Sub-optimum Receivers

Equally likely messages are to be transmitted over a channel which adds white Gaussian noise to the transmitted signal. The following signal set is employed:

1.

Draw and accurately label a block diagram of the optimum receiver and compute the probability of error for your receiver.

2.

Consider now the following receiver:

where the function g(t) is given by

$$g(t) = \left\{ \begin{array} {cl} 1 & \mbox{for $0 \leq t < ... ...box{for $\tau \leq t < T$}\\ 0 & \mbox{else.}\end{array}\right.$$

Compute the probability of error as a function of $\tau$.

3.

Compute the probability of error of the receiver in (b) for the special cases:

• $\tau = 0$
• $\tau = \frac{T}{4}$
• $\tau = \frac{T}{2}$
• $\tau = \frac{3T}{4}$.

Which value of $\tau$ yield the minimum and maximum probability of error, respectively? Explain!

4.

Compute the probability of error for the receiver in part (b), if g(t) is given by

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

Hint: You may need

$$\int x \cos(ax) dx = \frac{\cos ax}{a^2} + \frac{x \sin ax}{a}.$$