Achievable Operating Points (AOPs)

The previous explanations now allow us to come up with the notion of "achievable operating points" (AOPs). An AOP is defined as the tuple $(\text{SNR},R)$ for which reliable communication is possible in an asymptotical setting, i.e., we use the information theoretic property of $R_\text{BMD}(P_X,\text{SNR})$ being an achievable rate.

We choose the desired spetcral efficiency (SE) $R$ and code rate $c$. Using the PAS Scheme it can be shown that $R$ is given by $$R = \text{H}(X) - (1-c)\log_2(M).$$

This means that the SE can be modified by selecting $P_X$ accordingly. In te following, we denote the corresponding input distribution as $P_X^R$ and use the Maxwell-Boltzmann distribution family.

The tuple $(\text{SNR}, R)$ is then obtained as the solution to the following equation: $$R_\text{BMD}(P_X^R,SNR) \stackrel{!}{=} R.$$ Recalling the system model, we remember that the $\text{SNR}=\text{E}[(\Delta X)^2]$ can be set by the constellation scaling $\Delta$.