Discrete Stochastic Processes and Optimal Filtering

5.6. Exercises for Chapter 5

Exercise 5.1.

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

The measures y_K = x_K + n_K of the process Y_K are filtered by a Wiener filter of response h.

The noise N_K is orthogonal to the signal X_K and:

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

2) Give the least mean square error obtained.

Solution 5.1.

1) ;

2) with .

Exercise 5.2.

We propose calculating a 2^nd order FIR filter.

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

Knowing the statistical autocorrelation:

and knowing:

With:

1) Give ...