## 4 Asymptotic Analysis of Proposed Enhanced Distributed Fountain Codes
### Asymptotic Analysis of Decoding Performance  
In the proposed EDFC shown in Figure 3, the destination starts decoding when it receives enough symbols and recovers the source information. The destination can treat the two-layer fountain coding process of the sources and relays as equivalent to a single-layer fountain code and performs decoding using the BP algorithm. The asymptotic performance of EDFC can also be analyzed theoretically using the And-Or tree method [4; 16]. We can obtain the following lemma. **Lemma 2**. : _For EDFC with a BP decoder used at destination, the unrecovered rate of source \(S_{i}\) converges to \(y_{i}=\lim_{l\to\infty}y_{i,l}\) as \(k_{i}\to\infty\), where \(y_{i,l}\) is expressed as follows:_  
\[y_{i,l}=\left\{ \begin{aligned} & 1,& l=0\\ &\exp\left(-\frac{1}{n}(1+\varepsilon_{i})\Phi^{\prime}{}_{i}(1- y_{i,l-1})\Gamma^{\prime}{}_{1}\left(\frac{1}{n}\sum\limits_{j=1}^{n}\Phi_{j} \left(1-y_{j,l-1}\right)\right)\right),& l\geq 1\end{aligned} \right. \tag{4}\]  
_when \(n-t+1\leq i\leq n\),_  
\[y_{i,l}=\left\{\exp\left(-\frac{(1+\varepsilon_{i})}{2}\Phi^{\prime}{}_{i}(1- y_{i,l-1})\left(\frac{1}{n}\Gamma^{\prime}{}_{1}\left(\frac{1}{n}\sum \limits_{j=1}^{n}\Phi_{j}\left(1-y_{j,l-1}\right)\right)+\frac{1}{l}\Gamma^{ \prime}{}_{2}\left(\frac{1}{l}\sum\limits_{j=n-t+1}^{n}\Phi_{j}\left(1-y_{j,l- 1}\right)\right)\right)\right),& l\geq 1\end{aligned}\right. \tag{5}\]  
_where \(l\) is the number of decoding iterations._  
**Proof. ** The destination can regard the two-layer fountain coding process of the sources and relays as a single-layer fountain code. According to [4], the unrecovered rate of source \(S_{i}\) can be represented by  
\[y_{i,l}=\left\{ \begin{aligned} & 1,& l=0\\ &\exp\left(-(1+\varepsilon_{i})\frac{\partial\Omega_{i}(\overset {\rightharpoonup}{x})}{\partial x_{i}}\Big{|}_{1-y_{i,l-1}}\right),& l\geq 1\end{aligned} \right. \tag{6}\]  
which can be regarded as the unrecovered rate of a single-layer fountain code. By substituting the overall degree distribution in Lemma 1 into (6), we can derive the asymptotic performance of \(S_{i}\) in the following two cases. * Case 1: The source \(S_{i}\) is connected to relay \(R_{1}\) only. The partial derivative of \(\Omega_{i}\left(\overset{\rightharpoonup}{x}\right)\) can be calculated as \[\frac{\partial\Omega_{i}\left(\overset{\rightharpoonup}{x}\right)}{\partial x _{i}}=\frac{1}{n}\Phi^{\prime}{}_{i}(x_{i})\Gamma^{\prime}{}_{1}\Bigg{(}\frac{1 }{n}\sum\limits_{j=1}^{n}\Phi_{j}(x_{j})\Bigg{)}.\] (7) The average output degree \(\beta_{i}\) is the partial derivative of \(\Omega_{i}\left(\overset{\rightharpoonup}{x}\right)\) at \(\overset{\rightharpoonup}{x}=1\) as follows \[\beta_{i}=\left.\frac{\partial\Omega_{i}\left(\overset{\rightharpoonup}{x} \right)}{\partial x_{i}}\right|_{\overset{\rightharpoonup}{x}=1}=\frac{1}{n} \Phi^{\prime}{}_{i}(1)\Gamma^{\prime}{}_{1}(1).\] (8)The average input degree \(\alpha_{i}\) can be represented by  
\[\alpha_{i}=(1+\varepsilon_{i})\beta_{i}=\frac{\sum_{j=1}^{n}k_{j}}{nk_{i}}(1+ \varepsilon)\Phi^{\prime}{}_{i}(1)\Gamma^{\prime}{}_{1}(1). \tag{9}\]  
Thus, the edge-perspective degree distribution of \(\Omega_{i}\Big{(}\overset{\rightharpoonup}{x}\Big{)}\) can be expressed as  
\[\omega_{i}\Big{(}\overset{\rightharpoonup}{x}\Big{)}=\frac{\partial\Omega_{i} \Big{(}\overset{\rightharpoonup}{x}\Big{)}}{\partial x_{i}}/\frac{\partial \Omega_{i}\Big{(}\overset{\rightharpoonup}{x}\Big{)}}{\partial x_{i}}\Big{|} \overset{\rightharpoonup}{x}_{=1}=\phi_{i}(x_{i})\gamma_{1}\Bigg{(}\frac{1}{n} \underset{j=1}{\sum}^{n}\Phi_{j}(x_{j})\Bigg{)}. \tag{10}\]  
By substituting (7) into (6), we can obtain the expression of asymptotic performance in (4). * Case 2: The source \(S_{i}\) is connected to relay \(R_{1}\) and \(R_{2}\) simultaneously.