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Part of a series on Statistics
Correlation and covariance
For random vectors Autocorrelation matrix Cross-correlation matrix Auto-covariance matrix Cross-covariance matrix
For stochastic processes Autocorrelation function Cross-correlation function Autocovariance function Cross-covariance function
For deterministic signals Autocorrelation function Cross-correlation function Autocovariance function Cross-covariance function
v t e

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

${\displaystyle \operatorname {R} _{\mathbf {X} \mathbf {Y} }\triangleq \ \operatorname {E} }$

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

${\displaystyle \operatorname {R} _{\mathbf {X} \mathbf {Y} }={\begin{bmatrix}\operatorname {E} &\operatorname {E} &\cdots &\operatorname {E} \\\\\operatorname {E} &\operatorname {E} &\cdots &\operatorname {E} \\\\\vdots &\vdots &\ddots &\vdots \\\\\operatorname {E} &\operatorname {E} &\cdots &\operatorname {E} \\\\\end{bmatrix}}}$

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 ${\displaystyle \operatorname {E} }$.

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

${\displaystyle \operatorname {R} _{\mathbf {Z} \mathbf {W} }\triangleq \ \operatorname {E} }$

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

Uncorrelatedness

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

${\displaystyle \operatorname {E} =\operatorname {E} \operatorname {E} ^{\rm {T}}.}$

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

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

${\displaystyle \operatorname {E} =\operatorname {E} \operatorname {E} ^{\rm {H}}}$

and

${\displaystyle \operatorname {E} =\operatorname {E} \operatorname {E} ^{\rm {T}}.}$

Properties

Relation to the cross-covariance matrix

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

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

Respectively for complex random vectors:

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

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

References

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

Further reading

• 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.

• Covariance and correlation
• Time series
• Spatial analysis
• Matrices
• Signal processing