# Analysis and Design of Rateless Two-way Relay Networks Based on a Multiply-and-Forward Scheme
## 3 Analysis on Uncoded TWRN
### MF Scheme  
The relay node will amplify the noise in the AF scheme, which may cause severe error expansion problems, though the processing of the AF scheme is relatively simple. Therefore, this section presents a more efficient MF scheme and the performance in the flat Rayleigh fading channel is studied. In the MF scheme, relay node R multiplies signals from different source nodes before broadcasting. Therefore, at the third stage, the signal \(\mathbf{x}_{\text{R}}\) processed by relay node R is shown as  
\[\mathbf{x}_{\text{R}}=\rho_{\text{MF}}(\mathbf{y}_{\text{AR}}\circ\mathbf{y}_ {\text{BR}}) \tag{28}\]  
where \(\circ\) represents the Hadamard product of vectors, which is a product operation performed on corresponding elements in vectors of the same dimensions, and \(\rho_{\text{MF}}\) denotes the power normalization factor, so that  
\[\rho_{\text{MF}}^{2}\text{E}\Big{[}\|\mathbf{y}_{\text{AR}}\circ\mathbf{y}_{ \text{BR}}\|_{2}^{2}\Big{]}=1 \tag{29}\]  
Substituting (6) and (7) into (29), then we have  
\[\rho_{\text{MF}}=\sqrt{\frac{1}{\Big{(}h_{\text{A,R}}^{2}P+\sigma_{\text{n}}^{ 2}\Big{)}\Big{(}h_{\text{B,R}}^{2}P+\sigma_{\text{n}}^{2}\Big{)}}} \tag{30}\]  
Hence, in the MF scheme, the signal received from relay node R at node A can be described as  
\[\mathbf{y}_{\text{RA}}= \mathbf{n}_{\text{RA}}+\rho_{\text{MF}}h_{\text{A,R}}^{2}h_{ \text{B,R}}P^{\frac{3}{2}}\mathbf{x}_{\text{A}}\circ\mathbf{x}_{\text{B}}+ \rho_{\text{MF}}h_{\text{A,R}}^{2}P\mathbf{x}_{\text{A}}\circ\mathbf{n}_{ \text{BR}}\] \[+\rho_{\text{MF}}h_{\text{A,R}}h_{\text{B,R}}P\mathbf{x}_{\text{ B}}\circ\mathbf{n}_{\text{AR}}+\rho_{\text{MF}}h_{\text{A,R}}\sqrt{\rho} \mathbf{n}_{\text{AR}}\circ\mathbf{n}_{\text{BR}}\]  
To eliminate its own source signal \(\mathbf{x}_{\text{A}}\) of node A, simply multiply \(\mathbf{y}_{\text{RA}}\) with the conjugate signal of \(\mathbf{x}_{\text{A}}\) and we have  
\[ \begin{split}\mathbf{y}^{\prime}_{\text{RA}}=\rho_{\text{MF}}h_{ \text{A,R}}^{2}h_{\text{B,R}}P^{\frac{3}{2}}\mathbf{x}_{\text{B}}+\rho_{\text{ MF}}h_{\text{A,R}}^{2}P\mathbf{n}_{\text{BR}}+\rho_{\text{MF}}h_{\text{A,R}}h_{ \text{B,R}}P\mathbf{x}_{\text{B}}\circ\mathbf{n}_{\text{AR}}\circ\mathbf{x}^{ \prime}_{\text{A}}\\ +\rho_{\text{MF}}h_{\text{A,R}}\sqrt{\rho}\mathbf{n}_{\text{AR}} \circ\mathbf{n}_{\text{BR}}\circ\mathbf{x}^{\prime}_{\text{A}}+\mathbf{n}_{ \text{RA}}\circ\mathbf{x}^{\prime}_{\text{A}}\end{split}  \tag{32}\]  
Based on (32), we can observe that all the terms are noise-related except the one containing \(\mathbf{x}_{\text{B}}\). The noise terms are with mean zero and variance \(\Big{(}1+\rho_{\text{MF}}^{2}h_{\text{A,R}}^{4}P^{2}+\rho_{\text{MF}}^{2}h_{ \text{A,R}}^{2}h_{\text{B,R}}^{2}P^{2}+\rho_{\text{MF}}^{2}h_{\text{A,R}}^{2} P\sigma_{\text{n}}^{2}\Big{)}\sigma_{\text{n}}^{2}\). Since the product of two Gaussian noise is trivial when the noise variance \(\sigma_{\text{n}}^{2}\) is much less than the transmit power \(P\), the noise terms can be roughly approximated as Gaussian in the high SNR. BP decoding algorithm is then utilized to recover the original symbol \(\mathbf{s}_{\text{B}}\).