Signal Set Design

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

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

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:

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

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 use the same average bit energy $E_b=\frac{E_0+E_1}{2}$. Then compare their performance with respect to the minimum probability of error. Explain any differences. Which signal set do you prefer?