Multi-Path Channels

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

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

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 transmitted signal is filtered by a linear filter with impulse response

$$h(t) = \frac{1}{2}\delta(t)-\frac{1}{2}\delta(t-\frac{T}{2})+ \frac{1}{2}\delta(t-T)$$

before the noise is added. Such a channel model is appropriate to describe multi-path effects in radio communication systems. The following block diagram summarizes this channel.

Sketch the two possible received signals under the assumption that there is no noise.

3.

If the impulse response is unknown at the receiver, a reasonable approach might be to continue to use the receiver from part (a) which assumes that $h(t)=\delta(t)$. Compute the probability of error that your receiver from part (a) achieves if the actual channel impulse response h(t) is as given above and the noise spectral height is $\frac{N_0}{2}$.

4.

If the channel impulse response h(t) 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.