Multi-Path Channel

Equally likely messages are transmitted over an additive white Gaussian noise channel using binary phase shift keying (BPSK). Hence the signal set consists of the two signals

$$\begin{array}{cc} s_0(t) = \left\{ \begin{array}{cl} \s... ...q t < T$}\\ 0 &\mbox{else.} \end{array} \right. \end{array}$$

1. Draw the simplest possible block diagram of the optimum receiver and compute its error probability.
2. For the remainder of the problem assume that the transmitted signal passes through a (multi-path) channel with impulse response
   
   $$h(t) = \delta(t) + \delta(t-T)$$
   
   
   before the Gaussian noise is added. Also, assume that two consecutive bits are transmitted. In other words one of the two signals above is transmitted during the interval $[0,T)$ and one of the two signals above is transmitted during the interval $[T,2T)$. Nothing is transmitted before or after these two bits.

   Accurately sketch and label all four possible signals that can be observed at the out of the channel (before noise is added).

3. If your receiver from part (a) is used to demodulate the two consecutive bits which were subject to filtering with $h(t)$ what are the probabilities of error for the first and the second transmitted bit?
4. Assuming the channel impulse response $h(t)$ is known, design a better receiver for demodulating the two transmitted bits. Explain your approach and draw a block diagram of the resulting receiver.
5. If possible compute the probability of error of your receiver. If you feel that it is not possible to express the probability of error of your receiver in closed form explain why it is not possible to do so.