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.
"Diplomacy is the art of saying 'nice doggie' until you can find a stick."
- 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).
- Compute ÂMLE(R). Be sure to consider R < 0 as a separate case.
- 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.
- Means should equal the C-R bound for smaller s, and lie BELOW the bound for larger s.
Explain BOTH of these results.
- 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:
- Parseval's theorem, convolution with a sinc, and the Cauchy-Schwarz inequality;
- The results of problem #2 above and a little thought (your choice!)
- 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).
- 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.
- 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:
- Expand r(t) in a Karhunen-Loeve expansion whose first basis function is s(t).
- Show that coordinate random variables ri, i > 1, are irrelevant to this detection problem.
- Derive the LRT for r1, and re-derive the matched filter from Problem Set #4.