# Analysis and Design of Rateless Two-way Relay Networks Based on a Multiply-and-Forward Scheme
## 3 Analysis on Uncoded TWRN
### MF Scheme
For \(\mathbf{y}_{\text{RA}}\) in (32), the received SNR at source node A in the MF scheme is defined as  
\[\begin{split}\gamma_{\text{A}}^{\text{MF}}&=\frac{ \rho_{\text{MF}}^{2}h_{\text{A,R}}^{4}h_{\text{B,R}}^{2}P^{3}}{\Big{(}1+\rho_{ \text{MF}}^{2}h_{\text{A,R}}^{4}P^{2}+\rho_{\text{MF}}^{2}h_{\text{A,R}}^{2} P^{2}+\rho_{\text{MF}}^{2}h_{\text{A,R}}^{2}P\sigma_{\text{n}}^{2} \Big{)}\sigma_{\text{n}}^{2}}\\ &=\frac{h_{\text{A,R}}^{4}h_{\text{B,R}}^{  
**Theorem 2**: _In the high SNR regime, when source nodes transmit uncoded BPSK symbols in a three-stage MF-TWRN over the flat Rayleigh fading channel, the mean SER of MF-TWRN is derived as_  
\[P_{\text{SER,MF}}=\frac{3\big{(}d_{\text{A,R}}^{2\eta}+d_{\text{B,R}}^{2\eta} \big{)}}{8\gamma\sigma_{\text{R}}^{2}} \tag{35}\]  
**Proof:** For the convenience of theoretical analysis, an approximation is performed on (33) to simplify the complicated expression of received SNR in the MF scheme. In the high SNR regime, the quadratic term of power \(P\) is much larger than the linear term so that the received SNR at source node A can be approximated as  
\[\gamma_{\text{A}}^{\text{MF}}\approx\frac{l_{\text{A,R}}^{2}l_{\text{B,R}}^{2 }P}{\big{(}h_{\text{A,R}}^{2}+2l_{\text{B,R}}^{2}\big{)}\sigma_{\text{n}}^{2}} \tag{36}\]  
in which the linear term of \(P\) and noise term are ignored. By means of the harmonic mean, (36) can be reformed as  
\[\gamma_{\text{A}}^{\text{MF}}=\frac{1}{2}\cdot\frac{h_{\text{A,R}}^{2}\gamma \cdot 2h_{\text{B,R}}^{2}\gamma}{h_{\text{A,R}}^{2}\gamma+2l_{\text{B,R}}^{2} \gamma} \tag{37}\]  
As the SER derived in (19) for the AF scheme, the mean SER at source node A in the flat Rayleigh fading channel for the MF scheme is given by  
\[P_{\text{SER,A}}^{\text{MF}}=\frac{1}{\pi}\int_{0}^{\pi/2}M_{\gamma_{\text{A }}^{\text{MF}}}\bigg{(}\frac{1}{2\sin^{2}\theta}\bigg{)}d\theta \tag{38}\]  
Based on (37), the MGF of \(\gamma_{\text{A}}^{\text{MF}}\) in the high SNR regime can be approximately expressed by  
\[M_{\gamma_{\text{A}}^{\text{MF}}}(\tau)\approx\frac{2d_{\text{A,R}}^{2\eta}+d_ {\text{B,R}}^{2\eta}}{2\gamma\sigma_{\text{R}}^{2}\tau} \tag{39}\]  
Substituting (39) into (38), the mean SER at source node A in the MF scheme is derived as  
\[P_{\text{SER,A}}^{\text{MF}}=\frac{1}{\pi}\int_{0}^{\pi/2}\frac{2d_{\text{A, R}}^{2\eta}+d_{\text{B,R}}^{2\eta}}{\gamma\sigma_{\text{R}}^{2}}\sin^{2}\theta d \theta=\frac{2d_{\text{A,R}}^{2\eta}+d_{\text{B,R}}^{2\eta}}{4\gamma\sigma_{ \text{R}}^{2}} \tag{40}\]  
Similarly, the mean SER at source node B in the MF scheme is derived as  
\[P_{\text{SER,B}}^{\text{MF}}=\frac{d_{\text{A,R}}^{2\eta}+2d_{\text{B,R}}^{2 \eta}}{4\gamma\sigma_{\text{R}}^{2}} \tag{41}\]  
Therefore, the mean SER of a BPSK-modulated MF-TWRN is given by  
\[P_{\text{SER,MF}}=\frac{1}{2}\Big{(}P_{\text{SER,A}}^{\text{MF}}+P_{\text{ SER,B}}^{\text{MF}}\Big{)}=\frac{3\big{(}d_{\text{A,R}}^{2\eta}+d_{\text{B,R}}^{2 \eta}\big{)}}{8\gamma\sigma_{\text{R}}^{2}} \tag{42}\]  
A comparison between (22) and (35) indicates  
\[P_{\text{SER,MF}}<P_{\text{SER,AF}} \tag{43}\]  
which shows that, with the same SNR, the performance of MF scheme proposed in this section is better than that of traditional AF scheme in the flat Rayleigh fading channel.