Binary Receiver

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

1.

(5 pts) Draw and accurately label a block diagram of the optimum receiver for the above signal set.

2.

(10 pts) Compute the probability of error of the optimum receiver for $0 \leq \alpha \leq 1$.

3.

(5 pts) What value of $\alpha$ minimizes the probability of error in part (b).

4.

(5 pts) For $\alpha = \frac{1}{2}$, draw and accurately label a signal space diagram for the above signal set. State explicitly which basis functions $\phi_0(t)$ and $\phi_1(t)$ you have used to arrive at your diagram. Indicate the decision boundary.

5.

(5 pts) Repeat part (d) for $\alpha=0$.