Discrete Stochastic Processes and Optimal Filtering

6.10. Exercises for Chapter 6

Exercise 6.1. [WID 85]

An adaptive filter is characterized by:

– correlation matrix of data

– intercorrelation vector

and D_K being the desired output

1) Give the cost expression .

2) Calculate the optimal vector .

3) Give the expression of minimum cost .

4) Calculate the values proper to R.

5) Determine the proper vectors in such a way that matrix Q of the proper vectors be “normalized” (that is to say QQ^T = I), these vectors representing the principal axes of the family of ellipses.

6) Give the limits of μ convergence parameter used in the algorithm of stochastic gradient.

Solution 6.1.

1) .

2) .

3) .

4) .

5) .

6) 0<μ<1/3.