Sub-Optimum Receiver

Equally likely messages are transmitted over an additive white Gaussian noise channel (spectral height $\frac{N_0}{2}$) by means of the following signal set:

$$\begin{array}{cl} s_0(t) = & \left\{ \begin{array}{cl} ... ... for $\frac{T}{2} < t \leq T$} \end{array} \right. \end{array}$$ (2)

1. Sketch and accurately label the two signals.
2. Draw and accurately label the simplest possible block diagram for the receiver which achieves the minimum probability of error.
3. Compute the probability of error that you receiver achieves.
4. Now assume that a receiver is used which simply integrates the received signal between $t=0$ and $t=T$. If the result is positive the receiver decides that $s_0(t)$ was transmitted, otherwise, if the integral is negative a decision in favor of $s_1(t)$ is made.
   What is the probability of error for this receiver.
5. Clearly, the receiver in part (d) is the matched filter receiver if antipodal square pulses are used as the signal set. Give a good reason, why the designers of a communication system might prefer the signal set (2) over square pulses even though it leads to a somewhat more complex receiver.