Autocorrelation matrix: Difference between revisions

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	\end{bmatrix}		\end{bmatrix}
	</math>		</math>
			This is clearly a ]. More specifically because <math>R_{xx}(j) = R_{xx}(\!-j) = R_{xx}(N-j)</math>, it is a ].

	Note that this is a ].

	== References ==		== References ==

Revision as of 19:34, 29 August 2010

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}} ={\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 Toeplitz matrix. More specifically because $R_{xx}(j)=R_{xx}(\!-j)=R_{xx}(N-j)$ , it is a circulant matrix.

References

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