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

For two random vectors $\mathbf{𝐗}=(X_1,\dots ,X_m{)}^{\mathrm{T}}$ and $\mathbf{𝐘}=(Y_1,\dots ,Y_n{)}^{\mathrm{T}}$, each containing random elements whose expected value and variance exist, the cross-correlation matrix of $\mathbf{𝐗}$ and $\mathbf{𝐘}$ is defined by

${\mathrm{R}}_{\mathbf{𝐗}\mathbf{𝐘}}\triangleq \mathrm{E}[\mathbf{𝐗}{\mathbf{𝐘}}^{\mathrm{T}}]$

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

${\mathrm{R}}_{\mathbf{𝐗}\mathbf{𝐘}}=\left[\begin{array}{cccc}\mathrm{E}[X_1{Y}_1]& \mathrm{E}[X_1{Y}_2]& \cdots & \mathrm{E}[X_1{Y}_n]\\ \\ \mathrm{E}[X_2{Y}_1]& \mathrm{E}[X_2{Y}_2]& \cdots & \mathrm{E}[X_2{Y}_n]\\ \\ ⋮& ⋮& \ddots & ⋮\\ \\ \mathrm{E}[X_m{Y}_1]& \mathrm{E}[X_m{Y}_2]& \cdots & \mathrm{E}[X_m{Y}_n]\\ \\ \end{array}\right]$

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

Example

For example, if $\mathbf{𝐗}={(X_1,X_2,X_3)}^{\mathrm{T}}$ and $\mathbf{𝐘}={(Y_1,Y_2)}^{\mathrm{T}}$ are random vectors, then ${\mathrm{R}}_{\mathbf{𝐗}\mathbf{𝐘}}$ is a $3\times 2$ matrix whose $(i,j)$-th entry is $\mathrm{E}[X_i{Y}_j]$.

Complex random vectors

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

${\mathrm{R}}_{\mathbf{𝐙}\mathbf{𝐖}}\triangleq \mathrm{E}[\mathbf{𝐙}{\mathbf{𝐖}}^{\mathrm{H}}]$

where ${}^{\mathrm{H}}$ denotes Hermitian transposition.

Uncorrelatedness

Two random vectors $\mathbf{𝐗}=(X_1,\dots ,X_m{)}^{\mathrm{T}}$ and $\mathbf{𝐘}=(Y_1,\dots ,Y_n{)}^{\mathrm{T}}$ are called uncorrelated if

$\mathrm{E}[\mathbf{𝐗}{\mathbf{𝐘}}^{\mathrm{T}}]=\mathrm{E}[\mathbf{𝐗}]\mathrm{E}[\mathbf{𝐘}{]}^{\mathrm{T}}.$

They are uncorrelated if and only if their cross-covariance matrix ${\mathrm{K}}_{\mathbf{𝐗}\mathbf{𝐘}}$ matrix is zero.

In the case of two complex random vectors $\mathbf{𝐙}$ and $\mathbf{𝐖}$ they are called uncorrelated if

$\mathrm{E}[\mathbf{𝐙}{\mathbf{𝐖}}^{\mathrm{H}}]=\mathrm{E}[\mathbf{𝐙}]\mathrm{E}[\mathbf{𝐖}{]}^{\mathrm{H}}$

and

$\mathrm{E}[\mathbf{𝐙}{\mathbf{𝐖}}^{\mathrm{T}}]=\mathrm{E}[\mathbf{𝐙}]\mathrm{E}[\mathbf{𝐖}{]}^{\mathrm{T}}.$

Properties

Relation to the cross-covariance matrix

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

${\mathrm{K}}_{\mathbf{𝐗}\mathbf{𝐘}}=\mathrm{E}[(\mathbf{𝐗}-\mathrm{E}[\mathbf{𝐗}])(\mathbf{𝐘}-\mathrm{E}[\mathbf{𝐘}]{)}^{\mathrm{T}}]={\mathrm{R}}_{\mathbf{𝐗}\mathbf{𝐘}}-\mathrm{E}[\mathbf{𝐗}]\mathrm{E}[\mathbf{𝐘}{]}^{\mathrm{T}}$

Respectively for complex random vectors:

${\mathrm{K}}_{\mathbf{𝐙}\mathbf{𝐖}}=\mathrm{E}[(\mathbf{𝐙}-\mathrm{E}[\mathbf{𝐙}])(\mathbf{𝐖}-\mathrm{E}[\mathbf{𝐖}]{)}^{\mathrm{H}}]={\mathrm{R}}_{\mathbf{𝐙}\mathbf{𝐖}}-\mathrm{E}[\mathbf{𝐙}]\mathrm{E}[\mathbf{𝐖}{]}^{\mathrm{H}}$

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

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.