- Joe observes y(t)=x(t)+v(t) where v(t) is 0-mean white noise with S
_{v}(w)=½

and x(t) is 0-mean WSS with E[x(t)v(s)]=0 and autocorrelation R_{x}(t)=e^{-|t|}.

Joe is too lazy to compute the causal Wiener filter, so he simply uses a*suboptimal*filter

h(t)=ke^{-at},t> 0, and chooses k and a to minimize the steady-state mean square error.

- Show that x(t) and Joe's estimate z(t) of x(t) satisfy the vector state equation

d_|x(t)| = |-1 0||x(t)| + |1 0||w(t)| dt|z(t)| |k -a||z(t)| |0 k||v(t)|

where w(t) is a 0-mean white process which when filtered gives x(t). - Compute the steady-state error covariance matrix for this,

and from that read off the mean square error E[(x(t)-z(t))^{2}] in terms of k and a. - Show that this is minimized for a=2.236 and k=1.24 and compute it.
- Compute the mean square error for the infinite smoothing filter for estimating x(t).

Compare this to your answer to (c). Why is it smaller?

- Show that x(t) and Joe's estimate z(t) of x(t) satisfy the vector state equation
- A satellite is spinning at constant angular rate w radians/second (w is very small).

We observe y(n)=x(n)+v(n), n=1,2... where v(n) is 0-mean WGN with variance r,

x(n) is angular position of the satellite, x(0)=0, and w is N(m,s^{2}).- Formulate this as a Kalman filtering problem by augmenting the state x(n) with w.
- Write out the Kalman filtering equations for the augmented-state problem.

- A dynamic system is described by x(n+1)=x(n)+u(n),n=0,1,2...where in fact

u(n)=x(n-1)+w(n-1) where w(n) is iid and N(0,1) and x(-1)=0 and x(0)=10.

We observe y(n)=x(n)+v(n) where v(n) is iid and N(0,3) and E[w(i)v(j)]=0.- Augment the state x(n) with u(n) and write out the Kalman filter equations.

These should be specifically the ones for estimating x(n) from {y(0)...y(n)}. - Write out the algebraic Riccati equation for the steady-state error covariance matrix.
*Solve*these equations. You should get P=[6 3;3 3] (this is a 2× 2 matrix).

- Augment the state x(n) with u(n) and write out the Kalman filter equations.