# Design and Analysis of Joint Source-Channel Code System with Fixed-Length Code
## 4 Irregular Channel Code for Iterative Source Channel Decoding
### Irregular Channel Code Design  
Irregular convolutional codes as inner codes have been successfully employed in the context of iterative source-channel decoding in [10]. Here, we use the recursive non-systematic convolutional (RNSC) codes with rate 1 as the subcodes. Considering the computational of the RNSC decoder, we limit the maximum constraint length to \(D\leq 3\). There are \(2^{2D-1}\) kinds of different RNSC codes. In addition, the RNSC subcodes should have diverse shapes in the EXIT curves, so that the EXIT curve of the irregular channel code is expected to match the EXIT curve of the source code more accurately. Thus, we select five kinds of RNSC codes having the most dissimilar EXIT curves as shown in Figure 4. Their octal generator polynomials are \((3,2)_{8}\), \((5,7)_{8}\), \((6,7)_{8}\), \((7,4)_{8}\), and \((7,6)_{8}\).  
Denote the weight vector \(\mathbf{\alpha}~{}=~{}(\alpha_{1,}\alpha_{2,}\ldots\alpha_{J})\) of the different RNSC subcodes. The overall EXIT characteristic of the channel code is the weighted superposition of the EXIT characteristic of different RNSC subcodes, i.e.:  
\[\Gamma_{CD}(I)~{}=~{}\sum\limits_{j~{}=~{}1}^{J}\alpha_{j}\Gamma_{CD,j}(I) \tag{12}\]  
Then, the EXIT curves of encoders should satisfy \(\sum\limits_{j~{}=~{}1}^{J}\alpha_{j}\Gamma_{CD,j}(I)>\Gamma_{SD}^{-1}(I)\) for \(I\in[0,1)\) at a low SNR. Considering \(P\) sample points of the EXIT curves, the curve-fitting problem can be formulated as the optimization problem:  
\[\alpha_{\text{opt}}~{}=~{}\underset{\mathbf{\alpha}}{\text{argmin}}\left\|\mathbf{C }_{\text{CD}}\cdot\mathbf{\alpha}-\mathbf{c}_{\text{SD,inv}}\right\|_{2} \tag{13}\]  
subject to  
\[e_{i}>0,\forall i~{}=~{}1,\ldots,P \tag{14}\]  
\[\sum\limits_{j~{}=~{}1}^{J}\alpha_{j}~{}=~{}1 \tag{15}\]  
\[0\leq\alpha_{j}\leq 1,\forall j\in\{1,\ldots,J\} \tag{16}\]  
The matrix \(\mathbf{C}_{\text{CD}}\) of dimension \(P\times J\) contains \(P\) sample points of each RNSC decoder EXIT curve. The vector \(\mathbf{c}_{\text{SD,inv}}\) contains \(P\) sample points of the SDSD inverse EXIT curves. The width of the decoding tunnel is \(\mathbf{e}~{}=~{}\mathbf{C}_{\text{CD}}\cdot\mathbf{\alpha}-\mathbf{c}_{\text{SD, inv}}\). Constraint (14) ensures an open decoding tunnel while constraints (15) and (16) guarantee the validity of the weights. As all subcodes have the rate 1, no additional rate constraint needs to be considered.  
The optimization problem can be solved by the steepest descent algorithm, referred to in [23]. As an example, the optimum weights for the source codes with length \(L~{}=~{}6\) are summarized in Table 2, and the EXIT curves of the irregular and regular RNSC decoder and SBSD are illustrated in Figure 5. This shows that an open tunnel between the inverse EXIT curve of SBSD and the EXIT curve of the irregular channel decoder is obtained at SNR \(E_{s}/N_{0}~{}=~{}7.8\) dB. Meanwhile, for the regular channel code, an open tunnel is obtained until the 8.2 dB. The waterfall performance of ISCD is improved by employing the irregular channel code.  
Figure 4: EXIT curves of different RNSC codes, AWGN channel, SNR = 8 dB, 16 QAM Gray mapping.