# Performance Analysis of Dual-Hop AF Relaying with Non-Linear/Linear Energy Harvesting
## 6 Results and Discussion  
In this section, the simulation and theoretical results are presented for different system parameters, which provides a comprehensive insight into the analysis of the WP DH AF relaying system. In all figures, the simulation and theoretical curves are denoted by symbols and lines, respectively. The numerical results obtained from simulations and theoretical derivations are in perfect match with each other at high SNR values, which verifies our theoretical analysis. Moreover, throughout the paper, the integrals in (32), and (A21) are numerically calculated using MATLAB and Wolfram Mathematica Computer Software. Furthermore, the numerical results for the derived summations in analytical expressions are obtained for \(\mathrm{Y}=200\), which makes them sufficiently reliable. The BER results are provided with respect to various system parameters. The numerical results for BEP are obtained from (15) and (30) and for throughput by replacing (35) and (55) in (34), for the linear and non-linear EH models, respectively. Unless otherwise stated, we assume \(d_{v}=1\) and \(\eta_{s}=\eta_{r}=0.9\). The path loss coefficient is taken as \(\
u=2.7\)[57]. All channel gains and noise powers are fixed as \(\Omega=1\) and \(N_{0}=1\), respectively. Moreover, R is located at the middle of S and D, so that \(d_{sr}=d_{rd}=d_{sd}/2\). It is assumed that S, R, and D are located co-linearly, and PB moves along a trajectory parallel to S-R-D. Then, \(d_{st}=\sqrt{d_{v}^{2}+d_{h}^{2}}\) and \(d_{rt}=\sqrt{d_{v}^{2}+(d_{sr}-d_{h})^{2}}\). Please note that the above assumptions are for numerical examples; the derived analytical expressions are obtained for the general case, and the nodes can be randomly located. The channel parameters for the links S\(\rightarrow\)PB and R\(\rightarrow\)PB are chosen as \(m_{z}=2\) and \(m_{w}=2\), respectively. For the BEP analysis, we provide the results for 4-QAM modulation, and for rates \(R=1,2,3\) bit/sec/Hz, which are valid for low energy power-constraint nodes. Please note that the approximation in (6) provides a tight upper bound at medium and high SNR values for the following BER results. Figure 3 represents the relative approximation error of the throughput with respect to the parameter Y. Specifically, we define  
\[\varepsilon=\left|\frac{\text{analytical value}-\text{simulation value}}{ \text{simulation value}}\right| \tag{56}\]  
as the amount of relative error. Here the simulation value is obtained by the Monte Carlo method while the analytical value is calculated from (35). As seen from Figure 3, for both \(R=1\) and \(R=3\) bits/sec/Hz, by increasing the value of Y, the error is decreased and tends to zero. Moreover, the results are sufficiently accurate when they are obtained for \(\mathrm{Y}=7\), which provides an error value of \(\varepsilon=0.0002\). However, we guaranteed the results by taking \(\mathrm{Y}=200\). In addition, the BER results are depicted in Figure 4 for different values of Y considering (15). From Figure 4, it is concluded that \(\mathrm{Y}=10\) provides more accurate upper bound values compared to \(\mathrm{Y}=1\). The BER performance of the proposed WP DH AF relaying system with respect to \(P_{T}\) is presented in Figure 5. For \(P_{T}\) values lower than 40 dB, the linear and non-linear models provide the same performance, while for \(P_{T}\) higher than 40 dB, the BER performance for the non-linear EH model converges to the error floor, which verifies that the harvested power is higher than the threshold power. In other words, for higher SNR values, \(P_{s}=P_{r}=P_{th}\). It is seen from Figure 5 that the linear EH model overestimates the system performance compared to the realistic results obtained for the non-linear EH model. Moreover, considering the linear EH model, approximately, 2 dB, and 6 dB SNR losses are obtained for the BER of \(10^{-3}\) by increasing the value of \(d_{sd}\) from 1 to 2 and 3, respectively. Figure 4: BER performance of the WP DH AF relaying system versus \(P_{T}\) dB for \(P_{th}=35\) dB.