## 3 Numerical Simulation Results  
_VPI Transmission Master 9.8_ was utilized to undertake a series of numerical simulations and extensively verify the proposed MFI scheme. The length of the pseudo-random bit sequence (PRBS) was \(2^{15}-1\). Then, 28 GBaud PDM-QPSK/-8 QAM/-16 QAM/-32 QAM/-64 QAM signals were generated in the transmitter. The sample rate was 56 GSa/s, and the roll-off factor of the square root raised cosine (SRRC) filter was 0.1. The wavelength and linewidth of the laser were 1550 nm and 100 kHz, respectively. The modulated signals were then passed through an additive Gaussian white noise (AWGN) channel with adjustable OSNRs, calculated at a 0.1 nm ASE noise bandwidth. These PDM signals were finally detected, and their modulation formats were identified by a coherent receiver utilizing an off-line DSP module based on the proposed MFI scheme, as shown in Figure 1. The OSNR ranges of the PDM-QPSK/-8 QAM/-16 QAM/-32 QAM/-64 QAM signals were 5-26 dB, 5-31 dB, 14-33 dB, 16-37 dB and 16-38 dB, respectively, with an OSNR incremental of 1 dB. In order to evaluate the performance of the proposed scheme, for each modulation format, 100 independent simulations were conducted for each OSNR value. For an MFI scheme, the minimum required number of symbols (corresponding to the lowest OSNR required for the considered modulation format) is one of the most important factors, because it determines the response speed and computational complexity of the MFI algorithm [28; 32]. The minimum required OSNR values, versus the numbers of symbols for the five modulation formats, are shown in Figure 8, wherein the symbol variation ranged from 1000 to 7000 at intervals of 1000. For QPSK, 8 QAM, 16 QAM and 32 QAM, when the number of symbols was greater than or equal to 2000, the minimum required OSNR value to achieve a correct identification rate of 100% remained almost constant. Meanwhile, since 64 QAM was the highest order of modulation format, with more amplitude levels, an efficient feature extraction required more symbols to be employed. If the number of symbols was lower than 6000, the minimum required OSNR value increased substantially. Since the required number of symbols for the proposed MFI scheme was determined by the highest order modulation format, the numbers of symbols for all of the considered modulation formats were set at 6000 in the following numerical simulations and proof-of-concept experiments. It should be noted that, when the number of symbols was lower than 2000 or 5000, for 8 QAM or 64 QAM respectively, a correct identification rate of 100% could not be obtained. The number of histogram bins was another factor that determined the computational complexity of the proposed MFI scheme. Similar to the number of symbols, an optimal number of the histogram bin was also identified, as it gave rise to the minimum required OSNR value, as shown in Figure 9. If the number of histogram bins was less than 40, the minimum required OSNR values increased gradually, especially for 64 QAM. As a result, in order to achieve an optimum tradeoff between computational complexity and MFI performance, the numbers of histogram bins for the five modulation formats were all taken to be 40 in the following numerical simulations and proof-of-concept experiments. Figure 7: Flow chart of the proposed MFI scheme. In order to analyze the influence of carrier phase noise on the performance of the proposed MFI scheme, simulations (i.e., correct identification rate versus OSNR value under different linewidths) were undertaken, as shown in Figure 10a,b, respectively. Compared with Figure 10a, when the linewidth of the laser was 100 kHz, the constellation of 16 QAM rotated, as shown in Figure 10b, due to the existence of carrier phase noise. However, since the amplitude histogram depended on amplitude information only, the effect of carrier phase noise on the amplitude histogram was almost negligible.