## 5 Numerical and Simulation Results
### Single SWIPT Relay System  
Table 1 lists the average channel power gains used for the single-relay SWIPT system. In the figures presented for this scenario:  
* The label 'Theory' corresponds to the numerical results derived from (17) using the exact MGF in (14). * 'Theory, App' represents the results based on the approximated ASER expression in (18) and the MGF bound in (15). * 'Theory, Asym' denotes the high-SNR asymptotic expressions obtained from (19) and (21). * 'Theory, AF-App' refers to the approximated AF-based BER results given in (24). Additionally:  
* 'Theory, \(\bar{\gamma}_{\text{sr}}\)', 'Theory, \(\bar{\gamma}_{\text{rd}}\)', and 'Theory, \(\bar{\gamma}_{\text{id}}\)' correspond to the analytical expressions for \(\bar{\gamma}_{\text{sr}}\) and \(\bar{\gamma}_{\text{rd}}\) from (11) and \(\bar{\gamma}_{\text{id}}\) from (13), respectively. The simulation results labeled as 'Simulation' are obtained by directly using the amplification gain \(G\) as defined in (6). Figures 1-5 illustrate the BER performance as a function of the direct link SNR for various system parameters, including the energy conversion efficiency \(\eta\), power-splitting ratio \(\rho\), and channel power gains \([\Omega_{\text{sd}},\Omega_{\text{sr}},\Omega_{\text{rd}}]\). Figure 1: Performance comparison with respect to SNR (\([\Omega_{\text{sd}},\Omega_{\text{sr}},\Omega_{\text{nd}}]=[1,1,1]\), BPSK, \(\eta=1.0\), \(\rho=0.5\)): (**a**) BER vs. SNR (dB) and (**b**) Link SNR (dB) vs. SNR (dB). From Figure 1a, it can be observed that the indirect link exhibits inferior performance compared to the direct link, mainly due to the lack of a dedicated power source at the relay. However, when the direct and indirect links are combined, the SWIPT relay system outperforms the direct link alone across the entire SNR range. Furthermore, the numerical results labeled as 'Theory', 'Theory, App', and 'Theory, AF-App' show a strong agreement with the simulation results for all SNR values. In particular, 'Theory, Asym' accurately reflects the asymptotic BER behavior in the high SNR regime. In Figure 1b, it is shown that the theoretical link SNRs closely match the simulation results, with only minor discrepancies caused by the approximation used in (9), especially in the low SNR region. Additionally, it is confirmed that the SNR of the indirect link, \(\tau_{\rm id}\), is limited by the weaker of the SR and RD links, which is consistent with typical behavior in conventional AF relaying systems. Assuming a path-loss exponent of \(\mu=2\)[18; 23], the average channel power gains can be characterized in terms of the relative distances between nodes. Let \(d_{\text{sd}}\), \(d_{\text{sr}}\), and \(d_{\text{rd}}\) denote the distances of SD, SR, and RD links, respectively. Under this model, the ratio of the average power gains for the RD and SR links is given by \(\Omega_{\text{rd}}/\Omega_{\text{sr}}=d_{\text{sr}}^{2}/d_{\text{rd}}^{2}\). Moreover, when \(d_{\text{sd}}=d_{\text{sr}}\), it directly follows that \(\Omega_{\text{sd}}=\Omega_{\text{sr}}\). As summarized in Table 1, a change in the ratio \(\Omega_{\text{rd}}/\Omega_{\text{sr}}\) from 4 to 1 corresponds to an increase in the RD link distance from one-half to twice the SR link distance. As a result, the modified RD link of (20) becomes increasingly weaker. Following this trend, we aim to investigate how the optimization of the power-splitting factor \(\rho\) in the SWIPT relay system impacts the system BER performance. Figures 2-5 illustrate the performance variations resulting from the optimization of the power-splitting ratio \(\rho\). For the case of 'Theory', the optimal power-splitting (PS) factor was determined by performing an exhaustive search over the interval \([0,1]\), based on the analytical BER expression \(P_{\text{B,id}}^{\text{Asym}}\) given in (19). In contrast, for the case of 'Simulation', we evaluated the numerical BER for all possible PS factors and selected the value that yielded the minimum BER.