M-ary Signal Sets

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

$$s_{i,j} = \sqrt{\frac{2E_2}{T}}\cdot i \cdot \cos(2\pi f_c ... ...n(2\pi f_c t) \;\; \mbox{for $0 \leq t \leq T$, $i,j=-1,1$.}$$

Thus the signal set consists of $M=4$ 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. How many bits $N_b$ are sent with each transmission for this signal set?
4. Define the average bit-energy $E_b$ as
   
   $$E_b = \frac{1}{N_b} \sum_{i} \sum_{j} E_{i,j} \pi_{i,j},$$
   
   
   where $E_{i,j}$ denotes the energy of signal $s_{i,j}$ and $\pi_{i,j}$ denotes the corresponding a priori probabilities. Compute the average bit-energy $E_b$ for this signal set.
5. Repeat parts (a)-(c) for the following signal set with $M=16$ signals:
   
   $$s_{i,j} = \sqrt{\frac{2E_4}{T}}\cdot i \cdot \cos(2\pi f_c ... ...\pi f_c t) \; \mbox{for $0 \leq t \leq T$, $i,j=-3,-1,1,3$.}$$
   
   

6. If both signal sets would be using the same bit-energy $E_b$, which signal set yields the smaller probability of error? Explain.
7. Which other very important parameter in a communication system may lead a system designer to choose the second signal set over the first signal set, particularly if the signal-to-noise ratio is large? Explain.