# Wireless Energy Harvesting Two-Way Relay Networks with Hardware Impairments
## 2 System and Signal Model
### Instantaneous Achievable Sum Rate: DF Relaying  
Referring to [26; 27], we obtain the capacity region of the MA phase based on (5) as  
\[\begin{split}\mathcal{C}_{\text{MA}}(P_{1},P_{2},h_{1},h_{2})\\ =&\big{\{}(R_{1},R_{2})|R_{1}\leq C\left(Y_{1R} \right),R_{2}\leq C\left(Y_{2R}\right),\\ &\quad R_{1}+R_{2}\leq C\left(Y_{\text{MA}}\right)\big{\}},\end{split} \tag{9}\]  
where \(C(x)=\frac{1}{2}\log_{2}(1+x)\), \(Y_{iR},i=\{1,2\}\), are the end-to-end SNDRs from source node \(S_{i}\) to relay \(R\), \(Y_{\text{MA}}\) is the multiple access signal-to-noise-plus-distortion ratio (SNDR), which are presented as follows\[\mathrm{Y}_{1R} = \frac{(1-\rho)\gamma_{1}}{(\kappa_{1}^{2}(1-\rho)+\kappa_{2}^{2}) \gamma_{1}+1} \tag{10a}\] \[\mathrm{Y}_{2R} = \frac{(1-\rho)\gamma_{2}}{(\kappa_{2}^{2}(1-\rho)+\kappa_{2}^{2}) \gamma_{2}+1}\] (10b) \[\mathrm{Y}_{\mathrm{MA}} = \frac{(1-\rho)(\gamma_{1}+\gamma_{2})}{(\kappa_{1}^{2}(1-\rho)+ \kappa_{2}^{2})(\gamma_{1}+\gamma_{2})+1} \tag{10c}\]  
where \(\gamma_{1}=\frac{|h_{1}|^{2}P_{1}}{\sigma_{r}^{2}}\), \(\gamma_{2}=\frac{|h_{2}|^{2}P_{2}}{\sigma_{r}^{2}}\).  
With substituting (4) and (7) into (6), we can derive the capacity region of the BC phase as follows referring to [26; 27].  
\[\mathcal{C}_{\mathrm{BC}}(P_{1},P_{2},h_{1},h_{2}) \tag{11}\] \[= \bigg{\{}(R_{1},R_{2})|R_{1}\leq C\left(\mathrm{Y}_{R2}\right),R_ {2}\leq C\left(\mathrm{Y}_{R1}\right)\bigg{\}}\]  
where \(\mathrm{Y}_{Ri},i=\{1,2\}\), are the end-to-end SNDRs from relay \(R\) to source node \(S_{i}\) written as  
\[\mathrm{Y}_{R2} = \frac{\eta\rho\gamma_{2}(\gamma_{1}+\gamma_{2})}{\kappa^{2}\eta \rho\gamma_{2}(\gamma_{1}+\gamma_{2})+V_{2}} \tag{12a}\] \[\mathrm{Y}_{R1} = \frac{\eta\rho\gamma_{1}(\gamma_{1}+\gamma_{2})}{\kappa^{2}\eta \rho\gamma_{1}(\gamma_{1}+\gamma_{2})+V_{1}} \tag{12b}\]  
and \(V_{1}=\frac{P_{1}}{\sigma_{1}^{2}}\), \(V_{2}=\frac{P_{2}}{\sigma_{r}^{2}}\). For the noise components in the source and relay nodes, it is reasonable to assume \(\sigma_{1}^{2}=\sigma_{2}^{2}=\sigma_{r}^{2}=\sigma^{2}\), which will be adopted in the following derivation.  
The capacity region of a two-way relay network is constrained by both MA and BC transmissions, therefore, the capacity region of this network exploiting the DF protocol can be presented by  
\[\mathcal{R}= \mathcal{C}_{\mathrm{MA}}(P_{1},P_{2},h_{1},h_{2})\cap\mathcal{C }_{\mathrm{BC}}(P_{R},h_{1},h_{2}) \tag{13}\] \[= \big{\{}(R_{1},R_{2})|,R_{1}+R_{2}\leq R_{\mathrm{MA}},\] \[\quad 0\leq R_{1}\leq I_{1},\ 0\leq R_{2}\leq I_{2}\big{\}},\]  
where \(I_{1}=\min\big{(}\mathcal{C}(\mathrm{Y}_{1R})\,,\mathcal{C}(\mathrm{Y}_{R2}) \big{)}\), \(I_{2}=\min\big{(}\mathcal{C}(\mathrm{Y}_{2R})\,,\mathcal{C}(\mathrm{Y}_{R1}) \big{)}\), and \(R_{\mathrm{MA}}=C\left(\mathrm{Y}_{\mathrm{MA}}\right)\).  
The total achievable sum rate of the network with the DF protocol is expressed as  
\[\mathrm{R}_{sum}^{\mathrm{DF}} = R_{1}+R_{2} \tag{14}\] \[= \min\big{(}I_{1}+I_{2},R_{\mathrm{MA}}\big{)}.\]  
From (10) and (12), note that under the impact of hardware impairments, a continuous increase of the transmit power does not lead to a continuous increase of SNDR and achievable rate performance for DF-TWR scheme.