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Part of a series on Statistics
Correlation and covariance
For random vectors Autocorrelation matrix Cross-correlation matrix Auto-covariance matrix Cross-covariance matrix
For stochastic processes Autocorrelation function Cross-correlation function Autocovariance function Cross-covariance function
For deterministic signals Autocorrelation function Cross-correlation function Autocovariance function Cross-covariance function
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The auto-correlation matrix (als called second moment) of a random vector ${\displaystyle \mathbf {X} =(X_{1},\ldots ,X_{n})^{\rm {T}}}$ is a ${\displaystyle n\times n}$ matrix containing as elements the correlations of all pairs of elements of the random vector ${\displaystyle \mathbf {X} }$. The autocorrelation matrixis used in various digital signal processing algorithms.

Definition

For a random vector ${\displaystyle \mathbf {X} =(X_{1},\ldots ,X_{n})^{\rm {T}}}$ containing random elements whose expected value and variance exist, the auto-correlation matrix is defined by

${\displaystyle \operatorname {R} _{\mathbf {X} \mathbf {X} }\triangleq \ \operatorname {E} }$

Eq.1

where ${\displaystyle {}^{\rm {T}}}$ denotes transposition and has dimensions ${\displaystyle n\times n}$.

Written component-wise:

${\displaystyle \operatorname {R} _{\mathbf {X} \mathbf {X} }={\begin{bmatrix}\operatorname {E} &\operatorname {E} &\cdots &\operatorname {E} \\\\\operatorname {E} &\operatorname {E} &\cdots &\operatorname {E} \\\\\vdots &\vdots &\ddots &\vdots \\\\\operatorname {E} &\operatorname {E} &\cdots &\operatorname {E} \\\\\end{bmatrix}}}$

If ${\displaystyle \mathbf {Z} }$ is a complex random vector, the autocorrelation matrix is instead defined by

${\displaystyle \operatorname {R} _{\mathbf {Z} \mathbf {Z} }\triangleq \ \operatorname {E} }$.

Here ${\displaystyle {}^{\rm {H}}}$ denotes Hermitian transposition.

Example

For example, if ${\displaystyle \mathbf {X} =\left(X_{1},X_{2},X_{3}\right)^{\rm {T}}}$ is a random vectors, then ${\displaystyle \operatorname {R} _{\mathbf {X} \mathbf {X} }}$ is a ${\displaystyle 3\times 3}$ matrix whose ${\displaystyle (i,j)}$-th entry is ${\displaystyle \operatorname {E} }$.

Properties

• The autocorrelation matrix is a Hermitian matrix for complex random vectors and a symmetric matrix for real random vectors.
• The autocorrelation matrix is a Toeplitz matrix.
• The autocorrelation matrix is a positive semidefinite matrix, i.e. ${\displaystyle \mathbf {a} ^{\mathrm {T} }\operatorname {R} _{\mathbf {X} \mathbf {X} }\mathbf {a} \geq 0\quad {\text{for all }}\mathbf {a} \in \mathbb {R} ^{n}}$ for a real random vector respectively ${\displaystyle \mathbf {a} ^{\mathrm {T} }\operatorname {R} _{\mathbf {Z} \mathbf {Z} }\mathbf {a} \geq 0\quad {\text{for all }}\mathbf {a} \in \mathbb {C} ^{n}}$ in case of a complex random vector.
• All eigenvalues of the autocorrelation matrix are real and positive.
• The autocovariance matrix is related to the autocorrelation matrix as follows:

${\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {X} }=\operatorname {E} )(\mathbf {X} -\operatorname {E} )^{\rm {T}}]=\operatorname {R} _{\mathbf {X} \mathbf {X} }-\operatorname {E} \operatorname {E} ^{\rm {T}}}$

Respectively for complex random vectors:

${\displaystyle \operatorname {K} _{\mathbf {Z} \mathbf {Z} }=\operatorname {E} )(\mathbf {Z} -\operatorname {E} )^{\rm {H}}]=\operatorname {R} _{\mathbf {Z} \mathbf {Z} }-\operatorname {E} \operatorname {E} ^{\rm {H}}}$

References

• 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.

• Matrices
• Signal processing