## 3 ICI Problem in Underground Mining VLC Environments and ADR-Based Solutions
### ADR with the Hemi-Dodecahedron Structure
It is possible to express \(\beta_{PD_{i}}\) in terms of \(\omega\), taking this case, as the pentagon a polygon of \(N=5\). \[\beta_{PD_{i}}=\omega(i-1)\qquad for\ \ 1\leq i\leq 5. \tag{25}\]  
Thus, as \(PD_{0}\) is located on the upper face of the hemi-dodecahedron, it will not have any rotation, so the azimuth angle \(\beta_{PD_{0}}\), is not considered. Note that \(\alpha_{PD_{i}}\) and \(\beta_{PD_{i}}\) are important parameters in the physical design of the ADR. Therefore, and based on the ADR geometry, expressions that relate \(PD_{0}\)\(PD_{i}\), \(\alpha_{PD_{i}}\) and \(\beta_{PD_{i}}\) angles can be derived. To present these expressions, we will use the Cartesian to spherical coordinate transformation tool, whose derivations and assumptions are presented below. First, we will graphically establish the positions in the space of the PDs, with their respective angular and Cartesian coordinates, as we can see in Figure 6c; thus, the objective is to find an equivalence expressed as \((x_{PD},y_{PD},z_{PD})\equiv(r_{h},\alpha_{PD},\beta_{PD})\). We had previously defined the position of the \(PD_{0}\), so we will now define the position of a \(PD_{i}\) located on any of the ADR side faces, such as \(P_{i}\)=\((x_{PD_{i}},y_{PD_{i}},z_{PD_{i}})\). In addition, we will define the Euclidean distance between \(P_{0}\) and \(P_{i}\) as \(R=(x_{PD_{0}}-x_{PD_{i}},y_{PD_{0}}-y_{PD_{i}},z_{PD_{0}}-z_{PD_{i}})\). According to the relation between Cartesian and spherical coordinates observed in Figure 6c, it is possible to express the coordinates of \(P_{i}\) in terms of alpha and beta, as follows  
\[P_{i}=(x_{PD_{i}},y_{PD_{i}},z_{PD_{i}})=\left(r_{h}cos(\beta_{PD_{i}}),r_{h} sen(\beta_{PD_{i}}),Rcos(\alpha_{PD_{i}})\right). \tag{26}\]  
Based on Figure 6a, a relation between \(r_{h}\) and \(ap\) is obtained. We define \(r_{h}^{\prime}\) as a segment of \(r_{h}\) that is taken from \(P_{i}\) to the downward projection of the junction between the side face where \(P_{i}\) is located and the top face of the ADR. By trigonometric relationship, similarity of triangles and Thales theorem, we can obtain the following relation  
\[r_{h}^{\prime}=apcos(\alpha_{PD_{i}}), \tag{27}\]  
\[r_{h}=r_{h}^{\prime}+ap, \tag{28}\]  
\[r_{h}=ap\left[1+cos(\alpha_{PD_{i}})\right]. \tag{29}\]  
As expression (26) still has the component \(z_{PD_{i}}\) in terms of \(R\), it is necessary to find a relation between \(R\) and \(r_{h}\), which is obtained by  
\[R=\frac{r_{h}}{sen(\alpha_{PD_{i  
\[P_{l}=(x_{PD_{l}},y_{PD_{l}},z_{PD_{l}})= \tag{32}\] \[\left(x_{PD_{0}}+\frac{L}{2 tan(\omega/2)}\left[1+cos(\alpha_{PD_{l}} )\right]cos(\beta_{PD_{l}}),y_{PD_{0}}+\frac{L}{2 tan(\omega/2)}\left[1+cos( \alpha_{PD_{l}})\right]sen(\beta_{PD_{l}}),z_{PD_{0}}\right.\] \[-\left.\frac{L}{2 tan(\omega/2)}\left[1+cos(\alpha_{PD_{l}})\right] \frac{1}{tan(\alpha_{PD_{l}})}\right).\]  
According to this expression, and by finding the Cartesian and angular relationship, it is possible to optimize the size of the proposed ADR based solely on its \(\alpha_{PD_{l}}\) and \(\beta_{PD_{l}}\) angles; thus, trying to make it as compact and small as possible with \(L\), considering that its application and location physics will be in a helmet of mining workers. However, this optimization is not the focus of this research, but it can be developed in future work. A factor to analyze is due to the existence of multiple PDs in the ADR; each of them will receive a light signal of LoS and NLoS paths with specific and different channel gains according to their location. Therefore, in conjunction with the ADRs analyzed in this section, mechanisms are needed to select the best of the received signals or combine them according to the parameters of the communication system. These mechanisms are called signal combining schemes, and we will present in the next section.