Cross-correlation matrix
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| Correlation and covariance |
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Correlation and covariance of random vectors |
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Correlation and covariance of stochastic processes |
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Correlation and covariance of deterministic signals
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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 [math]\displaystyle{ \mathbf{X} = (X_1,\ldots,X_m)^{\rm T} }[/math] and [math]\displaystyle{ \mathbf{Y} = (Y_1,\ldots,Y_n)^{\rm T} }[/math], each containing random elements whose expected value and variance exist, the cross-correlation matrix of [math]\displaystyle{ \mathbf{X} }[/math] and [math]\displaystyle{ \mathbf{Y} }[/math] is defined by[1]:p.337
[math]\displaystyle{ \operatorname{R}_{\mathbf{X}\mathbf{Y}} \triangleq\ \operatorname{E}[\mathbf{X} \mathbf{Y}^{\rm T}] }[/math]
and has dimensions [math]\displaystyle{ m \times n }[/math]. Written component-wise:
- [math]\displaystyle{ \operatorname{R}_{\mathbf{X}\mathbf{Y}} = \begin{bmatrix} \operatorname{E}[X_1 Y_1] & \operatorname{E}[X_1 Y_2] & \cdots & \operatorname{E}[X_1 Y_n] \\ \\ \operatorname{E}[X_2 Y_1] & \operatorname{E}[X_2 Y_2] & \cdots & \operatorname{E}[X_2 Y_n] \\ \\ \vdots & \vdots & \ddots & \vdots \\ \\ \operatorname{E}[X_m Y_1] & \operatorname{E}[X_m Y_2] & \cdots & \operatorname{E}[X_m Y_n] \\ \\ \end{bmatrix} }[/math]
The random vectors [math]\displaystyle{ \mathbf{X} }[/math] and [math]\displaystyle{ \mathbf{Y} }[/math] need not have the same dimension, and either might be a scalar value.
Example
For example, if [math]\displaystyle{ \mathbf{X} = \left( X_1,X_2,X_3 \right)^{\rm T} }[/math] and [math]\displaystyle{ \mathbf{Y} = \left( Y_1,Y_2 \right)^{\rm T} }[/math] are random vectors, then [math]\displaystyle{ \operatorname{R}_{\mathbf{X}\mathbf{Y}} }[/math] is a [math]\displaystyle{ 3 \times 2 }[/math] matrix whose [math]\displaystyle{ (i,j) }[/math]-th entry is [math]\displaystyle{ \operatorname{E}[X_i Y_j] }[/math].
Complex random vectors
If [math]\displaystyle{ \mathbf{Z} = (Z_1,\ldots,Z_m)^{\rm T} }[/math] and [math]\displaystyle{ \mathbf{W} = (W_1,\ldots,W_n)^{\rm T} }[/math] are complex random vectors, each containing random variables whose expected value and variance exist, the cross-correlation matrix of [math]\displaystyle{ \mathbf{Z} }[/math] and [math]\displaystyle{ \mathbf{W} }[/math] is defined by
- [math]\displaystyle{ \operatorname{R}_{\mathbf{Z}\mathbf{W}} \triangleq\ \operatorname{E}[\mathbf{Z} \mathbf{W}^{\rm H}] }[/math]
where [math]\displaystyle{ {}^{\rm H} }[/math] denotes Hermitian transposition.
Two random vectors [math]\displaystyle{ \mathbf{X}=(X_1,\ldots,X_m)^{\rm T} }[/math] and [math]\displaystyle{ \mathbf{Y}=(Y_1,\ldots,Y_n)^{\rm T} }[/math] are called uncorrelated if
- [math]\displaystyle{ \operatorname{E}[\mathbf{X} \mathbf{Y}^{\rm T}] = \operatorname{E}[\mathbf{X}]\operatorname{E}[\mathbf{Y}]^{\rm T}. }[/math]
They are uncorrelated if and only if their cross-covariance matrix [math]\displaystyle{ \operatorname{K}_{\mathbf{X}\mathbf{Y}} }[/math] matrix is zero.
In the case of two complex random vectors [math]\displaystyle{ \mathbf{Z} }[/math] and [math]\displaystyle{ \mathbf{W} }[/math] they are called uncorrelated if
- [math]\displaystyle{ \operatorname{E}[\mathbf{Z} \mathbf{W}^{\rm H}] = \operatorname{E}[\mathbf{Z}]\operatorname{E}[\mathbf{W}]^{\rm H} }[/math]
and
- [math]\displaystyle{ \operatorname{E}[\mathbf{Z} \mathbf{W}^{\rm T}] = \operatorname{E}[\mathbf{Z}]\operatorname{E}[\mathbf{W}]^{\rm T}. }[/math]
Properties
Relation to the cross-covariance matrix
The cross-correlation is related to the cross-covariance matrix as follows:
- [math]\displaystyle{ \operatorname{K}_{\mathbf{X}\mathbf{Y}} = \operatorname{E}[(\mathbf{X} - \operatorname{E}[\mathbf{X}])(\mathbf{Y} - \operatorname{E}[\mathbf{Y}])^{\rm T}] = \operatorname{R}_{\mathbf{X}\mathbf{Y}} - \operatorname{E}[\mathbf{X}] \operatorname{E}[\mathbf{Y}]^{\rm T} }[/math]
- Respectively for complex random vectors:
- [math]\displaystyle{ \operatorname{K}_{\mathbf{Z}\mathbf{W}} = \operatorname{E}[(\mathbf{Z} - \operatorname{E}[\mathbf{Z}])(\mathbf{W} - \operatorname{E}[\mathbf{W}])^{\rm H}] = \operatorname{R}_{\mathbf{Z}\mathbf{W}} - \operatorname{E}[\mathbf{Z}] \operatorname{E}[\mathbf{W}]^{\rm H} }[/math]
See also
- Autocorrelation
- Correlation does not imply causation
- Covariance function
- Pearson product-moment correlation coefficient
- Correlation function
- Correlation function (statistical mechanics)
- Correlation function (quantum field theory)
- Mutual information
- Rate distortion theory
- Radial distribution function
References
- ↑ Gubner, John A. (2006). Probability and Random Processes for Electrical and Computer Engineers. Cambridge University Press. ISBN 978-0-521-86470-1.
Further reading
- Hayes, Monson H., Statistical Digital Signal Processing and Modeling, John Wiley & Sons, Inc., 1996. ISBN:0-471-59431-8.
- 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.
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