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 and , each containing random elements whose expected value and variance exist, the cross-correlation matrix of and is defined by
and has dimensions . Written component-wise:
The random vectors and need not have the same dimension, and either might be a scalar value.
Example For example, if and are random vectors, then is a matrix whose -th entry is .
Complex random vectors If and are complex random vectors, each containing random variables whose expected value and variance exist, the cross-correlation matrix of and is defined by
where denotes Hermitian transposition.
Uncorrelatedness Two random vectors and are called uncorrelated if
They are uncorrelated if and only if their cross-covariance matrix matrix is zero.
In the case of two complex random vectors and they are called uncorrelated if
and
Properties Relation to the cross-covariance matrix The cross-correlation is related to the cross-covariance matrix as follows:
Respectively for complex random vectors:
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
Further reading Hayes, Monson H., Statistical Digital Signal Processing and Modeling, John Wiley & Sons, Inc., 1996. . 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.