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Autocorrelation matrix: Difference between revisions

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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: 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:


:<math>\mathbf{R_x} = \begin{bmatrix} :<math>\mathbf{R}_x = E = \begin{bmatrix}
R_{xx}(0) & R_{xx}(1) & R_{xx}(2) & \cdots & R_{xx}(N-1) \\ 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}(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) \\ R_{xx}(2) & R_{xx}(1) & R_{xx}(0) & \cdots & R^*_{xx}(N-3) \\
\vdots & \vdots & \vdots & \ddots & \vdots \\ \vdots & \vdots & \vdots & \ddots & \vdots \\
R_{xx}(N-1) & R_{xx}(N-2) & R_{xx}(N-3) & \cdots & R_{xx}(0) \\ R_{xx}(N-1) & R_{xx}(N-2) & R_{xx}(N-3) & \cdots & R_{xx}(0) \\
\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 ]. 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 ].

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

:<math>\begin{align}
\mathbf{C}_x &= E\\
&= \mathbf{R}_x - \mathbf{m}_x\mathbf{m}_x^H\\
\end{align}
</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 == == References ==

Revision as of 23:33, 5 September 2010

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 = E [ x x H ] = [ 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}=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 x {\displaystyle \mathbf {x} } is a real valued function, then it is a circulant matrix since R x x ( j ) = R x x ( − j ) = R x x ( N − j ) {\displaystyle R_{xx}(j)=R_{xx}(\!-j)=R_{xx}(N-j)} . Finally if x {\displaystyle \mathbf {x} } is wide-sense stationary then it's autocorrelation matrix will be nonnegative definite.

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

C x = E [ ( x − m x ) ( x − m x ) H ] = R x − m x m x H {\displaystyle {\begin{aligned}\mathbf {C} _{x}&=E\\&=\mathbf {R} _{x}-\mathbf {m} _{x}\mathbf {m} _{x}^{H}\\\end{aligned}}}

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