Suboptimum Receivers

A binary communication system employs the following signals two communicate equally likely messages over an additive white Gaussian noise channel with spectral height $\frac{N_0}{2}$:

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

1.

Draw a block diagram of the optimum receiver. (2 pts.)

2.

Compute the probability of error achieved by your receiver from part (a). (2 pts.)

3.

Compute the probability of error achieved by the following receiver: (4 pts)

where g(t) is given by

$$g(t) = \left\{ \begin{array} {cl} \alpha & \mbox{for $0 \leq... ... $\frac{T}{2} \leq t < T$}\\ 0 & \mbox{else.}\end{array}\right.$$

4.

For what value of $\alpha$ is the probability of error of the above receiver minimized? Explain. (2 pts.)

Hint: $\sin^2 x = \frac{1}{2}(1-\cos 2x).$