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

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

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

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

${\begin{aligned}\mathbf {C} _{x}&=E[(\mathbf {x} -\mathbf {m} _{x})(\mathbf {x} -\mathbf {m} _{x})^{H}]\\&=\mathbf {R} _{x}-\mathbf {m} _{x}\mathbf {m} _{x}^{H}\\\end{aligned}}$

Where $\mathbf {m} _{x}$ is a vector giving the mean of signal $\mathbf {x}$ 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.