# Achievable Rate Maximization for Multi-Relay AF Cooperative SWIPT Systems with a Nonlinear EH Model
## 4 Numerical Results
Moreover, it is shown that the maximum achievable rate for the PS scheme decreases more rapidly than the TS scheme when \(\beta\) increases, which means that the PS scheme is more susceptible to the variation of path loss exponent. As for the performance degradation due to the resource allocation mismatch brought by using linear EH model instead of the nonlinear one, it is demonstrated that for the same value of \(\beta\), the performance gap between the linear EH and nonlinear EH models for the TS scheme is more larger than that for the PS scheme, and that when \(\beta\) gets larger, the gap becomes smaller until it disappears for both of the TS and PS schemes. The reason is that the larger path loss exponent brings much more transmission loss and means smaller SNR, so that the input RF power dose not saturate the EH receiver. As mentioned before, this leads to the same optimal PS ratio \(\rho_{I}^{*}\) or TS ratio \(\alpha^{*}\), and thus, brings the same system performance for the nonlinear and linear EH models. It can be observed that for smaller \(\beta\), the PS scheme outperforms the TS scheme, whereas for larger \(\beta\), the result is opposite. This is in line with the observation in Figure 3  
Figure 4: Maximum achievable rate versus the Rician factor \(K_{R}\) for SWIPT multi-relay cooperative system with PS and TS schemes when \(P_{s}/N_{0}=20\) dB, \(\beta=2\), \(d_{sc}=2\) m, \(d_{cd}=4\) m.  
Figure 3: Maximum achievable rate versus the transmit power \(P_{s}\) (dBm) for the SWIPT multi-relay cooperative system with PS and TS schemes when \(N_{0}=10\) dBm, \(K_{R}=0\) dB, \(\beta=2\), \(d_{sc}=2\) m, \(d_{cd}=4\) m.  
that for larger SNR, the PS scheme is superior to the TS scheme, whereas for smaller SNR, the TS scheme is superior, since a smaller \(\beta\) means a higher received SNR. Figure 6 demonstrates the impact of the location of the relay nodes denoted by \(d_{sc}\) on the maximum achievable rate. It can be observed that the performance gets worse as \(d_{sc}\) decreases for both the PS and TS schemes. This observation is aligned with that in [31], where single relay with nonlinear EH model is considered for the SWIPT relay systems. It is also shown that the PS scheme is more susceptible to the variation of \(d_{sc}\) than the TS scheme, i.e., the maximum achievable rate for the PS scheme increases more rapidly than the TS scheme when \(d_{sc}\) increases, whereas the performance gap between the linear and nonlinear EH models for the TS scheme is larger than that for the PS scheme, especially for smaller \(d_{sc}\). **Remark 4**. : _Channel susceptibility of the PS and TS schemes. Figures 3-6 show that, for given channel parameters Rician factor \(K_{R}\), path loss exponent \(\beta\), and relays' location denoted by \(d_{sc}\), using a linear EH model in modeling the practical EH circuit brings larger performance degradation for the system with the TS scheme than that with the PS scheme, implying that the TS scheme is more susceptible to the resource allocation mismatch caused by using the linear EH model than the PS scheme. Conversely, for the variation of the relays' location, Rician factor \(K_{R}\), and path loss  
Figure 5: Maximum achievable rate versus the path loss exponent \(\beta\) for SWIPT multi-relay cooperative system with PS and TS schemes when \(P_{s}/N_{0}=20\) dB, \(K_{R}=0\) dB, \(d_{sc}=2\) m, \(d_{cd}=4\) m.  
Figure 6: Maximum achievable rate versus \(d_{sc}\) for SWIPT multi-relay cooperative system with PS and TS schemes when \(P_{s}/N_{0}=20\) dB, \(K_{R}=0\) dB, \(\beta=2\), \(d_{sc}+d_{cd}=6\) m.  
exponent \(\beta\), more rapid performance change can be observed for the PS scheme, implying that the PS scheme is more susceptible to the variation of the channels than the TS scheme._