# Energy-Efficient Power Allocation in Non-Linear Energy Harvesting Multiple Relay Systems
## 2 System Model and Optimization Problem  
### System Model  
As shown in Figure 1, we considered the two-hop multi-relay cooperative forwarding system, including a source node \(S\), a destination node \(D\) and \(N\) relay nodes \(R_{i}\) for \(\mathrm{i}=1,2,\ldots,N\). All the \(R_{i}\) can harvest energy, while \(S\) and \(D\) cannot harvest energy. Assuming that there is no direct link between \(S\) and \(D\), the corresponding channel of each relay is independent of each other, where \(h_{i}\) represents the channel gain from \(S\) to \(R_{i}\) and \(g_{i}\) represents the channel gain from \(R_{i}\) to \(D\). Compared with PS, time switching (TS) cannot obtain a better trade-off between the transmission rate and the RF energy, and it was proven that the PS scheme achieves higher throughput than the TS scheme in [11]. Therefore, we assumed that \(R_{i}\) could only harvest energy from the signal sent by \(S\), and each transmission block period was divided into two time slots in the PS scheme of [13]. In the first time slot, \(R_{i}\) receives the signal sent by \(S\) and divides the received signal power into two parts, according to the ratio of \(1-\rho_{i}:\rho_{i}\). One part enters the information processing module to decode the information, expressed as  
\[y_{R_{i}}^{ID}(k)=\sqrt{1-\rho_{i}}\cdot\left(\sqrt{P_{S}h_{i}}\cdot s(k)+n_{i} ^{a}(k)\right)+n_{i}^{ID}(k) \tag{1}\]  
where \(\mathrm{E}\left[\left|s(k)\right|^{2}\right]=1\); \(P_{S}\) is the transmit power of \(S\), \(\rho_{i}\) is the PS ratio of \(R_{i}\), \(n_{i}^{a}(k)\sim CN\Big{(}0,\sigma_{a_{i}}^{2}\Big{)}\) is the additive white Gaussian noise of the relay receiving antenna and \(n_{i}^{ID}(k)\sim CN\Big{(}0,\sigma_{ID_{i}}^{2}\Big{)}\) is the additive baseband Gaussian noise. Therefore, the SNR of the receiver of \(R_{i}\) can be expressed as \(\gamma_{r}^{i}=\frac{(1-\rho_{i})\cdot P_{S}h_{i}}{(1-\rho_{i})\cdot\sigma_{a _{i}}^{2}+\sigma_{ID_{i}}^{2}}\). The other part enters the EH module to provide energy for the information decoding of this time slot and the information forwarding of the next time slot. The received RF signal at \(R_{i}\) used for energy harvesting is  
\[y_{R_{i}}^{EH}(k)=\sqrt{\rho_{i}}\cdot\left(\sqrt{P_{S}h_{i}}\cdot s(k)+n_{i} ^{a}(k)\right) \tag{2}\]  
Because of \(P_{S}h_{i}\gg\sigma_{a_{i}}^{2}\), the power used by \(R_{i}\) for EH is \(P_{RF}^{i}(\rho_{i})=\rho_{i}P_{S}h_{i}\). In many cases, because of the nonlinearity of the EH model, the problems studied are often too complex to be solved, so the piecewise linear model can not only keep the nonlinearity of the original model to a maximum extent but also solve the problem well [15]. For EH, we used a piecewise linear model, which can be expressed as  
\[ \begin{split} P_{H}^{i}(\rho_{i})&=\left\{\begin{array} []{ll}0,P_{RF}^{i}(\rho_{i})<P_{th}^{1}\\ a_{l}\cdot P_{RF}^{i}(\rho_{i})+b_{l},P_{RF}^{i}(\rho_{i})\in\Big{[}P_{th}^{l },P_{th}^{l+1}\Big{]},l=1,2,\ldots,L-1\\ P_{m},P_{RF}^{i}(\rho_{i})>P_{th}^{l}\end{array}\right.\\ &=a_{l}\cdot\rho_{i}P_{S}h_{i}+b_{l},P_{RF}^{i}(\rho_{i})\in\Big{[}P_{th}^{l },P_{th}^{l+1}\Big{]},l=0,1,2,\ldots,L\end{split}  \tag{3}\]  
where \(P_{RF}^{i}\) is the receiving power of the EH circuit, \(P_{th}=\Big{\{}P_{th}^{l}\Big{|}0\leq l\leq L+1\Big{\}}\) is the dividing point separating \(P_{RF}^{i}\) into \(L+1\) segments, satisfying \(P_{th}^{0}=0\), \(P_{th}^{L+1}=+\infty\), \(a_{l}\) and \(b_{l}\) are the slope and intercept of the linear function in the \(l-th(0\leq l\leq L)\) segment, respectively, satisfying \(a_{0}=b_{0}=a_{L}=0\),\(b_{L}=P_{m}\), and \(P_{m}\) denotes the maximum harvesting power. When it has enough segments, it can be approximated to the nonlinear EH model, but this will also increase its complexity. Therefore, the real measurement data [16] of the  
Figure 1: The system model. energy harvester can be fitted by the piecewise linear function to replace the nonlinear EH model [10]. Figure 2 compares the harvested power in different EH models.