# Reconfigurable Intelligent Surface in Wireless-Powered Interference-Limited Communication Networks
## 4 System Performance Analysis
### Outage Probability Analysis
#### 4.3.2 For M-Ary Coherent Modulation  
The average BER for the concerned system under the \(M\)-ary coherent modulation can be defined as [32; 33]:  
\[P_{b} =\varphi\sum\limits_{i=1}^{t}\int\limits_{0}^{\infty}f_{\gamma}( \gamma)Q\Big{(}\sqrt{\Xi_{i}\gamma}\Big{)}d\gamma, \tag{30}\]  
where \(\varphi\), \(t\) and \(\Xi_{i}\) are the parameters that determine the type of \(M\)-ary coherent modulation, and they are defined in [7; 33] for M-PSK and M-QAM.  
Invoking (18) into (30), the ABER under this modulation can be expressed as:  
\[P_{b} =\frac{\Psi^{k_{q}+m_{q}}\xi^{-(k_{q}+m_{q})/2}\bigg{(}\frac{m_{ p}\overline{\gamma}_{I}}{m_{1}\overline{\gamma}_{P}}\bigg{)}^{(k_{q}+m_{q})/2}}{ \Gamma\left(k_{q}\right)\Gamma\left(m_{q}\right)\Gamma\left(m_{I}N_{I}\right) \Gamma\left(m_{P}N_{P}\right)(\overline{\gamma}_{R}/D_{R})^{(k_{q}+m_{q})/2}} \sum\limits_{i=1}^{t}\int\limits_{0}^{\infty}\gamma^{((k_{q}+m_{q})/2-1)}Q \big{(}\sqrt{\xi_{i}\gamma}\big{)} \tag{31}\] \[\times G_{1,3}^{3,1}\Bigg{(}\frac{\Psi^{2}\overline{\gamma}_{I} m_{P}D_{R}\gamma}{m_{1}\overline{\gamma}_{R}\overline{\gamma}_{P}\xi}\Bigg{|} \begin{array}{c}\left(k_{q}+m_{q}\right)/2+m_{I}N_{I}\\ m_{P}N_{P}-\big{(}\left(k_{q}+m_{q}\right)/2\big{)},\big{(}k_{q}+m_{q}\big{)}/2, \big{(}m_{q}-k_{q}\big{)}/2\end{array}\Bigg{)}d\gamma\ \,\]  
\[\text{If }Q\big{(}\sqrt{\Xi_{i}\gamma}\big{)}=\frac{1}{2}erfc\bigg{(}\sqrt{ \frac{\Xi_{i}}{2}\gamma}\bigg{)}\text{ and is further converted to the Meijer-G function using the identity given in ([30], Equation (8.4.14(2))), then (31) can be expressed as:  
Figure 3: Outage probability performance at various values of \(N_{1}\) and \(N_{p}\) at \(N=10\).  
Figure 2: Outage performance of the concerned system for various values of \(N\).  
Figure 4: Impact of \(P_{I}\) on the system outage probability performance at various values of \(m_{I}\) and \(m_{p}\) at \(N=10\).  
Figure 5: Effect of the number of interferers on the average throughput when \(N=10\).  
The impact of number of antennas \(N_{p}\) at the PB on the system throughput is demonstrated in Figure 6. The results illustrate that the system average throughput increases with an increase in \(N_{p}\). Further, it can be observed from the results that the system throughput improves with an increase in energy harvesting time \(\rho\). However, maximum throughput is achieved when \(\rho\) reaches its optimum value. This is due to the fact that more time is allocated to harvesting energy from the PB, which leads to an increase in the system throughput.  
Similarly, the impact of number of reflecting elements in RIS on the system throughput is depicted in Figure 7. The results show that the proposed system with an RIS achieves higher throughput compared to a conventional system without RIS as the \(N\) increases.  
In Figure 8, the impact of the number of reflecting elements in an RIS and the fading parameter on the PB and interferer links on the system ABER of the system under the BPSK binary modulation scheme is presented. It can be seen that the increase in the number of \(N\) significantly improved the ABER, with higher value offering the system better performance. The results also indicate that the increase in the system fading parameters on both the PB and interferer links leads to better system performance. Additionally, it can be seen that the conventional system without RIS offers poor performance when compared with the proposed system with RIS.  
The effects of the number of interferers on the system ABER under the M-QAM coherent modulation system is presented in Figure 9. It can be observed that the system ABER degraded as the number of interferers increased at the destination. However, as the number modulation order \(M\) decreased, the better the system ABER performance.  
Figure 8: Effect of number of reflecting elements in the RIS and fading parameters on the system average BER under binary modulation scheme.  
Figure 7: Impact of number of reflecting elements on the average throughput under different system scenarios.