Homework 6

ECE 630: Statistical Communication Theory
Prof. B.-P. Paris
Homework 6
Due March 25, 2003

Reading

Wozencraft & Jacobs: Chapter 4. Another reference for this material is: H.L. van Trees, "Detection Estimation, and Modulation Theory," pp. 239-257.

Problems

1. Let the covariance function of a wide sense stationary process be
   
   $$K_X(\tau) = \left\{\begin{array}{cl} 1-\vert\tau\vert & \tau \leq 1 \\ 0 & \mbox{otherwise.} \end{array} \right.$$
   
   
   Find the eigenfunctions and eigenvalues associated with the Karhunen-Loeve expansion of $X_t$ over $(0,T)$ with $T<1$.

2. Wozencraft & Jacobs: Problem 4.1
3. Wozencraft & Jacobs: Problem 4.2
4. Wozencraft & Jacobs: Problem 4.5
5. Binary Hypothesis Testing
   The two hypotheses are of the form:
   
   $$\begin{array}{cl} H_0: & \displaystyle{p_R(r) = \frac{1}{2} ... ...p_R(r) = \frac{1}{\sqrt{2 \pi}} e^{-\frac{r^2}{2}}} \end{array}$$
   
   
   1. Find the likelihood ratio.
   2. Compute the decision regions for various values of the threshold in the likelihood ratio test.