## 2 Derivation of 3D Array Factor
### Amplitude Tapering Method of Array Factor (Bernstein Polynomial Using GLPSO)
\begin{split} costfunction(SLL_{target}=-25)=|\,SLL_{target}- FirstSidelobelevel|^{3}\\ +|\,SLL_{target}-SecondSidelobelevel|^{2}\\ +max(-SLL_{target}+Sidelobelevel,(SLL_{target}-10)-Sidelobelevel )^{2}\end{split}  \tag{13}\]  
The optimization cost function was set as follows, because the SLL values of the first and second SSLs are very close to the SLL value of the desired level, and other SLLs were set to naturally have SLLs within a certain value. Figure 9 shows a plot of the array factors with respect to the optimized Bernstein polynomial variable. In the array factor graph, it can be seen that the array factor of a linear array with the same number of elements is similar to that of a conformal array. Analyzing the output data after verifying the performance of GLPSO and validity of the cost function, we confirmed that \(N_{0}\) and \(N_{1}\) correspond to a certain curvature, and the number of elements is fixed. When c changes, it has a specific inflection point and can be expressed generally for a value of c above a specific inflection point. Figure 10 shows the changes in \(c_{0}\) and \(c_{1}\) when dx = 0.5\(\lambda\) (15 mm), dz is randomly set to a \(1\times 16\) conformal array at 10 GHz, and \(N_{0}=N_{1}=5\). (Since it is a symmetric structure, we assume that \(c_{0}=c_{1}=c\).) Figure 9: Array factors (\(-25\) dB) of conformal array optimized based on GLPSO: (**a**) conformal array type 1; (**b**) conformal array type 2; comparison of radiation patterns of (**c**) \(1\times 16\) conformal array type 1 and \(1\times 16\) planar array, (**d**) \(1\times 16\) conformal array type 2 and \(1\times 16\) planar array, (**e**) \(1\times 24\) conformal array type 1 and \(1\times 24\) planar array, (**f**) \(1\times 24\) conformal array type 2 and \(1\times 24\) planar array, (**g**) \(1\times 30\) conformal array type 1 and \(1\times 30\) planar array, and (**h**) \(1\times 32\) conformal array type 2 and \(1\times 32\) planar array. By arranging the experimentally obtained data in Figure 10, we find that the case of a \(1\times 16\) conformal array can be approximated as a function of \(c\)-value according to SLL (Figure 11). We obtain an expression by analyzing the trend of the optimization results that fit the array case. Based on this, for a specific conformal array, the relation between \(c\) and SLL can be obtained based on \(N_{0}\) and \(N_{1}\), which are obtained through optimization. Figure 12a shows the actual shape of the \(1\times 16\) conformal array used for the CST simulation based on the information used for the MATLAB optimization. Figure 12b compares the radiation pattern of the conformal array without beam scanning and the planar-array radiation pattern to observe only the amplitude effect. The red solid line indicates \(\mathrm{dx}=0.5\lambda\) (15 mm) for the \(1\times 16\) planar array at 10 GHz and the radiation pattern when \(-25\) dB Taylor distribution is applied. The blue dash line indicates \(\mathrm{dx}=0.5\lambda\) (15 mm) at 10 GHz, and \(\mathrm{dz}\) represents the randomly set \(1\times 16\) conformal array radiation pattern. Variables of the Bernstein polynomial applied to the \(1\times 16\) conformal array were set to \(N_{0}=N_{1}=5\), \(c_{0}=c_{1}=0.3\) according to the expressions obtained above, and Figure 11 shows that the first SLL is \(-26.5\) dB. Simulation results show that the first SLL is \(-26.57\) dB in the case of conformal arrays (Figure 12b), which is comparable to \(-26.5\) dB. In addition, the radiation pattern of the \(1\times 16\) planar array shows a decreasing trend with similar values, not only the shape of the pattern and the first SLL, but all SLL values. Figure 11 and the simulated radiation patterns (Figure 12b) confirm that it is possible to derive a conformal-array amplitude tapering relationship that can be universally used depending on the situation through optimization results and variable control. Figure 10: Relation graph between sidelobe level and variable \(c\) of Bernstein polynomial.