## 3 Results and Analysis  
To verify the validity of the proposed MFI scheme, a series of simulations of 28 GBaud PDM-QPSK/-16 QAM/-32 QAM/-64 QAM/-128 QAM signals were conducted by using _VPI Transmission Maker 9.8_. As illustrated in Figure 7, the optical carrier with a wavelength of 1550 nm and a linewidth of 100 kHz was generated by an external cavity laser (ECL). Then, the optical carrier was split by a polarization beam splitter (PBS) and modulated by two I/Q modulators. After a polarization beam combiner (PBC), the PDM signals were launched into the back-to-back (BTB) or long-distance transmission link. The Set OSNR module was applied to adjust the value of OSNR in BTB case, and the OSNR ranges of these five modulation formats were 7-26 dB, 14-33 dB, 14-37 dB, 14-38 dB, and 19-42 dB, respectively. The long-distance transmission link was composed of \(M\times 80\) km (\(M=25\) for QPSK, \(M=13\) for 16 QAM, \(M=5\) for 32 QAM, \(M=3\) for 64 QAM, \(M=1\) for 128 QAM) spans of single-mode fibers (SMFs). The dispersion parameter, PMD parameter, attenuation, and nonlinear coefficient of the SMF were \(D=16\) ps/nm/km, \(D_{PMD}=0.1\) ps/km\({}^{1/2}\), \(\alpha=0.2\) dB/km, and \(\gamma=1.267\) km\({}^{-1}\)W\({}^{-1}\), respectively. The noise figure of the erbium doped fiber amplifier (EDFA) was 5 dB. At the receiving end, the received signals were detected with the assistance from a local oscillator (LO). Then, the digital signals were obtained after ADC and provided to an off-line DSP module. To verify the accuracy of the proposed MFI scheme, in the BTB case, the training set and the testing set were obtained by extracting 100 independent samples for each OSNR value of each modulation format according to the ratio of 8:2. In the long-distance transmission case, the same established training set was also applied, while 20 independent samples for each launch power of each modulation format were included in the testing set. It should be noted that the samples extracted in an excessive low OSNR case cannot be employed as training samples since deteriorated training samples can actually lead to decreased performance. Therefore, the samples from the 64 QAM signals with an OSNR of 14 dB and the 128 QAM signals with OSNRs of 19 dB and 20 dB were only applied for testing rather than training. The required minimum number of symbols is regarded as one of the most important factors since it determines the response speed and complexity of the MFI scheme. To explore the required minimum number of symbols, in Figure 8, the required minimum OSNR as a function of symbol number was plotted, where the range of the symbol number was from 5000 to 9000, and the step size was 1000. In theory, if the number of symbols is more, the features are more obvious and easy to be identified. However, since the proposed MFI scheme only depends on the features in local zones, for high-order modulation formats, excessive symbols not only enhance the feature in specific local zones but also result in confused features in other local zones, especially in the low OSNR case. This causes slight performance degradation in the 9000 symbols case. Therefore, considering the trade-off between the MFI performance and complexity, the optimal number of symbols is 8000, and thus the number of symbols is fixed at 8000 in the following numerical simulations. As mentioned in Section 2.2, \(k\) is a key parameter for the Gaussian weighted KNN algorithm. As illustrated in Figure 9, the required minimum OSNR as a function of \(k\) was shown to evaluate the optimal value of \(k\). The range of the \(k\) value was from 1 to 9. For QPSK, 16 QAM, 64 QAM, and 128 QAM, the required minimum OSNR remained unchanged over the entire \(k\) range. On the other hand, when the value of \(k\) was decreased to 1, the required minimum OSNR for 32 QAM was significantly increased. Under the condition of low OSNR, the extracted feature of the testing sample was seriously interfered by noise.