EECS 564________________________PROBLEM SET #5________________________Winter 1999

ASSIGNED: February 05, 1999
DUE DATE: February 12, 1999

READ: Section 2.5 (Karhunen-Loeve expansions) and Sections 4.3-4.6 (colored noise) of Srinath.
Read Section 4.5 on integral equations (pertinent now); skip the references to spectral factorization.
THIS WEEK: Karhunen-Loeve expansions. We will use these for detection/estimation of signals.

1. A simple but illustrative example of asymptotic efficiency of ML estimation:
We observe r=A2+n where A > 0 is an unknown constant and n~N(0,s).
1. Compute ÂMLE(R). Be sure to consider R < 0 as a separate case.
2. Compute the Cramer-Rao lower bound on unbiased estimates Â(R).
• Using MATLAB, and letting the actual value of A=5, do the following:
• Use MATLAB function "randn" to generate realizations of n;
• Compute the square error (A-Â(R))2 for 100 realizations of n;
• Plot EACH square error as a dot, and their MEAN as an X, for each of the following
• values of noise variance s: s=106,105,104,103,102,10,1. Plot on a log-log plot;
• Plot the Cramer-Rao bound from (b) as a line on the same plot.
3. Means should equal the C-R bound for smaller s, and lie BELOW the bound for larger s.
Explain BOTH of these results.

2. Prove that the largest eigenvalue of an integral equation is bounded below by
&int &int f(t)K(t,u)f(u)dtdu, where f(t) is any function such that ||f(t)||=1.
HINT: What projection &int f(t)x(t)dt of x(t) has the biggest variance (energy)?
• Let x(t) have total energy ||x(t)||2=&int |x(t)|2dt=E. Let X(f) be the Fourier transform of x(t).
• Time-limit x(t) to |t| < T/2; call this xT(t)=x(t) for |t| < T/2; 0 for |t| > T/2.
• Band-limit xT(t) to W Hertz; call this xDB(t). XDB(f)=XT(f) for |f| < W; 0 for |f| > W.
• Let EDB=||xDB(t)||2 < E be the energy in xDB(t). What f(t) maximizes the ratio EDB/E?
HINT: Use EITHER of the following:
1. Parseval's theorem, convolution with a sinc, and the Cauchy-Schwarz inequality;
2. The results of problem #2 above and a little thought (your choice!)

3. Find the eigenvalues and eigenfunctions over the interval 0 < t < T < 1 of the covariance function
K(t,s)=1-|t-s| for 0 < |t-s| < 1; K(t,s)=0 for |t-s| > 1. That is, solve the integral equation (ugh).
4. Prove that the eigenfunctions of a covariance function are orthogonal if their eigenvalues differ.
(This is required to show solving the integral equation is sufficient to find the K-L expansion.)
HINT: Modify the proof of the analogous result for matrices.
5. We observe random process r(t)=s(t)+n(t) under H1; r(t)=n(t) under H0, 0 < t < T,
where s(t) is a known signal, and n(t) is a zero-mean white Gaussian noise random process.
We now formally derive the matched filter for additive white Gaussian noise:
1. Expand r(t) in a Karhunen-Loeve expansion whose first basis function is s(t).
2. Show that coordinate random variables ri, i > 1, are irrelevant to this detection problem.
3. Derive the LRT for r1, and re-derive the matched filter from Problem Set #4.

"Diplomacy is the art of saying 'nice doggie' until you can find a stick."