Correlation functions

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 $\mathbf = (X_1,\ldots,X_m)^$ and $\mathbf = (Y_1,\ldots,Y_n)^$, each containing random elements whose expected value and variance exist, the cross-correlation matrix of $\mathbf$ and $\mathbf$ is defined by

and has dimensions $m \times n$. Written component-wise:

:\operatorname_ =
\begin
\operatorname _1 Y_1& \operatorname _1 Y_2& \cdots & \operatorname _1 Y_n\\ \\
\operatorname _2 Y_1& \operatorname _2 Y_2& \cdots & \operatorname _2 Y_n\\ \\
\vdots & \vdots & \ddots & \vdots \\ \\
\operatorname _m Y_1& \operatorname _m Y_2& \cdots & \operatorname _m Y_n\\ \\
\end

The random vectors $\mathbf$ and $\mathbf$ need not have the same dimension, and either might be a scalar value.

Example

For example, if $\mathbf = \left( X_1,X_2,X_3 \right)^$ and $\mathbf = \left( Y_1,Y_2 \right)^$ are random vectors, then
$\operatorname_$ is a $3 \times 2$ matrix whose $(i,j)$-th entry is \operatorname _i Y_j/math>.

Complex random vectors

If $\mathbf = (Z_1,\ldots,Z_m)^$ and $\mathbf = (W_1,\ldots,W_n)^$ are complex random vectors, each containing random variables whose expected value and variance exist, the cross-correlation matrix of $\mathbf$ and $\mathbf$ is defined by

:\operatorname_ \triangleq\ \operatorname mathbf \mathbf^/math>

where $^$ denotes Hermitian transposition.

Uncorrelatedness

Two random vectors $\mathbf=(X_1,\ldots,X_m)^$ and $\mathbf=(Y_1,\ldots,Y_n)^$ are called uncorrelated if
:\operatorname mathbf \mathbf^= \operatorname mathbfoperatorname mathbf.

They are uncorrelated if and only if their cross-covariance matrix $\operatorname_$ matrix is zero.

In the case of two complex random vectors $\mathbf$ and $\mathbf$ they are called uncorrelated if
:\operatorname mathbf \mathbf^= \operatorname mathbfoperatorname mathbf
and
:\operatorname mathbf \mathbf^= \operatorname mathbfoperatorname mathbf.

Properties

Relation to the cross-covariance matrix

The cross-correlation is related to the ''cross-covariance matrix'' as follows:
:\operatorname_ = \operatorname \mathbf - \operatorname[\mathbf(\mathbf - \operatorname[\mathbf">mathbf.html" ;"title="\mathbf - \operatorname[\mathbf">\mathbf - \operatorname[\mathbf(\mathbf - \operatorname[\mathbf^] = \operatorname_ - \operatorname[\mathbf] \operatorname mathbf
: Respectively for complex random vectors:
:\operatorname_ = \operatorname \mathbf - \operatorname[\mathbf(\mathbf - \operatorname[\mathbf">mathbf.html" ;"title="\mathbf - \operatorname[\mathbf">\mathbf - \operatorname[\mathbf(\mathbf - \operatorname[\mathbf^] = \operatorname_ - \operatorname[\mathbf] \operatorname mathbf

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

References

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.

{{DEFAULTSORT:Correlation Function
Covariance and correlation
Time series
Spatial analysis
Matrices (mathematics)
Signal processing