## 5 Numerical and Simulation Results
### Single SWIPT Relay System
Considering the case where \([\Omega_{\text{sd}},\Omega_{\text{sr}},\Omega_{\text{rd}}]=[1,1,4]\), \(\rho=0.1\) (i.e., prior to \(\rho\) optimization), and \(\eta=1.0\), Figure 2b indicates that the RD link acts as the weak channel, while the SR link is dominant, leading to \(\tilde{\gamma}_{\text{id}}<\tilde{\gamma}_{\text{rd}}\). Upon applying the optimized \(\rho\) (i.e., \(\rho_{\text{opt}}<0.6\) in the high SNR region as seen in Figure 2c, the link roles are reversed, and the SNR gap increases, enhancing \(\tilde{\gamma}_{\text{id}}\) and reducing the BER. Specifically, Figure 2a shows approximately 4 dB and 1.6 dB SNR gains at \(\text{BER}=10^{-2}\) and \(\text{BER}=10^{-4}\) for the indirect and combined links, respectively. Similar trends are observed for \([\Omega_{\text{sd}},\Omega_{\text{sr}},\Omega_{\text{rd}}]=[1,1,1]\). As shown in Figure 3, with \(\rho=0.1\), the RD link remains the bottleneck. Optimization leads to \(\rho_{\text{opt}}\in(0.6,0.7)\) at high SNR in Figure 3c, improving \(\tilde{\gamma}_{\text{id}}\) and reducing the BER. Corresponding SNR gains are approximately 7.5 dB and 3.2 dB at near \(\text{BER}=10^{-2}\) and \(10^{-4}\), respectively. For the case of \([\Omega_{\text{sd}},\Omega_{\text{sr}},\Omega_{\text{rd}}]=[1,1,1/2]\), Figure 4 again highlights the RD link as the limiting factor. Optimization with \(\rho_{\text{opt}}\in(0.7,0.8)\) improves the RD link SNR and results in approximately 6 dB and 3 dB gains at near \(\text{BER}=10^{-2}\) and \(10^{-4}\), respectively. When \([\Omega_{\text{sd}},\Omega_{\text{sr}},\Omega_{\text{rd}}]=[1,1,1/4]\), the optimization process leads to a high power-splitting ratio, with \(\rho_{\text{opt}}>0.8\), resulting in a significant improvement in the RD link SNR (i.e., bringing it closer to that of the SR link). Nevertheless, improvements are still achieved, with about 2.5 dB gain in combined link performance at \(\text{BER}=10^{-4}\), as shown in Figure 5a. Across all scenarios, Figures 4a and 5a confirm that the analytical expressions \(P_{\text{B,id}}^{\text{AF-App}}\) and \(P_{\text{B,CD}}^{\text{AF-App}}\) in (24) provide a lower bound on the BER in the low SNR region, and an upper bound that closely approximates the asymptotic BER performance in the high SNR region. It is noteworthy that in the derivation of (19) from (A4), the second term associated with the RD link is obtained by applying an approximation process to the SR link SNR term. Without the optimization for the power-splitting ratio \(\rho\), this approximation introduces inaccuracy in scenarios where the RD link is the dominant weak channel. As a result, discrepancies between the simulation and numerical results for both (19) and (21) are observed in Figures 2a, 3a, 4a and 5a. On the other hand, when \(\rho=\rho_{\text{opt}}\) is employed as shown in Figures 2-5, the SR link becomes the weaker channel. In this case, the asymptotic BER bounds given by (19) and (21) show excellent agreement with the simulation results in the high SNR region. Moreover, although minor mismatches exist, the approximations in (18), obtained via (22), demonstrate good consistency with simulations, affirming the effectiveness of the proposed \(\rho\)-optimization strategy. Consequently, from Figures 2-5, it is confirmed that optimizing \(\rho\) according to (22) effectively enhances the system performance by reinforcing the weaker link, particularly by increasing \(\rho\) as the SNR increases and \(\Omega_{\text{rd}}\) decreases (i.e., as \(d_{\text{rd}}\) increases). Note that in the low SNR region, as shown in Figures 2-5, significant discrepancies can be observed between the theoretical and simulation results. These discrepancies arise primarily because the theoretical predictions are based on asymptotic BER expressions, which are derived under high SNR assumptions and are thus not accurate in the low SNR regime.