Homework 7

ECE 630: Statistical Communication Theory
Prof. B.-P. Paris
Homework 7
Due April 1, 2003

Reading

Wozencraft & Jacobs: Chapter 4.

Problems

1. Wozencraft & Jacobs: Problem 4.7
2. Wozencraft & Jacobs: Problem 4.8
3. Wozencraft & Jacobs: Problem 4.11
4. Wozencraft & Jacobs: Problem 4.13
5. Wozencraft & Jacobs: Problem 4.14
6. The following signals are used to transmit equally likely messages over a channel corrupted by additive, white Gaussian noise of spectral height $\frac{N_0}{2}$:

   
   
   

   Compute the probability of error attained by the following receivers.

   1. 
      
      

   2. 
      
      

   3. 
      
      

   4. 
      
      

   5. Let $g(t) = 1- \vert t-1\vert$, $0 \leq t \leq 2$.

      
      
      

7. In this problem, we analyze the dependence of the probability of error on the threshold of the comparator in the optimum receiver. Assume one of two equally likely messages is transmitted using the following signals,
   
   $$\begin{array}{ccll} s_0(t) & = & 0 & \mbox{for $0 \leq t \le... ... \sqrt{\frac{E}{T}} & \mbox{for $0 \leq t \leq T$.} \end{array}$$
   
   
   The channel is corrupted by additive, white Gaussian noise of spectral height $\frac{N_0}{2}$.
   1. Draw the block diagram of the optimum receiver.
   2. Compute the probability of error attained by this receiver.
   3. The threshold of the optimum receiver is given by $\gamma = \frac{1}{2} (\vert\vert s_1\vert\vert^2 - \vert\vert s_0\vert\vert^2)$. What is the probability of error if instead this threshold were chosen as $\gamma = \lambda \vert\vert s_1\vert\vert^2 - (1-\lambda) \vert\vert s_0\vert\vert^2$?
   4. Plot the probability of error computed in part (c) for $-1 \leq \lambda \leq 1$. Use $\frac{N_0}{2} = 1$, $E = 1$, $T = 1$. You may approximate $Q(x)$ by $\frac{1}{2} \exp(\frac{-x^2}{2})$.