# Analysis and Design of Rateless Two-way Relay Networks Based on a Multiply-and-Forward Scheme
## 4 Fountain-coded MF-TWRN
### Asymptotic Analysis
: _In the high SNR regime, with SLT coding and BPSK modulation, the mean lower bound on BER of the three-stage MF-TWRN in the flat Rayleigh fading channel is derived as_  
\[P_{\text{lb,MF,HSNR}}=\frac{\Big{(}d_{\text{A,R}}^{2\eta}+d_{\text{B,R}}^{2\eta }\Big{)}\beta\varepsilon e^{-\beta\varepsilon}}{16\gamma\sigma_{\text{R}}^{ 2}} \tag{51}\]  
**Proof:** With given \(h_{\text{A,R}}\) and \(h_{\text{B,R}}\), the lower bound on BER at node A can be approximated by [24]  
\[P_{\text{lb,MF,A,HSNR}}^{h_{\text{A,R},h_{\text{B,R}}}}=\frac{1}{12}\beta \varepsilon e^{-(\beta\varepsilon+\gamma_{\text{A}}^{\text{MF}})} \tag{52}\]  
Then, the MGF described in (39) can be used to derive the mean lower bound in the high SNR regime, i.e.,  
\[ \begin{array}{ll}P_{\text{lb,MF,A,HSNR}}&=\int_{0}^{+\infty}\int_{0}^{+ \infty}\frac{1}{12}\beta\varepsilon e^{-(\beta\varepsilon+\gamma_{\text{A}}^ {\text{MF}})}p(g_{\text{A,R}})p(g_{\text{B,R}})dg_{\text{A,R}}dg_{\text{B,R}} \\ &=\frac{1}{12}\beta\varepsilon e^{-\beta\varepsilon}M_{\gamma_{\text{A}}^{ \text{MF}}}(1)}{24\gamma\sigma_{\text{R}}^{2}}\\ \end{array}  \tag{53}\]  
Similarly, the mean lower bound on BER at node B in the high SNR regime is described as  
\[P_{\text{lb,MF,B,HSNR}}=\frac{\Big{(}d_{\text{A,R}}^{2\eta}+2d_{\text{B,R}}^{2 \eta}\Big{)}\beta\varepsilon e^{-\beta\varepsilon}}{24\gamma\sigma_{\text{R} }^{2}} \tag{54}\]  
Therefore, the mean lower bound on BER of SLT-coded three-stage MF-TWRN is given by  
\[P_{\text{lb,MF,HSNR}}=\frac{P_{\text{lb,MF,A,HSNR}}+P_{\text{lb,MF,B,HSNR}}}{2 }=\frac{\Big{(}d_{\text{A,R}}^{2\eta}+d_{\text{B,R}}^{2\eta}\Big{)}\beta \varepsilon e^{-\beta\varepsilon}}{16\gamma\sigma_{\text{R}}^{2}} \tag{55}\]  
\(\Box\)  
Because the channels between two source nodes and the relay node are asymmetric, nodes A and B may decode the received signals correctly with different overheads. In order to consider the overall efficiency of information exchanges in the TWRN, the corresponding definition of throughput is introduced as the following. **Definition 1**. : _During an information exchange, assume the overheads that nodes A and B need for successful decoding are \(\varepsilon_{A}\) and \(\varepsilon_{B}\), respectively. Then, the throughput of TWRN, i.e., the number of useful information bits contained in a one-bit transmission, can be defined as_  
\[T=1/(1+\varepsilon_{\text{max}}) \tag{56}\]  
_where \(\varepsilon_{\text{max}}=\text{max}(\varepsilon_{A},\varepsilon_{B})\)._  
With given \(h_{\text{A,R}}\) and \(h_{\text{B,R}}\), the equivalent channel capacity between nodes A and B can be derived by substituting the received SNRs \(\gamma_{\text{A}}^{\text{MF}}\) and \(\gamma_{\text{B}}^{\text{MF}}\) into [18]  
\[C=1-\frac{1}{2\sqrt{2\pi\gamma^{\prime}}}\int\limits_{-\infty}^{+\infty}\log_{ 2}(1+e^{-x})\exp\Biggl{(}-\frac{(x-2\gamma^{\prime})^{2}}{8\gamma^{\prime}} \Biggr{)}dx \tag{57}\]  
where \(C\) represents the BIAWGN channel capacity. Then, the capacities of equivalent channel between nodes A and B are shown as  
\[C_{\text{A}}(h_{\text{A,R}},h_{\text{B,R}})=1-\frac{1}{2\sqrt{2\pi\gamma_{ \text{A}}^{\text{MF}}}}\int\limits_{-\infty}^{+\infty}\log_{2}(1+e^{-x})\exp \Biggl{(}-\frac{(\tau-2\gamma_{\text{A}}^{\text{MF}})^{2}}{8\gamma_{\text{A} }^{\text{MF}}}\Biggr{)}dx \tag{58}\]  
and  
\[C_{\text{B}}(h_{\text{A,R}},h_{\text{B,R}})=1-\frac{1}{2\sqrt{2\pi\gamma_{ \text{B}}^{\text{MF}}}}\int\limits_{-\infty}^{+\infty}\log_{2}(1+e^{-x})\exp \Biggl{(}-\frac{(\tau-2\gamma_{\text{B}}^{\text{MF}})^{2}}{8\gamma_{\text{B} }^{\text{MF}}}\Biggr{)}dx \tag{59}\]  
where \(C_{\text{A}}(h_{\text{A,R}},h_{\text{B,R}})\) represents the equivalent channel capacity from node B to A, while \(C_{\text{B}}(h_{\text{A,R}},h_{\text{B,R}})\) represents the reverse one.