## 3 Convergence-Performance Analysis and Re-Design of PLDPC Channel Codes
### M-JPEXIT Algorithm
\[\sigma_{\text{c}}^{\text{Ev\_2}}=\sqrt{\sum_{k\
eq i}b_{\text{c}}^{kj}[J^{-1} \big{(}I_{\text{c}}^{\text{Av}}(k,j)\big{)}]^{2}+\Big{(}b_{\text{c}}^{ij}-1 \big{)}[J^{-1}\big{(}I_{\text{c}}^{\text{Av}}(i,j)\big{)}]^{2}+[J^{-1}\big{(}I _{\text{BCJR\_OUT}}\big{)}]^{2}}. \tag{3}\]  
\[\sigma_{\text{c}}^{\text{Ev\_2}}=\sqrt{\sum_{k\
eq i}b_{\text{c}}^{kj}[J^{-1} \big{(}I_{\text{c}}^{\text{Av}}(k,j)\big{)}]^{2}+\Big{(}b_{\text{c}}^{ij}-1 \big{)}[J^{-1}\big{(}I_{\text{c}}^{\text{Av}}(i,j)\big{)}]^{2}+[J^{-1}\big{(} I_{\text{s\to c}}(j-(n_{c}-m_{s}))\big{)}]^{2}}. \tag{4}\]  
\[\sigma_{\text{c}}^{\text{Ec}}=\sqrt{\sum_{k\
eq j}b_{\text{c}}^{kj}[J^{-1} \big{(}1-I_{\text{c}}^{\text{Ac}}(i,k)\big{)}]^{2}+\Big{(}b_{\text{c}}^{ij}-1 \Big{)}[J^{-1}\big{(}1-I_{\text{c}}^{\text{Ac}}(i,j)\big{)}]^{2}}. \tag{5}\]  
\[\sigma_{\text{BCJR\_IN}}=\sqrt{\sum_{k}b_{\text{c}}^{kj}[J^{-1}\big{(}I_{\text {c}}^{\text{Av}}(k,j)\big{)}]^{2}}. \tag{6}\]  
\[\sigma_{\text{c\to s}}=\sqrt{\sum_{k}b_{\text{c}}^{kj}[J^{-1}(I_{\text{c}}^{ \text{Av}}(k,j))]^{2}+[J^{-1}\big{(}I_{\text{BCJR\_OUT}}\big{)}]^{2}}. \tag{7}\]  
\[\mu_{\text{s}}^{\text{Ev}}=\frac{1}{2}\sum_{k\
eq i}b_{\text{s}}^{kj}\Big{[}J^ {-1}\Big{(}I_{\text{s}}^{\text{Av}}(k,j)\Big{)}\Big{]}^{2}+\frac{1}{2}\Big{(}b _{\text{s}}^{ij}-1\Big{)}\Big{[}J^{-1}\Big{(}I_{\text{s}}^{\text{Av}}(i,j) \Big{)}\Big{]}^{2}. \tag{8}\]  
\[\sigma_{\text{s}}^{\text{Ec}}=\sqrt{\sum_{k\
eq j}b_{\text{s}}^{kj}[J^{-1} \big{(}1-I_{\text{s}}^{\text{Ac}}(i,k)\big{)}]^{2}+\Big{(}b_{\text{s}}^{kj}-1 \Big{)}[J^{-1}\big{(}1-I_{\text{s}}^{\text{Ac}}(i,j)\big{)}]^{2}+[J^{-1}(1-I_{ \text{c\to s}}(i+n_{c}-m_{s}))]^{2}}. \tag{9}\]  
\[\sigma_{\text{s\to c}}=\sqrt{\sum_{k}b_{\text{s}}^{jk}[J^{-1}\big{(}1-I_{\text {s}}^{\text{Ac}}(i,k)\big{)}]^{2}}. \tag{10}\]  
Figure 2:0  
The M-JPEXIT algorithm for the proposed system is detailed in Algorithm 1. The iterative MI function \(J(\cdot)\) of the channel decoder of Algorithm 1 is defined as  
Figure 2:1  
where, \(J(\sigma_{\text{ch}})\) represents the MI between a binary bit and channel LLR value \(L_{\text{ch}}\sim N(\sigma_{\text{ch}}^{2}/2,\sigma_{\text{ch}}^{2})\). The inverse \(J^{-1}(I)\) function is also given by  
\[J^{-1}(I)=\left\{ \begin{array}{ll}a_{1}I^{2}+b_{1}I+c_{1}\sqrt{I},&0\leq I\leq 0.3646,\\ -a_{2}\ln[b_{2}(1-I)]-c_{2}I,&0.3646<I<1,\end{array} \right. \tag{13}\]  
where, \(a_{1}=1.09542,b_{1}=0.214217,c_{1}=2.33727,a_{2}=0.706692,b_{2}=0.386013,\)\(c_{2}=-1.75017.\)  
The iterative MI function \(J_{\rm B}(\cdot)\) of the source decoder of Algorithm 1 is  
\[J_{\rm B}(\mu,p)=(1-p)\times I(V;\chi^{(1-p)})+p\times I(V;\chi^{p}), \tag{14}\]  
where, \(I(V;\chi^{(1-p)})\) is the MI between the VN \(V\) of \({\rm B_{s}}\) and \(\chi\), \(\mu\) represents the average LLR value obtained by V, \(\chi^{(1-p)}\sim N(\mu+L_{s},2\mu)\), \(\chi^{(p)}\sim N(\mu-L_{s},2\mu)\). In Algorithm 1, the method to evaluate the MI \(I_{\rm BCJR\_OUT}\) in line 9 is described as follows: for a given \(E_{s}/N_{0}\), calculate the channel initial LLRs \(L_{\rm ch}\) for the bit sequence output from the demodulator by Monte Carlo simulation and denote these LLR values as \(\left\{L_{\rm ch}\right\}\). In addition, for a given a prior MI \(I_{\rm BCJR\_IN}\) (set the initial \(I_{\rm BCJR\_IN}\) as zero), the standard deviation \(\sigma_{\rm BCJR\_IN}\) of the corresponding _a priori_ LLRs \(L_{\rm BCJR\_IN}\) can be calculated using (13) and generate the LLR sequence \(\left\{L_{\rm BCJR\_IN}\right\}\) following the symmetric Gaussian distribution \(N(\sigma_{\rm BCJR\_IN}^{2}/2,\sigma_{\rm BCJR\_IN}^{2})\). Passing the sequences \(\left\{L_{\rm ch}\right\}\) and \(\left\{L_{\rm BCJR\_IN}\right\}\) into the BCJR detector, we can measure the extrinsic LLR sequence of the BCJR detector \(\left\{L_{\rm BCJR\_OUT}\right\}\) as (15). We then calculate the standard deviation \(\sigma_{\rm BCJR\_OUT}\) of the sequence \(\left\{L_{\rm BCJR\_OUT}\right\}\).