EECS 564________________________PROBLEM SET #7________________________Winter 1999

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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.

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1. We observe r(t)=E^½s_i(t)+w(t), 0< t< T under H_i, i=1...M. ∫ s_i(t)s_j(t)dt=p_ij; s_i(t) equal energies.
   
   w(t) is 0-mean WGN with power spectral density N₀/2. A priori probabilities: Pr[H_i]=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 [p_ij] is nonsingular.
   3. Compute Pr[error|H₁] in terms of p_ij. HINT: r is Gaussian with covariance K_r =N₀[p_ij]/2.
   4. Compute Pr[error] in terms of the matrix [p_ij] using your answer to #3 above.
   5. Show we can't do #3 and #4 above for simplex signals, since then [p_ij] is singular.
   6. HINT FOR THIS PROBLEM: See back side of the Problem Set #6 handout.

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2. 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 p_b(B)=N(0,s_b²).
   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 b_MAP in particular.

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3. 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.
   1. Let M be nonrandom. Compute M_MLE based on {R(t),0< t< T}. Draw a block diagram.
   2. Now let M be random with pdf p_M(M)=N(0,s_M²). Compute a quartic equation for M_MAP.
   3. Show M_MAP approaches M_MLE as s_M blows up. MLE commutes with nonlinear functions.

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4. 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 S_n(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.

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   1. Compute an explicit argmax formula for Â_MLE. 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).

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