Continuing problem 1145 of the noisy predictor generate

Question

Continuing Problem 11.4.5 of the noisy predictor, generate sample paths of X_n and Y_n for n = 0, 1,..., 50 with the following parameters:

(a) c = 0.9, d = 10

(b) c = 0.9, d = 1

(c) c = 0.9, d = 0.1

(d) c = 0.6, d = 10

(e) c = 0.6, d = 1

(f) c = 0.6, d = 0.1

In each experiment, use η = σ = 1. Use the analysis of Problem 11.4.5 to interpret your results.

Problem 11.4.5

Suppose X_n is a random sequence satisfying

Where Z₁, Z₂,... is an iid random sequence with E[Zn] = 0 and Var[Zn] = σ² and c is a constant satisfying |c| ₀] = 0 and Var[X=] = σ²/(1 - c²). We make the following noisy measurement

Where W₁, W₂,... is an iid measurement noise sequence with E[W_n] = 0 and Var[W_n] = η² that is independent of X_n and Z_n.

(a) Find the optimal linear predictor, _n(Y_n-1), of X_n using the noisy observation Y_n-1.

(b) Find the mean square estimation error