Optimum Receiver Design

The following binary signal set is used to transmit equally likely messages over an additive white Gaussian noise channel (spectral height $\frac{N_0}{2}$),

$$\begin{array} {cc} s_0(t) = & \left\{ \begin{array} {cl} 5 ... ... \leq t \leq T$}\\ 0 & \mbox{else}\end{array}\right.\end{array}$$

1.

Draw a block diagram of the optimum receiver.

2.

Compute the probability of error for this receiver.

3.

Repeat parts (a) and (b) for the following signal set:

$$\begin{array} {cc} s_0(t) = & \left\{ \begin{array} {cl} 7 ... ... \leq t \leq T$}\\ 0 & \mbox{else}\end{array}\right.\end{array}$$

4.

Sketch and accurately label a signal space diagram, including decision boundaries, for each of the two signal sets above.

5.

Verify that both of the above signal sets require the same average power to transmit a bit. Then compare their performance with respect to the minimum probability of error. Which signal set do you prefer?