# Achievable Rate Maximization for Multi-Relay AF Cooperative SWIPT Systems with a Nonlinear EH Model
## 4 Numerical Results  
In this section, numerical results are presented to demonstrate the performance of the SWIPT multi-relay cooperative system with nonlinear EH model for the proposed PS scheme (14) and TS scheme (25). For comparison, the numerical results for the system with traditional linear EH model are also depicted. With the linear EH model, for the PS scheme (14), the energy harvested at the relay nodes is given as (3) rather than (4), while for the TS scheme (25), the power that can be used for the transmission at \(R_{l}\) is given as \(P_{R_{l}}^{\mathrm{TS}}=\frac{2\alpha}{1-\alpha}\eta_{l}P_{\mathrm{ER}}^{ \mathrm{TS}}\) rather than (10). We consider that all nodes are located in a two-dimensional plane, and that the \(L\) relay nodes are randomly but uniformly distributed within a circle (called the relay nodes circle) with a radius of 1 meter. The center of the circle is located on the straight line between \(S\) and \(D\). The distance between \(S\) and the center of the relay nodes circle, and between the center of the relay nodes circle and \(D\), are denoted as \(d_{sc}\) and \(d_{cd}\), respectively. Unless otherwise specified, the simulation parameters for the numerical examples are set as follows. The number of the relay nodes is equal to 5. The transmission power of \(S\) is set to \(P_{s}=30\) dBm. For each relay node \(R_{l},\forall l\in\{1,2,\ldots,L\}\), the harvested saturation power \(M_{l}\) and the energy conversion efficiency factor \(\eta_{l}\) when the EH circuit is not saturated, is assume to be the same and set to \(M_{l}=24\) mW and \(\eta_{l}=0.7\) as in [25; 31], respectively. The results in this section are obtained by averaging over 1000 channel including small-scale Rician fading and large-scale path loss realizations. In Figure 3, the maximum achievable rate is depicted for the SWIPT wireless multi-relay systems with various transmit power \(P_{s}\). It can be observed that when the nonlinear EH model is used for the optimization, for larger input SNR, the PS scheme outperforms the TS scheme, whereas for smaller input SNR, the opposite result can be observed, where the input SNR is denoted as \(P_{s}/N_{0}\). This observation is the same as the results in [21; 37] for the SWIPT relay systems with the linear EH model. It shows that for the nonlinear EH model, both the maximum achievable rates of the system with the TS and PS schemes monotonically increase with the transmit power \(P_{s}\), whereas when the linear EH model is applied to the system, for the PS scheme, the maximum achievable rate is still a monotonically increasing function with respect to \(P_{s}\), while for the TS scheme, it is a concave function of \(P_{s}\). From another perspective, the use of the linear EH model leads to different degrees of performance degradation for the TS and PS schemes. Specifically, the nonlinear EH model is significantly superior to the linear EH model in the system performance for the proposed TS scheme in the region of both medium and high SNRs, whereas it slightly outperforms the linear EH model for the PS scheme only in the range of medium SNR. Compared with the nonlinear EH model, the system performance degradation of the linear EH model is due to the resource allocation mismatch, which is caused by using the linear EH model, while it does not account for the nonlinear characteristic of the EH circuits and the saturation of the input RF power [25]. **Remark 2**. : _Asymptotic Performance of the nonlinear EH model. It can be observed in Figure 3 that in the low input SNR region, the system performance of the nonlinear EH model is same as that of the linear EH model for both the TS and PS schemes, while in the region of high input SNR, the system performance of the nonlinear EH model, compared to that of the linear EH model, is also the same for the PS scheme, whereas it is much different for the TS scheme.