# Improving Polar-Coded SCMA System by Information Coupling and Parity Check
## 5 Optimization of Partially Information-Coupled Polar Code Based on EXIT Idea
### Transfer Characteristics of Base Polar Decoder  
As shown in Figure 1, the MPA detector converts the channel information and the prior information into the extrinsic information, which is interpreted as the prior information of the coupled polar decoder. This messages passing process complies with \"Turbo Principle,\" which is very common in iterative system. The EXIT chart can predict the behavior of the iterative decoder by solely analyzing the individual constituent decoders. The transmission characteristic curve of channel decoder reflects the relationship between its input mutual information and output mutual information. The output mutual information of channel decoder is corresponding to the input mutual information of MPA detector. Importantly, the higher the output mutual information of channel decoder is, the higher the input mutual information of MPA detector is, and the better the performance of SCMA system. Based on the EXIT idea, in this section, we focus on the design of PIC polar code by the TC chart.  
We use \(I_{A}\) to denote the mutual information between a prior information \(Z\) and BPSK-modulated symbols \(X\) and use \(I_{E}\) to denote the mutual information between a prior information \(L_{e}\) and BPSK-modulated symbols \(X\), respectively. According to [29], \(Z\) can be seen as a Gaussian-distributed variable with the mean \(\mu_{z}=\sigma_{z}^{2}/2\) and the variance \(\sigma_{z}^{2}\). The conditional probability density function can be expressed as  
\[p_{z}(\tau|X=x)=\frac{1}{\sqrt{2\pi}\sigma_{z}}\exp\left(\frac{-(\tau-\mu_{z}x) ^{2}}{2\sigma_{z}^{2}}\right). \tag{23}\]  
So the input mutual information \(I_{A}\) can be calculated as follows [29]  
\[I_{A}(\sigma_{z})=1-\int_{-\infty}^{+\infty}\frac{1}{\sqrt{2\pi}\sigma_{z}}\exp \left(-(\tau-\mu_{z})^{2}/2\sigma_{z}^{2}\right)\cdot\log_{2}(1+e^{-\tau})d\tau. \tag{24}\]  
The output mutual information \(I_{E}\) is denoted as  
\[I_{E}=\frac{1}{2}\sum_{x=\pm 1}\int_{-\infty}^{+\infty}p_{z}(\tau|x)\log_{2} \frac{2\cdot p_{z}(\tau|x)}{\sum_{x^{\prime}=\pm 1}p_{z}(\tau|x^{\prime})}d\tau. \tag{25}\]  
Viewing \(I_{E}\) as a function of \(I_{A}\), the extrinsic information transfer characteristics of polar decoder can be defined as  
\[I_{E}=\Omega(I_{A}). \tag{26}\]  
Figure 9a,b shows the TC chart and CBER curves of uncoupled systematic PCCA-polar code with SCAN decoder over AWGN channel, respectively. Base polar code is constructed by Bhattacharyya parameter-bound method proposed in [8], weight factor \(\alpha\) is taken from 0 to 1, and the number of out loop iterations is 5. The information bits length \(K=100\) and code rate \(R=1/2\). From Figure 9a, the TC curve is closer to the right, the better performance of decoder is (\(I_{E}=1\) represents the largest output mutual information). As shown in Figure 9b, the best CBER performance is the decoder with \(\alpha=0.2\), and its corresponding TC curve is also the rightmost in Figure 9a. In addition, both the TC chart and the CBER curves show that the decoders with \(\alpha=0.1\), 0.3, and 0.4 have very similar performance. It can be seen, the performance of CBER is consistent with the analysis of TC chart. However, compared with the growth of the code length \(N\), finding the optimal \(\alpha\)-value based on TC chart analysis is less complex.  
The extrinsic message of SCL decoder is constructed by HC algorithm in Section 4. In HC algorithm, when the CRC check fails, the SCAN algorithm is used to construct the extrinsic messages. Therefore, the iterative performance of the SCL decoder is also affected by the \(\alpha\)-value in SCMA system. The transfer characteristics of uncoupled systematic PCCA-polar code with SCL decoder is given by Figure 10. From the figure, the optimal value of \(\alpha\) is 0.2.