# A Novel and Adaptive Angle Diversity-Based Receiver for 6G Underground Mining VLC Systems
## 3 Adaptive Orientation Receiver
### Methodology Used to Estimate the Angle of Incidence  
To obtain the proper orientation that the receiving geometric structure will be given, we must have a fairly accurate estimate of the direction of the LoS optical link. Therefore, the suitable variable to estimate is the angle of incidence. In this research, we propose the estimation of \(\theta_{ij}^{tilt}\) based on the RSSR metric implemented and analyzed from the reference of the secondary coordinate system.  
According to the mathematical expression of received power derived in (14), we define in a general way that the approximate RSSR between PD \(j\) and any PD \(q\), where \(j\) and \(q\) belong to the set of PDs \(J\) corresponding to the receiving geometric structure, is expressed as  
\[\frac{P_{r,i,j}}{P_{r,i,q}}\approx\frac{\frac{\mathbb{C}}{d_{i}^{m+3}}\left(V_{ i}\cdot\mathbf{n}_{i}^{tilt}\right)^{m}\left(V_{i}\cdot\mathbf{n}_{j}^{tilt}\right)P_{i}+R_{ PD}P_{t}H_{NLoS,i}^{(1)}+R_{PD}P_{t}H_{sca,i}+N_{H}}{\frac{\mathbb{C}}{d_{i}^{m+3}} \left(V_{i}\cdot\mathbf{n}_{i}^{tilt}\right)^{m}\left(V_{i}\cdot\mathbf{n}_{q}^{tilt} \right)P_{i}+R_{PD}P_{t}H_{NLoS,i}^{(1)}+R_{PD}P_{t}H_{sca,i}+N_{H}}, \tag{19}\]  
In order to obtain simplified analytic expressions based on the direction cosines of the incidence angles between LED \(i\) and PDs \(j\) and \(q\) together with their normal vectors, and expressing them as a function of the Cartesian secondary coordinate system with respect to the receiver, we can rewrite Equation (19) as  
\[\frac{P_{r,i,j}}{P_{r,i,q}}\approx\frac{\cos(\theta_{ij}^{tilt})}{\cos(\theta _{iq}^{tilt})}=\frac{\mathbf{n}_{j}^{tilt(H)}\cdot d_{ij}^{H}}{\mathbf{n}_{q}^{tilt( H)}\cdot d_{iq}^{H}}, \tag{20}\]  
where \(d_{ij}^{H}\) and \(d_{iq}^{H}\) can be considered approximately equal, since they would be almost the same distance from LED \(i\). Therefore, we will define this general distance as \(d_{i}^{H}\). As mentioned before, to generalize, we define \(J\geq 3\) PDs. Therefore, we can obtain for each LED in the system (\(L-1\)) independent RSSR equations similar to expression (20). For simplicity in the calculations, we consider the RSSR between the first PD and the other PDs in the array to obtain the following expression:  
\[\frac{P_{r,i,j}}{P_{r,i,1}}\approx\frac{\mathbf{n}_{j}^{tilt(H)}\cdot d_{i}^{H}}{ \mathbf{n}_{1}^{tilt(H)}\cdot d_{i}^{H}}. \tag{21}\]  
This expression can be restructured as  
\[\left(\mathbf{n}_{j}^{tilt(H)}-\frac{P_{r,i,j}}{P_{r,i,1}}\mathbf{n}_{1}^{tilt(H)} \right)^{\text{T}}d_{i}^{H}=0. \tag{22}\]  
Since the module of \(d_{i}^{H}\) does not directly influence the equation, only the direction of \(d_{i}^{H}\) is needed. Therefore, we can define the independent RSSR set of equations in matrix form as  
\[\mathbf{A}_{i}\triangleq\begin{bmatrix}\left(\mathbf{n}_{2}^{tilt(H)}-\frac{P_{ r,i,2}}{P_{r,i,1}}\mathbf{n}_{1}^{tilt(H)}\right)^{T}\\ \vdots\\ \left(\mathbf{n}_{L}^{tilt(H)}-\frac{P_{r,i,L}}{P_{r,i,1}}\mathbf{n}_{1}^{tilt(H)} \right)^{T}\end{bmatrix}_{(L-1)\times 3}, \tag{23}\]  
\[\mathbf{x}_{i}\triangleq d_{i}^{H}. \tag{24}\]  
We can express these matrices as a group of homogeneous linear equations defined as  
\[\mathbf{A}_{i}\mathbf{x}_{i}=0;\quad subject\ to\ ||\mathbf{x}_{i}||=1. \tag{25}\]  
Finally, we obtain the linear solution of (25) by applying the method of least squares \((\hat{x}_{i})\). This expression is given by the eigenvector corresponding to the smallest eigenvalue of \(\mathbf{A}_{i}^{T}\mathbf{A}_{i}\). Furthermore, since \(0\leq\theta_{ij}^{tilt}\leq\Theta\), we can infer that \(\cos(\theta_{ij}^{tilt})\geq 0\). Therefore, the estimate incidence angle can be calculated as  
\[\theta_{ij}^{tilt}=\arccos\left(\left|\mathbf{n}_{j}^{tilt(H)}\hat{\mathbf{x}}_{ i}\right|\right). \tag{26}\]With the estimate \(\theta_{ij}^{\hat{tilt}}\) value and through mechanical rotation mechanisms, we can optimally orient the PD structure so that it can maximize the power received by the LED.