Autocorrelation matrix

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. Furthermore, if \mathbf{x} is a real valued function, then it is a circulant matrix since R_{xx}(j) = R_{xx}(\!-j) = R_{xx}(N-j). Finally 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{align} \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{align}

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

References