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 $\frac{N_0}{2}$,

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

Thus, this signal set consists of $M=3$ signals. Notice that the coefficient $i_1$ is squared!

1. Draw and accurately label a block diagram for the optimum receiver for this signal set.
2. Draw and accurately label the signal constellation in an appropriately chosen signal space and indicate the decision boundaries formed by the optimum receiver.
3. Compute the probability of error achieved by the optimum receiver.
4. Now consider the two-dimensional signal set:
   
   $$\begin{array}{ll} s_{i_1,i_2} = \sqrt{\frac{2E}{T}} \cdot ... ...;\mbox{for $0 \leq t \leq T$, $i_1,i_2=0,1,2$.} \end{array}$$
   
   
   Draw and accurately label the signal constellation in an appropriately chosen signal space and indicate the decision boundaries formed by the optimum receiver.
5. Compute the probability of error achieved by the optimum receiver.