# Energy Harvesting over Rician Fading Channel:
## 2 System Model
### Energy Harvesting and Information Transfer Protocols  
The energy harvesting protocol for this system is illustrated in Figure 2. In this protocol, the transmission is divided into blocks of length \(T\). Each transmission block consists of three time slots. In the first time slot, which has the duration of \(\alpha T\), the relay harvests energy from the nodes \(S_{1}\) and \(S_{2}\). The two remaining time slots are used for information transferring. First, \(S_{1}\) and \(S_{2}\) simultaneously transmit to \(R\) at the second time slot with the transmitted power P. Then, at the third time slot, \(R\) amplifies the signal that it received during the second time slot and broadcasts it to \(S_{1}\) and \(S_{2}\). Both the second and third time slots have the same length of \((1-\alpha)T/2\). Let \(P\) denote the average power transmitted by each node \(S_{1}\) and \(S_{2}\) during the first time slot. After the first time slot, the amount of the harvested energy at node \(R\) can be computed as  
\[E_{r}=\left(P|g_{1}|^{2}+P|g_{2}|^{2}\right)\eta\alpha T, \tag{1}\]  
where \(\eta\) is the energy harvesting efficiency. Let \(s_{1}\) and \(s_{2}\) denote the information-bearing symbols transmitted from \(S_{1}\) and \(S_{2}\), respectively. Again, assume that both \(s_{1}\) and \(s_{2}\) have the same average power \(P\), i.e. \(\mathrm{E}\left[\left|s_{i}\right|^{2}\right]=P\). Thus, the received signal at \(R\) at the second time slot can be written as [28]  
\[y_{r}=\sum_{i=1}^{2}g_{i}(s_{i}+\eta_{i})+n_{r},i=1,2, \tag{2}\]  
where:  
* \(\eta_{i}\) denotes the hardware distortion noise at \(S_{i}\) with zero mean and variance \(\kappa^{2}P\). Here, \(\kappa\) is sufficient to characterize the aggregate level of impairments of the channel [1]. * \(n_{r}\) is the additive white Gaussian noise (AWGN) at \(R\) with zero mean and variance \(\sigma_{r}^{2}\). At the second time slot, \(R\) amplifies the signal \(y_{r}\) with an amplifying coefficient \(\beta\) and then retransmits it to \(S_{1}\) and \(S_{2}\). The received signal \(y_{i}\) at node \(S_{i}\) at the third time slot is  
\[y_{i}=g_{i}(\beta y_{r}+\eta_{r})+n_{i},i=1,2, \tag{3}\]  
where \(\eta_{r}\) denotes the distortion noise with zero mean and variance \(\kappa_{r}^{2}P_{r}\), \(n_{i}\) is the additive white Gaussian noise (AWGN) at \(S_{i}\) with zero mean and variance \(\sigma_{i}^{2}\), for \(i=1,2\). According to the law of energy conservation, the total energy that node \(R\) uses to transmit the amplified signal to \(S_{1}\) and \(S_{2}\) must be equal to the energy that \(R\) received from \(S_{1}\) and \(S_{2}\) at the first time slot, \(E_{r}\). Now, because the transmission duration of the relay is \(\frac{(1-\alpha)T}{2}\), the average power of the signal transmitted by \(R\) at the third time slot can be computed as  
\[P_{r}=\frac{E_{r}}{(1-\alpha)T/2}=\psi\left[P|g_{1}|^{2}+P|g_{2}|^{2}\right], \tag{4}\]  
Figure 2: The energy harvesting and information processing in the system model. where \(\psi=\frac{2\eta\alpha}{1-\alpha}\). In order to ensure that the transmission power at \(R\) is \(P_{r}\), the amplifying coefficient \(\beta\) can be chosen as  
\[\beta=\frac{r}{y_{r}}=\sqrt{\frac{P_{r}}{X_{1}P+X_{1}P\kappa^{2}+X_{2}P+X_{2}P \kappa^{2}+\sigma_{r}^{2}}}, \tag{5}\]  
where \(X_{i}=\left|g_{i}\right|^{2}\) denotes the square of channel gain amplitude between \(S_{i}\) and \(R\). Note that the relay node \(R\) does not need to estimate the hardware noise parameters as well as the individual channel gains \(g_{i}\) because the denominator of Equation (5) is the received symbol power during the 2 nd time slot. Now, because of the similarity between the roles of \(S_{1}\) and \(S_{2}\), we can focus only on node \(S_{1}\).