## 3 ICI Problem in Underground Mining VLC Environments and ADR-Based Solutions
### ADR with the Pyramid Structure  
A pyramidal ADR distribution was chosen based on its simplicity as well as design feasibility, complying with the principle that the FoV of the PDs must point to different directions for a greater area of total coverage at reception. Besides, the overlap between the FoVs of the PDs is minimized and its structure is compact, thus making the separation between PDs small. This distribution is known as pyramidal because it consists of four PDs characterized by having the same FoV [22; 43]. These PDs are located in the center of the triangular side faces of the pyramid, leaving the base free for its placement in the helmet of the miners. Also, the PDs point in the direction of the normal vectors of the pyramid lateral surfaces, as we can observe in Figure 5a. Furthermore, it is also possible to observe in Figure 5b that the PDs are uniformly organized in a circle of radius \(r_{p}\), which is located in the horizontal plane that intersects in the center of the pyramidal distribution. For a better understanding of the orientation and location of the PDs in the ADR, based on normal PD positioning vectors \(\mathbf{n}_{PD}\). We will define a hybrid coordinate system, \((x,y,z,\alpha,\beta)\), where \((x,y,z)\) are the Cartesian coordinates of the position of the normal vector; \(\alpha\) is the angular coordinate of position with respect to the \(z\)-axis, which we will call the elevation angle and will be delimited in \(\alpha\in[0,\pi]\); and \(\beta\) is the angular coordinate of position with respect to the \(x\)-axis, which we will call  
Figure 4: ICI presented in a multi-cell VLC system. the azimuth angle and will be delimited in \(\beta\in[0,2\pi]\)[22; 43]. Therefore, the \(\mathbf{n}_{PD}^{i}\) of a \(PD_{i}\) located at \((x_{PD_{i}},y_{PD_{i}},z_{PD_{i}})\), will be given by [22; 43]  
\[\mathbf{n}_{PD}^{i}=(x_{PD_{i}},y_{PD_{i}},z_{PD_{i}},\alpha_{PD_{i}},\beta_{PD_{i}} )\qquad for\ \ 1\leq i\leq 4. \tag{14}\]  
Details of the coordinate system and its position are depicted in Figure 5c. In order to define the orientation of the PDs in the ADR, following the pyramidal geometry, it is possible to deduce that all the PDs will have the same elevation angle, which is also the same elevation angle of each of the pyramid side faces. As it is a regular polyhedron, we can mention that \(\forall i\in[1,4]\): \(\alpha_{PD_{i}}\)=\(\alpha_{PR}\), where \(\alpha_{PR}\) is the elevation angle of the side faces of the pyramid. As well as for pyramid symmetry, the normal vectors of the pyramid's lateral faces will be equidistant from the center of the radius circumference \(r_{p}\); therefore, \(\beta_{PD_{i}}\) for each \(PD_{i}\) will be evenly distributed, according to the following expression [2; 18],  
\[\beta_{PD_{i}}=\frac{2\pi(i-1)}{4}. \tag{15}\]  
Therefore, the PDs will be specifically oriented in 0, \(\pi/2\), \(\pi\), and \(3\pi/2\). By considering the center of the pyramid distribution as a reference point, it is possible to define the Cartesian coordinates for the \(PD_{i}\), as follows [2; 18]  
\[(x_{PD_{i}},y_{PD_{i}},z_{PD_{i}})=\left(x_{PD}+r_{p}cos(\beta_{PD_{i}}),y_{PD }+r_{p}sen(\beta_{PD_{i}}),z_{PD}\right), \tag{16}\]  
where \((x_{PD},y_{PD})\) are the coordinates in \((x,y)\) measured from the center of the pyramid, and \(z_{PD}\) is the ADR height. Using as reference the hybrid coordinate system illustrated in Figure 5c, it is then possible to obtain the relation between \(r_{p}\) and \(\alpha_{PD_{i}}\), given by \(r_{p}=sin(\alpha_{PD_{i}})\). The relation between \(z_{PD}\) and \(\alpha_{PD_{i}}\) given by \(z_{PD}=cos(\alpha_{PD_{i}})\).