## 4 Numerical Results  
This section analyzes the accuracy of the proposed approximations and discusses the improvements in capacity provided by LIS. In unspecified cases, this study adopts, by default, the Nakagami-\(m\) shape parameters \(m_{SL}=m_{LD}=m=2\). The spread parameters \(\Omega_{SL}=\Omega_{LD}=\Omega\) were chosen to make the variances \(\sigma_{SL}^{2}=\sigma_{LD}^{2}=1\), the Von Mises concentration parameter \(\kappa=2\), \(K=16\) antennas at the source, the size of the \(M\)-QAM constellation is \(M=16\), and the number of iterations is \(10^{6}\) for each Monte Carlo simulation. For each iteration of the Monte Carlo method, we generate the coefficients \(h_{k}\) of (6), the magnitudes are generated using the Nakagami-\(m\) distribution and the phase errors with the Von Mises distribution. Given the coefficients, it is easy to estimate the bit error rate, spectral efficiency, and the SOP in each realization of the random variables and approximating the simulated results by the mean value of these quantities. We compare each of the simulated results in several iterations with the theoretical formulas described in terms of channel parameters. Figure 3 shows the simulated and theoretical BERs considering the Von Mises and uniformly distributed phase errors. The theoretical BER is obtained assuming that the overall fading channel has a Gamma distribution. Note that the larger the number of reflectors, \(N\), the smaller the error probability for any SNR value. When the phase errors are uniformly distributed (\(\kappa=0\)), the error probability is higher than in the Von Mises scenario. This result shows the importance of accurately estimating the phases and channel gains and choosing the optimization method to find the best LIS phase shifts. Uniformly distributed phase noise indicates that the algorithm has equal chances to present significant phase errors (close to \(\pm\pi\)) or small phase errors (close to zero). That implies greater bit error probabilities, which can be compensated only with a large number of antennas at the transmitter or with a large number of reflectors at the LIS. In its turn, Figure 4 confirms that large reflecting surfaces can produce an LoS link between the transmitter and the user even in a far-field Rayleigh fading channel. However,  
Figure 3: Bit error probability varying the SNR. in a near-field scenario, a stronger LoS link (higher Nakagami\(-m\) parameter) implies a lower probability of error. Even in weak LoS scenarios, LIS can decrease the probability of bit error by creating an LoS that is the result of beamforming toward the target user. Figure 3 shows that, for an environment with a fixed value of \(m\), the increase in the number of reflectors (\(N\)), or the improvement of the phase adjustment performed by the LIS (related to the concentration parameter \(\kappa\)) can reduce the bit error rate in an aided LIS system. The upper bound (15) for the error probability proposed by [PERSON] et al. [19] is very close to the bit error rate as shown in Figure 5, even when the fading coefficients are Nakagami-\(m\) distributed. Regarding spectral efficiency, this study considers two scenarios. The first one has uniformly distributed phase noise, and the second one has a Von Mises distributed phase, as shown in Figure 6. Notably, the spectral efficiency increases when the LIS has a more significant number of reflectors, thus indicating a better sharing of the spectrum for the transmission of signals for a multiuser scenario. Moreover, the efficiency is higher for the case in which the phase errors have a Von Mises distribution and lower when the phase errors are uniformly distributed, which means that the phase adjustment of the LIS is a highly relevant factor in improving the spectral efficiency, reinforcing the importance of channel estimation and choice of the phase correction applied to reflectors. When the phase error distribution is more concentrated around zero (higher \(\kappa\) values), then the spectral efficiency is higher.