## 3 Best Relay Selection Design and Transmission Rate Analysis
### Transmission Rate Analysis  
In this paper, we consider the bidirectional transmission situation to improve the network SE and time slot utilization, thus the analysis of bidirectional signal transmission is considered. In such case, the total transmission rate in this paper is defined as \(R_{T}=R_{1}+R_{2}\), where \(R_{1}\) and \(R_{2}\) are the transmission rates in two directions [28]. Then, the total transmission rate of the BRT is \(R_{T}^{s}=R_{1}^{s}+R_{2}^{s}\), and the total transmission rate of the BDT is \(R_{T}^{d}=R_{1}^{d}+R_{2}^{d}\). With the \(R_{T}^{s}\) and \(R_{T}^{d}\), we can further define a function of \(f(P_{T})\) with \(f(P_{T})=R_{T}^{s}(P_{T})-R_{T}^{d}(P_{T})\). The \(P_{T}\) is the network total transmit power. Based on the function of \(f(P_{T})\) and also to achieve the adaptive transmission, in this paper, when \(f(P_{T})>0\), it means the total transmission rate of the BRT is higher, then we can let the network use the BRT to transmit the signal; otherwise, we can let the network use the BDT to transmit the signal. With the definition of function \(f(P_{T})\), the specific analysis of transmission rate can be given in the following parts. Figure 2: Time slots model. \(1\). When \(f(P_{T})>0\), from Figures 1 and 2, we can know that in time slot 1, SU\({}_{1}\) and SU\({}_{2}\) respectively transmit signals to the SR\({}_{\text{b}}\). The signal received by the SR\({}_{\text{b}}\) can be expressed as  
\[y_{b}=\sqrt{P_{1}}f_{b}x_{1}+\sqrt{P_{2}}g_{b}x_{2}+\sqrt{P_{p}}l_{r}x_{p}+n_{r} \tag{1}\]  
where \(n_{r}\) is the noise at the SR\({}_{\text{b}}\) with \(n_{r}\sim CN\big{(}0,\sigma_{r}^{2}\big{)}\). With the SWIPT-PS protocol, the power splitting coefficient of the SR\({}_{\text{b}}\) is \(\rho\) with \(0<\rho<1\)[24]. Therefore, according to (1), the harvested energy by the SR\({}_{\text{b}}\) can be expressed as  
\[E_{R}=\frac{1}{2}\rho\eta\Big{(}P_{1}|f_{b}|^{2}+P_{2}|g_{b}|^{2}+P_{P}|l_{r}| ^{2}+\sigma_{r}^{2}\Big{)} \tag{2}\]  
where \(\eta\) is the energy conversion efficiency with \(0<\eta<1\)[24]. As in (2), to simplify the analysis, the linear energy harvesting model as in [23; 24] is considered in this paper. Actually, the non-linear energy harvesting model can reflect the practical non-linear energy harvesting structures more accurately, and the non-linear one can be considered in our further work. According to [29], the whole harvested energy \(E_{R}\) by the SR\({}_{\text{b}}\) can be used for signal transmission in time slot 2. Therefore, the transmit power at the SR\({}_{\text{b}}\) in time slot 2 can be expressed as  
\[P_{R}=\frac{E_{R}}{1/2}=\rho\eta\Big{(}P_{1}|f_{b}|^{2}+P_{2}|g_{b}|^{2}+P_{P} |l_{r}|^{2}+\sigma_{r}^{2}\Big{)} \tag{3}\]  
From (3), we can find that with the SWPIT-PS protocol, the SR\({}_{\text{b}}\) does not increase the network total transmit power. In such a case, without the OPA, \(P_{1}=P_{2}=\frac{1}{2}P_{T}\) can be obtained. Then, with the same network total transmit power, the transmission rate can be increased because more transmit powers can be allocated to SU\({}_{1}\) and SU\({}_{2}\) with (3). In time slot 2, the SR\({}_{\text{b}}\) transmits the amplified signal to the SU\({}_{1}\) and SU\({}_{2}\) with AF protocol. In such case, the signals received by SU\({}_{1}\) and SU\({}_{2}\) can be expressed as  
\[ \begin{split} y_{1}=& f_{b}\widehat{y}_{b}+\sqrt{P_ {p}}l_{1}x_{p}=\beta\sqrt{1-\rho}\sqrt{P_{1}}f_{b}{}^{2}x_{1}+\beta\sqrt{1- \rho}\sqrt{P_{2}}f_{b}g_{b}x_{2}\\ &+\beta\sqrt{1-\rho}\sqrt{P_{p}}f_{b}l_{r}x_{p}+\sqrt{P_{p}}l_{1 }x_{p}+\beta\sqrt{1-\rho}f_{b}n_{r}+\beta f_{b}n_{b}+n_{1}\end{split}  \tag{4}\]  
\[ \begin{split} y_{2}=& g_{b}\widehat{y}_{b}+\sqrt{P_ {p}}h_{2}x_{p}=\beta\sqrt{1-\rho}\sqrt{P_{2}}g_{b}^{2}x_{2}+\beta\sqrt{1-\rho }\sqrt{P_{1}}f_{b}g_{b}x_{1}\\ &+\beta\sqrt{1-\rho}\sqrt{P_{p}}g_{b}l_{r}x_{p}+\sqrt{P_{p}}l_{2 }x_{p}+\beta\sqrt{1-\rho}g_{b}n_{r}+\beta g_{b}n_{b}+n_{2}\end{split}