}${\displaystyle \mathbf {X} =(X_{1},\ldots ,X_{m})^{\rm {T}}} and ${\displaystyle \mathbf {Y} =(Y_{1},\ldots ,Y_{n})^{\rm {T}}}${\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 ${\displaystyle \mathbf {X} }${\displaystyle \mathbf {X} } and ${\displaystyle \mathbf {Y} }${\displaystyle \mathbf {Y} } is defined by^{[1] : p.337}

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

${\displaystyle \operatorname {R} _{\mathbf {X} \mathbf {Y} }={\begin{bmatrix}\operatorname {E} [X_{1}Y_{1}]&\operatorname {E} [X_{1}Y_{2}]&\cdots &\operatorname {E} [X_{1}Y_{n}]\\\\\operatorname {E} [X_{2}Y_{1}]&\operatorname {E} [X_{2}Y_{2}]&\cdots &\operatorname {E} [X_{2}Y_{n}]\\\\\vdots &\vdots &\ddots &\vdots \\\\\operatorname {E} [X_{m}Y_{1}]&\operatorname {E} [X_{m}Y_{2}]&\cdots &\operatorname {E} [X_{m}Y_{n}]\\\\\end{bmatrix}}}${\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 ${\displaystyle \mathbf {X} }${\displaystyle \mathbf {X} } and ${\displaystyle \mathbf {Y} }${\displaystyle \mathbf {Y} } need not have the same dimension, and either might be a scalar value.

For example, if ${\displaystyle \mathbf {X} =\left(X_{1},X_{2},X_{3}\right)^{\rm {T}}}${\displaystyle \mathbf {X} =\left(X_{1},X_{2},X_{3}\right)^{\rm {T}}} and ${\displaystyle \mathbf {Y} =\left(Y_{1},Y_{2}\right)^{\rm {T}}}${\displaystyle \mathbf {Y} =\left(Y_{1},Y_{2}\right)^{\rm {T}}} are random vectors, then ${\displaystyle \operatorname {R} _{\mathbf {X} \mathbf {Y} }}${\displaystyle \operatorname {R} _{\mathbf {X} \mathbf {Y} }} is a ${\displaystyle 3\times 2}${\displaystyle 3\times 2} matrix whose ${\displaystyle (i,j)}${\displaystyle (i,j)}-th entry is ${\displaystyle \operatorname {E} [X_{i}Y_{j}]}${\displaystyle \operatorname {E} [X_{i}Y_{j}]}.

If ${\displaystyle \mathbf {Z} =(Z_{1},\ldots ,Z_{m})^{\rm {T}}}${\displaystyle \mathbf {Z} =(Z_{1},\ldots ,Z_{m})^{\rm {T}}} and ${\displaystyle \mathbf {W} =(W_{1},\ldots ,W_{n})^{\rm {T}}}${\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 ${\displaystyle \mathbf {Z} }${\displaystyle \mathbf {Z} } and ${\displaystyle \mathbf {W} }${\displaystyle \mathbf {W} } is defined by

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

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

Two random vectors ${\displaystyle \mathbf {X} =(X_{1},\ldots ,X_{m})^{\rm {T}}}${\displaystyle \mathbf {X} =(X_{1},\ldots ,X_{m})^{\rm {T}}} and ${\displaystyle \mathbf {Y} =(Y_{1},\ldots ,Y_{n})^{\rm {T}}}${\displaystyle \mathbf {Y} =(Y_{1},\ldots ,Y_{n})^{\rm {T}}} are called uncorrelated if

${\displaystyle \operatorname {E} [\mathbf {X} \mathbf {Y} ^{\rm {T}}]=\operatorname {E} [\mathbf {X} ]\operatorname {E} [\mathbf {Y} ]^{\rm {T}}.}${\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 cross-covariance matrix ${\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {Y} }}${\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {Y} }} matrix is zero.

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

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

and

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

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

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

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

• 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

• 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.