## 2 System Model  
As shown in Figure 1, this study considers a base station (BS) equipped with an antenna array of \(K\) antennas transmitting the same signal to a unique single antenna user, the destination. Additionally, a large intelligent surface system with \(N\) reflecting elements aids the system. Both channels BS to LIS and LIS to the user are modeled by the Nakagami-\(m\) distribution. There is a direct link between the user and the BS and between an eavesdropper and the BS whose channel is also Nakagami-\(m\) distributed. The signal that arrives at the destination antenna is given by  
\[r=(\mathbf{h}_{SL}^{H}\mathbf{\Phi}^{H}\mathbf{h}_{LD}^{H}+\mathbf{h}_{SD}^{H} )\boldsymbol{\
abla}+\eta, \tag{1}\]  
where \(\mathbf{h}_{SL}\in\mathbb{C}^{N\times 1}\) is the link between the source and the LIS, \(\mathbf{h}_{LD}\in\mathbb{C}^{K\times N}\) is the link between the LIS and the destination and \(\mathbf{h}_{SD}\in\mathbb{C}^{K\times 1}\) is the direct link between the sourceand the destination. The term \(\mathbf{\Phi}\in\mathbb{C}^{N\times N}\) is a diagonal matrix, whose elements are the phase shifts \(e^{-j\phi_{1}}\ldots e^{-j\psi_{N}}\) applied by the LIS to the incident electromagnetic waves. The LIS's phases, \(\phi_{n}\ \forall n\), are assumed continuous in the interval of 0 to \(2\pi\) radians. The term \(\mathbf{\
abla}=\mathbf{v}s\) represents the precoded signal, where the data symbol is \(s\sim\mathcal{CN}(0,1)\) and the optimal precoding vector is applied by BS, according to the MRT (maximum ratio transmission) criterion, i.e.,  
\[\mathbf{v}=\frac{\mathbf{h}^{H}}{\left\|\mathbf{h}\right\|}. \tag{2}\]  
Finally, the term \(\eta\sim\mathcal{CN}(0,1)\) is additive white Gaussian noise (AWGN) with zero mean and unit variance. Suppose that there is no LoS in the direct link and that it is modeled as a complex normal random variable, with zero mean and variance \(\sigma_{SD}^{2}\). Additionally, the magnitude of the channels \(\mathbf{h}_{i}=|\mathbf{h}_{i}|e^{j\phi_{i}}\) with \(i\in\{SL,LD\}\) are Nakagami-\(m\) distributed with probability density function (PDF) given by  
\[f_{X}(x)=\frac{2m_{i}^{m_{i}}}{\Gamma(m_{i})\Omega_{i}^{m_{i}}}x^{2m_{i}-1}e^ {-\frac{m_{i}}{\Omega_{i}^{m}}x^{2}}. \tag{3}\]  
In this work, the parameters \(m_{i}\) and \(\Omega_{i}\) refer to the shape and spread of the Nakagami-\(m\) PDF, respectively. The distribution of the phases is not specified since, for this model, these phases are not relevant. Then, the overall channel, including the LIS and the antenna array, can be defined as  
\[\mathbf{h}=\mathbf{h}_{SL}^{H}\mathbf{\Phi}^{H}\mathbf{h}_{LD}^{H}+\mathbf{h}_{SD} ^{H}, \tag{4}\]  
whose representation in scalar form is  
\[h_{k}=\sum_{i=1}^{N}|h_{ki}^{LD}|\left|h_{i}^{SL}\right|e^{j(\phi_{i}-\phi_{i} ^{SL}-\phi_{ki}^{LD})}+h_{k}^{SD}.\ k\in\mathbb{N} \tag{5}\]  
Perfect phase cancelling occurs when \(\phi_{i}-\phi_{i}^{SL}-\phi_{ki}^{LD}=0\). Therefore, in this scenario, it follows that \(\phi_{i}=\phi_{i}^{SL}+\phi_{ki}^{LD}\). However, the task of removing the overall channel phase is unfeasible. Some residual phase noise is left behind, in this case, \(\phi_{i}-\phi_{i}^{SL}-\phi_{ki}^{LD}=\theta_{ki}\), where \(\theta_{ki}\) is the phase noise, which, in this work, is modeled as a Von Mises random variable with concentration parameter \(\kappa\). Therefore, the overall channel can be written as  
\[h_{k}=\sum_{i=1}^{N}|h_{ki}^{LD}|\left|h_{i}^{SL}\right|e^{j\theta_{ki}}+h_{k} ^{SD}. \tag{6}\]  
Figure 1: System model with eavesdropper link. It is expected that there is no phase error in the best case analysis, but this situation is entirely unfeasible. However, it is possible to estimate an optimal phase adjustment matrix that provides a performance as good as possible, so it is expected that, on average, the phase errors are zero. The zero mean Von Mises circular distribution can be proper to model the phases of each antenna's fading coefficients [20].