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 matrix is used in various digital signal processing algorithms.

Definition[edit]

For two random vectors and , each containing random elements whose expected value and variance exist, the cross-correlation matrix of and is defined by[1]:p.337

and has dimensions . Written component-wise:

The random vectors and need not have the same dimension, and either might be a scalar value.

Example[edit]

For example, if and are random vectors, then is a matrix whose -th entry is .

Cross-correlation matrix of complex random vectors[edit]

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[edit]

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[edit]

Relation to the cross-covariance matrix[edit]

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

Respectively for complex random vectors:

See also[edit]

References[edit]

  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[edit]