EECS 564________________________PROBLEM SET #9________________________Winter 1999

ASSIGNED: March 29, 1999 (Monday) READ: Sections 6.4-6.7 of Srinath (on Kalman filtering).
DUE DATE: April 05, 1999 (Monday) THIS WEEK: Causal Wiener and basic Kalman filtering.

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

    1. 0-mean WSS y(t) has Ry(t)=(3/2)e-|t|+(11/3)e-3|t|.
    2. Show that Sy(s)=[49-25s2]/[(1-s2)(9-s2)].
    3. Show the LLSE filter for predicting (log 2) seconds ahead is h(t)=(1/5)d(t)+(3/25)e-7t/51(t), where d(t)=impulse and 1(t)=unit step function.
    4. Show that the mean square error for this causal prediction filter is 117/24.

  2. We observe r(t)=a(t)+n(t) where E[a(t)n(s)]=0 and everything is 0-mean WSS.
    Power spectra: Sa(w)=2ks2/(w2+k2) and Sn(w)=Nw2 for known constants k,s,N.
    Do the following for general k,s,N (you will get some messy expressions):
    1. Compute the causal Wiener filter for estimating a(t) from {R(s),s< t}.
    2. Compute the mean square error for this filter.
    3. Compute the infinite smoothing filter and its mean square error.

  3. 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.
    1. Write out the Kalman filter equations for estimating x(n) from {y(1)..y(n)}.
    2. Use them to compute the LLSE of x(2) from y(1) and y(2).
    3. 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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