Cross-correlation matrix

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

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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

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

Complex random vectors

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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

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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

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Relation to the cross-covariance matrix

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

 
Respectively for complex random vectors:
 

See also

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References

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  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

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