Binary Signals in AWGN

The following signal set is used to transmit equally likely messages over an AWGN channel with spectral height N₀/2:

$$\begin{array} {l} s_0(t) = \left\{ \begin{array} {cl} 1 &... ...eq t < T$}\\ 0 & \mbox{else.} \end{array} \right. \end{array}$$

The amplitude A is positive, A>0.

1.

Draw a block diagram of the optimum receiver.

2.

Compute the probability of error achieved by your receiver from part (a) as a function of the amplitude A.

3.

Consider now the following receiver:

In the diagram above, the two functions $\phi_0(t)$ and $\phi_1(t)$ are chosen as:

$$\begin{array} {l} \phi_0(t) = \left\{ \begin{array} {cl} ... ...t < T$}\\ 0 & \mbox{else.} \end{array} \right. \end{array}$$

Compute the conditional distributions of the two integrator outputs. Notice that you will have to compute four such distributions: f_R0|b=0, f_R0|b=1, f_R1|b=0 and f_R1|b=1.

4.

Is it possible to chose coefficients a and b such that the receiver in part (c) achieves the same probability of error as the optimum receiver?

If your answer is yes, compute the values of a and b that minimize the probability of error and explain why optimum performance is achieved.

If your answer is no, compute the probability of error of the receiver in part (c) as a function of the coefficients a and b.