“Cross-correlation matrix”的意思、由来-开放百科全书

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Cross-correlation matrix

释义

Definition
Example
Cross-correlation matrix of complex random vectors
Uncorrelatedness
Properties
Relation to the cross-covariance matrix
See also
References
Further reading

{{Correlation and covariance}}{{Other uses2|Correlation function}}{{disputed|date=December 2018}}{{More citations needed|date=December 2009}}

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 matrixis 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]{{rp|p.337}}

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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:

Respectively for complex random vectors:

References

1. ^{{cite book |first=John A. |last=Gubner |year=2006 |title=Probability and Random Processes for Electrical and Computer Engineers |publisher=Cambridge University Press |isbn=978-0-521-86470-1}}