Binary Signal Sets

The following signal set is used to transmit equally likely message over an AWGN channel with spectral height $\frac{N_0}{2}$:

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

1.

Draw and accurately label the simplest possible block diagram of the receiver which minimizes the probability of error with this signal set.

2.

Compute the probability of error for your receiver as a function of the parameter $\tau$.

3.

Which value of $\tau \in [0,T]$ minimizes the probability of error? Which value of $\tau \in [0,T]$ maximizes the probability of error?

4.

Illustrate the signal constellations which lead to minimum and maximum probability of error in suitably chosen signal space diagrams. Be precise: indicate clearly the basis functions of your signal spaces, the locations of the signals, decision boundaries, etc.