## 2 Antenna Array Design
### Array Simulation  
The antenna array was designed in CST resorting to the Array Task tool and it was studied with its main beam pointing towards the following set of \(\varphi\) and \(\theta\) angles: (0,0), (0, 45), (0,90), (45, 45), (45, 90), (90, 45), and (90, 90). This selection of directions allows for the study of the array's characteristics and performance when pointing to the zenith (broadside) and to the horizon (endfire) in two orthogonal planes and with a step in-between. As the  
Figure 7: 3D radiation pattern of the 4 \(\times\) 4 array with spacing d with the beam pointing to different (\(\varphi\), \(\theta\)) angles. (**A**) d = 0.4\(\lambda_{0}\) and (0, 90). (**B**) d = 0.5\(\lambda_{0}\) and (0, 90). (**C**) d = 0.5\(\lambda_{0}\) and (0, 0). (**D**) d = 0.4\(\lambda_{0}\) and (0, 0). (**E**) d = 0.33\(\lambda_{0}\) and (0, 0). antennas in the array are symmetrically positioned, the performance in this 90\({}^{\circ}\) segment of the array is representative of the array's performance in the remaining directions. The 3D model of the square array of d = 0.4\(\uplambda_{0}\) and size of 305 \(\times\) 305 mm\({}^{2}\) is presented in Figure 8. The ground plane of each antenna element was extended so it is a single structure common to all antennas. The simulated 3D radiation patterns of the square array pointing to the previously enumerated directions are presented in Figure 9, and the 2D plots in the elevation plane cut are presented in Figure 10. As can be seen, the gain is highest in the broadside direction (14.9 dBi), as expected, since both the antenna element gain and the array factor's directivity are highest in this direction. When \(\theta\) = 45\({}^{\circ}\), the gain is slightly lower (13.3 dBi for \(\varphi\) = 0\({}^{\circ}\)), as both the antenna element gain and the array factor directivity are lower at this angle. This fact is also the reason why the array is not capable of adequately pointing the beam towards \(\theta\) = 90\({}^{\circ}\): The antenna element has much lower gain towards the horizon, thus hindering the array's performance in this plane. However, the array is still capable of achieving a gain of 6.2 dBi for \(\varphi\) = 0\({}^{\circ}\) and \(\theta\) = 90\({}^{\circ}\) and 5.2 dB for \(\varphi\) = 90\({}^{\circ}\) and \(\theta\) = 90\({}^{\circ}\) while the single-antenna element can only provide a gain of \(-\)3.4 dBi and \(-\)3.0 dBi, respectively, thus accomplishing the goal of improving signal reception at low elevations. Figure 8: CST model (**A**) and photograph (**B**) of the 4 \(\times\) 4 array. Figure 9: 3D radiation pattern of the 4 \(\times\) 4 array with the beam pointing to different (\(\varphi\), \(\theta\)) angles. (**A**) (0, 0). (**B**) (0, 45). (**C**) (0, 90). (**D**) (45, 45). (**E**) (45, 90). (**F**) (90, 45). (**G**) (90, 90). Figure 11 presents the axial ratio of the array when the main beam is directed towards the aforementioned directions. It is possible to observe that the array maintains reasonable circular polarization (AR close to 3 dB) in the directions the main beam points to. This statement holds even for beam directions of \(\theta=90^{\circ}\), except for the case where \(\varphi=0^{\circ}\), where the AR at the \(90^{\circ}\) mark is over 10 dB. This could be due to the inherent asymmetry of the antenna element on the square array, and it is a possible path to be explored in future work. Figure 11: Axial ratio in the elevation plane of the \(4\times 4\) array with the beam pointing to different (\(\varphi\), \(\theta\)) angles. (**A**) (0, 0). (**B**) (0, 45). (**C**) (0, 90). (**D**) (45, 45). (**E**) (45, 90). (**F**) (90, 45). (**G**) (90, 90). The red dashed lines represent the defined beamforming direction. The blue dashed circles correspond to a 3 dB AR. Figure 10: Elevation plane radiation pattern of the \(4\times 4\) array with the beam pointing to different (\(\varphi\), \(\theta\)) angles. (**A**) (0, 0). (**B**) (0, 45). (**C**) (0, 90). (**D**) (45, 45). (**E**) (45, 90). (**F**) (90, 45). (**G**) (90, 90).