Multi-Path Channels

The following two signals are used to transmit equally likely messages over a channel with AWGN of spectral height $\frac{N_0}{2}$.

$$\begin{array}{cc} s_0(t) = \left\{ \begin{array}{cl} A & \mb... ...eq t \leq T$}\\ 0 & \mbox{else} \end{array}\right. \end{array}$$

1. Sketch the simplest possible block diagram of the optimum receiver and compute the probability of error attained by your receiver.
2. For the remainder of the problem, assume that the channel has an impulse response
   
   $$h(t) = 4\delta(t)-3\delta(t-\frac{T}{2})+2\delta(t-T).$$
   
   
   Further, to avoid problems with overlapping of signals (intersymbol interference), consecutive transmissions are spaced $2T$ seconds.
   Sketch the two possible received signals if the channel introduces no noise.
3. Compute the probability of error that your receiver from part (a) achieves if the channel impulse is as given above and the noise spectral height is $\frac{N_0}{2}$.
4. If the channel impulse response is known at the receiver, can you design a better receiver than the one in part (a)? If yes, draw a block diagram of the new receiver and compute the associated probability of error. If no, explain why not.