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.

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