# Analysis and Design of Rateless Two-way Relay Networks Based on a Multiply-and-Forward Scheme
## 4 Fountain-coded MF-TWRN
### Asymptotic Analysis  
According to (32), the MF-TWRN channel from node B to node A is equivalent to a point-to-point fading channel shown in Figure 2. The BPSK-modulated symbol \(\mathbf{x}_{\text{B}}\) is transmitted with unit power and the channel fading coefficient is given by \(\rho_{\text{MF}}h_{\text{A,R}}^{2}h_{\text{B,R}}p^{3/2}\). The equivalent Gaussian noise satisfies  
\[ \begin{split}\mathbf{n}_{\text{eqv}}=\rho_{\text{MF}}h_{\text{A, R}}^{2}P\mathbf{n}_{\text{B}\text{R}}+\rho_{\text{MF}}h_{\text{A,R}}h_{\text{B,R}}P \mathbf{x}_{\text{B}}\circ\mathbf{n}_{\text{AR}}\circ\mathbf{x}_{\text{A}}^{ \prime}\\ +\rho_{\text{MF}}h_{\text{A,R}}\sqrt{\mathbf{p}}\mathbf{n}_{\text{ AR}}\circ\mathbf{n}_{\text{BR}}\circ\mathbf{x}_{\text{A}}^{\prime}+\mathbf{n}_{ \text{RA}}\circ\mathbf{x}_{\text{A}}^{\prime}\end{split}  \tag{44}\]  
with mean zero and variance  
\[\sigma_{\text{n,eqv}}^{2}=\Big{(}1+\rho_{\text{MF}}^{2}h_{\text{A,R}}^{4}P^{2 }+\rho_{\text{MF}}^{2}h_{\text{A,R}}^{2}h_{\text{B,R}}^{2}P^{2}+\rho_{\text{MF }}^{2}h_{\text{A,R}}^{2}P\sigma_{\text{n}}^{2}\Big{)}\sigma_{\text{n}}^{2} \Big{)}\sigma_{\text{n}}^{2} \tag{45}\]  
In the flat Rayleigh fading channel, coefficients \(h_{\text{A,R}}\) and \(h_{\text{B,R}}\) remain invariant during an information exchange between nodes A and B. Therefore, with known \(h_{\text{A,R}}\) and \(h_{\text{B,R}}\), the method we proposed in [24] can be used for the decoding and performance analysis of SLT codes in the channel model shown in Figure 2. With the received signal in(32), when the logarithm domain BP decoding algorithm is utilized for the decoder, the initial log-likelihood ratio (LLR) message is given by  
\[\mathbf{z}=\frac{2\rho_{\text{MF}}h_{\text{A,R}}^{2}h_{\text{B,R}}p^{3/2} \mathbf{y^{\prime}}_{\text{RA}}}{\sigma_{\text{n,eqv}}^{2}} \tag{46}\]  
When all-zero codewords are transmitted, the initial LLR message in (46) follows the symmetric Gaussian distribution \(\mathcal{N}\Big{(}2\gamma_{\text{A}}^{\text{MF}},4\gamma_{\text{A}}^{\text{ MF}}\Big{)}\). Therefore, with known \(h_{\text{A,R}}\) and \(h_{\text{B,R}}\), the lower bound on BER at node A is given by [24]  
\[p_{\text{IB,MF,A}}^{h_{\text{A,R}}h_{\text{B,R}}}=\frac{1}{12}e^{-\beta \varepsilon-\gamma_{\text{A}}^{\text{MF}}/2}\bigg{(}\delta^{\varepsilon \varepsilon-\gamma_{\text{A}}^{\text{MF}}/2}-1\bigg{)} \tag{47}\]  
where \(\beta\) represents the mean degree of degree distribution used in LT codes for source nodes. Employing Rayleigh density in (3), the mean lower bound on BER at node A can be calculated by means of  
\[P_{\text{IB,MF,A}}=\int_{0}^{+\infty}\int_{0}^{+\infty}\frac{1}{12}e^{-\beta \varepsilon-\gamma_{\text{A}}^{\text{MF}}/2}\bigg{(}\delta^{\varepsilon \varepsilon-\gamma_{\text{A}}^{\text{MF}}/2}-1\bigg{)}p(g_{\text{A,R}})p(g_{ \text{B,R}})dg_{\text{A,R}}dg_{\text{B,R}} \tag{48}\]  
Figure 2: The equivalent point-to-point fading channel model from node B to node A.  
while the mean lower bound at node B is given by  
\[P_{\text{lb,MF,B}}=\int_{0}^{+\infty}\int_{0}^{+\infty}\frac{1}{12}e^{-\beta \varepsilon-\gamma_{\text{B}}^{\text{MF}}/2}\bigg{(}e^{\beta\varepsilon e^{- \gamma_{\text{B}}^{\text{MF}}/2}}-1\bigg{)}p(g_{\text{A,R}})p(g_{\text{B,R}}) dg_{\text{A,R}}dg_{\text{B,R}} \tag{49}\]  
Then, the mean lower bound on BER of SLT-coded MF-TWRN in the flat Rayleigh fading is derived as  
\[P_{\text{lb,MF}}=\frac{P_{\text{lb,MF,A}}+P_{\text{lb,MF,B}}}{2} \tag{50}\]  
Specifically, an approximation of the mean lower bound in a three-stage MF-TWRN with respect to high SNRs is proposed in Theorem 3. **Theorem 3**.