## Operating Principle
Nevertheless, as the effects of noise increased for the low OSNR cases, the symbols of 32 QAM in part C were dispersed, which resulted in an increased value of \(H_{3}\). Incorrect decisions could thus be made between \(H_{3}\) of 16 QAM for high OSNR cases and \(H_{3}\) of 32 QAM for low OSNR cases. Therefore, while relying solely on a single ratio \(H_{3}\), one could not accurately distinguish 32 QAM from 16 QAM over a wide OSNR range. To address this problem, an additional ratio was also required, as detailed below. As seen in Figure 5, the third partition of the amplitude histogram divided the first \(N/4\) symbols into three parts. The numbers of symbols in part A, B and C were \(N/16\), \(N/8\) and \(N/16\), respectively. As listed in Table 1, the first \(N/4\) symbols in the amplitude histogram corresponded to only one amplitude value for 16 QAM, while the symbols for 32 QAM and 64 QAM corresponded to two and three amplitude values, respectively. Even under the effects of noise, for 16 QAM, the distribution of amplitudes in part A and C still concentrated around 0.447. By comparison, the distributions of part A and C spread more widely for 32 QAM and 64 QAM. Therefore, the ratio \(H_{4}\) was introduced and defined as  
\[H_{4}=\frac{H_{\text{C4}}}{H_{A4}} \tag{5}\]\[H_{A4} =m_{1}\times c_{1}+\cdots+m_{i}\times c_{i}+(\frac{N}{16}-\sum\limits_ {x=1}^{i}m_{x})\times c_{i+1}\] Figure 2:0 Figure 2:1 Figure 2:2 Figure 2:3  
Due to the more concentrated distribution, \(H_{A4}\) of 16 QAM was relatively larger than those related to 32 QAM and 64 QAM, while \(H_{\text{C4}}\) was relatively smaller than those related to 32 QAM and 64 QAM, resulting in a smaller \(H_{4}\). As a direct result, 16 QAM and 32 QAM could be identified over a wide OSNR range by making use of both \(H_{3}\) and \(H_{4}\). Meanwhile, although 16 QAM and 64 QAM could be distinguished by relying on \(H_{4}\) only, as seen in Figure 6, the utilization of both \(H_{3}\) and \(H_{4}\) was necessary for identifying 16 QAM. As mentioned above, the different modulation formats gave rise to different ratios. Figure 6 shows the values of \(H_{1}\), \(H_{2}\), \(H_{3}\) and \(H_{4}\) for the five modulation formats in different OSNR cases. The number of symbols was 6000, and the bin number of the amplitude histogram was 40. By selecting appropriate thresholds, the incoming PDM signals could be identified as QPSK, 8 QAM, 16 QAM, 32 QAM or 64 QAM signals. The corresponding thresholds of \(H_{1}\), \(H_{2}\), \(H_{3}\) and \(H_{4}\) were 1.984, 2.64, 2.02 and 2.4/2.637, respectively, as indicated by the dotted red lines in Figure 2:4 It should be noted that there were two thresholds of \(H_{4}\) for distinguishing 16 QAM from 32 QAM (2.4) and 64 QAM (2.637). The thresholds for \(H_{1}\), \(H_{2}\), \(H_{3}\) and \(H_{4}\) were determined to balance the optimal identification performance of each identified modulation format. Figure 7 illustrates a flow chart of the proposed MFI scheme. If \(H_{1}\) of the incoming signals was greater than 1.984, the signals were recognized as 8 QAM signals, otherwise, if \(H_{2}\) of the incoming signals was less than 2.64, the signals were identified as QPSK signals. The remaining 16 QAM, 32 QAM and 64 QAM signals could be further distinguished by making use of both \(H_{3}\) and \(H_{4}\). Figure 2:5 Ratios varies with OSNR in back-to-back simulation: (**a**) \(H_{1}\); (**b**) \(H_{2}\); (**c**) \(H_{3}\); (**d**) \(H_{4}\) (for separating 16 QAM and 32 QAM); (**e**) \(H_{4}\) (for separating 16 QAM and 64 QAM). The dotted red lines indicate the corresponding thresholds.