## 4 Discussion
### Sensitivity Analysis  
In this section, we investigate the sensitivity of the proposed GS constellation structures when confronted with varying parameters between the training and testing channels. Specifically, we simulate and test the MI performance of the GS-32 QAM and GS-128 QAM constellations, which were trained and derived under a 200 kHz system line width condition, across different channel conditions where the combined line widths are 100 kHz and 500 kHz, respectively. The resulting simulation outcomes are depicted in Figure 12. First, the results in Figure 12a,b show that GS-32 QAM and GS-128 QAM can improve the MI performance of the PCPE algorithm by up to 0.24 bit/symbol and 1.04 bit/symbol, respectively, when the linewidth value of the testing channel is 100 kHz. Although the MI improvement that can be achieved by the GS scheme is slightly reduced compared with the scenario with a testing linewidth of 200 kHz, the fact that the GS constellation structures can effectively adapt to the PCPE algorithm does not change. Then, as shown in Figure 12c,d, the two GS constellations deliver up to 0.50 bit/symbol and 1.62 bit/symbol MI performance improvements, respectively, at a linewidth of 500 kHz for the testing system. Drawing upon the aforementioned findings, it is evident that despite variations in parameter settings between the testing and training channels, the proposed GS constellation structures consistently enhance the MI performance of the PCPE algorithm across all assessed SNR ranges. Furthermore, as the line width of the testing systems increases, the MI performance gains are observed through the adoption of the proposed GS constellation structures in lieu of traditional cross-shaped constellation configurations. In addition, the tolerance and sensitivity analysis of the proposed scheme for chromatic dispersion (CD) effect, amplified spontaneous emission (ASE) noise, and Kerr nonlinear effect are also studied after transmission through the fiber channel, to verify the feasibility of the proposed scheme in actual communication scenarios. The transmission system evaluation is still based on the VPI simulation software. The difference compared to the system setup in Figure 7 is that the Set OSNR component is removed and replaced with a fiber link consisting of a single-mode fiber (SMF) spanning 80 km, an EDFA with a gain of 16 dB, and a noise figure of 4 dB, and a recirculation fiber loop. The attenuation value  
Figure 11: Results of performing the PO process on 32 QAM under the condition that the SNR is 25 dB. (**a**) The constellation diagram of the obtained GS-32 QAM; (**b**) Measured MI versus SNR curves corresponding to the GS-32 QAM constellation structures obtained at SNR of 13 dB and 25 dB, respectively. of the fiber is set to 0.2 dB/km, the dispersion coefficient is set to 16 ps\(\cdot\)nm\({}^{-1}\)km\({}^{-1}\), and the nonlinearity coefficient is 2.6e\({}^{-20}\) m\({}^{2}\)/W. Moreover, in the DSP chain at the receiving end, the CD compensation algorithm based on frequency domain filtering is supplemented after the GSOP algorithm. Firstly, the transmission performance of GS-32 QAM signal is analyzed under the condition that the total length of the fiber link is 400 km (that is, the number of recirculating loops is 5). For the launched optical power in the range of \(-\)5 dBm to 5 dBm, the MI performance corresponding to the two modulation formats is tested by adjusting the output power of the EDFA connected to the output of the IQ modulator, and the results are shown in Figure 13. The MI performance of the GS-32 QAM is better than that of the conventional R-32 QAM after the recovery of the PCPE algorithm over the entire measured launched optical power range. Specifically, in the process of increasing the launched power from \(-\)5 dBm to 1 dBm, that is, in the range where the system performance is mainly dominated by the ASE noise, the MI values of the two modulation formats increase with the increase of optical power.