## 5 Simulation Results
### Simulation Results  
In the following, we will discuss the simulation results of the proposed system. It has been shown that for each SNR, there is an optimum PMF, which maximizes the R\({}_{\text{BMD}}\) over the Gamma-Gamma channel. Hence, the distribution of constellation points with different SNRs and the uniform distributions are illustrated in Figure 3. The specific probabilities are listed in Table 2. When the probability distribution deviates from the uniform distribution, the entropy decreases. Moreover, it can be seen that in the case of lower SNR, the probabilities of the constellation points near the origin are higher than that far away from the origin. Figure 3 illustrates that the PMF begins as a Gaussian-like shape; with an increase of the SNR, PMF eventually tends to a uniform distribution. This is because when the SNR is lower, more probabilistic shaping is required. Specifically, there is a great difference between the constellation points with high probabilities and those with low probabilities. Figure 2: Simulation. An example for 16 QAM. Theoretically, each SNR has a corresponding optimal PMF. The effect of optimizing the PMF at a corresponding SNR is shown in Figure 4. The gap to the Shannon limit is closed by the optimized PS. We compare the performances of the optimized solutions and uniform distributions under different turbulence conditions. Under week turbulence, the corresponding optimized PS scheme is closer to the Shannon limit. With the increase of the SNR, the shaping gain between the optimized PS and the uniform distribution decreases gradually. Eventually, the PMF approaches the uniform distribution. These phenomena are consistent with the tendency of the constellation with the input distribution, shown in Figure 3. The PMF tends to a uniform distribution, as the effect of the noise is decreasing. Hence, the shaping gain obtained by the optimized PS becomes smaller. \begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline \multirow{2}{*}{**Schemes**} & \multicolumn{3}{c}{**The Probability of Signal Constellation Point Distribution**} \\ \cline{2-9}  & \([\) 0.0247 & 0.0539 & 0.0247 & 0.0539 & 0.0539 & 0.1175 & 0.0539 & 0.1175 \\ PS, \(H=3.7964\) & 0.0247 & 0.0539 & 0.0247 & 0.0539 & 0.0539 & 0.1175 & 0.0539 & 0.1175 ] \\  & \([\) 0.0406 & 0.0601 & 0.0406 & 0.0601 & 0.0601 & 0.0892 & 0.0601 & 0.0892 \\ PS, \(H=3.9451\) & 0.0406 & 0.0601 & 0.0406 & 0.0601 & 0.0601 & 0.0892 & 0.0601 & 0.0892 \\ PS, \(H=3.9932\) & \([\) 0.0542 & 0.0622 & 0.0542 & 0.0622 & 0.0622 & 0.0713 & 0.0622 & 0.0713 ] \\ Uniform Distribution & \([\) 0.0625 & 0.0625 & 0.0625 & 0.0625 & 0.0625 & 0.0625 & 0.0625 & 0.0625 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Probability of signal constellation point distribution with or without the PS. Figure 3: Signal constellation point distribution. (**a**) PS at SNR = 2 dB, \(H=3.7964\). (**b**) PS at SNR = 10 dB, \(H=3.9451\). (**c**) PS at SNR = 18 dB, \(H=3.9932\). (**d**) Uniform distribution. The result of applying PS with a fixed entropy is shown in Figure 5. It can be seen that the PS with \(H=3.7964\) (resp. PS with \(H=3.9451\)) outperforms the uniform distribution by 0.4 dB (resp. 0.3 dB) at the achievable information rate of about 1.8 bit/symbol. Furthermore, the PS with \(H=3.9451\) is superior to others at the medium SNR. When the \(E_{s}/N_{0}\) is about 8 dB, the performance of the PS with \(H=3.9451\) is better than the PS with \(H=3.7964\). Besides, after 16 dB, the uniform distribution has more outstanding performance than others. In addition, we compare the performance of the Maxwell-Boltzmann (MB) distribution with fixed entropy. As we can see, when the \(E_{s}/N_{0}\) is less than 6 dB, there is little difference between the PS scheme and MB distribution, however when \(E_{s}/N_{0}\) is greater than 6 dB, it can be seen that the PS scheme performance is gradually better than MB distribution. As we can see, the PS with \(H=3.7964\) is worst at a higher SNR. This is because its PMF has a marked difference with the uniform distribution.