Autocorrelation matrix: Difference between revisions

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	\end{bmatrix}		\end{bmatrix}
	</math>		</math>
	This is clearly a ] and a ]. Furthermore, if <math>\mathbf{x}</math> is a real valued function, then it is a ] since <math>R_{xx}(j) = R_{xx}(\!-j) = R_{xx}(N-j)</math>. Finally if <math>\mathbf{x}</math> is ] then ~~it's~~ autocorrelation matrix will be ].		This is clearly a ] and a ]. Furthermore, if <math>\mathbf{x}</math> is a real valued function, then it is a ] since <math>R_{xx}(j) = R_{xx}(\!-j) = R_{xx}(N-j)</math>. Finally if <math>\mathbf{x}</math> is ] then its autocorrelation matrix will be ].

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

Revision as of 09:58, 1 November 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}=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}}

This is clearly a Hermitian matrix and a Toeplitz matrix. Furthermore, if $\mathbf {x}$ is a real valued function, then it is a circulant matrix since $R_{xx}(j)=R_{xx}(\!-j)=R_{xx}(N-j)$ . Finally 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:

{\begin{aligned}\mathbf {C} _{x}&=E\\&=\mathbf {R} _{x}-\mathbf {m} _{x}\mathbf {m} _{x}^{H}\\\end{aligned}}

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.

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