# Reduction of Phase Shifters in Planar Phased Arrays Using Novel Random Subarray Techniques
## 2 Random Subarray Design Methodology
### Fusions of Subarrays
The latter was due to the advantages of having subarrays of size \(j\in[1,2]\) near the center of the array. In this case, each fusion could contain up to 6 antenna elements per subarray, and the probability of fusion used for the 2 nd row was 75% and a probability of 90% for the 3 rd row. The design of the planar array obtained for this case is illustrated in Figure 8. This solution was obtained after 8351 iterations with a computational time of 1.20 h. A reduction of 62% in the number of phase shifters was accomplished assuring a scan range of the mainlobe of \(150^{\circ}\) in the azimuth plane and \(40^{\circ}\) in the elevation plane. \begin{table}
\begin{tabular}{c c c c} \hline \hline \(\phi\) & \(SLL\) **(dB)** & **FNBW (rad.)** & **Color** \\ \hline \(75^{\circ}\) & \(-10.84\) & \(0.5694\) & Red \\ \(90^{\circ}\) & \(-10.66\) & \(0.7566\) & [PERSON] \\ \(105^{\circ}\) & \(-11.90\) & \(0.7962\) & Magenta \\ \(120^{\circ}\) & \(-10.19\) & \(0.7789\) & Blue \\ \hline \end{tabular}
\end{table} 
Table 3: Results of SLL and FNBW for the planar phased array shown in Figure 7.

Figure 7: Radiation pattern and design of the planar phased array obtained in Case III. The natural response is shown in black dotted lines. The scanning range in the elevation plane is illustrated by the red rectangle (\(-40^{\circ}\leq\theta\leq 0^{\circ}\)) with an azimuth scanning range of \([75^{\circ}\leq\phi\leq 120^{\circ}]\). The color palette for different radiation patterns indicates each step angle \(\Delta\phi\) shown in Table 3.

## 4 Discussion and Conclusions

A novel method for the design of phased planar antenna arrays was presented in this work, in order to perform a reduction of up to 70% in the number of phase shifters used by the array, while maintaining the desired radiation characteristics. This method consists of creating fusions of subarrays to generate random sequences that construct the best feeding network configuration for planar phased arrays. It was shown that each solution was created based on search criteria that influenced the morphology of the array in terms of subarray size and orientation. This also could be visualized using the grey gradient, which showed the smoothness provided by the raised cosine function amplitudes of each subarray and its contribution to the whole array. One of the main reasons for choosing a random approach for the search algorithms is that the number of solutions in each case increases to an intractable number. Consider that for Case I depicted in Section 3.1 there was an exhaustive search time of approximately two days, which is attainable in terms of computational time. However, as mentioned in Case II (Section 3.2), the execution time increases exponentially due to the number of different possible rows (462) that can be created randomly from a set of given subarrays due to (9). Then, the number of possible solutions is given by

\[\mathcal{S}=\frac{(462+16-1)!}{16!(462-1)!}\approx 2.662\times 10^{29}. \tag{17}\] \begin{table}
\begin{tabular}{c c c c} \hline \(\phi\) & \(SLL\) (dB) & \(FNBW\) (rad.) & **Color** \\ \hline \(-75^{\circ}\) & \(-13.14\) & \(0.9498\) & Purple \\ \(-60^{\circ}\) & \(-13.11\) & \(1.7294\) & Pink \\ \(-45^{\circ}\) & \(-14.39\) & \(0.6319\) & Red \\ \(-30^{\circ}\) & \(-10.40\) & \(0.5891\) & Green \\ \(-15^{\circ}\) & \(-11.45\) & \(0.5184\) & Magenta \\ \(0^{\circ}\) & \(-15.61\) & \(0.5397\) & Orange \\ \(15^{\circ}\) & \(-10.90\) & \(0.5496\) & Cyan \\ \(30^{\circ}\) & \(-11.11\) & \(0.6428\) & Yellow \\ \(45^{\circ}\) & \(-10.61\) & \(0.7880\) & Blue \\ \(60^{\circ}\) & \(-11.15\) & \(0.7891\) & Aqua \\ \(75^{\circ}\) & \(-14.24\) & \(0.8223\) & Beige \\ \hline \end{tabular}
\end{table} 
Table 4: Results of SLL and FNBW for the planar phased array shown in Figure 8.

Figure 8: Radiation pattern and design of the planar phased array obtained in Case IV. The natural response is shown in black dotted lines.