EECS 564________________________PROBLEM SET #7________________________Winter 1999

ASSIGNED: March 08, 1999 (Monday) READ: Sections 3.9, 5.6, 7.1 and 8.2 of Srinath. And catch up.
DUE DATE: March 15, 1999 (Monday) THIS WEEK: Detection and estimation in WGN and NWGN.

1. We observe r(t)=E½si(t)+w(t), 0< t< T under Hi, i=1...M. ∫ si(t)sj(t)dt=pij; si(t) equal energies.
w(t) is 0-mean WGN with power spectral density N0/2. A priori probabilities: Pr[Hi]=1/M.
1. Draw a block diagram of the optimal receiver, which uses a bank of matched filters.
2. Show that the number of basis functions N equals M if the matrix [pij] is nonsingular.
3. Compute Pr[error|H1] in terms of pij. HINT: r is Gaussian with covariance Kr =N0[pij]/2.
4. Compute Pr[error] in terms of the matrix [pij] using your answer to #3 above.
5. Show we can't do #3 and #4 above for simplex signals, since then [pij] is singular.
6. HINT FOR THIS PROBLEM: See back side of the Problem Set #6 handout.

• We observe r(t)=bs(t-a)+w(t), -T< t< T. s(t) is a known short pulse with duration < < T.
• w(t) is WGN as in Problem #1. Assume the energy in s(t-a) is independent of the delay a.
• a is an unknown time delay random variable, uniformly distributed over interval [-T,T].
• b is an unknown gain random variable with Gaussian pdf pb(B)=N(0,sb2).
Draw a block diagram of the receiver for computing the joint MAP estimates of a and b.
It should include a matched filter, a peak detector, and a gain factor for bMAP in particular.
1. We observe r(t)=(1/M)f(t)+w(t), 0< t< T. w(t) is WGN as above. f(t) is known; ||f(t)||2=E.
2. Let M be nonrandom. Compute MMLE based on {R(t),0< t< T}. Draw a block diagram.
3. Now let M be random with pdf pM(M)=N(0,sM2). Compute a quartic equation for MMAP.
4. Show MMAP approaches MMLE as sM blows up. MLE commutes with nonlinear functions.

• How much difference does prewhitening of estimation in non-white Gaussian noise make?
• We observe r(t)=s(t-a)+n(t) for 0 < t< 10, where a is an unknown (constant) time delay.
• n(t) is 0-mean non-white Gaussian noise with power spectrum Sn(w)=0.04/(w2+1).
• s(t)=(t2-1)3 for |t|< 1, 0 for |t|> 1. Let the actual value of a=3 below.

1. Compute an explicit argmax formula for ÂMLE. What is Q(t)*s(t) for this problem?
2. Explain why you can still neglect the second term here in the formula from the handout.
3. Write a Matlab program that generates 10 realizations of n(t) (filter wgn with 0.2e-t,t> 0),
plots the correlator output for each realization, matching to both s(t) (wrong) and Q(t)*s(t),
and computes the mean and standard deviation of the estimated a for both matched filters.
10 realizations is insufficient, but it does give you some feel about the correlator outputs.
4. Re-run the program several times. Your result should be that matching to Q(t)*s(t) usually
(not always) works better, sometimes a LOT better, than matching (incorrectly) to s(t).

"Acting is a profession in which one day you're delivering soliloquies,
and the next day you're delivering pizzas"--Lord Lawrence Olivier