# Statistical Modeling of Wall Roughness and Its Influence on NLOS VLC Channels in Underground Mining
## 2 UM-VLC System Model
Furthermore, \(\theta_{lq}^{tilt}\) is the angle between the normal of the transmitter (\(l\)) and the ray that travels to the wall (\(q\)), \(\theta_{lq}^{tilt}\) is the angle between the receiver's normal and the ray transmitted from the wall, \(\phi_{qp}^{tilt}\) is the angle between the normal of the wall and the ray of the transmitter, \(\theta_{qp}^{tilt}\) is the angle between the normal of the wall and the ray to the receiver, which depends on their respective horizontal tilt and random azimuth rotation angles, \(g(\theta_{qp}^{tilt})\) is the unit gain of the LED, and \(m\) corresponds to the Lambertian order of transmission, which are related by the following form:  
\[g(\theta_{qp}^{tilt})= \begin{cases}\frac{n^{2}}{\sin^{2}(\Theta)},&0\leq \theta_{qp}^{tilt}\leq\Theta\\ 0,&\theta_{qp}^{tilt}>\Theta.\end{cases}  \tag{2}\]  
In this case, \(n\) is the refractive index of the concentrator. Finally, \(m\) is defined by the following expression:  
\[m\ =\ -\ \frac{\ln(2)}{\ln(\cos(\phi_{1/2}))}. \tag{3}\]  
On the other hand, the angle of incidence and the irradiance are calculated by a normal distribution to add an irregular geometry and to show how this affects system performance. This distribution gives a specific roughness that will be the default configuration. Then, the same scenarios are repeated to these angles to represent the roughness that affects the reflections in terms of geometry. Initially, when a light ray strikes a smooth and regular surface, such as a flat wall, specular reflection occurs. However, when roughness is introduced, due to either the material composition or the geometric features of the wall, the reflection becomes diffuse, resulting in random behavior. Consequently, the trajectory of the ray ceases to be deterministic. Angles are deterministic with a normal vector that divides these. In irregular walls, reflections are diffuse, but the angles calculus is kept. However, the angles exhibit random behavior as a specular case in this context. Then, the angles between two vectors can be defined as follows:  
The angle of incidence of radiation is represented by [5]:  
\[cos(\phi_{qp}^{tilt})\ =\ \frac{V_{q-p}\cdot\mathbf{n_{p}^{tilt}}}{\|V_{q-p}\| \|\mathbf{n_{p}^{tilt}}\|}, \tag{4}\]  
where \(V_{q-p}\) is the vector transmitter wall and \(\mathbf{n_{p}}\) is the normal transmitter vector. The incidence angle is given by [5]:  
\[cos(\theta_{lq}^{tilt})\ =\ \frac{V_{l-q}\cdot\mathbf{n_{q}^{tilt}}}{\|V_{l-q}\| \|\mathbf{n_{q}^{tilt}}\|}, \tag{5}\]  
where \(V_{l-q}\) is the vector wall receiver and \(\mathbf{n_{q}}\) is the normal receiver vector. Equation (1) (NLOS typical expression) represents a regular wall with a unique reflection coefficient. For the case of rough walls, the proposed NLOS equation is as follows:\[H_{NLOS}=\frac{A_{p}(m+1)}{2\pi}\sum_{q=1}^{Q}\frac{\Delta A_{q}\xi_{q}}{2\pi d_{q} ^{2}}cos^{m}(\phi_{lq}^{tilt})cos(\theta_{lq}^{tilt})cos(\zeta_{qp}^{tilt}(y)) cos(\eta_{qp}^{tilt})g(\eta_{qp}^{tilt})rect(\frac{\eta_{qp}^{tilt}}{\Theta}), \tag{6}\]  
where \(\xi_{q}\) corresponds to the component of a randomly calculated coefficients matrix representing a material with specific characteristics. It uses uniform, normal, exponential, and gamma distributions to simulate a varied wall. Therefore, it can apply different configurations that represent the composition of the material in specific situations. The difference from the NLOS equation in the literature is mainly \(\xi_{q}\), \(\zeta_{qp}^{tilt}\), and \(\theta_{qp}^{tilt}\). The association of these three represents a component of the wall matrix with roughness. However, it is crucial to note that the reflection coefficient may not directly reveal details about the material itself, though it can offer an approximation of its nature. Moreover, \(\xi_{q}\) has random values generated in an interval of [0, 1]. It is important to mention that \(\xi_{q}\) is a ratio between the light received and the light reflected as a percentage.