Assume that the state-space model xn = xn−1+vn, yn = xn+wn for the one-dimensional state xn is given, where vn ∼ N(0, τ 2 ), wn ∼ N(0,1) and x0 ∼ N(0,102 ). (1) Show the Kalman filter algorithm for...

Assume that the state-space model xn = xn−1+vn, yn = xn+wn for the one-dimensional state xn is given, where vn ∼ N(0, τ 2 ), wn ∼ N(0,1) and x0 ∼ N(0,102 ).

(1) Show the Kalman filter algorithm for this model.

(2) Show the relation between Vn+1|n and Vn|n−1 .

(3) If Vn|n−1 → V as n → ∞, show that V satisfies the equation V 2 − τ 2V −τ 2 = 0.

 (4) Consider the Kalman filter algorithm as n → ∞ (stationary Kalman filter).