# Achievable Rate Maximization for Multi-Relay AF Cooperative SWIPT Systems with a Nonlinear EH Model
## 2 System Model
By using the similar way to [11; 31; 35], \(\gamma^{\text{PS}}_{sr,l}\) can be derived as :  
\[y_{S,R_{l}}(n)=\sqrt{P_{s}}h_{S,R_{l}}x_{s}(n)+N_{R_{l},a}(n), \tag{1}\]1where \(\sigma_{R_{l},D}^{2}\) is the variance of the overall noise \(N_{R_{l},D}\) (consisting of the receive antenna noise and the RF to baseband signal conversion noise) over the link (\(R_{l}\to D\)) at \(D\) and \(\gamma_{S,R_{l}}^{\text{PS}}\) and \(\gamma_{R_{l},D}^{\text{PS}}\) denote the instantaneous SNRs for the \(S\to R_{l}\) and \(R_{l}\to D\) links given by, respectively:  
\[y_{S,R_{l}}(n)=\sqrt{P_{s}}h_{S,R_{l}}x_{s}(n)+N_{R_{l},a}(n), \tag{1}\]2  
and:  
\[y_{S,R_{l}}(n)=\sqrt{P_{s}}h_{S,R_{l}}x_{s}(n)+N_{R_{l},a}(n), \tag{1}\]3  
where \(N_{0}\) is the power spectral density of complex additive white Gaussian noise (AWGN) and it is assumed that for all relay nodes, the power of the receive antenna noise equals to the RF to baseband signal conversion noise, and is the half of the power of total noise at the destination node [12; 36], i.e., \(\sigma_{R_{l},D}^{2}=2\sigma_{R_{l},a}^{2}=2\sigma_{R_{l},c}^{2}=N_{0}, \forall l\in\{1,2,\cdots,L\}\). The achievable information rate for the PS scheme can be given as \(R^{\text{PS}}=\frac{1}{2}\text{log}_{2}\bigl{(}1+\gamma^{\text{PS}}\bigr{)}= \frac{1}{2}\text{log}_{2}\Bigl{(}1+\sum_{l=1}^{L}\gamma_{srl}^{\text{PS}} \bigr{)}\)[21; 33]. We then consider the same TS scheme as [20; 37] that is illustrated in Figure 2b, where the transmission time of _phase_-I is divided into two parts. The first part of the transmission time with a duration of \(\alpha T\) is used for EH from the source's RF signal received at \(R_{l}\), while the second part of the transmission time with a duration of \((1-\alpha)T/2\) is used for signal receiving at \(R_{l}\). For the TS scheme, the signals received at \(R_{l}\) in _phase_-I is given as \(y_{S,R_{l}}(n)=\sqrt{P_{s}}h_{S,R_{l}}x_{s}(n)+N_{R_{l}}(n)\), where \(N_{R_{l}}(n)\sim\mathcal{CN}\Bigl{(}0,\sigma_{R_{l}}^{2}\Bigr{)}\) is the overall noise consisting of the receive antenna noise and the RF to baseband signal conversion noise at \(R_{l}\), \(\sigma_{R_{l}}^{2}=N_{0}\). The input RF power and corresponding output DC power of the energy harvester at \(R_{l}\) are given as \(P_{\text{ER}_{l}}^{\text{TS}}=P_{s}\big{|}h_{S,R_{l}}\big{|}^{2}\) and \(P_{\text{ER}_{l}}^{\text{TS}}=\min\Bigl{\{}\eta_{l}P_{\text{ER}_{l}}^{\text{ TS}},M_{l}\Bigr{\}}\), respectively. Therefore, the energy harvested by \(R_{l}\) during _phase_-I is given as \(E_{\text{ER}_{l}}^{\text{TS}}=\alpha T_{\text{ER}_{l}}^{\text{TS}}=\alpha T \min\Bigl{\{}\eta_{l}P_{\text{ER}_{l}}^{\text{TS}},M_{l}\Bigr{\}}\), and the power that can be used for the transmission at \(R_{l}\) is then given by:  
\[y_{S,R_{l}}(n)=\sqrt{P_{s}}h_{S,R_{l}}x_{s}(n)+N_{R_{l},a}(n), \tag{1}\]4  
During _phase_-II, \(R_{l}\) amplifies the received information signal and forwards it to \(D\) using the harvested energy \(P_{R_{l}}^{\text{TS}}\). Similarly to the PS scheme, the equivalent instantaneous SNR of the \(l_{th}\) relay link (\(S\to R_{l}\to D\)) for the TS scheme can be derived as:  
\[y_{S,R_{l}}(n)=\sqrt{P_{s}}h_{S,R_{l}}x_{s}(n)+N_{R_{l},a}(n), \tag{1}\]5  
where \(\gamma_{S,R_{l}}^{\text{TS}}\) and \(\gamma_{R_{l},D}^{\text{TS}}\) represent the instantaneous SNRs of the \(S\to R_{l}\) and \(R_{l}\to D\) links given by, respectively:  
\[y_{S,R_{l}}(n)=\sqrt{P_{s}}h_{S,R_{l}}x_{s}(n)+N_{R_{l},a}(n), \tag{1}\]6  
and:  
\[y_{S,R_{l}}(n)=\sqrt{P_{s}}h_{S,R_{l}}x_{s}(n)+N_{R_{l},a}(n), \tag{1}\]7  
As illustrated in Figure 2b, the time duration for the information transmission from each of the relay nodes to the destination is \((1-\alpha)T/2\). Hence, the achievable information rate for the TS scheme is \(R^{\text{TS}}=\frac{1-\alpha}{2}\log_{2}\Bigl{(}1+\sum_{l=1}^{L}\gamma_{sr_{l} }^{\text{TS}}\Bigr{)}\)[21; 33]. since it is the pointwise minimum of linear functions \(C_{2,l}(1-\rho_{l})\) and \(C_{3,l}\) with respect to \(\rho_{l}\).