## 2 System Model  
The considered system model and the TS architecture are shown in Figure 1. One source wants to transfer its own information to a far-end source. Due to the long-distance information transmission, two sources cannot communication directly. Thus, two sources use the selected TS relay node to transmit information and the system model can be abstracted as Figure 1. The abstracted system model consists of two source nodes \((S_{1},S_{2})\) and a set of \(N\) relay nodes \((R_{i},i=1,2\ldots N)\). The relay nodes are merely powered through wireless energy transfer from two sources and all relay nodes adopt amplify-and-forward mode. There is no direct link between the two source nodes \(S_{1}\) and \(S_{2}\) due to deep fading. Each node is equipped with a single antenna and operated in a half-duplex mode. \(h_{i}\) and \(g_{i}\) are donated as the instantaneous channel gains between \(S_{1}\) to \(R_{i}\) (or \(R_{i}\) to \(S_{1}\)) and \(S_{2}\) to \(R_{i}\) (or \(R_{i}\) to \(S_{2}\)), respectively. \(h_{i}\) and \(g_{i}\) are assumed to be independent and distributed complex Gaussian random variables (CGRVs) with zero-mean and variances \(\sigma_{hi}^{2}\) and \(\sigma_{gi}^{2}\). According to TSR protocol, the relay node uses a switch-like structure to acquire energy or signals in two time phases. Thus, the whole transmission block \(T\) is divided into \(\alpha T\), \((1-\alpha)T/2\) and \((1-\alpha)T/2\), where \(\alpha\in[0,1]\) is donated as TS factor. \(\alpha T\) part is called energy harvesting phase, \((1-\alpha)T/2\) part is called multiple-access phase, and the remainder \((1-\alpha)T/2\) part is called broadcast phase, as shown in Figure 2.  
In the energy harvesting phase, the relay nodes receive the energy signal from two source nodes \(S_{1}\) and \(S_{2}\) through RF transmission. The signal \(y_{R_{i},E}\) received by the relay nodes can be written as  
\[y_{R_{i},E}=\sqrt{P_{S1}}h_{i}s_{1}+\sqrt{P_{S2}}g_{i}s_{2}+n_{R_{i}} \tag{1}\]  
where the parameters \(P_{S1}\) and \(P_{S2}\) are the transmit power from two sources, and the parameter \(n_{R_{i}}\sim CN\big{(}0,\sigma^{2}\big{)}\) is the additive noise. Since the energy of the noise signal \(n_{R_{i}}\) is negligible as compared to that harvested energy from the RF signal [16], the harvested energy \(E_{R_{i}}\) at relay node \(R_{i}\) can be written as  
\[E_{R_{i}}=\eta\alpha T\Big{(}P_{S1}\big{|}h_{i}\big{|}^{2}+P_{S2}\big{|}g_{i} \big{|}^{2}\Big{)} \tag{2}\]  
Figure 1: Illustration of a two-way AF relay system and the TS relay architecture. Figure 2: TSR protocol at the energy harvesting relay. where the parameter \(\eta(0<\eta<1)\) is the conversion efficiency of the energy receiver and \(\alpha\) is the TS factor. Then, the received energy is used for the broadcast phase, and we can describe the relay transmit power \(P_{R_{i}}\) as  
\[P_{R_{i}}=\frac{E_{R_{i}}}{(1-\alpha)/2}=\frac{2\eta(P_{S1}|h_{i}|^{2}+P_{S2}|g_ {i}|^{2})\alpha}{1-\alpha} \tag{3}\]  
In multiple-access phase, \(S_{1}\) and \(S_{2}\) transmit the information signal \(s_{1}\) and \(s_{2}\) simultaneously to the selected \(i\)th relay (\(R_{i}\)), so the received signal at the relay node \(R_{i}\) can be denoted as  
\[y_{R_{i}}=\sqrt{P_{S1}}h_{i}s_{1}+\sqrt{P_{S2}}g_{i}s_{2}+n_{R_{i}} \tag{4}\]  
In the broadcast phase, the \(i\)th relay node \(R_{i}\) amplifies the received signal then broadcasts it to two source nodes \(S_{1}\) and \(S_{2}\). The received signal at \(S_{1}\) and \(S_{2}\) can be presented as  
\[y_{S1}=h_{i}\big{(}\beta y_{R_{i}}\big{)}+n_{1} \tag{5}\]  
Figure 1.0  
where the parameters \(n_{1}\) and \(n_{2}\) follow \(CN\big{(}0,\sigma_{1}^{2}\big{)}\) and \(CN\big{(}0,\sigma_{2}^{2}\big{)}\), respectively.