# Achievable Rate Maximization for Multi-Relay AF Cooperative SWIPT Systems with a Nonlinear EH Model
## 2 System Model
The received signal \(y_{S,R_{l}}(n)\) at \(R_{l}\) is split into two signal streams during _phase_-I. One signal stream is fed into the EH receiver with a power ratio of \(1-\rho_{l}\), the other is used for the information decoding with a power ratio of \(\rho_{l}\). The signal received at the information receiver (IR) at \(R_{l}\) can be expressed as [11; 12]:  
\[y_{\text{IR}_{l}}(n) =\sqrt{\rho_{l}}y_{S,R_{l}}(n)+N_{R_{l},c}(n)\] \[=\sqrt{\rho_{l}P_{s}}h_{S,R_{l}}x_{s}(n)+N_{S,R_{l}}(n), \tag{2}\]  
where \(N_{S,R_{l}}(n)=\sqrt{\rho_{l}}N_{R_{l},a}(n)+N_{R_{l},c}(n)\) is the overall noise at \(R_{l}\) and \(N_{R_{l},c}(n)\sim\mathcal{CN}\left(0,\sigma_{R_{l},c}^{2}\right)\) is the RF to baseband signal conversion noise of \(R_{l}\). For the EH receiver of \(R_{l}\), the received signal is given by \(y_{\text{ER}_{l}}^{\text{PS}}(n)=\sqrt{1-\rho_{l}}y_{S,R_{l}}(n)\). Therefore, the input RF power at the EH receiver of \(R_{l}\) can be derived as \(y_{\text{ER}_{l}}^{\text{PS}}=\left(1-\rho_{l}\right)P_{s}\big{|}h_{S,R_{l}} \big{|}^{2}\). When the linear EH model is considered, the energy conversion efficiency of the EH receiver at \(R_{l}\) is assumed to be a constant value \(\eta_{l}(0<\eta_{l}<1)\) and the energy harvested at \(R_{l}\) is therefore given as [11; 25]:  
\[P_{\text{EH}_{l}}^{\text{PS}}=\eta_{l}P_{\text{ER}_{l}}^{\text{PS}}=\eta_{l}(1 -\rho_{l})P_{s}\big{|}h_{S,R_{l}}\big{|}^{2}. \tag{3}\]  
However, as pointed out before, the linear EH model is inaccurate in modeling realistic nonlinear EH circuit. In fact, it is shown that the output power of the EH receiver first linearly increases then keeps invariant (reaches to saturation) when the input power increases [25]. Hence, here we consider the nonlinear EH model and use the piece-wise linear model that captures the nonlinear saturation characteristic for the EH receiver. The output direct-current (DC) power of the EH receiver at \(R_{l}\) is then given by [3; 29; 31]:  
\[P_{\text{EH}_{l}}^{\text{PS}}=\min\Big{\{}\eta_{l}P_{\text{ER}_{l}}^{\text{PS }},M_{l}\Big{\}}, \tag{4}\]  
where \(M_{l}\) is the maximum (saturation) power that can be harvested at the EH receiver of \(R_{l}\) and \(\eta_{l}\) is the energy conversion efficiency factor when the EH receiver of \(R_{l}\) is not saturated. The power that can be used for the transmission at \(R_{l}\) is then given as \(P_{R_{l}}^{\text{PS}}=P_{\text{EH}_{l}}^{\text{PS}}\). For _phase_-II transmission, \(R_{l}\) first amplifies \(y_{\text{IR}_{l}}(n)\), then forwards it to \(D\), where the amplification factor \(A_{l}\) satisfies the average power constraint and can be denoted as [11; 31]:  
\[A_{l}=\frac{P_{R_{l}}^{\text{PS}}}{E\Big{[}|y_{\text{IR}}(n)|^{2}\Big{]}}= \frac{P_{R_{l}}^{\text{PS}}}{\rho_{l}P_{s}\big{|}h_{S,R_{l}}\big{|}^{2}+\sigma _{S,R_{l}}^{2}}, \tag{5}\]  
where \(\sigma_{S,R_{l}}^{2}\) is the variance of \(N_{S,R_{l}}(n)\) and \(E[\cdot]\) means expectation operation. The signal transmitted from \(R_{l}\) is then given as:  
\[x_{R_{l}}(n) =\sqrt{A_{l}}y_{\text{IR}_{l}}(n)\] \[y_{S,R_{l}}(n)=\sqrt{P_{s}}h_{S,R_{l}}x_{s}(n)+N_{R_{l},a}(n), \tag{1}\]0  
Since the OSTBC is adopted so that the transmission are mutually orthogonal in time domain for the relays in _phase_-II, the end-to-end SNR at the receiver of the destination \(D\), \(\gamma^{\text{PS}}\), is equal to the summation of the SNR of all the relay links, i.e., \(\gamma^{\text{PS}}=\sum_{l=1}^{L}\gamma^{\text{PS}}_{sr,l}\)[34], where \(\gamma^{\text{PS}}_{sr,l}\) is the equivalent instantaneous SNR of the relay link (\(S\to R_{l}\to D\)).