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. If $\mathbf{x}$ is wide-sense stationary then its autocorrelation matrix will be positive definite.

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

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

Where $\mathbf{m}_x$ 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.
• 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.