M-ary Signal Sets

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

$$s_{i_1}(t) = \sqrt{2E} \cdot i_1 \cdot \cos(20 \pi t) \;\;\;\;\mbox{for $0 \leq t \leq 1$, $i_1=-2,0,1$.}$$

Thus, this signal set consists of M=3 signals.

1.

Draw and accurately label a block diagram for the optimum receiver for this signal set.
Note that the three possible values for i₁ are not evenly spaced.

2.

Draw and accurately label the signal constellation in an appropriately chosen signal space and indicate the decision boundaries formed by the optimum receiver. Then, compute the probability of error achieved by the optimum receiver.

3.

Repeat part (b) for the following signal set

$$\begin{array} {ll} s_{i_1,i_2}(t) = & \sqrt{2E} \cdot i_1 \... ... $0 \leq t \leq 1$, $i_1=-2,0,1$, $i_2=-2,0,1$.} \end{array}$$

Thus, this signal set consists of M=9 signals.
Note that the possible values for i₁ and i₂ are not evenly spaced.

4.

Assume now that the signal s_0,0(t) is removed from the signal set in part (c). For the resulting signal set (consisting of eight signals), draw and accurately label the signal constellation in an appropriately chosen signal space and indicate the decision boundaries formed by the optimum receiver.

5.

You do not need to compute the minimum probability of error for the signal set in part (d), however, you should indicate whether the probability of error is larger or smaller than the probability of error you found in part (c). Explain your answer.