## 4 Asymptotic Analysis of Proposed Enhanced Distributed Fountain Codes  
### Description of Degree Distributions  
The degree distribution is an important parameter in determining the performance of fountain codes. In this section, we will discuss the degree distributions of the proposed EDFC and derive the expressions of the asymptotic performance. We assume that the output node-perspective and edge-perspective degree distribution of source \(S_{i}\) are \(\Phi_{i}(x_{i})\) and \(\phi_{i}(x_{i})=\Phi^{\prime}_{i}(x_{i})/\Phi^{\prime}_{ii}(1)\), respectively. Similarly, we denote \(\Gamma_{i}(x)\) and \(\gamma_{i}(x)=\Gamma^{\prime}_{i}(x)/\Gamma^{\prime}_{i}(1)\) as the output node-perspective and edge-perspective degree distribution of relay \(R_{i}\), respec  
Figure 3: Encoding process for the proposed enhanced distributed fountain codes. tively. We assume that the input node-perspective and edge-perspective degree distribution of source \(S_{i}\) are \(\Lambda_{i}(x_{i})\) and \(\lambda_{i}(x_{i})=\Lambda^{\prime}_{i}(x_{i})/\Lambda^{\prime}_{i}(1)\), respectively. In the Tanner graph, the connections between an input symbol and encoded symbols are determined after encoding at the source. The re-encoding at the relay will change the connections of the output symbols in the Tanner graph but will not affect the connections of the input symbols. In other words, the re-encoding of the relay will change the output degree distribution of each source but will have no effect on the input degree distribution of each source. Therefore, the input degree distribution \(\Lambda_{i}(x_{i})\) still follows a binomial distribution [4; 40]. As \(n\) goes to infinity, the input degree distribution approaches a Poisson distribution asymptotically as follows  
\[\Lambda_{i}(x_{i})=\lambda_{i}(x_{i})=\exp(-\alpha_{i}(1-x_{i})) \tag{2}\]  
where \(\alpha_{i}\) is the average input degree of source \(S_{i}\). In point-to-point communication systems, the degree distribution of an LT code is usually a unary function. This is because all the input (or output) nodes follow only one degree distribution, and their degree distribution characteristics can be completely described by a unary function. In the multiple-access relay network with cooperative relay shown in Figures 2 and 3, there are differences among the sources. The number of information symbols \(k_{i}\) can be different for each source, and different degree distributions can be chosen for encoding. Therefore, the resulting degree distribution seen at the destination cannot be represented by a univariate function. In this case, we can introduce multivariate functions and use multivariate degree distributions to describe the overall degree distributions. **Lemma 1**. : _Let source \(S_{i}\) perform LT encoding with output degree distribution \(\Phi_{i}(x_{i})=\sum_{d=1}^{k_{i}}\Phi_{i,d}x_{i}^{d}\), relay \(R_{1}\) and \(R_{2}\) perform re-encoding with output degree distribution \(\Gamma_{1}(x)=\sum_{d=1}^{n}\Gamma_{1,d}x^{d}\) and \(\Gamma_{2}(x)=\sum_{d=1}^{t}\Gamma_{2,d}x^{d}\), respectively. The overall output degree distribution of source \(S_{i}\) for EDFC is as follows:_  
\[ \begin{array}{ll}\Omega_{i}(\stackrel{{\rightarrow}}{{x}})&= \Omega_{i}(x_{1},x_{2},\ldots,x_{n})\\ &=\begin{cases}\Gamma_{1}\left(\frac{1}{n}\sum\limits_{j=1}^{n}\Phi_{j}(x_{j}) \right),&1\leq i\leq n-t\\ \frac{1}{2}\Gamma_{1}\left(\frac{1}{n}\sum\limits_{j=1}^{n}\Phi_{j}(x_{j}) \right)+\frac{1}{2}\Gamma_{2}\left(\frac{1}{i}\sum\limits_{j=n-t+1}^{n}\Phi_{ j}(x_{j})\right),&n-t+1\leq i\leq n\end{array} \end{array} \tag{3}\]  
**Proof. ** As shown in Figure 2, the multiple-access relay network with a cooperative relay can be regarded as consisting of two distributed networks: One is with sources \(\{S_{1},S_{2},\ldots,S_{n}\}\), relay \(R_{1}\), and destination \(D\); the other one is with sources \(\{S_{n-t+1},S_{n-t+2},\ldots,S_{n}\}\), relay \(R_{2}\), and destination \(D\).