# Reduction of Phase Shifters in Planar Phased Arrays Using Novel Random Subarray Techniques
## 2 Random Subarray Design Methodology
### Fusions of Subarrays  
A novel technique called _fusion of subarrays_ is introduced in this work to reduce the number of phase shifters by combining two or more adjacent subarrays into a new subarray with more antenna elements and a new phase distribution. The idea behind this methodology is to randomize the process of creating new subarrays with geometric figures based on tiles, to reorder the grouping of the antenna elements in an initial configuration of a planar array. In order to construct an initial \(N\times M\) planar array composed of \(M\) rows, where each row corresponds to a linear array of integers, a starting sequence with a fixed number of subarrays of size \(K_{j}\) is randomly shuffled \(M\) times to create each row of the matrix. This allows each row to contain the same number of subarrays \(K_{j}\) but in random positions, ensuring \(PS_{m}\times M\) dimensions of the array. Then, a search for adjacency between subarrays with index \(i_{S_{m}}\) is performed, and if this condition is met, a new fusion with probability \(P\) may occur. \[i_{S_{m}}\in[1,2,\ldots,PS_{m}-1,PS_{m}] \tag{10}\]  
For the purpose of this work, the phase distribution in the fusioned subarrays is dictated by the previously assigned distribution of phase, starting from the subarray with the lowest row in the fusion and alternating with an upper row each time fusion is performed. This means that every time a fusion happens, it is necessary to trace the past index used as reference, in order to assign a different phase each time. The intention behind this is to have an equitable distribution of phases across the array. An example of the pseudo code for this method is shown in Algorithm 1. ``` \(seq=[1,1,1,1,2,2]\); // initializes seed sequence // \(N=M=sum(seq)\); // sum all elements in seq // \(PS_{m}=length(seq)\); for\(m<M/2\)do \(S_{(m)}=shuffle(seq)\); // shuffle subarray order // \(S_{(m+1)}=shuffle(seq)\); for\(i_{S_{m}}<PS_{m}\)do \(P=random(0,1)\); // random number between 0 and 1 // \(rand=0.75\); // 75% of fusion // if\(P<rand\)then  align-indices(\(i_{S_{m}},i_{S_{m+1}}\)); // check for contiguous indices // \(fusion(S_{m},S_{m+1},i_{S_{m}},i_{S_{m+1}})\); \(dnum(i_{S_{m}},i_{S_{m+1}})\); // compute new avg distances // else  align-indices(\(i_{S_{m}},i_{S_{m+1}}\));  end if  end for  end for \(Inn(dnum)\); // compute raised cosine // \(AF(\theta,\phi,Inn)\); // compute array factor //

#### 2.4.1 Two-Row Fusion

This case requires that the number of antenna elements per row, represented by \(M\), is an even number such that \((M\bmod 2)=0\). The later is because each subarray fusion happens between a pair of rows, e.g., \((S_{1},S_{2}),(S_{3},S_{4}),\ldots,(S_{M-1},S_{M})\). Restriction of the size of each subarray \(j\in[1,2]\) is required to make every fusioned subarray restricted to a maximum of 4 antenna elements per subarray, following a design criterion for linear arrays mentioned in [13]. The fusion of two subarrays contiguously located between rows happens with a probability of \(P\) and considering that subarrays with indices \(i_{S_{m}}\) and \(i_{S_{m+1}}\) have at least one antenna element adjacent between them. This allows expanding the size of the subarrays up to \(j\in[2,3,4]\) after fusion has been performed. For example, let \(S_{1}=[1,1,1,1,2,2]\) and \(S_{2}=[1,1,2,2,1,1]\) be a pair of rows that form a section of a planar array composed of a set of sequences \((S_{1},S_{2},\ldots,S_{m})\), as shown in Figure 2. The fusions of subarrays happen on indices \((i_{S_{1}},i_{S_{2}}=1,3)\) and indices \((i_{S_{1}}=5,\)\(i_{S_{2}}=4)\), but cannot be attainable for \((i_{S_{1}},i_{S_{2}}=5)\) due to the separation between antenna elements in both rows. #### 2.4.2 Three-Row Fusion

This case is similar to the previous case, with the option of incorporating a third row into the fusion process.