Misplaced Pages

This is an old revision of this page, as edited by Axnicho (talk | contribs) at 09:58, 1 November 2010 (possessive pronoun ‘its’). The present address (URL) is a permanent link to this revision, which may differ significantly from the current revision.

The autocorrelation matrix is used in various digital signal processing algorithms. It consists of elements of the discrete autocorrelation function, ${\displaystyle R_{xx}(j)}$ arranged in the following manner:

${\displaystyle \mathbf {R} _{x}=E={\begin{bmatrix}R_{xx}(0)&R_{xx}^{*}(1)&R_{xx}^{*}(2)&\cdots &R_{xx}^{*}(N-1)\\R_{xx}(1)&R_{xx}(0)&R_{xx}^{*}(1)&\cdots &R_{xx}^{*}(N-2)\\R_{xx}(2)&R_{xx}(1)&R_{xx}(0)&\cdots &R_{xx}^{*}(N-3)\\\vdots &\vdots &\vdots &\ddots &\vdots \\R_{xx}(N-1)&R_{xx}(N-2)&R_{xx}(N-3)&\cdots &R_{xx}(0)\\\end{bmatrix}}}$

This is clearly a Hermitian matrix and a Toeplitz matrix. Furthermore, if ${\displaystyle \mathbf {x} }$ is a real valued function, then it is a circulant matrix since ${\displaystyle R_{xx}(j)=R_{xx}(\!-j)=R_{xx}(N-j)}$. Finally if ${\displaystyle \mathbf {x} }$ is wide-sense stationary then its autocorrelation matrix will be nonnegative definite.

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

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

Where ${\displaystyle \mathbf {m} _{x}}$ is a vector giving the mean of signal ${\displaystyle \mathbf {x} }$ 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.

• Matrices
• Signal processing