## 2 Antenna Array Design
### Array Geometry  
In this section, the geometry of the antenna array will be discussed, where three key deign parameters were considered: first, the gain and half power beam width (HPBW) of the array, which changes proportionally with the number of antennas; second, the complexity of a future analog or digital beamforming system, as the number of required components, computational power, and cost also increases with the number of antennas in the array; third, the size of the array increases as more antennas are used. With this in mind, a 16-antenna square array was proposed, as it is a good compromise of the three previous figures, along with the square being the most common array geometry.  
The distance between elements was chosen to keep the array as small as possible, keeping in mind the limited space availability in the car, without significantly hindering performance, and considering the effect of mutual coupling. For all the accounted combinations of element spacing and array geometry that use 16 antennas, the array factor (AF) was analyzed for broadside, endfire, and in-between configurations using Matlab' s (version 2022b) Sensor Array Analyzer toolbox.  
Figure 5: Realized gain plot at 1.57542 GHz in the elevation plane for \(\varphi=0^{\circ}\) (**A**) and \(\varphi=90^{\circ}\) (**B**) for LHCP and RHCP signals.  
Figure 6: Phase versus theta angle for (**A**) \(\varphi=0^{\circ}\) and (**B**) \(\varphi=90^{\circ}\) at 1.57542 for LHCP and RHCP signals.  
Since the objective of the work was the development of an antenna array for GNSS signal spatial filtering, high directivity of the AF and low HPBW must be achieved. Analyzing the data concerning square arrays with different element spacing of 0.33\(\lambda_{0}\), 0.4\(\lambda_{0}\), 0.5\(\lambda_{0}\), and 0.75\(\lambda_{0}\), it was possible to verify that, as expected, the main lobe directivity increases as the distance d between the array elements increases. On the other hand, the half power beamwidth decreases. Given this, it would seem intuitive that since the goal is to obtain high directivity and low HPBW, d should be as high as possible. However, higher \(d\) values lead to more and higher grating lobes: as detailed in [21] side lobes are small compared to the main beam when the following criteria are met:  
\[d<\frac{\lambda}{1+|\cos\theta_{0}|} \tag{1}\]  
Because the intention is to direct the main beam to values of \(\theta\) ranging from 0\({}^{\circ}\) to 90\({}^{\circ}\), from horizon to zenith, ideally \(d\) should be less than \(\lambda_{0}/2\) to reduce the side lobe level (SLL), coinciding with the following statement from [PERSON] [22]: \"to have only one end-fire maximum and to avoid any grating lobes, the maximum spacing between elements should be less than \(d_{max}<\lambda_{0}/2\)\".  
A compromise between the directivity and HPBW of the array with d = 0.5\(\lambda_{0}\) and the low grating lobe level achieved with d = 0.33\(\lambda_{0}\) is achieved with d = 0.4\(\lambda_{0}\). Figure 7 shows why an element spacing of d = 0.5\(\lambda_{0}\) is not ideal due to having two main beams in opposite directions at \(\theta\) = 90\({}^{\circ}\) compared to a spacing of d = 0.4\(\lambda_{0}\). It is also presenting the radiation pattern for \(\theta\) = 0\({}^{\circ}\), where the directivity difference, the HPBW, and the reducing of side lobe levels are noticeable, as mentioned above.