Binary Signal Set

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

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

1.

Draw a block diagram of the receiver which achieves the minimum probability of error for this signal set. Be sure to specify the value of the threshold.

2.

What is the minimum probability of error achievable with this signal set?

3.

Consider the following receiver, where $\alpha$ is a positive constant.

Determine the distribution of the random variable R as a function of $\alpha$ for both cases, s₀(t) was transmitted and s₁(t) was transmitted. Sketch the two density functions and indicate the location of the threshold.

4.

Compute the probability of error achieved by this receiver as a function of the constant $\alpha$.

5.

For what value of $\alpha$ is the probability of error minimized? Compare with the result from part (b).