Problem 14

Equally likely messages are transmitted over an additive white Gaussian noise channel with spectral height $\frac{N_0}{2}$. The signal set is given by

$$\begin{array}{cc} s_0(t) = \left\{ \begin{array}{cl} A ... ...\leq T$}\\ 0 & \mbox{ else.} \end{array} \right. \end{array}$$

1. Draw and accurately label the simplest possible block diagram of the minimum probability of error receiver.
2. Compute the probability of error for your receiver.
3. For the remainder of the problem the following receiver is used:

   
   
   

   The nonlinearity in the receiver above has the following characteristic:

   

   and the decision device operates as follows:
   

   $$\hat{b} = \left\{ \begin{array}{cl} 0 & \mbox{ if $S=2$,}... ...undecided} & \mbox{if $\vert S\vert = 1$.} \end{array}\right.$$
   
   
   Find the threshold $\Delta$ in the non-linearity such that the probability that the receiver is ``undecided'' equals some constant $\alpha$.
4. Find the probability of error (do not consider an ``undecided'' decision as an error) as a function of $\Delta$.
5. Illustrate the operation of this receiver in a suitably chosen and accurately labeled signal space diagram.
6. Suggest a scenario where the proposed receiver might be useful.