## 4 Discussion  
### The Design Rationale of the GS Constellation Structure  
In this subsection, the design rationale behind the selected geometric shapes for odd-bit QAM constellations is discussed. Based on the analysis in Section 2.2, it can be seen that the core reason why the traditional cross-shaped odd-bit QAM signals are not compatible with the PCPE algorithm, is the lack of constellation points at the four corners of the constellation diagram, which leads to the decrease in the obviousness of the PC vector in the squared constellation diagram. Since the rotation angle of the PC corresponding to each block is extracted by only one PIM iteration in the PCPE algorithm, the estimation accuracy of the PCPE algorithm will be significantly deteriorated when the obviousness of the principal component decreases. To address the compatibility challenges between the PCPE algorithm and odd-bit QAM signals, we employ the geometric shaping technique, specifically the PO algorithm, to reconfigure the constellation structure of odd-bit QAM. The previous literature primarily utilized the PO algorithm to optimize constellation designs for achieving maximal MI or BER performance in channels affected solely by AWGN. Consequently, the typical constellation structures designed tend to approximate a circular shape, ensuring that the distribution of the entire constellation closely resembles the Maxwell-Boltzmann distribution when the occurrence probability of each constellation point is uniform [28; 29; 30]. However, such a constellation design scheme cannot effectively solve the incompatibility problem between the PCPE algorithm and the odd-bit QAM signal. In order to achieve the adaptation between the two, it is necessary to try to approximate the constellation structure to a square, so as to ensure that the PIM can quickly and accurately capture the PC vector. Therefore, in this paper, we select the MI value after PCPE recovery of odd-bit QAM signals as the objective function of the PO iteration, and obtain the constellation structures of GS-32 QAM and GS-128 QAM that tend to be square. To elucidate the impact of the proposed constellation structure on the recovery performance of the PCPE algorithm, we conducted a comparative analysis of the iteration counts  
Figure 9: Results of 128 QAM constellations using the PCPE algorithm for phase recovery in the case of back-to-back transmission. (**a**) Measured MI curves versus OSNR; (**b**) Measured MAE of residual phase error curves versus OSNR. required for PIM convergence using ideal GS-32 QAM and R-32 QAM signals subject to a \(\pi/6\) phase shift, as depicted in Figure 10. Our findings demonstrate that the employment of the GS-32 QAM constellation structure, as devised in this study, effectively expedites the convergence process of PIM, achieving highly accurate phase estimation after only four iterations. Conversely, for the R-32 QAM, approximately sixteen times more iterations are necessary to attain comparable recovery accuracy. This analysis underscores that the GS scheme devised herein markedly enhances the clarity of the PC vector in the constellation diagram following squaring of the odd-bit QAM signal. It notably accelerates the convergence speed of PIM and diminishes the residual phase error magnitude, thereby enhancing the recovery efficacy of the PCPE algorithm. Furthermore, it is imperative to consider the comprehensive impact of both LPN and AWGN on signal performance when determining the SNR for executing the pair-wise optimization (PO) iteration. This consideration involves balancing the adaptability of the designed constellation structure to both the principal PCPE algorithm and the goal of approximating a square shape. Simultaneously, it is crucial to ensure that the designed constellation structure maintains reasonable MI performance even under the sole influence of AWGN, emphasizing the need for a sufficient minimum Euclidean distance in the constellation diagram.