## 3 Convergence-Performance Analysis and Re-Design of PLDPC Channel Codes  
### M-JPEXIT Algorithm  
Aiming to facilitate the asymptotic-performance analysis for the proposed system, the M-JPEXIT algorithm is provided in this subsection. Specifically, the M-JPEXIT algorithm is employed to investigate the MI evolution among the BCJR detector, the channel decoder, the source decoder and thus can be used to calculate the decoding thresholds of this system. In particular, the threshold value is assumed to be the minimum signal-to-noise ratio (SNR) per source bit (i.e., \(E_{s}/N_{0}\)) that allows to achieve error-free transmission. To show the differences between the conventional JPEXIT algorithm and the proposed M-JPEXIT algorithm, the Tanner graphs of two algorithms are given in Figure 2. In Figure 2a, it is found that the initial log-likelihood ratios (LLRs) of the channel decoder in the JPEXIT algorithm can be derived from the output extrinsic LLRs of the AWGN channel. However, in M-JPEXIT algorithm, they are calculated by the LLR processor of the BCJR detector, which is shown Figure 2b. Furthermore, other than examining the evolution of the MIs exchanging between the channel decoder and the source decoder, the MIs exchanging between the channel decoder and the BCJR detector are taken into account in M-JPEXIT algorithm. Specifically, in Figure 2b, \(\textbf{B}_{c}\) with size of \(m_{c}\times n_{c}\) and \(\textbf{B}_{s}\) with size of \(m_{s}\times n_{s}\) represent the base matrices of the channel PLDPC code and the source PLDPC code, respectively. In addition, the \((i,j)\)-th element of \(\textbf{B}_{c}(\textbf{B}_{s})\), denoted by \(b_{c}^{i,j}\) (\(b_{s}^{i,j}\)), represents the number of edges connecting the VN \(v_{j}\) to the CN \(c_{i}\) in \(\textbf{B}_{c}(\textbf{B}_{s})\). The black squares represent the CNs of \(\textbf{B}_{s}\) and \(\textbf{B}_{c}\). The black circles in \(\textbf{B}_{c}\) indicate the source VNs for source coding and the black circles in \(\textbf{B}_{s}\) indicate the transmitted channel VNs for channel coding. In particular, the punctured channel VNs are denoted by empty circles. For the BCJR detector, the gray circles are used to represent the OD-ISI MR channel outputs and the squares denoted as trellis nodes are used to represent the state structure of the BCJR detector. Figure 2: The Tanner graph of (**a**) the conventional JPEXIT algorithm, (**b**) the M-JPEXIT algorithm with MIs updating. For the sake of introducing the M-JPEXIT algorithm in detail, nine types of MI are defined by:  
(1) \(I_{\text{BJCR\_OUT}}\): the extrinsic MI from the BCJR detector to VNs in \(\textbf{B}_{\text{c}}\). (2) \(I_{\text{BJCR\_IN}}\): the a prior MI from VNs in \(\textbf{B}_{\text{c}}\) to the BCJR detector. (3) \(I_{\text{s(c)}}^{\text{Ev}}(i,j)\): the extrinsic MI from the \(j\)-th VN to the \(i\)-th CN in \(\textbf{B}_{\text{s(c)}}\). (4) \(I_{\text{s(c)}}^{\text{Av}}(i,j)\): the a prior MI from the \(i\)-th CN to the \(j\)-th VN in \(\textbf{B}_{\text{s(c)}}\). (5) \(I_{\text{s(c)}}^{\text{Ev}}(i,j)\): the extrinsic MI from the \(i\)-th CN to the \(j\)-th VN in \(\textbf{B}_{\text{s(c)}}\). (6) \(I_{\text{s(c)}}^{\text{Ac}}(i,j)\): the a prior MI from the \(j\)-th VN to the \(i\)-th CN in \(\textbf{B}_{\text{s(c)}}\). (7) \(I_{\text{c\to s}}(j)\): the extrinsic MI from the \(j\)-th VN in \(\textbf{B}_{\text{c}}\) to the connected CN in \(\textbf{B}_{\text{s}}\). (8) \(I_{\text{s\to c}}(i)\): the extrinsic MI from the \(i\)-th CN in \(\textbf{B}_{\text{s}}\) to the connected VN in \(\textbf{B}_{\text{c}}\). (9) \(I_{\text{s}}^{\text{APP}}(j)\): the MI between the a posteriori LLR evaluated at \(j\)-th VN in \(\textbf{B}_{\text{s}}\) and the corresponding source bit.