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| #REDIRECT ] | |||
| The '''autocorrelation matrix''' is used in various digital signal processing algorithms. It consists of elements of the discrete ] function, <math>R_{xx}(j)</math> arranged in the following manner: | |||
| {{R with history}} | |||
| :<math>\mathbf{R}_x = E = \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} | |||
| </math> | |||
| This is clearly a ] and a ]. If <math>\mathbf{x}</math> is ] then its autocorrelation matrix will be ]. | |||
| The ''autocovariance matrix'' is related to the autocorrelation matrix as follows: | |||
| :<math> | |||
| \mathbf{C}_x &= E\\ | |||
| &= \mathbf{R}_x - \mathbf{m}_x\mathbf{m}_x^H\\ | |||
| </math> | |||
| Where <math>\mathbf{m}_x</math> is a vector giving the mean of signal <math>\mathbf{x}</math> 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. | |||
| ] | ] | ||
| ] | |||
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