## 3 Results and Analysis
If the \(k\) was set to be 1, the identification of KNN completely relied on the one training sample closest to the testing sample. In such a case, the chance of misjudgment is increased by the single training sample. Therefore, the optimal value of \(k\) was set to be 3, which was employed in the following numerical simulations. The correct identification rate versus OSNR is illustrated in Figure 10, and the vertical dashed lines are the OSNR thresholds corresponding to the 20% FEC for the five modulation formats. Except for 16 QAM, the required minimum OSNR for the other four modulation formats were significantly lower than the corresponding thresholds. Even for 16 QAM, the required minimum OSNR (15 dB) was less than the theoretical 20% FEC limit (15.85 dB). The results indicate the outstanding performance of the proposed MFI scheme. Figure 8: The required minimum OSNR versus different number of symbols for five modulation formats. Figure 7: The simulation setup of the PDM communication system. OBPF: optical band-pass filter, LPF: low-pass filter, PRBS: pseudo-random bit sequence. Meanwhile, the tolerance against residual CD and PMD for the proposed MFI scheme is also plotted in Figures 11 and 12, respectively. The testing set contained 20 independent samples for each value of CD (five modulation formats, 2020 testing samples in total). Considering the 100% correct identification rate as a metric to evaluate the MFI performance, the tolerable ranges of the residual CD for QPSK (12 dB), 16 QAM (19 dB), 32 QAM (22 dB), 64 QAM (24 dB), and 128 QAM (25 dB) were \(-\)1920 ps/nm\(\sim\)1920 ps/nm, \(-\)720 ps/nm\(\sim\)360 ps/nm, \(-\)1200 ps/nm\(\sim\)1680 ps/nm, \(-\)600 ps/nm\(\sim\)360 ps/nm, and \(-\)600 ps/nm\(\sim\)480 ps/nm, respectively. The results indicate that the proposed scheme has good tolerance against residual CD. The results illustrated in Figure 12 also show the tolerance to PMD. The range of differential-group delay (DGD) was from 0 ps to 34 ps with a step size of 2 ps. A total of 20 independent samples for each value of DGD (five modulation formats, 1160 testing samples in total) were employed for testing. The results demonstrate the maximum tolerable DGD for the QPSK (12 dB), 16 QAM (19 dB), 32 QAM (22 dB), and 64 QAM (24 dB) signals were 34 ps, 16 ps, 20 ps, and 6 ps, respectively. From Figure 12b, it can also be seen that the maximum tolerable DGD for 128 QAM was much less than that of other lower-order modulation formats when the value of OSNR was 25 dB. For 128 QAM, when the value of OSNR was low, features that had already been blurred by noise were very sensitive to the influence of DGD. On the other hand, it was noteworthy that once the OSNR value increased, the tolerable DGD for the 128 QAM was able to be significantly raised. Figure 10: The correct identification rate versus OSNR for five modulation formats. Figure 9: The required minimum OSNR versus different values of \(k\) for five modulation formats. To evaluate the impact of fiber nonlinearities, a series of long-distance transmission simulations were conducted for the QPSK (2000 km), 16 QAM (1040 km), 32 QAM (400 km), 64 QAM (240 km), and 128 QAM (80 km) signals. The correct identification rates of these five modulation formats in the long-distance transmission links are illustrated in Figure 13. For QPSK, 16 QAM, 32 QAM, and 64 QAM, the decrease in identification accuracy caused by the nonlinear effects occurred only when their corresponding launch powers were increased to 8 dBm, 8 dBm, 6 dBm, and 10 dBm, respectively. In particular, the correct identification rate of 128 QAM still remained at 100% even when its launch power was increased to 10 dBm. These results indicate that the proposed MFI scheme is robust against the fiber nonlinearities.