# Reconfigurable Intelligent Surface in Wireless-Powered Interference-Limited Communication Networks
## 2 System and Channel Models  
A wireless powered interference-limited communication network that consists of a power beacon (PB), access point (AP), RIS, a destination (D) and multiple \(N_{I}\) interferer nodes is illustrated in Figure 1. The PB and RIS are equipped with \(N_{P}\) antennas and \(N\) reflecting elements respectively while the AP, destination and interferer nodes are equipped with a single antenna. Owing to blockage and fading issues, it is assumed that there is no direct link between the AP and the destination. In the network, the AP harvests RF energy from the PB using a time-splitting protocol and uses the energy to transmit its information to the destination via the RIS. Based on time-splitting, the overall transmission time \(T\) is divided into two phases. During the first transmission phase, the AP harvests RF energy from the PB with duration \(\rho T\) and the \(\rho(0<\rho<1)\) is the time-splitting ratio. During the second transmission phase with duration \((1-\rho)T\), the AP utilizes the harvested RF energy to transmit its information to the destination. Thus, the signal received at the AP can then be expressed as:  
\[y_{AP}=\sqrt{P_{s}d_{p}^{-\zeta}}h_{p}x_{p}+z_{p}, \tag{1}\]  
where \(P_{s}\) denotes the PB transmit power, \(d_{p}\) is the distance between the PB and AP with \(\zeta\) signifying the path loss-exponent, \(h_{p}\) is the channel gain between the PB and AP, \(x_{p}\) is the unit energy signal and \(z_{p}\) denotes the additive white Gaussian noise (AWGN) with noise variance \(\sigma_{p}^{2}\) and zero mean. Moreover, the amount of energy harvested by the AP at the energy period \(\rho T\) can be thus expressed as:  
\[E_{p}=\frac{\beta\rho TP_{s}}{d_{p}^{\zeta}}, \tag{2}\]  
where \(\beta\) signifies the energy conversion efficiency at \(0<\beta<1\). Therefore, the transmit power at the AP during the information transmission period \((1-\rho)T\) can be expressed as:  
\[P_{t}=\frac{E_{p}}{(1-\rho)T}\triangleq\frac{\beta\rho P_{s}}{(1-\rho)d_{p}^ {\zeta}}, \tag{3}\]  
At the system destination, the received signal can then be given as:  
\[y_{D}=\sqrt{P_{t}D_{R}^{-\zeta}}\left[\sum_{i=1}^{N}\alpha_{i}exp^{i\phi_{i}} \eta_{i}\right]x_{t}+\sum_{j=1}^{N_{I}}\sqrt{P_{t_{j}}d_{t_{j}}^{-\zeta}}h_{t _{j}}x_{t_{j}}, \tag{4}\]where \(x_{t}\) is the transmit information, \(\phi_{i}\) denotes the phase shift produced by \(i\)-th reflecting elements of the RIS, \(\alpha_{i}=\omega_{i}exp^{-\phi_{i}}/\sqrt{L_{AR}}\) and \(\eta_{i}=v_{i}exp^{-\phi_{i}}/\sqrt{L_{RD}}\) are the channel gain of RIS with \(\omega_{i}\) and \(v_{i}\) are the Rayleigh and Rician random variable, respectively. \(L_{AR}=10 log10\left(d_{AR}^{\xi}\right)+\lambda\) and \(L_{RD}=10 log10\left(d_{RD}^{\xi}\right)+\lambda\) are the RIS link path loss where \(d_{AR}\) and \(d_{RD}\) are, respectively, the distances between the AP-to-RIS and RIS-to-D links, and \(\lambda\) denotes the constant that depends on the signal frequency and transmission environment. Moreover, the channel gain and the transmit power of the \(j\)-th interferer are denoted respected as \(h_{I_{j}}\) and \(P_{I_{j}}\). Additionally, \(d_{I_{j}}\) signifies the distance between the destination and the \(j\)-th interferer. The maximum SNR for the system can be obtained by setting the \(\phi_{i}=\psi_{i}+\phi_{i}\), then the (4) can be further expressed as:  
\[y_{D}=\sqrt{P_{t}D_{R}^{-\xi}}\left[\sum\limits_{i=1}^{N}\omega_{i}v_{i} \right]x_{t}+\sum\limits_{j=1}^{N_{t}}\sqrt{P_{t_{j}}d_{I_{j}}^{-\xi}}h_{I_{j }}x_{t_{j}}, \tag{5}\]  
Through (1), (3) and (5), the received signal-to-interference ratio (SIR) at the destination can be expressed as:  
\[\gamma=\ \xi\ \|h_{P}\|^{2}\frac{\|h_{R}\|^{2}}{\|h_{I_{j}}\|^{2}}. \tag{6}\]  
where \(\xi=\frac{\beta\rho P_{s}}{(1-\rho)^{\alpha}d_{P}^{\xi}D_{R}^{\xi}\rho d_{I}^{ -\xi}}\) with \(D_{R}=L_{AR}L_{RD}\) and \(h_{R}=\sum\limits_{i=1}^{N}\omega_{i}v_{i}\).