## 4 Simulation Results
### Generalized Mutual Information and BER Results  
The Generalized mutual information (GMI) can be used to express the achievable information rate for bit-metric decoding. In this section, the generalized mutual information and bit error rate performance of the probability shaping signal using parallel distributed matcher are analyzed using the PAS scheme. The advantages of the proposed structure are demonstrated when compared with the PS-QAM signal using CCDM. The GMI for bit-interleaved coded modulation is represented by (8) (Equation (13) of [3] and Equation (7) of [28]). In Monte Carlo simulations of N samples, GMI can be expressed as (9) (Equation (8) of [28]). Considering the rate loss of DM, the achievable information rate is shown in (10) (Equation (15) of [24]). \[GMI(X;Y)=H(X)-\sum_{j=1}^{m}H(B_{j}|Y) \tag{8}\]  
\[ \begin{split} GMI\approx\frac{1}{N}\sum_{k=1}^{N}\left[-\log_{2 }P_{X}(x_{k})\right]-\frac{1}{N}\sum_{k=1}^{N}\sum_{i=1}^{m}\left[\log_{2}(1+e^ {(-1)^{k_{i}\lambda_{k,j}}})\right.\\ \Lambda_{k,i}=\log\frac{\sum_{x\in\lambda_{i}^{j}}p_{Y|X}(y_{k}|x )P_{X}(x)}{\sum_{x\in\lambda_{i}^{j}}p_{Y|X}(y_{k}|x)P_{X}(x)}=\log\frac{p_{Y| B_{i}}(y_{k}|1)}{p_{Y|B_{i}}(y_{k}|0)}+\log\frac{p_{B_{i}}(1)}{P_{B_{i}}(0)} \end{split}  \tag{9}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**n**} & **Parallel DM \(p_{1}\)** & **Parallel DM \(p_{1}\)** & **Parallel DM \(p_{1}\)** & **Parallel DM \(p_{1}\)** & **Parallel DM \(p_{1}\)** & **CCDM** \\  & **= 8** & **= 7** & **= 6** & **= 5** & **= 4** & **\\ \hline
10 & – & – & 1.447 & 1.458 & 1.4375 & 1.3 \\
20 & – & – & 1.612 & 1.606 & 1.589 & 1.5 \\
30 & – & 1.677 & 1.673 & 1.665 & 1.653 & 1.6 \\
40 & 1.713 & 1.711 & 1.706 & 1.699 & 1.69 & 1.65 \\
50 & 1.732 & 1.73 & 1.725 & 1.72 & 1.712 & 1.68 \\
60 & 1.7491 & 1.7462 & 1.7425 & 1.7352 & 1.7287 & 1.7 \\
70 & 1.7604 & 1.7569 & 1.7521 & 1.747 & 1.743 & 1.7143 \\
80 & 1.7685 & 1.7649 & 1.7611 & 1.7563 & 1.75 & 1.725 \\
90 & 1.7749 & 1.7721 & 1.7679 & 1.7636 & 1.7604 & 1.733 \\
100 & 1.7807 & 1.7767 & 1.7734 & 1.7702 & 1.7653 & 1.75 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Rate of CCDM and parallel DM under different block lengths \(n\). Figure 3: Rate over block length n for CCDM and parallel distribution matcher (Rmax and Rmin); the probability distribution is \(P_{A}=\{0.4,0.3,0.2,0.1\}\). \[GMI_{DM}=GMI-R_{loss} \tag{10}\]  
We consider a dual-polarization PS-64 QAM modulation system with P(1, 3, 5, 7) = (0.4, 0.3, 0.2, 0.1), the baud rate per polarization is 28 GBaud. The simulation block diagram and parameters in VPI are shown in Figure 4. In Figure 5, GMI in bits per symbol is shown over OSNR in dB for PS-64 QAM. Under the same OSNR, the greater n (n = 30, 50, 100) is, the smaller the rate loss is, and the larger the generalized mutual information is for the PS-64 QAM signal utilizing the parallel distribution matcher. The OSNR required by the parallel distribution matcher is less than that using CCDM when n is 100 and the GMI is the same. When GMI is 4 bit/symbol, compared with the uniformly distributed 64 QAM signal, the OSNR of the PS-64 QAM signal using CCDM is reduced by 0.35 dB, and the OSNR of the PS-64 QAM signal using the parallel distribution matcher is reduced by 0.47 dB, which is a 0.12 dB improvement compared with CCDM. When the block length of CCDM increases from n = 100 to n = 50,000, the OSNR only reduces by 0.33 dB; so, the parallel structure has a relatively large improvement of 0.12 dB. Figure 6 demonstrates the maximum and minimum GMI values for PS-64 QAM signals using this parallel distribution matcher at different block lengths when the OSNR is 15, 16, and 17 dB. When OSNR = 15 dB, GMI = 3.655 bit/symbol; OSNR = 16 dB, GMI = 3.948 bit/symbol; and OSNR = 17 dB, GMI = 4.23 bit/symbol. The needed block length n of the parallel distribution matcher is only 60, although the required block length of the CCDM is 100; so, the block length can be saved by 40%. The matcher can attain the same performance as CCDM while using fewer blocks.