Binary Signal Sets

The following signal set is employed to transmit equally likely signals over an additive white Gaussian noise channel with spectral height $\frac{N_0}{2}$.

1. Sketch and accurately label the block diagram of the receiver which minimizes the probability of error.
2. Compute the probability of error of your receiver from part (a).
3. Consider now the following receiver:
   
   
   
   Compute the probability of error of this receiver if $g(t)$ is given by
   
   $$g(t) = \left\{ \begin{array}{cl} 1 & \mbox{for $0 \leq t < 1$}\\ -1 & \mbox{for $1 \leq t < 2$} \end{array} \right.$$
   
   

4. Determine the signal set, $s_0(t)$ and $s_1(t)$, such that the receiver in part (c) is optimum and the average signal energy is equal to $E=2$.
5. Compute the probability of error achieved by the receiver in part (c) and your signal set from part (d).