## 4 Results and Discussion  
### Simulation Results  
The optimization problem in this paper is related to four variables: SE, SNR, NGMI, and QAM order. The simulation results in the AWGN channel are illustrated in Figure 2. Here we drew the surface plot of the SE, SNR, and NGMI of 64, 128, 256, and 512 QAM. If we draw a plane with NGMI = 0.92, then the intersection of the surface of this plane is the maximum SE of different SNRs that satisfy the threshold. In this figure, the NGMI rises with an increase in SNR for each SE. When the SNR is at a high level and the SE is at a low level, the NGMI saturates to 1, which means that the FEC code is not necessary for error-free decoding. In this study, we investigated the PS QAM order and SE in three scenarios:  
1. The NGMI threshold is 0.92, which corresponds to the FEC code rate 9/10;
2. The NGMI threshold is 0.86, which corresponds to the FEC code rate 5/6;
3. The code rate \(R_{c}~{}=~{}NGMI-0.02\). In these scenarios, we try to find the QAM order and SE which own big enough AIR at each SNR with low computation complexity. The criteria are listed as follows: when the AIR gap between the alternative QAM orders is less than 0.1 bits per QAM symbol (bpQs), we choose the lower QAM order, which owns a much lower computation complexity in the CCDM-based PAS system. The reason is listed as follows: take the SE = 5.3 bpQs as an example. The innermost and outermost probability ratio of PS-64 QAM, PS-128 QAM, PS-256 QAM, and PS-512 QAM is 107.23, \(8.86\times 10^{3}\), \(3.50\times 10^{10}\), \(4.99\times 10^{17}\), respectively. The probability ratio of the innermost and outermost is proportional to the storage depth of the digital to analog and the computation complexity of CCDM [16; 21]. Therefore, it is hard to realize a perfect PS constellation by the higher QAM order. Furthermore, according to  
Figure 2: Relationship between normalized generalized mutual information (NGMI), source entropy (SE), and signal-to-noise ratio (SNR) with different probabilistic shaping (PS) QAM order of 16, 32, 64, 128, 256, and 512. Equation (8), the system AIR without redundancy is closely related to the QAM order and SE at a specific NGMI threshold. The higher the QAM order, the more redundant bits the FEC code introduces, thereby introducing a higher system delay. Therefore, it is more reasonable and practical to realize a PS constellation with the lowest possible QAM order. First, the simulation results are carried out on the AWGN channel. The optimal AIR and SE that satisfy the 0.92 NGMI threshold at different SNR are shown in Figure 3a,b, respectively. As Figure 3a shows, higher order QAM can realize better AIR performance. However, as raising one order will inevitably increase the computation complexity significantly, it is not sensible to always take the highest order as the PS order. In Figure 3a, the AIR gap bigger than 0.1 bpQs between 64 QAM and 128 QAM first appears at the SNR of 16.67 dB. Meanwhile, the AIR gap between 128 QAM and 256 QAM first appears at 18.79 dB. As the AIR gap between 256 QAM and 512 QAM is below 0.1 bpQs, the 512 QAM is not taken into consideration. The optimal AIR and SE that satisfy the 0.86 NGMI threshold at different SNR are shown in Figure 4a,b, respectively. The transition SNR from QAM orders 6-7 and 7-8 are 15.65 dB and 17.63 dB, respectively. As the FEC code rate decreases, higher SE is permitted to transmit more bits in the channel. Furthermore, the AIR performances of 256 QAM and 512 QAM are very close to each other. Similar to the results in Figure 3, it is not necessary to implement 512 QAM in the simulation. The optimal HAIR performance is shown in Figure 5a. The corresponding SE is shown in Figure 5b. As the target is to achieve the optimal AIR at the FEC code rate \(R_{c}~{}=~{}NGMI-0.02\), we choose to change the SE by 0.1 bpQs for each QAM order. This operation is different from scenario 1 and 2 because \(R_{c}\) is a variable as well. The transfer point from 64 to 128 QAM and 128 to 256 QAM of SNR are 15.78 dB and 17.51 dB, respectively.