## 3 Convergence-Performance Analysis and Re-Design of PLDPC Channel Codes
### A Re-Design Scheme for PLDPC Channel Codes  
By taking advantage of the M-JPEXIT algorithm, a re-design method for channel codes of this system is provided in Algorithm 2. As is shown in [8], the edge connection between the VNs of the channel code and the CNs of the source code plays an important role in the error performance. Algorithm 2 also takes this edge connection into consideration. ```
1:Input: \(p\), \(\mathbf{B}_{\mathrm{s}}\), \(\mathbf{B}_{v}\), \(m_{\mathrm{s}}\), \(n_{\mathrm{s}}\), \(m_{\mathrm{c}}\), \(n_{\mathrm{c}}\); the initial value of SNR \(x_{j}\); the number of Turbo equalization iteration \(N_{t}\) and the maximum number of JSC iteration \(N_{j}\);
2:Start:
3:for\(N_{\mathrm{i},min}=1\) to \(N_{t}\)do
4:for\(j=1\) to \(n_{\mathrm{c}}\)do
5:BCJR detector:
6:Evaluate the MI \(I_{\mathrm{BCJR\_OUT}}\);
7:endfor
8:for\(N_{\mathrm{i},min}=1\) to \(N_{j}\)do
9:Channel decoder:
10:for\(i=1\) to \(m_{\mathrm{c}}\), \(j=1\) to \(n_{\mathrm{c}}\)do
11:Channel LLR initialization:
12:The channel initial LLRs are defined by \(\sigma_{\mathrm{BCJR\_OUT}}\). If \(v_{j}\) is punctured, \(I_{\mathrm{BCJR\_OUT}}=0\);
13:The MI update between CNs and VNs:
14:for\(j=1\) to \(n_{\mathrm{c}}-m_{\mathrm{s}}\)do
15:if\(b_{\mathrm{c}}^{j,i}=0\)then
16:\(I_{\mathrm{c}}^{\mathrm{Ev}}(i,j)=0\);
17:else
18:\(I_{\mathrm{c}}^{\mathrm{Ev}}(i,j)=I(\sigma_{\mathrm{c}}^{\mathrm{Ev\_1}})=I_ {\mathrm{c}}^{\mathrm{Ac}}(i,j)\), \(\sigma_{\mathrm{c}}^{\mathrm{Ev\_1}}\) is calculated as (3);
19:endif
20:endfor
21:for\(j=n_{\mathrm{c}}-m_{\mathrm{s}}+1\) to \(n_{\mathrm{c}}\)do
22:if\(b_{\mathrm{c}}^{j,i}=0\)then
23:\(I_{\mathrm{c}}^{\mathrm{Ev}}(i,j)=0\);
24:else
25:\(I_{\mathrm{c}}^{\mathrm{Ev}}(i,j)=J(\sigma_{\mathrm{c}}^{\mathrm{Ev\_2}})=I_ {\mathrm{c}}^{\mathrm{Ac}}(i,j)\), \(\sigma_{\mathrm{c}}^{\mathrm{Ev\_2}}\) is calculated as (4);
26:endif
27:endfor
28:if\(b_{\mathrm{c}}^{j,i}=0\)then
29:\(I_{\mathrm{c}}^{\mathrm{Ec}}(i,j)=0\);
30:else
31:\(I_{\mathrm{c}}^{\mathrm{Ec}}(i,j)=1-I(\sigma_{\mathrm{c}}^{\mathrm{Ec}})=I_ {\mathrm{c}}^{\mathrm{Aw}}(i,j)\), \(\sigma_{\mathrm{c}}^{\mathrm{Ec}}\) is calculated as (5);
32:endif
33:The MI update from the channel decoder to the BCJR detector and the source decoder:
34:\(I_{\mathrm{BCJR\_IN}}=\frac{1}{n_{\mathrm{c}}}\sum_{j=1}^{n_{\mathrm{c}}}I( \sigma_{\mathrm{BCJR\_IN}})\), \(\sigma_{\mathrm{BCJR\_IN}}\) is calculated as (6);
35:\(I_{\mathrm{c}\to\mathrm{s}}(j)=I(\sigma_{\mathrm{c\to s}})\), \(\sigma_{\mathrm{c\to s}}\) is calculated as (7);
36:endfor
37:Source decoder:
38:for\(i=1\) to \(n_{\mathrm{s}}\)do
39:Source LLR initialization:
40:\(I_{\mathrm{s}}=\ln\left((1-p)/p\right)\);
41:The MI update between CNs and VNs:
42:if\(b_{\mathrm{c}}^{j,i}=0\)then
43:\(I_{\mathrm{c}}^{\mathrm{Ev}}(i,j)=0\);
44:else
45:\(I_{\mathrm{c}}^{\mathrm{Ev}}(i,j)=I_{\mathrm{c}}(n_{\mathrm{s}}^{\mathrm{Ev}},p)=I_{\mathrm{s}}^{\mathrm{Ac}}(i,j)\), \(\mu_{\mathrm{s}}^{\mathrm{Ev}}\) is calculated as (8);
46:endif
47:if\(b_{\mathrm{c}}^{j,i}=0\)then
48:\(I_{\mathrm{c}}^{\mathrm{Ec}}(i,j)=1-J(\sigma_{\mathrm{c}}^{\mathrm{Ec}})=I_ {\mathrm{s}}^{\mathrm{Aw}}(i,j)\), \(\sigma_{\mathrm{s}}^{\mathrm{Ec}}\) is calculated as (9);
49:endif
50:The MI update from the source decoder to the channel decoder:
51:\(I_{\mathrm{s}\to\mathrm{c}}(i)=1-I(\sigma_{\mathrm{c\to c}})\), \(\sigma_{\mathrm{s\to c}}\) is calculated as (10);
52:Aposteriori MI evaluation:
53:\(I_{\mathrm{s}}^{\mathrm{Aw}}(j)=I_{\mathrm{s}}(\mu_{\mathrm{s}}^{\mathrm{Aw}},p)\), \(\mu_{\mathrm{s}}^{\mathrm{Aw}}\) is calculated as (11);
54:endfor
55:if\(I_{\mathrm{s}}^{\mathrm{Aw}}(1)=1,\cdots,I_{\mathrm{s}}^{\mathrm{Aw}}(n_{ \mathrm{s}})=1\)then
56:goto Output;
57:endif
58:endfor
59:endfor
60:endfor
61:endfor
62:Set \(x=x-0.1\) and goto Start;
63:Output:
64:\((E_{\mathrm{s}}/N_{0})_{\mathrm{min}}=x\). ```  
**Algorithm 1** M-JPEXIT algorithm for the proposed system. Based on Algorithm 2, the best optimized channel code for PR channels in the separate coding system, which denoted as IARA2 in [22] is redesigned.