## 3 Simulation System and Results Discussion
### Results and Discussion  
To study the performance of channel capacity of different orders of IPM and QAM modulation format, we quantify it by using the physical amount of mutual information. MI versus SNR of different modulation formats after 100 km transmission is shown in Figure 4a. The line with black diamonds is the Shannon capacity. We can find out that MI in IPM outperforms QAM modulation. Moreover, the larger the modulation order is, the closer the distance IPM is to the Shannon Capacity, especially in IPM-256, the purple line in Figure 4a. The MI versus recirculation loops are shown in Figure 4b. We consider the range of the loop is 1 to 10, length of each loop is 100 km. The results show that when employing IPM-16 and QAM 16, the MI is almost the same at loop = 1 and 2. When transmission distance is small, the nonlinear effect is not accumulated too much; when increasing the loop, the MI of IPM-16 is larger than QAM-16, which indicates that IPM-16 could carry more information. When M = 64, the capacity of IPM outperforms QAM even at the larger loop. It indicates that the average power of IPM-64 has a much lower value and has good tolerance to the dispersion, nonlinear effect, and other noises of SMF. When M = 256, the MI of IPM outperforms QAM at loop = 1 to 4; the performance of IPM is not better than QAM when loop > 4. However, as we all know, the distance between DC is usually no more than 100 km. Therefore, the IPM-256 is also suitable for the data center interconnection network. The fiber channel model can be represented as a linear dispersion noise, additive white Gaussian noise (AWGN) and nonlinear phase noise channel [28; 29]. The iterative polar quantization process considers AWGN, such as thermal and ASE noise, rendering the position distribution of constellation points is more subject to the channel characteristics. Hence, the MI can reach its maximum value and the geometric shaping gain is realized. The aforementioned are the major reasons which explain how the suggested technique can effectively narrow the gap with the Shannon limit. Figure 4: (**a**) MI versus SNR of different modulation format; (**b**) MI versus Loop. We evaluate the BER performance of IPM-16 and QAM-16 with K-means or BPS in different SNR ranges at loop = 5, as shown in Figure 5a, and the recovery constellations SNR = 16 dB are shown in Figure 5b. The BER decreases as the SNR increases, and it is evident that IPM-16 with K-means algorithm outperforms IPM without K-means and QAM-16. The hard-decision FEC (HD-FEC) threshold is \(3.8\times 10^{-3}\), which is mainly used to evaluate the BER performance. Specifically, when SNR is small, the performance of these three schemes is approximately equal. Even the performance of QAM-16 is better than IPM at SNR = 10 to 13. With the increase in SNR, the advantages of the IPM scheme gradually appear. It is clear that IPM with K-means reach the HD-FEC threshold at SNR = 16, while the SNR of reaching the threshold of IPM without K-means and QAM-16 is 16.9 dB, 17.7 dB, respectively. Therefore, the gain of proposed schemes is 0.9 dB, 1.7 dB compared with IPM-16 without K-means cluster algorithm and QAM-16. In the diagram of recovery constellations, the black cross is the centroid locations after a serious iteration, and the centroid could provide a reference for subsequent demodulation. To study the BER performance of IPM-64 and QAM-64, we draw the curves of BER versus SNR, as shown in Figure 6a. The BER gradually decreases as the SNR increases, and there is a positive correlation between BER and SNR. Additionally, it could be seen that the proposed scheme is superior to IPM-64 without K-means and QAM-64 throughout the entire range of SNR. The proposed scheme reaches the threshold of HD-FEC at SNR = 23.7 dB, while the IPM-64 without K-means cluster algorithm is at 24 dB and QAM-64 is 24.8 dB. The gap between the proposed scheme and the other two schemes is 0.3 dB, 1.1 dB, respectively.