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The autocorrelation matrix is used in various digital signal processing algorithms. It consists of elements of the discrete autocorrelation function, ${\displaystyle R_{xx}(j)}$ arranged in the following manner:

${\displaystyle \mathbf {R} _{x}=E={\begin{bmatrix}R_{xx}(0)&R_{xx}^{*}(1)&R_{xx}^{*}(2)&\cdots &R_{xx}^{*}(N-1)\\R_{xx}(1)&R_{xx}(0)&R_{xx}^{*}(1)&\cdots &R_{xx}^{*}(N-2)\\R_{xx}(2)&R_{xx}(1)&R_{xx}(0)&\cdots &R_{xx}^{*}(N-3)\\\vdots &\vdots &\vdots &\ddots &\vdots \\R_{xx}(N-1)&R_{xx}(N-2)&R_{xx}(N-3)&\cdots &R_{xx}(0)\\\end{bmatrix}}}$

This is a Hermitian matrix and a Toeplitz matrix. If ${\displaystyle \mathbf {x} }$ is real and wide-sense stationary then its autocorrelation matrix will be positive definite.

The autocovariance matrix is related to the autocorrelation matrix as follows:

${\displaystyle \mathbf {C} _{x}=\operatorname {E} =\mathbf {R} _{x}-\mathbf {m} _{x}\mathbf {m} _{x}^{H}}$

Where ${\displaystyle \mathbf {m} _{x}}$ is a vector giving the mean of signal ${\displaystyle \mathbf {x} }$ at each index of time.

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