Binary Signaling over a Random Amplitude Channel

The following signal set is used to transmit equally likely messages over an additive white Gaussian noise channel:

$$\begin{array}{cc} s_0(t) = \begin{array}{cl} 0 & \mbox{fo... ...{1}{T}} & \mbox{for $0 \leq t \leq T$.} \end{array}\end{array}$$

1. Draw and accurately label a block diagram of the receiver which minimizes the probability of an incorrect decision.
2. Compute the probability of error of your receiver.
3. For the remainder of the problem assume that during transmission the signal is subject to a random gain $A$. The density function of the random variable $A$ is given by:
   
   $$f_A(a) = \frac{1}{\sqrt{2\pi}} \exp(-\frac{a^2}{2})$$
   
   
   Show that the optimum decision rule for deciding which of the two signals was transmitted in the presence of the random gain is of the form
   
   $$\begin{array}{cc} R^2 > \gamma & \mbox{decide $s_1(t)$\ wa... ...\gamma & \mbox{decide $s_0(t)$\ was transmitted,} \end{array}$$
   
   
   where
   
   $$R = \int_0^T R_t \cdot s_1(t) dt$$
   
   
   Be as specific as you can and try to determine the threshold $\gamma$.
4. Find an expression for the probability of error in the presence of the unknown gain. Is this probability of error larger or smaller than the probability of error you computed in part (b). Justify your answer!