## 2 System Model
### Influence of the Angular Diversity  
Using an angle diversity receiver enables the determination of orientation, i.e., the radiation angles of the optical transmitters, based on the signals received by multiple photodiodes. Because the photodiodes share the same elevation angle, power variations are determined by the azimuthal angle, which depends on the vector between the optical transmitter and each photodiode. This approach allows us to determine the orientation angle of the optical transmitters, assigning a label to each of them.  
The first step in determining the orientation of the transmitters is to identify the photodiode that maximizes the received signal. When the receiver lacks angular diversity, this is generally the photodiode nearest to the LED. However, with angular diversity, this may not always be the case.  
In situations where several photodiodes maximize the received signal within a given area, a method is proposed to identify the photodiode directed toward the desired LED. Specifically, this photodiode is determined by evaluating the intensity received for it and its neighboring photodiodes. In this way, the photodiode that provides the maximum sum of its intensity and the intensity of the neighboring photodiode is selected as the valid one (main photodiode).  
Once the photodiode pointing to the LED is found, the next step is to determine the angular variation concerning the transmitter. This is carried out by considering the intensities and angles of the \"main\" photodiode, a photodiode to its right, and a photodiode to its left. First, a straight line is drawn through the detected points of maximum and minimum intensity. Then, a line of inverse slope is drawn through the resting point (the second with the highest intensity). The intersection of these two lines allows us to determine the orientation angle of the LED, as shown in Figure 2.  
At this point, four possible scenarios emerge based on the deployment of the LEDs in the transmitter plane and the positioning of the angular diversity receiver. The angular and dimensional characteristics that apply in each case, depending on the quadrant in which the transmitter is located relative to the receiver, are described in Table 3.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Scenario** & **Optical Transmitters** & **Algorithm Iterations** & **Description** \\ \hline
1 & 4 & \(l=\{1,2,3\}\), \(l=\{1,3,4\}\), & More favorable case, there are more transmitters than \\  & & \(l=\{1,2,4\}\), \(l=\{2,3,4\}\) & unknowns to be solved. Higher resolution, \\ \hline
2 & 3 & \(l=\{1,2,3\}\), \(l=\{1,2\}\), & The number of transmitters is the \\  & & \(l=\{1,3\}\), \(l=\{2,3\}\) & same as the number of unknowns. \\ \hline
3 & 2 & \(l=\{1,2\}\) & Less favorable case, there are fewer transmitters than unknowns, \\  & & & so the height is adopted as a constant parameter. \\ \hline \hline \end{tabular}
\end{table}
Table 2: Different possible scenarios to determine positioning.  
Recall that \(\alpha\) is the radiation angle and \(\theta\) is the angle of the transmitter to the position in the plane so that \(\theta=\arctan\left(\frac{y_{p}-y_{\text{LED}}}{x_{p}-x_{\text{LED}}}\right)\), where the spatial position of the \(l\)-th transmitter is given in (6) and the spatial position of the receiver is \(P_{\text{pos}}=\begin{bmatrix}x_{p}&y_{p}&z_{p}\end{bmatrix}\). By calculating these angles and understanding their relationship to the reference plane, we can accurately determine the angular displacement of the photodiode's zero-axis by averaging the detected angular displacements, which yields a consistent and precise measurement for properly adjusting the orientation of the semi-sphere.