## 3 Numerical Simulation Results
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{2}{c}{**PS-16 QAM (3 Bit/Symbol)**} & \multicolumn{2}{c}{**PS-64 QAM (5 Bit/Symbol)**} \\ \hline
**Amplitude Value** & **Probability** & **Amplitude Value** & **Probability** \\ \hline  & & 0.363 & 28.70\% \\ \cline{3-4}
0.731 & 79.40\% & 0.812 & 34.68\% \\ \cline{3-4}  & & 1.09 & 10.38\% \\ \hline  & & 1.31 & 12.50\% \\ \cline{3-4}
1.634 & 19.40\% & 1.498 & 7.35\% \\ \cline{3-4}  & & 1.817 & 4.11\% \\ \hline  & & 1.957 & 1.65\% \\ \cline{3-4}
2.193 & 1.20\% & 2.21 & 0.58\% \\ \cline{3-4}  & & 2.543 & 0.05\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Amplitude value and associated probability for each level of PS-16 QAM and PS-64 QAM signals. Figure 13: Simulation results of correct identification rate versus OSNR value, under different baud rate: (**a**) 10 GBaud; (**b**) 20 GBaud. in both part A and part B. Since part A of PS-64 QAM concentrated around low amplitude values (the associated probability for 0.363 was 28.7%), and the amplitude distribution range of part B as wide, as shown in Figure 14c, the ratio \(H_{1}\) (mentioned in Equation (2)) of PS-64 QAM was greater than that of 8 QAM over a wide OSNR range. Therefore, two thresholds for \(H_{1}\) were set: Th1 was 2.23 and Th2 was 2.04. Based on these two thresholds of \(H_{1}\), 8 QAM as not only distinguishable from QPSK, PS-16 QAM and 32 QAM, but also from PS-64 QAM. The second partition operation for the amplitude histograms of the rest of these modulation formats (QPSK, PS-16 QAM and 32 QAM) is shown in Figure 14b. The numbers of symbols in part A, part B and part C were \(N/4\), \(N/2\) and \(N/4\), respectively. The distribution of PS-16 QAM in part B was more concentrated than those of QPSK and 32 QAM, while the distribution of PS-16 QAM in part C was dispersed. Therefore, as shown in Figure 14d, the value of \(H_{3}\) (mentioned in Equation (4)) for PS-16 QAM was greater than those for QPSK and 32 QAM. QPSK was also distinguishable from 32 QAM based on another threshold of \(H_{3}\). The two thresholds of \(H_{3}\) were 4.12 (Th3) and 2.3 (Th4). The correct identification rates for identifying QPSK, SQAM, PS-16 QAM, 32 QAM and PS-64 QAM are shown in Figure 14e. The proposed MFI scheme was still able to achieve a 100% correct identification rate for all of these five modulation formats over a wide OSNR range. The scalability of the proposed scheme was verified by the simulation results. It should also be noted that if the information entropy changed, the probability distribution features of PS-QAM also changed. As such, when PS-QAM (with different information entropy) needed to be identified, corresponding modifications needed to be made to the procedure of the proposed MFI scheme and their relevant thresholds. The proposed scheme was suitable for application scenarios wherein the set of the modulation formats that needed to be identified were made known before applying the MFI scheme. Figure 14: Operating principle and simulation results for identifying five modulation formats, including PS-QAM. (**a**) The first partition operation and (**b**) second partition operation for the amplitude histograms of QPSK, 8 QAM, PS-16 QAM, 32 QAM and PS-64 QAM; (**c**) \(H_{1}\) and (**d**) \(H_{3}\) vary with the OSNR of the back-to-back optical transmission system, and (**e**) the correct identification rate for identifying these five modulation formats.