Autocorrelation matrix

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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:


\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[edit]

  • Hayes, Monson H., Statistical Digital Signal Processing and Modeling, John Wiley & Sons, Inc., 1996. ISBN 0-471-59431-8.