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Introduction   ::   System Model   ::   Expectation   ::   Part 1   ::   Filters   ::   Part 2   ::   Submission

Estimating the expectations

The algorithm described above requires calculation of the autocorrelation matrices and the cross-correlation vectors. In practice, these are frequently estimated from sample averages. (In other words, we invoke an ergodic assumption.)

To illustrate this idea, suppose that we have a vector random process $ \xbf(t)$ , and that we have $ N$ vectors of measured sample data which we stack into columns as

$\displaystyle \begin{bmatrix}\xbf(1) & \xbf(2) & \xbf(3) & \ldots & \xbf(N)
\end{bmatrix}.
$

Then the autocorrelation matrix $ R_x = E[\xbf(t) \xbf(t)^T]$ may be estimated as

$\displaystyle \Rhat_x = \frac{1}{N-1} \sum_{i=1}^{N} \xbf(i) \xbf(i)^T.$ (12)

Let $ y(t)$ be a random process, with measured sample values $ y(1),
y(2), \ldots, y(n)$ . The cross correlation vector between a random process $ y(t)$ and $ \xbf(t)$ , $ \rbf = E[y(t) \xbf(t)]$ can be estimated as

$\displaystyle \rbfhat = \frac{1}{N} \sum_{i=1}^N y(t) \xbf(t).
$


Exercise 5
Let

$\displaystyle \Xbf = \begin{bmatrix}\xbf(1) & \xbf(2) & \xbf(3) & \ldots & \xbf(N)
\end{bmatrix}.
$

show that the estimated autocorrelation matrix $ R_x$ in ( 12 ) can be written as

$\displaystyle \Rhat_x = \frac{1}{N-1} \Xbf \Xbf^T.
$


Copyright 2008, by the Contributing Authors. Cite/attribute Resource . admin. (2006, June 13). Programming Assignments. Retrieved January 07, 2011, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Stochastic_Processes/sysid_3.htm. This work is licensed under a Creative Commons License Creative Commons License