## 3 Numerical Simulation Results
Therefore, as shown in Figure 10a,b, when the value of the linewidth was increased from 0 Hz to 100 kHz, the minimum required OSNR values to achieve a correct identification rate of 100% for the five modulation formats remained unchanged. In conclusion, the proposed MFI scheme was insensitive to carrier phase noise. To explore the OSNR-dependent correct identification performance rates of the proposed MFI scheme under typical values of linewidth, for different signal modulation formats, their numerically-simulated correct identification rates, as a function of OSNR, are shown in Figure 10b. The simulation results showed that 100% accurate identification could be obtained for 8 QAM even when the OSNR values were very low (5 dB). For QPSK, 16 QAM, 32 QAM and 64 QAM, the minimum OSNR values for achieving the correct identification rate of 100% were 7 dB, 16 dB, 18 dB and 19 dB, respectively. As seen in Figure 10b, the proposed MFI scheme achieved a correct identification rate of 100% for all five modulation formats when the OSNR values were higher than their thresholds (indicated by vertical dash lines), corresponding to 20% FEC correcting BER of \(2.4\times 10^{-2}\)  
Figure 8: Minimum required OSNR values, with different numbers of symbols for five modulation formats; 6000 symbols is the optimum selection. Figure 9: Minimum required OSNR values with different numbers of histogram bins for five modulation formats; 40 bins is the optimum selection. By making use of incoming signals identical to those presented above, the performance of the proposed scheme was further evaluated in comparison with schemes using DNN, SVM, modified particle swarm optimization (M-PSO) [20] and principal component analysis of Stokes parameters (PCASP) [8]. The SVM and DNN schemes also adopted the amplitude histogram as the proposed scheme. The numbers of symbols and histogram bins were fixed at 6000 and 40, respectively. To ensure the optimal performance, a large training data set for DNN was required. The training set comprised \(9200\) (\(18\times 100+23\times 100+15\times 100+17\times 100+19\times 100\)) amplitude histograms. In this work, a four-layer DNN was employed, in which the number of neurons in the input, first hidden, second hidden and output layer were 40, 40, 10 and 5, respectively. The activation functions of the hidden layer and output layer were _ReLU_ and _softmax_, respectively [6]. On the other hand, since the complexity of SVM was related to the number of support vectors, by considering the tradeoff between the identification performance and the computational complexity for SVM, the training set of SVM comprised 3640 (\(18\times 40+23\times 40+15\times 40+17\times 40+18\times 40\)) amplitude histograms. For SVM, the kernel function was the default radial basis function (RBF) kernel [38]. The minimum required OSNR comparisons between the proposed scheme and these four schemes are shown in Figure 11. The scheme based on M-PSO only identified three modulation formats, including QPSK, 8 QAM and 16 QAM, and the minimum required OSNR for identifying QPSK and 8 QAM was much higher than that of the proposed scheme. The minimum required OSNRs of the proposed scheme, in order to identify each modulation format, were both lower than that of the schemes based on PCASP, except for 32 QAM. For QPSK, 16 QAM and 32 QAM, the minimum required OSNR values for the proposed scheme were lower than those corresponding to DNN and SVM. For 8 QAM and 64 QAM, the proposed scheme offered minimum required OSNRs identical to DNN and SVM. The proposed scheme improved the performances because it primarily focused on identifying effective local features, while the MFI abilities of DNN and SVM came at the cost of higher complexity, focused on identifying global features. Furthermore, DNN and SVM also required more detailed amplitude histograms to complete the automatic feature extraction. Thus, 6000 symbols and 40 bins were not sufficient, giving rise to slower response speeds and increased computational complexity, as mentioned above.