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.

- A simple but illustrative example of
*asymptotic efficiency*of ML estimation:- We observe r=A
- Compute Â
_{MLE}(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=10
^{6},10^{5},10^{4},10^{3},10^{2},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.

^{2}+n where A > 0 is an unknown constant and n~N(0,s).

- Compute Â
- 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)||
- Time-limit x(t) to |t| < T/2; call this x
_{T}(t)=x(t) for |t| < T/2; 0 for |t| > T/2. - Band-limit x
_{T}(t) to W Hertz; call this x_{DB}(t). X_{DB}(f)=X_{T}(f) for |f| < W; 0 for |f| > W. - Let E
_{DB}=||x_{DB}(t)||^{2}< E be the energy in x_{DB}(t). What f(t) maximizes the ratio E_{DB}/E?

^{2}=&int |x(t)|^{2}dt=E. Let X(f) be the Fourier transform of x(t).- 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!)

- Time-limit x(t) to |t| < T/2; call this x
- 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 H
_{1}; r(t)=n(t) under H_{0}, 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 r
_{i}, i > 1, are irrelevant to this detection problem. - Derive the LRT for r
_{1}, and re-derive the matched filter from Problem Set #4.