## 3 Results
### VPItransmissionMaker Simulation Results  
Moreover, the proposed GS scheme is rigorously validated through a VPItransmissionMaker simulation, conducted within a 32 GBaud single polarization coherent detection system, as illustrated in Figure 7. In the transmission process, a co-simulation program is utilized at the transmitter end to generate odd-bit QAM signals corresponding to both cross-shaped constellations and GS constellations, and the generated digital sequences are converted into radio frequency (RF) signals at a rate of eight samples per symbol. These signals, after being amplified by identical electrical drivers, are then subjected to electrical-optical conversion using an IQ modulator with a half-wave voltage of 5 V, an insertion loss of 6 dB, and an extinction ratio of 35 dB. For the continuous wave (CW) laser that provides the optical carrier signal, its center wavelength is 193.1 THz, the linewidth is 100 kHz, and the average power is 5 dBm. The modulated signal's power is precisely adjusted to 0 dBm via an Erbium-doped fiber amplifier (EDFA). Control over the optical signal-to-noise ratio (OSNR) is achieved through the Set OSNR component for MI measurement. On the receiving end, the de-sired signal is combined with a local oscillator (LO) signal within the coherent receiver, which also has a center frequency of 193.1 THz and an output power of 0 dBm. Subsequently, the electrical signal is captured at a sampling rate twice the baud rate and with a resolution of 8 bits, and the stored data undergo a comprehensive digital signal processing program for signal recovery and demodulation. This process encompasses normalization, the Gram-Schmidt orthogonalization process (GSOP), frequency offset estimation (FOE), PCPE, and MI calculation. Figure 6: MI versus SNR for odd-bit QAM constellations with different CPR algorithms. (a) 32 QAM and (b) 128 QAM. Figure 8a gives the measured MI versus OSNR curve for 32 QAM obtained by VPI-transmissionMaker simulation. For regular 32 QAM signals using PCPE for phase recovery, even when the OSNR is as high as 30 dB, there is still an AIR loss of about 0.07 bit/symbol. In contrast, the AIR loss corresponding to the proposed GS-32 QAM signal after PCPE recovery is almost negligible when the OSNR is about 26 dB. It can be seen from the insets of constellation diagrams in Figure 8a that, under the same OSNR, the residual phase noise of the GS-32 QAM after PCPE phase recovery is significantly smaller than R-32 QAM. In addition, the mean absolute error (MAE) values of the residual phase error between the recovered R-32 QAM and GS-32 QAM signals and the original signals are also calculated under different OSNRs, as shown in Figure 8b. The measured results show that the MAE values corresponding to the GS-32 QAM are significantly lower than those of the R-32 QAM throughout the entire tested OSNR range, which indicates that the proposed constellation structure can indeed improve the accuracy of the PCPE algorithm. Moreover, as the value of OSNR increases, the MAE reduction percentage that can be achieved by replacing R-32 QAM with GS-32 QAM gradually increases from 15.48% to 39.05%, which once again proves that the proposed scheme can effectively solve the incompatibility problem of the traditional constellation structure and the PCPE algorithm especially in the high SNR range. The performance comparison between the proposed GS constellation and the regular constellation in 128 QAM modulation format is shown in Figure 9a. For the regular 128 QAM constellation, the MI performance cannot correctly converge to 7 bit/symbol. Even when the OSNR is high enough, there is still an AIR loss of about 1.06 bit/symbol. On the other hand, the GS-128 QAM constellation can achieve a much better MI performance, especially when the OSNR is high. Throughout the entire tested OSNR range, the MI performance can  
Figure 8: Results of 32 QAM constellations using the PCPE algorithm for phase recovery in the case of back-to-back transmission.