# Two-Dimensional Constellation Shaping in Fiber-Optic Communications
## 3 Performance Comparison in Gaussian-Noise-Limited Channels  
As a rule of thumb, HPGS-MQAM cannot show clear performance improvement over PS-MQAM and GS-MQAM when M \(<\) 32. In addition, if M \(\geq\) 64, PS-MQAM could closely approach the Shannon limit, thus it is not necessary to apply HPGS to higher order QAM formats. In this paper, we performed Monte Carlo simulations in MATLAB. The block-length of the CCDM was chosen more than 5000,  
Figure 6: Modified probabilistic shaping (PS) schemes based on (**a**) probabilistic fold shaping, (**b**) universal PS. so the normalized divergence of the encoder output and the desired distribution is negligible. The _awgn_ function provided by MATLAB was used to add white Gaussian noise to the 2D-MQAM signals. All the results were averaged over 1000 trials. In a numerical simulation, as shown in Figure 7, we compared the MI performances of the minimizing mean-square error of Gaussian-distribution-based GS-8/16/32 QAM, CCDM-based PS-8/16/32 QAM, and regular-8/16/32 QAM. Figure 7a1,a2,b1,b2,c1,c2 show the constellation diagrams of PS/GS-8/16/32 QAM. Here we chose MI as the metric for performance comparison, because MI can be measured without the consideration of the FEC coding performance. Given that most of the current nonbinary FEC coding and binary FEC coding schemes may vary in performance and implementation penalty, MI instead shows the upper limit of the capacity obtained with the \"ideal\" FEC coding scheme [61]. In order to easily describe the PS-MQAM with an entropy of \(A\) b/s, we adopt the notation of PS-MQAM\(A\). For example, we use PS-8 QAM2.3 to denote the PS-8 QAM with an entropy of 2.3 b/s. As we can see from Figure 7a,b, the best MI performances can be obtained by GS-8/16 QAM. GS-8/16 QAM can always outperform regular-8/16 QAM. PS-8 QAM can have comparable performance over GS-8 QAM when the SNR is less than 6.2 dB; the performances of PS-16 QAM and GS-16 QAM are quite similar when the SNR is less than 11.7 dB. In the region of high SNR region, PS-8/16 QAM cannot bring better MI performance. In Figure 7c, we find that the best performance can be achieved by PS/GS-32 QAM separately. In addition, GS-32 QAM is always better than regular-32 QAM in terms of MI performance. When the SNR is more than 15.7 dB, GS-32 QAM has the best performance, while PS-32 QAM formats can maximize the MI performance when the SNR is less than 15.7 dB. Figure 7: _Cont._  
Although PS-256 QAM and PS-1024 QAM have been investigated over fiber links [62], we still believe that PS-64 QAM is expected to be the most promising solution for high-order QAM-based fiber-optic communications, at least for the upcoming 400 G/800 G Ethernet. It is mainly because that PS-64 QAM has near-capacity-approaching performance, and its implementation penalty may also be well reduced to an acceptable level in the near future. While, unfortunately, regular-64 QAM and PS-64 QAM have been suffering a large implementation penalty so far. Here we compare the performances of PAS-64 QAM and GMI-optimized HPGS-32 QAM (referred to as opti-32 QAM in this paper). In another numerical simulation, Figure 8 shows the post-FEC bit error rate (BER) versus SNR performances of PAS-64 QAM, opti-32 QAM, PS-32 QAM, and regular-32 QAM. DVB-S2 irregular low-density parity-check (LDPC) codes were applied for FEC coding. The performance comparisons were executed under the same capacity levels, i.e., \(C\) = 3.33 b/s and \(C\) = 4 b/s. Table 1 lists the parameters used in the post-FEC BER analysis under the capacity levels of 3.33 b/s and 4 b/s. In Figure 8a, the performance of PS-32 QAM is slightly better than opti-32 QAM, but worse than PAS-64 QAM by 0.2 dB. Opti-32 QAM shows a 0.8-dB performance improvement over regular-32 QAM in case of \(C\) = 3.33 b/s. In Figure 8b, the performance of opti-32 QAM is better than PS-32 QAM by 0.2 dB, better than regular-32 QAM by 0.8 dB when the capacity is 4 b/s. However, PAS-64 QAM is also shown to outperform opti-32 QAM by 0.4 dB.