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Discrete Stochastic Processes and Optimal Filtering by Roger Ceschi, Jean-Claude Bertein

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5.6. Exercises for Chapter 5

Exercise 5.1.

Our task is to estimate a signal XK, whose autocorrelation function is:

images

The measures yK = xK + nK of the process YK are filtered by a Wiener filter of response h.

The noise NK is orthogonal to the signal XK and:

images

1) Give the response of the 2nd order Wiener filter (FIR).

2) Give the least mean square error obtained.

Solution 5.1.

1) image;

2) image with image.

Exercise 5.2.

We propose calculating a 2nd order FIR filter.

YK the input to the filter has the form YK = XK + WK where XK is the signal emitted and WK is a white noise orthogonal at XK (the processes are all wide sense stationary (WSS)).

Knowing the statistical autocorrelation:

images

and knowing:

images

With:

images

1) Give ...

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