- We observe r(t)=E
^{½}s_{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 N
- Draw a block diagram of the optimal receiver, which uses a bank of matched filters.
- Show that the number of basis functions N equals M if the matrix [p
_{ij}] is nonsingular. - Compute Pr[error|H
_{1}] in terms of p_{ij}. HINT: r is Gaussian with covariance K_{r}=N_{0}[p_{ij}]/2. - Compute Pr[error] in terms of the matrix [p
_{ij}] using your answer to #3 above. - Show we can't do #3 and #4 above for
*simplex*signals, since then [p_{ij}] is singular. - HINT FOR THIS PROBLEM: See back side of the Problem Set #6 handout.

_{0}/2.*A priori*probabilities: Pr[H_{i}]=1/M.

- 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_{b}^{2}).

It should include a matched filter, a peak detector, and a gain factor for b_{MAP}in particular.

- 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)||
- Let M be
*nonrandom*. Compute M_{MLE}based on {R(t),0< t< T}. Draw a block diagram. - Now let M be
*random*with pdf p_{M}(M)=N(0,s_{M}^{2}). Compute a*quartic*equation for M_{MAP}. - Show M
_{MAP}approaches M_{MLE}as s_{M}blows up. MLE commutes with nonlinear functions.

^{2}=E.

- Let M be
- 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/(w^{2}+1). - s(t)=(t
^{2}-1)^{3}for |t|< 1, 0 for |t|> 1. Let the actual value of a=3 below.

- Compute an explicit argmax formula for Â
_{MLE}. What is Q(t)*s(t) for this problem? - Explain why you can
*still*neglect the second term here in the formula from the handout. - 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.

- 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