# A Novel and Adaptive Angle Diversity-Based Receiver for 6G Underground Mining VLC Systems
## 2 VLC System Model Applied to Underground Mining Environments
### Optical Transmitters and Receivers  
As is widely known, in VLC systems, we assume that LEDs are a point source of light that follows a Lambertian pattern of radiation. In terms of position and orientation, LEDs in underground mines do not mount directly to the ceiling or point vertically downward. Therefore, in our work, we consider a set of \(I\) LEDs as optical transmitters. Furthermore, the physical position and the normal vector of an LED \(i\), where \(i=1,2,...,I\), are denoted by \((x_{i}^{T},y_{i}^{T},z_{i}^{T})\) and \(\mathbf{n}_{i}^{tilt}\), respectively, as we can see in Figure 1.  
On the other hand, we assume PDs as optical receptors, which will be located in an ADR that we will define in Section 3. The PDs are composed of a non-imaging concentrator (lens) and a physical active area \(A_{p}\). In terms of position and orientation, when we install a PD in a mine worker's helmet, it does not always point vertically upwards due to the very nature of the worker's movement. Furthermore, the physical position and the normal vector of an PD \(j\), where \(j=1,2,...,J\), are denoted by \((x_{j}^{R},y_{j}^{R},z_{j}^{R})\) and \(\mathbf{n}_{j}^{tilt}\), respectively, as we can see in Figure 1.  
We assume that all LEDs have the same generalized Lambertian radiation pattern; therefore, the radiation intensity pattern \(Ri(\phi_{ij}^{tilt})\) and the term \(\cos(\phi_{ij}^{tilt})\) can be modeled as follows [21]:  
\[Ri(\phi_{ij}^{tilt})= \begin{cases}\frac{m+1}{2\pi}\cos^{m}(\phi_{ij}^{tilt})& \text{if }-\pi/2\leq\phi_{ij}^{tilt}\leq\pi/2\\ 0&\text{otherwise}\end{cases} , \tag{1}\]  
\[\cos(\phi_{ij}^{tilt})=\frac{V_{i-j}\cdot\mathbf{n}_{i}^{tilt}}{\|V_{i-j}\|\|\bm {n}_{i}^{tilt}\|}, \tag{2}\]  
where \(\phi_{ij}^{tilt}\) is the radiance angle with respect to the normal vector to the LED \(i\), the vector from LED \(i\) to PD \(j\) is denoted by \(V_{i-j}\), the notation \(\|\cdot\|\) denotes the two-norm, and \(\cdot\) represents the product dot operation. In addition, for vector concepts, \(\|V_{i-j}\|=d_{ij}\), where \(d_{ij}\) is the Euclidean distance between the LED \(i\) and PD \(j\), \(\|\mathbf{n}_{i}^{tilt}\|=1\), and \(\mathbf{n}_{i}^{tilt}\) can be represented in terms of \(\alpha_{i}\) and \(\beta_{i}\) as \(\mathbf{n}_{i}^{tilt}=[\sin(\beta_{i})\cos(\alpha_{i}),\sin(\beta_{i})\sin(\alpha _{i}),-\cos(\beta_{i})]\), where \(\beta_{i}\) is the tilt angle with respect to the z-axis, which takes values in range of \([90^{\circ},180^{\circ})\) and \(\alpha_{i}\) is the rotation angle with respect to the x-axis, which is defined in the interval \([0^{\circ},360^{\circ})\). Furthermore, \(V_{i-j}=\left[x_{j}^{R}-x_{i}^{T},y_{j}^{R}-y_{i}^{T},-\Delta h_{ij}\right]\), assuming that \(\Delta h_{ij}\) is the difference height between LED \(i\) and PD \(j\), that is \(z_{i}^{T}-z_{i}^{R}=\Delta h_{ij}\). In the receptor side, the effective collection area of the PD \(j\) and the term \(\cos(\theta_{ij}^{tilt})\) acquire the form of [21]  
\[A_{eff}(\theta_{ij})= \begin{cases}A_{p}cos(\theta_{ij}^{tilt})&\text{if }-\Theta/2\leq\theta_{ij}^{tilt}\leq\Theta/2\\ 0&\text{otherwise}\end{cases} , \tag{3}\]  
\[\cos(\theta_{ij}^{tilt})=\frac{V_{j-i}\cdot\mathbf{n}_{j}^{tilt}}{\|V_{j-i}\|\| \mathbf{n}_{j}^{tilt}\|}, \tag{4}\]  
Figure 1: UM-VLC propagation scenario with the geometry of optical channel components. where \(\theta_{ij}^{tilt}\) is the incidence angle with respect to the normal vector to the PD \(j\), \(V_{j-i}\) is the vector from PD \(j\) to LED \(i\), \(\|V_{j-i}\|=d_{ij}\), \(\|n_{j}^{tilt}\|=1\), \(n_{j}^{tilt}=\big{[}\sin(\beta_{j})\cos(\alpha_{j}),\sin(\beta_{j})\sin(\alpha_ {j}),\cos(\beta_{j})\big{]}\big{]}\), where \(\beta_{j}\) is the tilt angle with respect to the z-axis, which takes values in range of \([0^{\circ},90^{\circ})\), and \(\alpha_{j}\) is the rotation angle with respect to the x-axis that can be values in range of \([0^{\circ},180^{\circ})\), \(V_{j-i}=\big{[}x_{i}^{T}-x_{j}^{R},y_{i}^{T}-y_{j}^{R},\Delta h_{ij}\big{]}\), and \(\Theta\) is the FoV of the PD.