# Performance Analysis of Dual-Hop AF Relaying with Non-Linear/Linear Energy Harvesting
## 3 System Model
#### Linear and Non-Linear EH Models  
In the literature, two EH models are considered: linear and non-linear. In the linear EH model, the transmit power of the power-constrained node is increased by increasing the harvested power at the considered node [2,4]. This causes a misrepresentation of the amount of harvested power [53] since, in practical EH circuits, the amount of the harvested power increases to a threshold level rather than the considered amount of harvested power in the linear EH model. In other words, in practice, due to the non-linear behavior of the diodes in EH circuits, non-linear EH models are more realistic compared to the linear EH model. Additionally, a maximum threshold power is defined for the non-linear EH model, such that, for harvested power greater than this threshold value, the transmit powers of S and R take this fixed threshold value [8, 9, 10, 11, 12, 13]. Specifically, we assume the non-linear EH model given in [8, 11] since the non-linear EH models proposed in [9, 10, 12, 13] are not mathematically tractable. Moreover, the considered model provides a good approximation of practical EH circuits at low and high amounts of harvested power [8, 53] which broadens the insight of EH system design along with the well-studied linear EH model. \begin{table}
\begin{tabular}{c l} \hline \hline
**Notation** & **Description** \\ \hline \(\Gamma(\cdot,\cdot)\) & Upper incomplete Gamma function ([50], 8.350) \\ \(F(\cdot)\) & Cumulative distribution function (CDF) \\ \(f(\cdot)\) & Probability distribution function (PDF) \\ \(\Gamma(\cdot)\) & Gamma function ([50], 8.310). \\ \(\gamma(\cdot,\cdot)\) & Lower incomplete Gamma function ([50], 8.350) \\ \((\cdot)\) & Pochhammer’s symbol ([51], 6.1-22) \\ \({}_{1}F_{1}(\cdot;\cdot)\) & Kummer Confluent Hyper-geometric function ([52], 07.20.07.0001.01) \\ \(G\,{}^{mn}_{pq}\left(\begin{array}{c}a_{p}\\ b_{q}\end{array}\right)\) & Meijer-G function ([50], 9.301) \\ K\({}_{x}(\cdot)\) & Modified Bessel function of the second kind with order \(x\)[50] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: List of notations. \begin{table}
\begin{tabular}{c l} \hline \hline
**Parameters** & **Description** \\ \hline \(P_{T}\) & PB transmit power \\ \(\eta_{s}\) & Energy conversion coefficient \((0<\eta_{s}<1)\) \\ \(\eta_{r}\) & Energy conversion coefficient \((0<\eta_{r}<1)\) \\ \(P_{s}\) & Harvested power at node S \\ \(P_{r}\) & Harvested power at node R \\ \(x\) & Transmitted signal from S with \(\mathrm{E}\{|x|^{2}\}=1\) \\ \(a,b\) & Modulation specific constants [4] \\ \(\Omega_{h_{i}}=\mathrm{E}\{|h_{i}|^{2}\}\) & Channel fading gains, \(i\in\{s,r\}\) \\ \(\Omega_{g_{j}}=\mathrm{E}\{|g_{j}|^{2}\}\) & Channel fading gains, \(j\in\{1,2\}\) \\ \(P_{h_{i}}\) & Threshold (saturation) power at source and relay nodes \(i\in\{s,r\}\) \\ \(\
u\) & Path loss exponent \\ \(L_{i}=1/\sqrt{1+d_{i}^{\
u}}\) & Pathloss \(i\in\{st,rt,sr,rd\}\)[1,4] \\ \(d_{sr}\) & S\(\rightarrow\)R link distance \\ \(d_{rd}\) & R\(\rightarrow\)D link distance \\ \(d_{st}\) & Distances from PB to S \\ \(d_{rt}\) & Distances from PB to R \\ \(d_{q}\) & Vertical distance from PB to node S \\ \(d_{k}\) & Horizontal distance from S to PB \\ \(N_{0}\) & Noise power spectral density \\ \(\vartheta=\log_{2}M\) & Number of bits per symbol \\ \(M\) & Modulation order \\ \hline \hline \end{tabular}
\end{table} 
Table 2: List of parameters. In the first time slot, S and R harvest energy from PB. For the linear EH model, the harvested power for S and R is given as [2]  
\[P_{i}=P_{h_{i}}=\eta_{i}P_{T}L_{tt}|h_{i}|^{2}, \tag{1}\]  
where \(P_{h_{i}}\) is the harvested power at node \(i\) with \(i\in\{s,r\}\). However, assuming the nonlinear EH model, the harvested power at S and R is given as [8,11]  
\[P_{i}= \begin{cases}P_{h_{i}},&P_{h_{i}}\leq P_{ht_{i}}\\ P_{h_{i}},&P_{h_{i}}>P_{ht_{i}},\end{cases}  \tag{2}\]  
where \(i\in\{s,r\}\).