# Reduction of Phase Shifters in Planar Phased Arrays Using Novel Random Subarray Techniques
## 2 Random Subarray Design Methodology
### Fusions of Subarrays
All cases were run on a _Linux Debian 12_ computer (Waltham, MA, USA) system with 8 processors _Intel Core i7-8565U @ 1.80 GHz_ and \(16 GB\) of RAM (Santa Clara, CA, USA). The elevation plane with range \([-\pi/2\leq\theta\leq\pi/2]\) is represented by the \(x\)-axis on each graph using a resolution of \(\pi/36\),000 points. The \(y\)-axis corresponds to the normalized array factor calculated by

\[y=20\log 10\bigg{(}\frac{|AF|}{\max(|AF|)}\bigg{)}. \tag{15}\]

For each case, a solution had three restrictions that had to be satisfied in order to qualify as a candidate solution, such as the ones presented in this section; starting with a minimum reduction in phase shifters or the number of fusions. Then, the performance of each array configuration was measured using a threshold value of SLL scanning at broadside or \([\theta,\phi]=[0^{\circ},0^{\circ}]\). Finally, the mainlobe was scanned into a desired elevation angle \(\theta\) and measurement of a new SLL threshold was performed with every step angle addition \(\Delta\phi\) in the azimuth plane given the desired scanning range for \(\phi\). ### Case I: Planar Array Design from Optimal Linear Arrays

An initial number of 8 sequences was used from a Pareto front compromising SLL and FNBW, which resulted from an exhaustive search of a linear phased array with 16 antenna elements and 8 subarrays scanning at \(\theta=12^{\circ}\) with an SLL threshold of \(-10\) dB. The number of possible solutions for this instance of the problem was given by the search space calculated by (8) and (9). Hence,

Figure 2.0

Consider that each solution was computed in an average time of 0.7696 s. Therefore, an exhaustive search of the whole search space was attainable in approximately 52.41 h.

Instead of searching through all possible solutions, a random approach was used to search for a solution with SLL below \(-28\) dB scanning at broadside in \([\theta,\phi]=[0^{\circ},0^{\circ}]\). Moreover, there was the capability of a maximum scanning of \(\theta=60^{\circ}\) in elevation and

Figure 2.1 An illustration of the amplitudes per antenna in a \(8\times 8\) planar array by applying a raised cosine tapering function. (**a**) Grey color gradient of amplitudes per antenna element. (**b**) Normalized values of the raised cosine function. Notice that the amplitude in each antenna element decreases progressively as it moves away from the center location of the array. a scanning range of \([30^{\circ}\leq\phi\leq 150^{\circ}]\) in azimuth with a step angle of \(\Delta\phi=15^{\circ}\), while maintaining a SLL below \(-10\) dB. The number of phase shifters used by the array could be calculated in advance due to the fixed number of 8 subarrays per row, therefore \(PS_{total}=8\times 16=128\) according to (7). This ensured a reduction of 50% of the phase shifters used in the conventional case, where a phase shifter is used for each antenna element in the array. A solution that satisfied the search criteria was obtained after 197 iterations with a computational time of 2.53 min. The planar array is illustrated in Figure 5, with SLL levels for each scanning angle in \(\phi\) shown in Figure 2.2

### Case II: Greedy Fusions of Subarrays in 2 Rows

In order to reduce the number of phase shifters used by the array to near 75%, according to (11), a greedy approach was performed for this instance of the problem. Each sequence for a row was created randomly using subarrays of size \(K_{1}=6\) and \(K_{2}=5\), and a maximum number of 4 antenna elements per subarray was allowed after a successful fusion. For this case, the search criteria were to find a solution with a natural response in broadside below \(-25\) dB scanning at \([\theta,\phi]=[0^{\circ},0^{\circ}]\) and a scanning capability of \(\theta=60^{\circ}\) in elevation and \([-20^{\circ}\leq\phi\leq 20^{\circ}]\) in azimuth, with a step angle of \(\Delta\phi=10^{\circ}\), while maintaining a SLL below \(-10\) dB.