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

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3.6. Exercises for Chapter 3

Exercise 3.1.

Study the stationarity of the Gaussian process

XimagesN(m(K), min(j, K)) where E(XK) = m(K) is constant.

Exercise 3.2.

We are considering the real sequence hn defined by:

images

1) Determine the convergence domain of the Laurent series images.

2) If h = {hn |nimages} is a digital filter, determine its transfer function H(z) by clarifying its definition domain.

Solution 3.2.

images

The series converges if images and if |z| < 4, thus in the annulus images.

images

The series converges if |z| > 2 and if |z| < 1/4, thus in the annulus images

In K′: .

Exercise 3.3.

Develop H (z)= in series (of Laurent) of ...

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