For other uses, see Correlation function (disambiguation).
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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.
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[]
For example, if and are random vectors, then
is a matrix whose -th entry is .
Cross-correlation matrix of 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: