## 3 Performance Analysis of WCwEH
### Exact Analysis  
The communications reliability is represented by the outage probability at D. Therefore, the lower the OP at D, the higher the reliability of the communication. The OP at D is expressed to be  
\[ \begin{split}\mathcal{O}&=\mathbb{P}\{\Lambda< \Lambda_{0}\}\\ &=\mathbb{P}\Big{\{}\frac{gP}{\rho gP+\sigma}<\Lambda_{0}\Big{\}} \\ &=\left\{\begin{array}{cc}\tilde{\mathcal{O}}&,1>\Lambda_{0} \rho\\ 1&,1\leq\Lambda_{0}\rho\end{array}\right.\end{split}  \tag{6}\]where (Since the duration of Phase 2 is \((1-\beta)U\), the channel capacity corresponding to the SINR \(\Lambda\) in Phase 2 is \((1-\beta)\text{log}_{2}(1+\Lambda)\). The outage event happens if this capacity is below \(R_{0}\) or the SINR \(\Lambda\) is lower than \(\Lambda_{0}\)) \(w=gP\), \(\Lambda_{0}=2^{R_{0}/(1-\beta)}-1\), and  
\[ \begin{split}\tilde{\mathcal{O}}&=\mathbb{P}\bigg{\{} w<\frac{\Lambda_{0}\sigma}{1-\Lambda_{0}\rho}\bigg{\}}\\ &=F_{w}\bigg{(}\frac{\Lambda_{0}\sigma}{1-\Lambda_{0}\rho} \bigg{)}.\end{split}  \tag{7}\]  
It is drawn from Equation (6) that since \(\Lambda_{0}=2^{R_{0}/(1-\beta)}-1\), choosing the target transmission rate \(R_{0}\), the time-splitting factor \(\beta\), and the HWi \(\rho\) may induce \(\Lambda_{0}\rho\geq 1\), causing \(\mathcal{O}\) to be \(1\) or leading WCwEH to be in a complete outage. However, this complete outage event can be prevented by selecting properly \(\{R_{0}\), \(\beta\), \(\rho\}\) such that \(\Lambda_{0}\rho<1\). This insight can be drawn from the condition \(\Lambda_{0}\rho<1\) to avoid the complete outage event as follows. It is seen that \(\Lambda_{0}\rho<1\) is equivalent to \(R_{0}<(1-\beta)\text{log}_{2}(1+\rho^{-1})\), which means that for WCwEH to prevent the complete outage, the target transmission rate must be upper-bounded properly. The upper bound on \(R_{0}\) depends on the parameter of energy-harvesting \(\beta\) and the HWi \(\rho\). The higher the \(\beta\) (or \(\rho\)), the lower target transmission rate the WCwEH achieves. The higher \(\beta\) means more time for energy harvesting while less time for signal transmission. Similarly, the higher \(\rho\) means the HWi is more severe. Furthermore, \(\mathcal{O}\) depends on the parameter set (\(R_{0}\), \(\beta\), \(P\), \(M\), \(\iota\), \(\eta\), \(\rho\)), which means that S can achieve the desired performance by properly setting this set. To complete the OP analysis, we must derive the CDF of \(w\), \(F_{w}(z)\), in Equation (7), which is addressed as  
\[F_{w}(z)=\mathbb{P}\{gP\leq z\}. \tag{8}\]  
Invoking \(P\) in Equation (3), the CDF of \(w\) is further represented to be  
\[ \begin{split} F_{w}(z)&=\mathbb{P}\Big{\{}g\leq \frac{z}{GH},H\leq\ R\Big{\}}+\mathbb{P}\bigg{\{}g\leq\frac{z}{J},H>R\Big{\}} \\ &=\int\limits_{0}^{R}F_{g}\Big{(}\frac{z}{Gx}\Big{)}f_{H}(x)dx+ \bar{F}_{H}(R)F_{g}\bigg{(}\frac{z}{J}\bigg{)}.\end{split}  \tag{9}\]  
The integral in Equation (9) is approximated exactly by employing the Gaussian-Chebyshev quadrature in [31] as  
\[F_{w}(z)=\frac{\pi R}{2I}\sum\limits_{i=1}^{I}\sqrt{1-\epsilon_{i}^{2}}f_{H}( \upsilon_{i})F_{g}\bigg{(}\frac{z}{\upsilon_{i}G}\bigg{)}+\bar{F}_{H}(R)F_{g} \bigg{(}\frac{z}{J}\bigg{)}, \tag{10}\]  
where \(\epsilon_{i}=\cos\Big{(}\frac{2i-1}{2I}\pi\Big{)}\), \(\upsilon_{i}=R(\epsilon_{i}+1)/2\), and \(I\) stands for the accuracy-complexity tradeoff of the Gaussian-Chebyshev quadrature. In Section 5, we demonstrate the results with \(I=200\) that guarantees very high preciseness. To complete the derivation of Equation (10), we must derive the PDF and the CCDF of \(H=\sum\limits_{m=1}^{M}g_{ms}\). To this end, it is noted that the MGF of the sum of \(M\) i.i.d random variables is a product of \(M\) individual MGFs. Applying this note, one obtains the MGF of \(H\) to be  
\[ \begin{split}\Psi_{H}(v)&=\prod\limits_{m=1}^{M} \Psi_{g_{ms}}(v)\\ &=\Big{[}\Psi_{g_{p}}(v)\Big{]}^{M}.\end{split}