## 2 Principles of Probabilistic Shaping  
The famous Shannon capacity of the channel can be expressed as:  
\[C\ =\ \log_{2}(1+SNR). \tag{1}\]  
The SNR is measured by the uniform power training sequence. It has been proven that for the QAM constellations, the optimal Gaussian source can be generated by making the constellation yield the Maxwell-Boltzmann distribution [11]. If we assume the M QAM constellations are taken from a set with complex values \(\chi~{}=~{}\{x_{0},x_{1},\ldots,x_{M-1}\}\), the Maxwell-Boltzmann (MB) distribution for PS M-QAM is expressed as:  
\[P(x_{i})~{}=~{}\frac{1}{\sum\limits_{\begin{subarray}{c}j~{}=~{}0\end{subarray} }^{M-1}e^{-\lambda|x_{j}|^{2}}}e^{-\lambda|x_{j}|^{2}}. \tag{2}\]  
The \(\lambda\) is the shaping factor of the PS signal. The larger the \(\lambda\) is, the greater the difference in probability between the inner and outer circles of the constellation points. In this paper, we utilize the famous probabilistic amplitude shaping (PAS) system with the MB distribution to realize the PS scheme. For more details of PAS, please refer to the article [14]. As the GMI and NGMI can adequately reflect the decoding performance of soft decision LDPC (SD-LDPC) in the PAS system, we utilize them as the metrics in this study. The GMI can be expressed as follows:  
\[GMI\approx\frac{1}{N}\sum\limits_{k~{}=~{}0}^{N-1}~{}[-\log_{2}(P_{k})]-\frac {1}{N}\sum\limits_{k~{}=~{}0}^{N-1}\sum\limits_{i~{}=~{}1}^{M}~{}[\log_{2}(1+e ^{(-1)^{b_{kj}}\Lambda_{kj}})]. \tag{3}\]  
The \(b_{k,i}\) are the transmitting bits of the \(\mathrm{k_{th}}\) symbol. \(\Lambda_{k,i}\) is the soft bit-wise log-likelihood ratio (LLR) output which can be expressed as:  
\[\Lambda_{k,i}~{}=~{}\log\frac{\sum_{x\in\chi_{1}^{i}}q_{\gamma|X}(y_{k}|x)P(x)} {\sum_{x\in\chi_{0}^{i}}q_{\gamma|X}(y_{k}|x)P(x)}. \tag{4}\]  
The \(\chi_{1}^{i}\) and \(\chi_{0}^{i}\) represent the subsets in \(\chi\) whose \(\mathrm{i_{th}}\) bit is 1 or 0. \(q_{\gamma|X}(y_{k}|x)\) is the channel probability mass function (PMF). For the additive white Gaussian noise (AWGN) auxiliary channel with bit interleaved coded modulation (BICM) and a noise variance of \(\sigma^{2}\), the PMF can be expressed as:  
\[q_{\gamma|X}(y_{k}|x)~{}=~{}\frac{1}{\sqrt{2\pi\sigma^{2}}}e^{-\frac{|y_{k}- \pi^{2}}{2\sigma^{2}}}. \tag{5}\]  
The normalized GMI (NGMI) reflects the ideal binary FEC rate of the system for error-free transmission. For M QAM PAS with an LDPC encoding procedure, the NGMI can be expressed as:  
\[NGMI~{}=~{}1-\frac{H-GMI}{\log_{2}M}. \tag{6}\]  
Here, \(H\) is the SE of M QAM, which is written as:  
\[H~{}=~{}\sum\limits_{i~{}=~{}0}^{M-1}-P(x_{i})\cdot\log_{2}(P(x_{i})). \tag{7}\]  
Although the NGMI represents the ideal maximum rate, which can realize error-free decoding, it is practical to build a gap between the NGMI threshold and the code rate. In this paper, we utilize the NGMI threshold of 0.86 for the FEC code rate 5/6, and 0.92 for the FEC code rate 9/10, to carry out our simulation and experiment [19]. If we define the FEC code rate as \(R_{c}\), the NGMI is above the corresponding threshold, the achievable information rate (AIR) of PAS can be expressed as:  
\[AIR~{}=~{}H-(1-R_{c})\cdot\log_{2}M. \tag{8}\]  
In this paper, we compare the simulation results of the NGMI for different PS QAM orders at the same AIR and preset NGMI threshold. Furthermore, we also consider optimizing the FEC code rate for each PS constellation and SNR to achieve a higher AIR. The SE of each PS QAM order is increased by 0.1 bit/symbol to find the highest AIR (HAIR) of each PS constellation when \(R_{c}~{}=~{}NGMI-0.02\)Although we cannot obtain the relationship between the arbitrary code rate and the corresponding NGMI threshold, according to the results in Table III in [19], we consider it reasonable to set the gap between the code rate and its corresponding NGMI threshold to 0.02. Therefore, the HAIR can be written as:  
\[HAIR\ =\ H-(1-(NGMI-0.02))\cdot\log_{2}M. \tag{9}\]