## 3 Numerical Simulation Results
Figure 10: Simulation results of correct identification rate versus OSNR value under different linewidths: (a) 0 Hz; (b) 100 kHz. The illustrations are the constellation diagrams and amplitude histograms of PDM-16 QAM when the linewidth is 0 Hz or 100 kHz, respectively. The vertical dash lines indicate the OSNR thresholds, corresponding to 20% FEC correcting BER of \(2.4\times 10^{-2}\). In order to evaluate the effects of residual CD and PMD on the identification performance of the proposed MFI scheme, numerical simulation was employed. It should be noted that, as shown in Figure 1, the proposed MFI scheme was placed after CMA equalization. The CMA could compensate residual CD and PMD. Therefore, the tolerance, with respect to residual CD and PMD of the proposed scheme, embedded in such DSP architecture, was enhanced. The tolerance values, with respect to residual CD and DGD of the proposed scheme, are shown in Figure 12a,b, respectively. The OSNR values of PDM-QPSK/-8 QAM/-16 QAM/-32 QAM/-64 QAM were 12 dB, 17 dB, 19 dB, 22 dB and 24 dB, respectively. The range of residual CD for PDM-QPSK/-8 QAM/-16 QAM/-32 QAM signals was from \(-\)1920 ps/nm to 1920 ps/nm, while the range of residual CD for PDM-64 QAM was \(-\)720 ps/nm-720 ps/nm. The steps for all modulation formats were 120 ps/nm. As shown in Figure 12a, the proposed MFI scheme was able to tolerate a relatively wide range of residual CDs for PDM-QPSK (\(-\)1920 ps/nm\(-\)1920 ps/nm), PDM-8 QAM (\(-\)1920 ps/nm-1920 ps/nm), PDM-16 QAM (\(-\)1440 ps/nm-1320 ps/nm) and PDM-32 QAM (\(-\)1680 ps/nm-1680 ps/nm) signals. Because of the greater amplitude levels, for PDM-64 QAM, the proposed scheme was able to tolerate residual CD from \(-\)240 ps/nm to 240 ps/nm. The range of DGD for PDM-QPSK/-8 QAM/-16 QAM/-32 QAM/-64 QAM signals was from 0 ps to 34 ps, with a step size of 2 ps. Analogous to residual CD, the tolerable DGD for PDM-64 QAM was much lower than the other four modulation formats. The proposed scheme was able to achieve a 100% correct identification rate for PDM-64 QAM when the DGD was 8 ps. For PDM-16 QAM and PDM-32 QAM, the tolerable D  
Figure 11: Minimum required OSNR for identifying different modulation formats for five MFI schemes. * indicates the modulation format is not identified by this scheme. Figure 12: The tolerance with respect to (**a**) residual CD and (**b**) DGD of the proposed scheme. respectively. The correct identification rate of PDM-QPSK and PDM-8 QAM remained at 100% over the whole DGD range. In order to analyze the relationship between MFI performance and baud rate, _VPI Transmission Maker 9.8_ was utilized to undertake numerical simulations for 10 GBaud and 20 GBaud transmission systems. The numerical simulation setup remained unchanged, except for baud rates. Simulated correct identification rates for the five modulation formats used in the 10 GBaud and 20 GBaud systems are shown in Figure 13a,b, respectively. These figures showed that, compared with the 28 GBaud case, shown in Figure 10b, the MFI performances were relatively better because the tolerance to noise was improved for the lower baud rate cases. To discuss the scalability of the proposed scheme, the MFI performance for probabilistic shaping (PS) of QAM signals was also investigated. The amplitude value and associated probability values for each level of the PS-QAM signals, including PS-16 QAM (3 bit/symbol) and PS-64 QAM (5 bit/symbol), are shown in Table 2. Since the probability distribution feature of the PS-QAM signals differed from those of the traditional QAM signals, the procedure and corresponding thresholds of the proposed scheme needed to be modified accordingly, as detailed below. The numerical simulation setup remained unchanged, except for the process used in generating the PS-16 QAM and PS-64 QAM signals. After power normalization, the first partition operations for the amplitude histograms of QPSK, 8 QAM, PS-16 QAM, 32 QAM and PS-64 QAM were employed. As shown in Figure 14a, the number of symbols was \(N/2\)