## 3 Dependence of Geometric Placement of Antenna Array Elements in Regards to Radio Link Improvements
### Geometric Assessment of Phased Antenna Arrays  
We propose two antenna arrays with lengths of \(3.5\lambda\) and \(7.5\lambda\). The number of elements required for each antenna array, according to its geometry, are follows: (1) planar (PA) of the \(8x8\) and \(16x16\) elements, respectively; (2) circular (CA) of 21 and 47 elements; and (3) 42 circular concentric rings (CR) (with three rings) and 196 (with seven rings) elements. The design and model of each array geometry were carried out by computing the array factor; PA [28; 29; 30; 31; 32], CA [28; 33; 29; 30; 31; 33], CR [34; 35; 36; 37; 38]. The numerical simulations of the antenna arrays were performed by the product of the array factor and the radiation pattern data of the antenna element. A patch was used as the antenna element and characterized in every interest frequency. The radiation performance is improved by using the antenna element. The results obtained by numerical results are similar to full-wave simulations. Figure 1 shows a comparison of the numerical simulation of the array factor and the radiation pattern (including the circular patch antenna). Figure 1: Comparison between the numerical simulations of the array factor and radiation pattern (including a patch antenna) of the CA of 21 elements at \(\theta=0^{\circ}\). The circular patch uses substrate laminates FR-4 (for 700 MHz and 3.5 GHz) and RT/duroid 5880 (for 26 and 48 GHz) with a dielectric constant (\(\epsilon_{r}\)) of 4.2 and 2.2, respectively. The FR-4 laminate considers a thickness of 1.6 mm, while the RT/duroid 5880 a laminate of 0.787 mm. The radiation efficiency of the circular patch is 65% in every frequency. An example of the radiation pattern of the circular patch antenna is shown in Figure 2.  
On the other hand, we defined an interelement spacing of 0.5\(\lambda\) with the aim of avoiding the mutual coupling between elements, and obtaining the least deformations in the radiation pattern, a main radiation beam direction at \(\theta_{0}=0^{\circ}\), with the samples of scanning angle \(\theta\) in a range of \(-90\) to 90 degrees. The numerical outcomes regarding the antenna beamwidth and gain for the different geometries under the proposed antenna setup are summarized in Table 1.  
The obtained results illustrate that increasing the number of elements can achieve higher directivity (Gain) values and smaller beamwidth. Additionally, the circular antenna array has the best performance, with a narrower beamwidth and higher directivity compared to the other geometries. Although each geometry has its pros and cons, another required feature associated with the antenna arrays is the scanning capacity, a vital parameter to perform beamforming links and directional tracking of the mobile user in the radio mobile. Figures 3 and 4 show a comparison of the beamwidth and gain levels at different scanning directions of the planar, circular, and concentric ring antenna arrays. From Figure 3, we can see that the planar geometry is the more significant array with the higher area, higher number of antenna elements, and narrow beamwidths (12.25 \(\lambda^{2}\) \begin{table}
\begin{tabular}{c c c} \hline \hline
**Array** & **BW (Degrees)** & **Gain (dB)** \\ \hline Planar \(8\times 8\) & 28.98 & 15.86 \\ \hline Planar \(16\times 16\) & 14.4 & 21.94 \\ \hline Circular & & \\
21 elements & 25.37 & 17.015 \\ \hline Circular & & \\
47 elements & 11.87 & 23.61 \\ \hline Concentric rings & & \\
42 elements (3 rings) & 32.4 & 14.4 \\ \hline Concentric rings & & \\
196 elements (7 rings) & 16.92 & 20.53 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Numeric results of the antenna arrays simulated at \(\theta_{0}=0^{\circ}\). Figure 2: Radiation pattern of the circular patch antenna; (**a**) radiation pattern data for numerical simulations, and (**b**) 3D radiation pattern from the CST MicroWave Studio Suite. for 8 \(\times\) 8 array and 56.25 \(\lambda^{2}\) for 16 \(\times\) 16 array), but its performance is not appropriate for large scanning directions.