- y(t) is 0-mean WSS and i=its
*integral*over the interval 0< t< T.

We observe only y(0) and y(T), and we want to compute î_{LLSE}{y(0),y(T)}=ay(0)+by(T).

- Use the orthogonality principle to compute a and b.
- Show if R
_{y}(t)=e^{-a|t|}then î=(1/a)tanh[½ aT](y(0)+y(T)). - Show that for very small T, î in general becomes simply T×[
*average*of y(0) and y(T)].

- 0-mean WSS y(t) has R
- Show that S
_{y}(s)=[49-25s^{2}]/[(1-s^{2})(9-s^{2})]. - Show the LLSE filter for predicting (log 2) seconds ahead is h(t)=(1/5)d(t)+(3/25)e
^{-7t/5}1(t), where d(t)=impulse and 1(t)=unit step function. - Show that the mean square error for this causal prediction filter is 117/24.

_{y}(t)=(3/2)e^{-|t|}+(11/3)e^{-3|t|}.

- Show that S
- We observe r(t)=a(t)+n(t) where E[a(t)n(s)]=0 and everything is 0-mean WSS.

Power spectra: S_{a}(w)=2ks^{2}/(w^{2}+k^{2}) and S_{n}(w)=Nw^{2}for known constants k,s,N.

Do the following for general k,s,N (you will get some messy expressions):- Compute the causal Wiener filter for estimating a(t) from {R(s),s< t}.

- Compute the mean square error for this filter.
- Compute the infinite smoothing filter and its mean square error.

- Compute the causal Wiener filter for estimating a(t) from {R(s),s< t}.
- We observe y(n)=x(n)+v(n) for n=1,2... where v(n) is iid with variance 1 and

x(n) is an independent increments process: x(n)=x(n-1)+w(n) where w(n) is iid with variance 3.

- x(0)=0 and E[v(i)w(j)]=0 for all i,j> 0.
- Write out the Kalman filter equations for estimating x(n) from {y(1)..y(n)}.
- Use them to compute the LLSE of x(2) from y(1) and y(2).
- Using the LLSE formula â(R)=E[a]+[cov(a,r)/var(r)](R-E[r]) directly,

compute the LLSE of x(2) from y(1) and y(2). Show that the answers to (b) and (c) agree.

This shows that the Kalman filter can be viewed as a recursive implementation of LLSE.

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