# Achievable Rate Maximization for Multi-Relay AF Cooperative SWIPT Systems with a Nonlinear EH Model
## 2 System Model
### Asymptotic Analysis for the Maximum Achievable End-to-End Rate
#### 3.3.1 Asymptotic Analysis for the PS Scheme  
By substituting (17)-(19) into (7), \(\gamma^{\text{PS}}_{srl}\) can be rewritten as:  
\[\gamma^{\text{PS}}_{srl}=\frac{C_{1,l}\frac{p_{l}}{l+p_{l}}\min\{C_{2,l}(1-p_{ l}),C_{3,l}\}}{C_{1,l}\frac{p_{l}}{l+p_{l}}+\min\{C_{2,l}(1-p_{l}),C_{3,l}\}+1}. \tag{26}\]  
From (14) and (26), the practical maximum achievable information rate is given by:  
\[R_{\max\text{ -PS}}=\frac{1}{2}\text{log}_{2}\left(1+\sum_{l=1}^{L}\frac{C_{1,l} \frac{\rho_{l}^{\ast}}{1+\rho_{l}^{\ast}}\min\{C_{2,l}\left(1-\rho_{l}^{\ast} \right),C_{3,l}\}}{C_{1,l}\frac{\rho_{l}^{\ast}}{1+\rho_{l}^{\ast}}+\min\{C_{2, l}\left(1-\rho_{l}^{\ast}\right),C_{3,l}\}+1}\right). \tag{27}\]  
Let \(x_{l}^{*}\) and \(y_{l}^{*}\) be the optimal PS ratio \(\rho_{l}^{\ast}\) obtained for relay \(R_{l}\) with the linear and nonlinear EH models, respectively. Since the practical EH circuits at the relay nodes are nonlinear, the practical maximum achievable information rate is given as (27) whether the linear and nonlinear EH models are used. Then, from (27), by substituting \(\rho_{l}^{\ast}\) with \(x_{l}^{*}\) and \(y_{l}^{*}\), the practical maximum achievable information rate, when the linear and nonlinear EH models are used, can be, respectively, expressed as:  
\[R_{\max-\text{PS}}^{linear}=\frac{1}{2}\text{log}_{2}\left(1+\sum_{l=1}^{L} \frac{C_{1,l}\frac{y_{l}^{*}}{1+y_{l}^{*}}\min\{C_{2j}\left(1-x_{l}^{*}\right)C _{3j}\}}{C_{1,l}\frac{y_{l}^{*}}{1+y_{l}^{*}}+\min\{C_{2j}\left(1-x_{l}^{*} \right)C_{3j}\}+1}\right) \tag{28}\]  
and:  
\[R_{\max-\text{PS}}^{nonlinear}=\frac{1}{2}\text{log}_{2}\left(1+\sum_{l=1}^{L} \frac{C_{1,l}\frac{y_{l}^{*}}{1+y_{l}^{*}}\min\{C_{2j}\left(1-y_{l}^{*}\right) C_{3j}\}}{C_{1,l}\frac{y_{l}^{*}}{1+y_{l}^{*}}+\min\{C_{2j}\left(1-y_{l}^{*} \right)C_{3j}\}+1}\right). \tag{29}\]  
Case 1 (Low Input SNR) with \(P_{s}\to 0\): When \(P_{s}\to 0\), it can be obtained from (17) that \(C_{1,l}\to 0\). Then, from (28) and (29), the practical maximum achievable information rate is equal to zero whether the nonlinear or linear EH model is used for the EH receivers at the relay nodes, which means that the maximum achievable rate is the same for both EH models when \(P_{s}\to 0\). Actually, when \(P_{s}\) is so small that \(C_{2,l}\ll C_{3,l}\) and \(\min\{C_{2,l}(1-\rho_{l}),C_{3,l}\}=C_{2,l}(1-\rho_{l})\), i.e., the EH receivers have small input power and are not saturated, the nonlinear EH model has the same effect on system performance as the linear EH model. In this case, the nonlinear EH model is essentially equivalent to traditional linear one, which results in the same optimal PS ratios \(x_{l}^{*}\) and \(y_{l}^{*}\) and, therefore, brings the same system performance for both EH models.  
Case 2 (High Input SNR) with \(P_{s}\to\infty\): when \(P_{s}\to\infty\), it can be obtained from (17) and (18) that \(C_{1,l}\to\infty\) and \(C_{2,l}\to\infty\) such that \(C_{2,l}\gg C_{3,l}\) and \(\min\{C_{2,l}(1-\rho_{l}),C_{3,l}\}=C_{3,l}\). Then, from (28) and (29), the practical maximum achievable information rate, when the linear or nonlinear EH models are used, can be derived as \(R_{\max-\text{PS}}^{linear}\approx\frac{1}{2}\text{log}_{2}\left(1+\sum_{l=1} ^{L}C_{3,l}\right)\) and \(R_{\max-\text{PS}}^{nonlinear}\approx\frac{1}{2}\text{log}_{2}\left(1+\sum_{l=1} ^{L}C_{3,l}\right)\), which means that for high SNR, the maximum achievable rate is the same for both EH models.