M-ary Signal Sets

Throughout this problem, two types of signal sets are used to transmit equally likely signals over an additive, white Gaussian noise channel with spectral height $\frac{N_0}{2}$. The first signal set uses quadrature amplitude modulation (QAM), i.e., the transmitted signal are of the form

$$s(t) = \sqrt{\frac{2}{T}} A_1 \left(a_c \cos(2 \pi f_c t) + a_s \sin(2 \pi f_c t) \right) \;\; \mbox{for $0 \leq t < T$,}$$

where the symbols $a_c$ and $a_s$ can take on values from the set $\{\pm 1, \pm 3, \ldots, \pm M-1\}$ and $M$ is an even integer. Hence, this signal set contains $M^2$ signals.

The second signal set uses orthogonal carriers to transmit one of $2^L$ signals of the form

$$s(t) = \sqrt{\frac{2}{T}} A_2 \sum_{i=1}^{L} b_i \cos(2 \pi (f_c + \frac{i}{T})t) \;\; \mbox{for $0 \leq t < T$,}$$

where the bits $b_i$ are drawn from the alphabet $\{1,-1\}$.

1. Consider the QAM signal set with $M=2$ and the orthogonal signal set with $L=2$, such that both signal sets consist of 4 signals. For each signal set determine
   • the amplitudes $A_1$ and $A_2$ such that the average signal energy equals unity,
   • orthonormal basis functions for depicting the constellation in an appropriate signal space; draw the signal space diagrams,
   • the minimum probability of error for this signal set.
   In your expression expression for the probability of error, indicate explicitly that both signal sets are supposed to have unit energy.
2. Repeat part (a) for the signal sets with 16 signals resulting when $M=4$ and $L=4$.
3. Compare the probability of error for the two types of constellations with equal numbers of signals as $M$ and $L$ increase. I.e., consider the case of 64 signals ($M=8$, $L=6$), 256 signals ($M=16$, $L=8$), etc. You do not need to compute $P_e$ explicitly but you should be as specific as possible.
4. Compare the bandwidth requirements for the two types of constellations with equal numbers of signals as $M$ and $L$ increase.