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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 x x ( j ) {\displaystyle R_{xx}(j)} arranged in the following manner:

R x = [ R x x ( 0 ) R x x ( 1 ) R x x ( 2 ) ⋯ R x x ( N − 1 ) R x x ( 1 ) R x x ( 0 ) R x x ( 1 ) ⋯ R x x ( N − 2 ) R x x ( 2 ) R x x ( 1 ) R x x ( 0 ) ⋯ R x x ( N − 3 ) ⋮ ⋮ ⋮ ⋱ ⋮ R x x ( N − 1 ) R x x ( N − 2 ) R x x ( N − 3 ) ⋯ R x x ( 0 ) ] {\displaystyle \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 x x ( j ) = R x x ( − j ) = R x x ( N − j ) {\displaystyle 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.
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