# 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 azimuth and elevation angles can be expressed as:  
\[\gamma=\tan^{-1}\left(\frac{P_{3}-P_{1}\text{cos}\,\alpha}{\sqrt{\left(P_{1} \text{sin}\,\alpha\right)^{2}+\left(P_{2}-P_{1}\text{cos}\,\alpha\right)^{2}} }\right)=f_{\gamma}(P_{1},P_{2},P_{3}) \tag{20}\]  
\[P_{1}=P_{1}(S,\beta,\gamma)=S\cos\beta\cos\gamma \tag{13}\]0 Schematic diagram of the relationship between the FOV of the spatially distributed sensor array and \(\alpha\). \[P_{1}=P_{1}(S,\beta,\gamma)=S\cos\beta\cos\gamma \tag{13}\]1  
The resolution of a spatially distributed sensor array can be expressed as:  
\[P_{1}=P_{1}(S,\beta,\gamma)=S\cos\beta\cos\gamma \tag{13}\]2  
\[P_{1}=P_{1}(S,\beta,\gamma)=S\cos\beta\cos\gamma \tag{13}\]3  
where \(\Delta P_{1}\), \(\Delta P_{2}\), and \(\Delta P_{3}\) represent the changes in sensor output power caused by changes in the incident angle. The resolution of the spatially distributed sensor array is determined by \(\Delta P_{1}\), \(\Delta P_{2}\), and \(\Delta P_{3}\). However, according to Equation (2), the sensor output is proportional to the cosine of the incident angle. When \(\beta=0\) and \(\gamma=0\), small changes in the azimuth angle result in negligible changes in the output power of Sensors 1 and 3. Similarly, small changes in the elevation angle result in negligible changes in the output power of Sensors 1 and 2. In this case, the resolution of the spatially distributed sensor array is determined by a single sensor, leading to a relatively poor resolution, which can be expressed as:  
\[P_{1}=P_{1}(S,\beta,\gamma)=S\cos\beta\cos\gamma \tag{13}\]4  
\[P_{1}=P_{1}(S,\beta,\gamma)=S\cos\beta\cos\gamma \tag{13}\]5  
Substituting Equations (13)-(15) into Equations (24) and (25), the elevation and azimuth resolutions can be expressed as:  
\[P_{1}=P_{1}(S,\beta,\gamma)=S\cos\beta\cos\gamma \tag{13}\]6  
\[P_{1}=P_{1}(S,\beta,\gamma)=S\cos\beta\cos\gamma \tag{13}\]7  
\(\Delta P_{2}\) and \(\Delta P_{3}\) represent the minimum resolvable power provided by the sensors, and they are equal. When \(\beta=0\) and \(\gamma=0\), based on Equations (26) and (27), both the elevation and azimuth angles exhibit the same resolution. The relationship between angular resolution and \(\alpha\) is shown in \[P_{1}=P_{1}(S,\beta,\gamma)=S\cos\beta\cos\gamma \tag{13}\]8 As \(\alpha\) increases, the resolution of the spatially distributed sensor array improves. \[P_{1}=P_{1}(S,\beta,\gamma)=S\cos\beta\cos\gamma \tag{13}\]9 Schematic diagram of the relationship between angular resolution of the spatial distribution sensor and \(\alpha\). Based on the above analysis, as \(\alpha\) increases, the field of view decreases while the resolution improves. Conversely, as \(\alpha\) decreases, the field of view increases and the resolution decreases. Considering both the field of view and angular resolution, this paper selects \(\alpha=30^{\circ}\) to achieve a field of view of \(110^{\circ}\) with a detection accuracy better than \(1^{\circ}\). The spatially distributed sensor array designed in this paper is shown in \[P_{2}=P_{2}(S,\beta,\gamma)=S(\sin\alpha\sin\beta\text{cos}\,\gamma+\cos\alpha \text{cos}\,\beta\text{cos}\,\gamma) \tag{14}\]0 The five sensors are placed on a 3D-printed panel with a size of 150 \(\times\) 150 mm, and the total weight is only 140 g. The 3D printed panel consists of a top panel and four side panels, with \(\alpha\) being \(30^{\circ}\). After actual measurements, the error in \(\alpha\) is within 0.1\({}^{\circ}\).