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.

  2. 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.

    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