# Reduction of Phase Shifters in Planar Phased Arrays Using Novel Random Subarray Techniques
## 2 Random Subarray Design Methodology
### Fusions of Subarrays
\begin{table}
\begin{tabular}{c c c c} \hline \hline \(\phi\) & \(\mathit{SLL}\) (dB) & \(\mathit{FNBW}\) (rad.) & \(\mathit{Color}\) \\ \hline \(30^{\circ}\) & \(-10.11\) & \(1.4142\) & Red \\ \(45^{\circ}\) & \(-12.74\) & \(1.2269\) & Green \\ \(60^{\circ}\) & \(-16.45\) & \(1.3133\) & Magenta \\ \(75^{\circ}\) & \(-22.91\) & \(0.8407\) & Cyan \\ \(90^{\circ}\) & \(-26.35\) & \(1.1809\) & Orange \\ \(105^{\circ}\) & \(-25.54\) & \(0.8523\) & Yellow \\ \(120^{\circ}\) & \(-15.47\) & \(1.3314\) & Blue \\ \(135^{\circ}\) & \(-13.70\) & \(0.8332\) & Pink \\ \(150^{\circ}\) & \(-10.09\) & \(0.8132\) & Aqua \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Results of SLL and FNBW for the planar phased array shown in Figure 5.

Figure 5: Radiation pattern and design of the planar phased array obtained in Case I. The natural response is shown in black dotted lines. The scanning range in the elevation plane is illustrated by the red rectangle (\(0^{\circ}\leq\theta\leq 60^{\circ}\)) with an azimuth scanning range of \([30^{\circ}\leq\phi\leq 150^{\circ}]\). The color palette for different radiation patterns indicates each step angle \(\Delta\phi\) shown in Table 1.

A reduction of 69% in the number of phase shifters with respect to the conventional case was accomplished using this approach. The solution shown in Figure 6 was found after 1576 iterations with a total computational time of 12.64 min. ### Case III: Fusions of Subarrays in 3 Rows

An initial condition of \(K_{1}=1,K_{2}=3,K_{3}=3\) was used to randomly create solutions with 112 initial subarrays. In this case, an additional row was added to the fusion process. This time, a combination of the greedy approach and a restriction of 6 antenna elements per subarray was introduced in order to keep some of the arrays out of the fusion process, with the latter to generate disturbances in terms of the variables used that might enrich the design process. The search criteria for this case corresponded to a natural response in broadside below \(-25\) dB scanning at \([\theta,\phi]=[0^{\circ},0^{\circ}]\) with an elevation angle of \(\theta=-40^{\circ}\) and a scanning range of \([75^{\circ}\leq\phi\leq 120^{\circ}]\) in the azimuth plane, with a step angle of \(\Delta\phi=15^{\circ}\) and a SLL below \(-10\) dB. A reduction of 70% in the number of phase shifters was accomplished by using this technique. The solution illustrated in Figure 7 was found after 17,602 iterations, with a total computational time of 2.45 h. \begin{table}
\begin{tabular}{c c c c} \hline \(\phi\) & \(SLL\) (dB) & \(FNBW\) (rad.) & **Color** \\ \hline \(-20^{\circ}\) & \(-11.41\) & \(1.1480\) & Red \\ \(-10^{\circ}\) & \(-12.69\) & \(0.8492\) & Green \\ \(0^{\circ}\) & \(-11.49\) & \(1.2844\) & Magenta \\ \(10^{\circ}\) & \(-12.38\) & \(1.3229\) & Cyan \\ \(20^{\circ}\) & \(-10.02\) & \(1.4058\) & Orange \\ \hline \end{tabular}
\end{table} 
Table 2: Results of SLL and FNBW for the planar phased array shown in Figure 6.

Figure 6: Radiation pattern and design of the planar phased array obtained in Case II. The natural response is shown in black dotted lines. The scanning range in the elevation plane is illustrated by the red rectangle (\(0^{\circ}\leq\theta\leq 60^{\circ}\)) with an azimuth scanning range of \([-20^{\circ}\leq\phi\leq 20^{\circ}]\). The color palette for different radiation patterns indicates each step angle \(\Delta\phi\) shown in Table 2.

### Case IV: Fusions of Subarrays with Mixed Techniques

This case demonstrated the potential capabilities of all the techniques previously discussed but mixed together. The search criteria for the planar array were to scan the mainlobe at \(\theta=-40^{\circ}\) in elevation with a range of scanning of \([-75^{\circ}<\phi<75^{\circ}]\) in azimuth, with a step angle of \(\Delta\phi=15^{\circ}\) while maintaining the SLL below \(-10\) dB. A new approach was introduced by reducing the probability of fusion to zero for antenna elements in columns \(6<n<11\) and rows \(6<m<11\), and the last row that was not fusionable was inserted at \(m=6\).