# Reduction of Phase Shifters in Planar Phased Arrays Using Novel Random Subarray Techniques
## 2 Random Subarray Design Methodology
### Fusions of Subarrays
This means that if a first fusion occurs between two rows, a second fusion may occur if the current total number of antenna elements in the new subarray complies with the restriction of size previously stated and if the probability condition is met. This process adds more diversity to the geometric figures and reduces the number of subarrays \(K_{j}\) per row, which also reduces the number of phase shifters used by the array system. Let \(S_{2}=[1,1,2,2,1,1]\) and \(S_{3}=[2,1,1,2,1,1]\) be the new pair of rows chosen to perform a fusion, as shown in Figure 3. The fusions of subarrays happen on indices \((i_{S_{2}},i_{S_{3}}=1,3,5)\), while maintaining the condition of size of the subarrays at \(j\in[2,3,4]\). A new phase distribution is assigned based on choosing a different phase alternating with the last fusions performed. Figure 2: An illustration of the fusion of subarrays process in 2 rows: (**a**) The sequences of subarrays \(K_{j}\) are presented as \(S_{1}=[1,1,1,1,2,2]\) with phase distribution marked in blue, and \(S_{2}=[1,1,2,2,1,1]\) with phase distribution marked in green. (**b**) Resultant subarrays with three fusions performed. A new phase distribution is assigned to the fusioned subarrays. #### 2.4.3 Greedy Fusion Approach

This instance of the problem eliminates the randomness of each fusion event and necessitates accomplishing every possible fusion in the array given a set of random sequences of subarrays in each row. One of the main advantages of this approach is the reduction of the search space, given that only one solution exists for each initial configuration of \(K_{j}\) subarrays. The number of phase shifters reduces approximately to

\[PS_{total}\approx(PS_{m}\times M)/2. \tag{11}\]

For example, suppose that \(K_{2}=8\) are all the subarrays in the array, then there is only one solution for this instance of the problem with \(8\times 16=128\) phase shifters before the fusion process and \(128/2=64\) subarrays after a greedy approach, considering that every tried fusion is successful. This results in a reduction of \(75\%\) in the number of phase shifters due to (11) in comparison with the conventional case. ### R raised Cosine Tapering

Consider that the array center serves as the phase reference and \(d_{n,m}\), the distance from the origin to the antenna element located at the \(n\)-column and \(m\)-row. Then, the raised cosine tapering function is given by

\[I_{n,m}=\frac{1+\cos(\frac{d_{n,m}\cos^{-1}(2n-1)}{0.5L})}{2}, \tag{12}\]

where \(L\) represents the array longitude and \(a=0.14\) according to [18], due to the symmetry of the array. Hence,

\[d_{n,m}=\sqrt{dx_{n,m}^{2}+dy_{n,m}^{2}}, \tag{13}\]

corresponds to the distances from the origin of the array to the antenna elements in the \(x\) and \(y\) axes. Then, \(d_{n,m}\) can be treated as the hypotenuse of a right triangle with \(dx_{n,m}\) and \(dy_{n,m}\) as its cathetos. Each subarray has a distance \(di_{n,m}\) to the center of the planar array given by (13) as shown in Figure 4; therefore, a new average distance \(d_{n,m}\) is calculated by computing the sum of the distances of each subarray involved in the fusion process divided by the number of rows \(R\) that contain those subarrays. Hence,

\[d_{n,m}=\frac{\sum_{i=1}^{R}di_{n,m}}{R}. \tag{14}\]

This can be visualized as the calculation of the centroid of the new geometric figure created from the fusion of subarrays. Figure 3: Example of a three-row fusion between the two-row fusion result shown in Figure 2 and a third row \(S_{3}=[2,1,1,2,1,1]\). New fusions of subarrays happen on indices \(i_{S_{3}}=1,3,5\). In addition, a new phase distribution is set for the resultant array. ## 3 Simulation Results

This section presents the results obtained for each case previously described. An algorithm programmed in R was used to perform various instances of the problem of the design of a \(16\times 16\) phased planar antenna array.