## 3 Numerical Results
### Bit Error Probability  
One possible way to demonstrate that the LIS can improve the SINR is by showing how the bit error probability varies, increasing the number of antennas at the base station, the number of reflecting elements, and the concentration parameter of the phase error distribution.  
Figure 3 illustrates how the bit error probability decreases when the number of reflectors increases. This scenario has four users, eight transmitting antennas, the Nakagami parameter \(m=2\), and the Von Mises \(\kappa=2\). Notably, although the bit error probability is low, the values do not change significantly from \(N=32\) to \(N=64\) as much as from \(N=16\) to \(N=32\) reflectors. This may be because the channel already has LoS. Additionally, and beyond a certain number of reflectors, the SINR becomes optimal with the assistance of the LIS, which is something this type of analysis can help discover.  
Figure 4 demonstrates that increasing the number of system users raises the bit error probability when keeping the number of LIS reflectors, base station antennas, and the  
Figure 3: Error probability varying with \(N\).  
Figure 2: Probability distribution function via Monte Carlo.  
Von Mises parameter constant. In this scenario, the transmission of 16-QAM signals was considered. One strategy to mitigate the increase of the error probability is to use many reflectors to enhance the LIS's capacity to serve more users with higher SINR and, consequently, reduce the bit error probability.  
For the scenario depicted in Figure 5, in which signals from the 4-QAM constellation are transmitted, the approximation of SINR by the gamma distribution continues to hold, and the bit error probability remains higher when the system accommodates more users.  
When there is no line of sight as in Figure 6, where the scenario with SINR channels following a Rayleigh distribution (Nakagami with \(m=1\)) is analyzed, it is noticeable that the exact bit error probability continues to approximate the simulated bit error rate closely. In this case, a higher SINR was required compared to Figure 7 to reduce the bit error rate promoted by the LIS. In this simulation, the number of reflectors was kept constant. However, it is worth noting that one case deals with the 4-QAM constellation and the other with the 16-QAM constellation.  
Figure 4: Error probability for \(m=2\), 16-QAM.  
Figure 5: Error probability for different \(\kappa\) values.  
Regarding the [PERSON] parameter, it is notable that the bit error probability decreases faster for larger values of \(\kappa\), as observed in Figure 5. The worst-case scenario is when \(\kappa=0\) and the error distribution is uniform. In this case, large and small phase errors have the same probability density and occur when the phase error has been poorly corrected.  
If the phase error is concentrated around the mean (in this study, the mean phase error is zero), the PDF of the phase error approaches a delta function at the origin. In this case, the SINR will be higher because the sum of phasors will be greater if their phase is zero, and the complex exponential vanishes, leaving only the magnitudes.