## 3 Simulation System and Results Discussion
### Results and Discussion
It is worth observing from the diagram of the recovery constellation that the points in the corner of QAM-64 mix together and are difficult to distinguish, while the outermost constellation points of IPM have higher discrimination, which indicates that IPM has good robustness to phase noise caused by self-phase modulation. Figure 5: (**a**) BER versus SNR at loop = 5; (**b**) recovery constellation at SNR = 16 dB; black intersections: ideal constellation points under impacting by NLPN (associated centroids); colored dots: the signals in the cluster associated with the nearest black intersections. Figure 7a shows the BER performance of the proposed scheme, IPM-256 without K-means and QAM-256 when loop = 1. We could observe that when SNR = 28 dB, the BER performance of QAM-256 is superior to IPM, yet not as good as the proposed scheme. When SNR > 28.8 dB, the IPM-256 without K-means has a lower BER. For comparison, our proposed scheme acquires 0.4 dB and 0.8 dB, respectively, at the HD-FEC threshold, compared with IPM-256 without K-means and QAM-256. The results fully show that although IPM-256 is not as good as QAM-256 in resisting the influence of nonlinear effect, the system's tolerance to nonlinear effect is improved after the application of the K-means algorithm. Figure 6: (**a**) BER versus SNR at loop = 5; (**b**) recovery constellation at SNR = 22 dB; black intersections: ideal constellation points under impacting by NLPN (associated centroids); colored dots: the signals in the cluster associated with the nearest black intersections. Figure 7: (**a**) BER versus SNR at loop = 1; (**b**) recovery constellation at SNR = 28 dB; black intersections: ideal constellation points under impacting by NLPN (associated centroids); colored dots: the signals in the cluster associated with the nearest black intersections. Due to the involvement of NLPN and the centroid judgement problem, the signal points are excessively divergent when the algorithm converges. Consequently, the BER performance is worse than the IPM employing BPS in the initial SNR range. When employing the K-means algorithm, the SNR of the IPM scheme at HD-FEC threshold is improved dramatically compared to the other two approaches, as shown by Table 2. K-means algorithm can directly process the constellation with phase noise according to the minimum Euclidean distance criterion which will bring classification gain to system. We firstly assign M (depending on the modulation order) initial centroid positions before running the K-means algorithm and generally use the coordinates of the constellation points of the ideal constellation as the initial centroid so that the K-means algorithm can converge quickly. The new centroid of each symbol is determined by calculating the Euclidean distance between the received N symbols with phase noise and M original centroids. Then, the new centroid location coordinates are obtained based on the current constellation point clustering. Repeat the procedure above until the algorithm converges and the final centroid location is the optimum constellation point position with phase offset. Finally, each computed centroid matches the initial ideal constellation point one by one, completing the demodulation step. In contrast, BPS procedure should recover the constellation to standard firstly before demodulating it. The recovery mistake may occur if the constellation was impacted by sufficient phase noise. That is the main reason why the proposed scheme has better BER performance than other schemes. Simultaneously, the proposed technique diminishes the computing complexity by avoiding the phase recovery process.