# Autocorrelation matrix

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:

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

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

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

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

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