# Wireless Energy Harvesting Two-Way Relay Networks with Hardware Impairments
## 2 System and Signal Model
### Information Transmission in TWR  
Information transmission in this paper consist of two phase, MA phase and BC phase. Let \(T\) indicate the duration of one round of transmission, from which half of the time, \(T/2\), is used for MA phase, and the remaining half, \(T/2\), is used for BC phase, which is shown in Figure 1b. In the MA phase, source nodes \(S_{1}\) and \(S_{2}\) transmit the information simultaneously. With hardware impairments distortion mingled, the received signal at the relay can be written as  
\[y_{R}=h_{1}\sqrt{P_{1}}(x_{1}+\tau_{tx,1})+h_{2}\sqrt{P_{2}}(x_{2}+\tau_{tx,2} )+\tau_{rx,r}+n_{r} \tag{2}\]  
where \(\tau_{tx,i}\sim\mathcal{CN}(0,\kappa_{1}^{2}),i=1,2\), is the HI caused by the transmitter of \(S_{i}\), \(\tau_{rx,r}\sim\mathcal{CN}(0,\kappa_{2}^{2}(|h_{1}|^{2}P_{1}+|h_{2}|^{2}P_{2} ))\) is the HI caused by the receiver of \(R\), \(h_{i}\) denotes the channel coefficient between \(S_{i}\) and \(R\), \(n_{r}=n_{a,r}+n_{b,r}\), \(n_{r}\sim\mathcal{CN}\left(0,\sigma_{r}^{2}\right)\) is the additive white noise generated at the relay. And \(n_{a,r}\sim\mathcal{CN}\left(0,\sigma_{a}^{2}\right)\) is the noise generated at the receiver antenna, and \(n_{b,r}\sim\mathcal{CN}\left(0,\sigma_{b}^{2}\right)\) is the noise generated in the down-conversion process of the received signal to baseband [11]. Since in practice the power of the noise generated at the antenna (\(\sigma_{a}^{2}\)) is generally much smaller than the noise power introduced by the receiver chain (\(\sigma_{b}^{2}\)), for simplicity, we neglect the noise term \(n_{a,r}\) and adopt \(n_{r}=n_{b,r}\) in the following analysis. Next, the relay then splits the received signal into two portions: \(\sqrt{\rho}y_{R}\) is direct to the energy harvester for EH, \(\sqrt{1-\rho}y_{R}\) is direct to the information processing (IP) circuit for information decoding (ID), if the relay works in DF protocol or information amplify (IA) if the relay works in AF protocol. Note that the distortion noises generated at the transmitter could contribute as a source of energy harvesting, while the distortion noises generated at the receiver and the noise \(n_{b,r}\) does not contribute to energy harvesting, since the power splitting process is done before the down-conversion process (the majority of the HI distortion noises of receiver is assumed to be caused after the down-conversion process). Thus, we can parameterize the acquired energy at the relay node as  
\[E=\eta\rho(P_{1}|h_{1}|^{2}+P_{2}|h_{2}|^{2}+\kappa_{1}^{2}(P_{1}|h_{1}|^{2}+ P_{2}|h_{2}|^{2})+\sigma_{r}^{2})T/2 \tag{3}\]  
where \(0\leq\eta\leq 1\) is the energy-conversion coefficient. The level of hardware impairments in the transmitter and receiver, i.e., \(\kappa_{1}\) and \(\kappa_{2}\), are required within the region \(\kappa_{i},i=\{1,2\}\in[0.08,0.175]\) in 3 GPP LTE [25]. Thus, (3) can be approximated as  
\[E=\eta\rho(P_{1}|h_{1}|^{2}+P_{2}|h_{2}|^{2}+\sigma_{r}^{2})T/2 \tag{4}\]And the signal delivered to the IP circuit can be expressed as  
\[ \begin{split} y_{IP}&=\sqrt{1-\rho}(h_{1}\sqrt{P_{1}}x_{ 1}+h_{2}\sqrt{P_{2}}x_{2})\\ &+\sqrt{1-\rho}(h_{1}\sqrt{P_{1}}\tau_{tx,1}+h_{2}\sqrt{P_{2}}\tau _{tx,2})+\tau_{rx,r}+n_{r}\end{split}  \tag{5}\]  
Assume that perfect channel state information is available for all nodes. With different relaying protocol, the received information \(y_{IP}\) will undergo different process which is described in the following and simply depict in Figure 1a. We describe the different process in the next. Decode-and-Forward ProtocolWhen applying the DF protocol, the relay first decodes \(y_{IP}\) to generate the messages \(x_{1}\) and \(x_{2}\), then applies a network coding function to construct the transmitted signal as \(x_{R}=x_{1}\oplus x_{2}\), and finally broadcasts it to both source nodes with the harvested energy.