# Rateless Coded Uplink Transmission Design for Multi-User C-RAN
## 4 Decoding Performance Analysis and Degree Profile Design
### EXIT Analysis of the Decoding Process  
Before the degree profile optimization, we first analyze the decoding process at the BBU pool to derive the successful decoding condition. The analysis is based on EXIT [42]. Explicitly, we trace the update of the extrinsic information (EI) transferred between output nodes and input nodes in the decoding graph during the decoding iterations. EI is defined as the mutual information between the LLR messages and the corresponding bits (i.e., input bits and output bits in this paper). It can be assumed that the LLR messages interchanged between the variable nodes and check nodes are symmetrically Gaussian distributed [42]. EI carried by LLR messages with a mean of \(\tau\) and a variance of \(2\tau\) is given by [42]:  
\[J(\tau)=1-\frac{1}{\sqrt{4\pi\tau}}\int_{-\infty}^{\infty}\log_{2}(1+e^{-\
u}) \exp\!\left(-\frac{(\
u-\tau)^{2}}{4\tau}\right)\!\!dv. \tag{14}\]  
Before the EXIT analysis, some definitions are given first. Recall the decoding graph in Figure 2. The LDPC variable node degree distribution is defined as \(\xi(x)=\sum_{d=1}^{d_{v}^{\prime}}\xi_{d}x^{d}\), where \(\xi_{d}\) is the probability of an LDPC variable node with degree \(d\), and \(d_{v}^{\prime}\) is the maximum degree of the variable nodes. (Note that both users employ the same LDPC code). The edge distribution of the LDPC check nodes can be defined as \(\widetilde{\rho}(x)=\sum_{d=1}^{d_{v}^{\prime}}\widetilde{\rho}_{d}x^{d-1}\), where \(\widetilde{\rho}_{d}\) is the probability of a randomly chosen edge being connected to a check node with degree \(d\), and \(d_{v}^{\prime}\) is the maximum degree of the check nodes. The distribution of the input node degree of user \(i\), \(i=1,2\), is defined as \(\alpha_{i}(x)=\sum_{d=1}^{d_{v}}\alpha_{i}\), \(x^{d}\), \(\alpha_{i,d}\) is the probability of the input nodes of user \(i\) with degree \(d\), and \(d_{v}\) represents the maximum degree of the input nodes. The edge distribution of the input nodes of the user \(i\) is defined as \(\widetilde{\alpha}_{i}(x)=\sum_{d=1}^{d_{v}}\widetilde{\alpha}_{i}x^{d-1}\), where \(\widetilde{\alpha}_{i,d}\) is the probability of an edge connected to an input node with degree \(d\). Both \(\alpha_{i}(x)\) and \(\widetilde{\alpha}_{i}(x)\) can be approximated by the Poisson distribution with an average of \(\widetilde{\alpha}_{i}\)[43]. The output node edge distribution of user \(i\) is defined as \(\omega_{i}(x)=\sum_{d=1}^{d_{v}}\omega_{i,d}x^{d-1}\), where \(\omega_{i,d}\) represents the probability of an edge connected to the output node with degree \(d\). The relationship between \(\omega_{i}(x)\) and the output node degree distribution \(\Omega_{i}(x)\) given in (2) is given below:  
\[\omega_{i}(x)=\frac{\Omega_{i}^{\prime}(x)}{\Omega_{i}^{\prime}(1)}. \tag{15}\]  
For ease of reading, we summarize the aforementioned definitions in Table 1, wherein \(i=1,2\). \begin{table}
\begin{tabular}{c c} \hline \hline
**Symbol** & **Annotation** \\ \hline \(\xi(x)\) & LDPC variable node degree distribution \\ \(\widetilde{\xi}_{d}\) & Probability of an LDPC variable node with degree \(d\) \\ \(\widetilde{\rho}(x)\) & Edge distribution of the LDPC check nodes \\ \(\widetilde{\rho}_{d}\) & Probability of an edge connected to an LDPC check node with degree \(d\) \\ \(\alpha_{i}(x)\) & Input node degree distribution for user \(i\) \\ \(\widetilde{\alpha}_{i,d}\) & Probability of input nodes with degree \(d\) for user \(i\) \\ \(\widetilde{\alpha}_{i}(x)\) & Edge distribution of the input nodes for user \(i\) \\ \(\widetilde{\alpha}_{i,d}\) & Probability of an edge connected to an input node with degree \(d\) for user \(i\) \\ \(\omega_{i}(x)\) & Output node edge distribution for user \(i\) \\ \(\omega_{i,d}\) & Probability of an edge connected to an output node with degree \(d\) of user \(i\) \\ \(d_{v}\) & Maximum degree of input nodes \\ \(d_{v}^{\prime}\) & Maximum degree of LDPC variable nodes \\ \(\widetilde{\alpha}_{i}\) & Average degree of the input nodes for user \(i\) \\ \hline \hline \end{tabular}
\end{table}