# A Novel and Adaptive Angle Diversity-Based Receiver for 6G Underground Mining VLC Systems
## 2 VLC System Model Applied to Underground Mining Environments
### Channel DC Gain
\[\cos(\theta_{iw}^{tilt})=\frac{V_{w-j}\cdot\mathbf{n}_{w}^{tilt}}{\|V_{w-i}\|\|n_{w }^{tilt}\|}, \tag{7}\]  
where \(V_{w-i}\) is the vector from \(w\) to LED \(i\), \(\|V_{w-i}\|=d_{iw}\), \(\|\mathbf{n}_{w}^{tilt}\|=1\), \(\mathbf{n}_{w}^{tilt}\) can be represented in terms of \(\alpha_{w}\) and \(\beta_{w}\), that is \(\mathbf{n}_{w}^{tilt}=[\sin(\beta_{w})\cos(\alpha_{w}),\sin(\beta_{w})\sin( \alpha_{w}),\cos(\beta_{w})]\) and \(V_{w-i}=\left[x_{i}^{T}-x_{w^{\prime}}^{S}y_{i}^{T}-y_{w^{\prime}}^{S},\Delta h _{iw}\right]\). \[\cos(\theta_{wj}^{tilt})=\frac{V_{w-j}\cdot\mathbf{n}_{w}^{tilt}}{\|V_{w-j}\|\|n _{w}^{tilt}\|}, \tag{8}\]  
where \(V_{w-j}\) is the vector from \(w\) to PD \(j\), \(\|V_{w-j}\|=d_{wj}\), where \(d_{wj}\) is the Euclidean distance between reflective element \(w\) and PD \(j\), and \(V_{w-j}=\left[x_{j}^{R}-x_{w}^{S},y_{j}^{S}-y_{w^{\prime}}^{S},-\Delta h_{wj}\right]\), assuming that \(\Delta h_{wj}\) is the difference height between \(w\) and PD \(j\). \[\cos(\theta_{wj}^{tilt})=\frac{V_{j-w}\cdot\mathbf{n}_{w}^{tilt}}{\|V_{j-w}\|\|n _{w}^{tilt}\|}, \tag{9}\]  
where \(V_{j-w}\) is the vector from PD \(j\) to \(w\), \(\|V_{j-w}\|=d_{wj}\), and \(V_{j-w}=\left[x_{w}^{R}-x_{j}^{R},y_{w}^{S}-y_{j}^{R},\Delta h_{wj}\right]\). By including the effect of non-regular walls in the UM-VLC channel model and also consider the shadowing effect, the DC gain of the non-LoS optical wireless channel can be calculated by adding all the components arriving at the PD \(j\) after being reflected in a surface, namely  
\[H_{NLoS,i,j}^{(1)}=\frac{(m+1)A_{p}}{2\pi}\sum_{w=1}^{W}\frac{ \Delta A_{w}\rho_{w}}{d_{iw}^{m+3}d_{wj}^{4}}\Big{\{}\Big{[}x_{w}^{S}-x_{i}^ {T},y_{w}^{S}-y_{i}^{T},-\Delta h_{iw}\Big{]}\cdot[\sin(\beta_{i})\cos(\alpha _{i}),\sin(\beta_{i})\sin(\alpha_{i}),-\cos(\beta_{i})]\Big{\}}^{m}\\ \times\Big{\{}\Big{[}x_{i}^{T}-x_{w^{\prime}}^{S}y_{i}^{T}-y_{w^{ \prime}}^{S}\Delta h_{iw}\Big{]}\cdot[\sin(\beta_{w})\cos(\alpha_{w}),\sin( \beta_{w})\sin(\alpha_{w}),\cos(\beta_{w})]\Big{\}}\\ \times\Big{\{}\Big{[}x_{j}^{R}-x_{w^{\prime}}^{S}y_{j}^{R}-y_{w^{ \prime}}^{S}-\Delta h_{wj}\Big{]}\cdot[\sin(\beta_{j})\cos(\alpha_{j}),\sin( \beta_{j})\sin(\alpha_{j}),\cos(\beta_{j})]\Big{\}}G(\theta_{wj}^{tilt})rect( \frac{\theta_{wj}^{tilt}}{\Theta})P_{iw}P_{wj}, \tag{10}\]  
where \(\Delta A_{w}\) denotes the \(w^{th}\) area of the considered reflective element \(w\), whose reflection coefficient is represented by \(\rho_{w}\), and \(W\) is the total number of reflective elements considered in the scenario. In addition, \(P_{iw}\) and \(P_{wj}\) are the weighted functions that consider the shadowing effect. These functions are obtained through the same statistical process to obtain \(P_{ij}\), and they represent possible blockages in the optical link between LED \(i\) and the reflective element \(w\) and the optical link between \(w\) and PD \(j\), respectively. The last optical component that is included in the UM-VLC channel is the one produced by scattering. This phenomenon is caused by dust particles generated by the work of an underground mine. Therefore, its effect on the UM-VLC channel is direct in terms of magnitude and temporal dispersion.