## 2 Derivation of 3D Array Factor
### Amplitude Tapering Method of Array Factor (Bernstein Polynomial Using GLPSO)  
We confirmed in the preceding subsections that a phase can be calculated for arbitrary arrays that can specify the element position in the 3D coordinate system using the 3D array factor in cases of the uniform amplitude feeding. However, as with planar arrays, it is necessary to adjust the amplitude for the desired side-lobe level (SLL) in conformal arrays. Thus, Equation (10) considering the amplitude \(w_{i}\) of the \(i\)th element with respect to the previously proposed \(AF_{uniform}\) expresses the array factor for the final 3D array. Figure 8: Comparison of \(8\times 8\) random grid array uniform feeding 2D radiation patterns: (**a**) array factor of conformal array calculated using MATLAB and (**b**) radiation pattern from EM simulation. Figure 7: \(8\times 8\) random grid array shape: (**a**) 3D view and (**b**) XY plane view. \[0\leq w_{i}\leq 1,i=1,2,3,\cdots,I \tag{9}\]  
\[ \begin{split} AF_{3D}=\sum_{i=1}^{I}w_{i}\times e^{|\frac{2\pi}{ \lambda}(x_{i}\sin(\theta)\cos(\phi))-\frac{2\pi}{\lambda}(x_{i}\sin(\theta_{i })\cos(\phi_{0}))|}\\ \times e^{|\frac{2\pi}{\lambda}(y_{i}\sin(\theta)\sin(\phi))-\frac {2\pi}{\lambda}(y_{i}\sin(\theta_{0})\sin(\phi_{0}))|}\times e^{|\frac{2\pi}{ \lambda}(z_{i}\cos(\theta))-\frac{2\pi}{\lambda}(z_{i}\cos(\theta_{0}))|}\end{split}  \tag{10}\]  
However, as with a phase, the conformal array cannot achieve the desired performance using only the amplitude method for planar arrays due to the structural differences and nonlinearity. Consequently, several amplitude tapering methods have been studied for applications in conformal arrays. Conformal-array amplitude tapering has been performed using the Particle Swarm Optimization (PSO) method on Bernstein polynomials based on the structure for conformal array situations, and only amplitude was optimized to increase radiation efficiency and lower SLL [2]. However, it can be used universally because phase-related parts are excluded. It can also be used for a particular case. Herein, based on phase compensation, we employed GLPSO, which is more advanced than PSO, GLSPO combines Genetic Algorithm (GA) and PSO, and whenever an iteration is performed, the entire cluster moves in the optimization direction and causes specific objects to mutate and converge to the global optimum at a high speed [17]. The GLPSO setting was performed using 2000 particles, probability of mutation = 1%, probability of tournament = 20%, and the maximum number of repetitions of 1000. By analyzing the optimized data, the trend of the variable was observed. We propose the generalizability of amplitude tapering of conformal arrays by analyzing the trend of the optimization results. The Bernstein polynomial used for optimization is expressed as follows. In this case, the objective is to optimize the Bernstein polynomial variables \(C_{0}\), \(C_{1}\), \(N_{0}\), and \(N_{1}\). \(A\) is set to 0.5 because the shape of the arrangement is symmetric, and beam steering is not performed. \[0\leq A\leq 1,0\leq C_{0}\leq 1,0\leq C_{1}\leq 1,0<N_{0},0<N_{1} \tag{11}\]  
\[f(u)= \begin{cases}C_{0}+\frac{1-C_{0}}{A^{N_{0}\lambda}(1-A)N_{0}^{(1-A)}} \cdot u^{N_{0}A}\cdot(1-u)^{N_{0}(1-A)},0\leq u\leq A\\ C_{1}+\frac{1-C_{1}}{A^{N_{1}\lambda}(1-A)N_{1}^{(1-A)}}\cdot u^{N_{1}A}\cdot(1 -u)^{N_{1}(1-A)},A\leq u\leq 1\end{cases}  \tag{12}\]  
For optimization, the element positions of \(1\times 16\), \(1\times 24\), \(1\times 30\), and \(1\times 32\) conformal arrays were used. All arrays had an x-direction element spacing of dx = 0.5\(\lambda\) (15 mm) at 10 GHz, and the z-direction element spacing dz was randomly set. The array factors were calculated using MATLAB for the given arrays and compared with the \(-25\) dB Taylor array factor using the Taylor distribution for the planar arrays. The cost function used to optimize the array factor is the same as Equation (13). \[