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

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 .

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

Summarize
Perspective

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

Summarize
Perspective

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

References

Further reading

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