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 $\frac{N_0}{2}$,

$$s_{i,j}(t) = \sqrt{\frac{2E}{T}}(i \cdot \cos(\omega t) + j \cdot \sin(\omega t))$$

for $0 \leq t \leq T$, i=-1,0,1 and j=-2,-1,0,1,2. Thus, this signal set consists of M=15 signals.

1.

Draw and accurately label the signal constellation in an appropriately chosen signal space and indicate the decision boundaries formed by the optimum receiver.

2.

Compute the probability of error achieved by the optimum receiver.

3.

Assume now that signal s₀₀ is removed from the above signal set. Draw and accurately label the new signal constellation in an appropriately chosen signal space and indicate the new optimum decision boundaries.

4.

Can you still express the resulting probability of error in terms of the Q-function? If your answer is yes, compute the probability of error; if it is no, explain why not and indicate whether the probability of error of the reduced signal set is larger or smaller than that of the original set.