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 $\frac{N_0}{2}$,

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

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.

Determine the number of bits N_b that is transmitted simultaneously with this signa 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:

$$s_{i,j} = \sqrt{\frac{2E_4}{T}}\cdot i \cdot \cos(2\pi f_c t... ...\pi f_c t) \; \mbox{for $0 \leq t \leq T$, $i,j=-3,-1,1,3$.}$$

6.

If both signal sets would be using the same bit-energy E_b, which signal set yields the smaller probability of error? Explain.