# Achievable Rate Maximization for Multi-Relay AF Cooperative SWIPT Systems with a Nonlinear EH Model
## 4 Numerical Results
The reason is that, as analyzed in Section 3, low input SNR makes the nonlinear EH model equivalent to the linear one, which results in the same optimal PS ratio \(\rho_{1}^{*}\) or TS ratio \(\alpha^{*}\), and therefore, brings the same system performance for nonlinear and linear EH models, and that in the region of high input SNR, for the PS scheme, the practical maximum achievable rate of the system is the same and tends to be invariable for both EH models, whereas for the TS scheme, it tends to be zero for the linear EH model and invariable for the nonlinear EH model._  
**Remark 3**. : _Resource allocation mismatch of the linear EH model. The performance of the system with the linear EH model, compared to that with the nonlinear EH model, is slightly changed for the PS scheme, whereas it significantly declined for the TS scheme, implying that the impact of the resource allocation mismatch brought by the use of traditional linear EH model is inconspicuous for the PS scheme, whereas this is much more serious for the TS scheme, especially in higher SNRs. Specifically, it is shown in Figure 3 that in the region of high SNR, the maximum achievable rate for the TS scheme even decreases for the linear EH model while keeps invariant for the nonlinear EH model when the SNR increases. The reason is that, as analyzed in Section 3, for high input SNR, the optimal TS ratio \(y^{*}\) for the nonlinear EH model keeps invariable as \(P_{s}\) changes, whereas the optimal TS ratio \(x^{*}\) for linear EH model gets smaller when the SNR increases (\(x^{*}\to 0\) when \(P_{s}\rightarrow\infty\)), as shown in Table 1. From (31), in the region of high input SNR, the decreasing optimal TS ratio \(x^{*}\) will lead to a more rapid decrease in the received SNR at \(D\) denoted as \(\frac{2x^{*}}{1-x^{*}}\sum_{l=1}^{L}C_{3,l}\) than the increase in the information transmission time ratio \(\frac{1-x^{*}}{2}\), and therefore, leading to the decrease in the practical maximum achievable rate for the system with nonlinear EH circuit._  
Figure 4 shows the impact of variation of Rician factor \(K_{R}\) on the maximum achievable rate for the system with a medium level of SNR (\(P_{s}/N_{0}=20\) dB). It can be observed that for both the TS and PS schemes, the rate increases as \(K_{R}\) increases whether the nonlinear EH model or linear EH model is considered, since a larger Rician factor \(K_{R}\) means a better channel condition. It is shown that compared with the conventional linear EH model, both the TS and PS schemes obtain performance gain despite the variation of Rician factor \(K_{R}\) when the the nonlinear EH model is adopted. Moreover, the performance gain is more larger for the TS scheme than that for the PS scheme when \(K_{R}\) is same. On the other hand, it is shown that the performance gap between the linear and nonlinear EH models does not change much for the TS scheme, while it obviously increases for the PS scheme when \(K_{R}\) increases. Specifically, the performance gap increases 35.47% (from 0.0203 bits/s/Hz to 0.0275 bits/s/Hz) for the PS scheme while it increases 9.17% (from 0.0338 bits/s/Hz to 0.0369 bits/s/Hz) for the TS scheme when \(K_{R}\) increases from 0 dB to 10 dB. \begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline
**Transmit Power (dBm)** & \(P_{s}=30\) & \(P_{s}=35\) & \(P_{s}=40\) & \(P_{s}=45\) & \(P_{s}=50\) & \(P_{s}=55\) & \(P_{s}=60\) \\ \hline Optimal TS ratios \(x^{*}\) for linear EH model & 0.4227 & 0.3390 & 0.2734 & 0.2229 & 0.1867 & 0.1584 & 0.1372 \\ Optimal TS ratios \(y^{*}\) for nonlinear EH model & 0.6054 & 0.6012 & 0.6006 & 0.5994 & 0.6008 & 0.6003 & 0.6003 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Optimal TS ratios for linear and nonlinear EH models. In Figure 5, the maximum achievable rate is plotted against the path loss exponent \(\beta\). Not surprisingly, it is shown that the rate decreases rapidly as \(\beta\) increases for both PS and TS schemes, since a larger path loss exponent means a much more transmission loss.