# Angle of Arrival for the Beam Detection Method of Spatially Distributed Sensor Array
## 2 Methods and Design
### Structural Design of the Spatially Distributed Sensor Array  
The number and location of sensors have a direct relationship with the field of view and measurement accuracy of the spatially distributed sensor array. We developed a support structure consisting of one top panel and \(M\) side panels, where the angle between the top panel and the side panels is \(\alpha\), and each panel is equipped with the same sensor. According to Equations (7) and (8), the rank of matrix \(\mathbf{K}\) should satisfy:  
\[r(\mathbf{K})\geq 3 \tag{10}\]  
It indicates that the minimum number of \(M\) is 2. The side panels are evenly distributed around the top panel, which is beneficial for analyzing the sensor field of view and error. The field of view of a single sensor is denoted as \(\phi\). In practical applications, the sensor is encapsulated, resulting in \(\phi<\pi\). The number of side panels \(M\) should satisfy:  
\[M>\frac{2\pi}{\phi}\ (\phi<\pi) \tag{11}\]  
The commercially packaged photodiodes have a field of view of \(120^{\circ}\), so we choose \(M=4\) to design the spatially distributed sensor array and improve the redundancy of the sensors. Figure 2a shows a schematic diagram of a spatially distributed sensor array when \(M=4\). At this time, the matrix \(\mathbf{K^{T}}\) can be expressed as:  
\[\mathbf{K^{T}}= \begin{bmatrix}\mathbf{n_{1}}\\ \mathbf{n_{2}}\\ \mathbf{n_{3}}\\ \mathbf{n_{4}}\\ \mathbf{n_{5}}\end{bmatrix} = \begin{bmatrix}1&0&0\\ \cos\alpha&\sin\alpha&0\\ \cos\alpha&0&\sin\alpha\\ \cos\alpha&-\sin\alpha&0\\ \cos\alpha&0&-\sin\alpha\end{bmatrix}  \tag{12}\]  
Figure 2: (**a**) Schematic diagram of a spatially distributed sensor array when \(M=4\). (**b**) Field of view of spatially distributed sensor array when \(M=4\). Substituting \(K^{T}\) into Equation (7), the output power of each sensor can be expressed as:  
\[P_{1}=P_{1}(S,\beta,\gamma)=S\cos\beta\cos\gamma \tag{13}\]  
\[P_{2}=P_{2}(S,\beta,\gamma)=S(\sin\alpha\sin\beta\text{cos}\,\gamma+\cos\alpha \text{cos}\,\beta\text{cos}\,\gamma) \tag{14}\]  
\[P_{3}=P_{3}(S,\beta,\gamma)=S(\sin\alpha\sin\gamma+\cos\alpha\text{cos}\,\beta \text{cos}\,\gamma) \tag{15}\]  
\[P_{4}=P_{4}(S,\beta,\gamma)=S(-\sin\alpha\sin\beta\text{cos}\,\gamma+\cos \alpha\text{cos}\,\beta\text{cos}\,\gamma) \tag{16}\]  
\[P_{5}=P_{5}(S,\beta,\gamma)=S(-\sin\alpha\sin\gamma+\cos\alpha\text{cos}\, \beta\text{cos}\,\gamma) \tag{17}\]  
The output power in the field of view of each sensor should be satisfied:  
\[P_{i}=P(S,\beta,\gamma,\alpha)=S(\vec{s}\cdot\vec{n}_{i})\geq\text{Scos}\, \frac{\phi}{2} \tag{18}\]  
The top view of the field of view of the spatially distributed sensor array when M = 4 is shown in Figure 2b. It can be seen that the field of view of the spatially distributed sensor array is not a regular area. The solid angle formed by the intersection of the fields of view of Sensor 3 and Sensor 5 with those of Sensor 2 and Sensor 4 is defined as the field of view of the spatially distributed sensor array in this paper. Substituting \(\gamma=0\) and Equation (15) into Formula (18), the field of view of the spatially distributed sensor array can be expressed as:  
\[FOV=2\text{cos}^{-1}\left(\frac{\cos\frac{\phi}{2}}{\cos\alpha}\right) \tag{19}\]  
The relationship between the FOV of the spatially distributed sensor array and \(\alpha\) of the spatially distributed sensor array is shown in the Figure 3. As \(\alpha\) increases, the field of view of a spatially distributed sensor array decreases. From Equation (8), we can see that the sensor combination matrix involved in the AOA solution needs to satisfy \(r(\vec{B})\geq 3\). The spatially distributed sensor is a symmetrical structure. Therefore, Sensors 1, 2, and 3 are selected to analyze the impact of \(\alpha\) on sensor measurement resolution.