Receivers for Binary Signal Sets

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

$$\begin{array} {cc} s_0(t) = & \left\{ \begin{array} {cl} \s... ...\begin{array} {cl} 0 & \mbox{for all $t$}\end{array}\end{array}$$

1.

Draw a block diagram of the optimum receiver.

2.

Compute the probability of error for your receiver.

3.

For the remainder of the problem, consider the following receiver:

The gain of the amplifier is determineded by the control signal applied at the inputc (see diagram); if the control signal is positive the amplifier's gain is g > 0, otherwise the gain is -g.

Find the probability of error for this receiver as a function of the gain g.

4.

For what value of g is the probability of error for the above receiver minimized? Compare with your result from part (b).