Cross-correlation matrix explained

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

X = (X _{1,\ldots,X} ^{\rm T} ^{m)} and Y = (Y _{1,\ldots,Y} ^{\rm T} ^{n)} , each containing random elements whose expected value and variance exist, the cross-correlation matrix of X and Y is defined by^[1]

and has dimensions

m x n . Written component-wise:\operatorname{R} _{XY} = \begin{bmatrix} \operatorname{E}[X _{1} Y _{1]} & \operatorname{E}[X _{1} Y _{2]} & … & \operatorname{E}[X _{1} Y _{n]} \ \\ \operatorname{E}[X _{2} Y _{1]} & \operatorname{E}[X _{2} Y _{2]} & … & \operatorname{E}[X _{2} Y _{n]} \ \\ \vdots & \vdots & \ddots & \vdots \ \\ \operatorname{E}[X _{m} Y _{1]} & \operatorname{E}[X _{m} Y _{2]} & … & \operatorname{E}[X _{m} Y _{n]} \ \\ \end{bmatrix}

The random vectors

X and Y need not have the same dimension, and either might be a scalar value.

Example

For example, if

X = \left( X _{1,X} _{2,X} _{3} \right) ^{\rm} and Y = \left( Y _{1,Y} _{2} \right) ^{\rm} are random vectors, then\operatorname{R} _{XY} is a 3 x 2 matrix whose (i,j) -th entry is \operatorname{E}[X _{i} Y _{j]} .

Complex random vectors

If

Z = (Z _{1,\ldots,Z} ^{\rm T} ^{m)} and W = (W _{1,\ldots,W} ^{\rm T} ^{n)} are complex random vectors, each containing random variables whose expected value and variance exist, the cross-correlation matrix of Z and W is defined by\operatorname{R} _{ZW} \triangleq \operatorname{E}[ Z W ^{\rm} ]

where

{} ^{\rm} denotes Hermitian transposition.

Uncorrelatedness

Two random vectors

X = (X _{1,\ldots,X} ^{\rm T} ^{m)} and Y = (Y _{1,\ldots,Y} ^{\rm T} ^{n)} are called uncorrelated if\operatorname{E}[ X Y ^{\rm} ] = \operatorname{E}[ X ]\operatorname{E}[ Y ] ^{\rm} .

They are uncorrelated if and only if their cross-covariance matrix

\operatorname{K} _{XY} matrix is zero.

In the case of two complex random vectors

Z and W they are called uncorrelated if\operatorname{E}[ Z W ^{\rm} ] = \operatorname{E}[ Z ]\operatorname{E}[ W ] ^{\rm} and\operatorname{E}[ Z W ^{\rm} ] = \operatorname{E}[ Z ]\operatorname{E}[ W ] ^{\rm} .

Properties

Relation to the cross-covariance matrix

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

\operatorname{K} _{XY} = \operatorname{E}[( X - \operatorname{E}[ X ])( Y - \operatorname{E}[ Y ]) ^{\rm} ] = \operatorname{R} _{XY} - \operatorname{E}[ X ] \operatorname{E}[ Y ] ^{\rm}

Respectively for complex random vectors:

\operatorname{K} _{ZW} = \operatorname{E}[( Z - \operatorname{E}[ Z ])( W - \operatorname{E}[ W ]) ^{\rm} ] = \operatorname{R} _{ZW} - \operatorname{E}[ Z ] \operatorname{E}[ W ] ^{\rm}

See also

• Autocorrelation
• Correlation does not imply causation
• Covariance function
• Pearson product-moment correlation coefficient
• Correlation function (astronomy)
• Correlation function (statistical mechanics)
• Correlation function (quantum field theory)
• Mutual information
• Rate distortion theory
• Radial distribution function

Further reading

• Hayes, Monson H., Statistical Digital Signal Processing and Modeling, John Wiley & Sons, Inc., 1996. .
• Solomon W. Golomb, and Guang Gong. Signal design for good correlation: for wireless communication, cryptography, and radar. Cambridge University Press, 2005.
• M. Soltanalian. Signal Design for Active Sensing and Communications. Uppsala Dissertations from the Faculty of Science and Technology (printed by Elanders Sverige AB), 2014.

Notes and References

1. Book: Gubner, John A. . 2006 . Probability and Random Processes for Electrical and Computer Engineers . Cambridge University Press . 978-0-521-86470-1.