## 4 Conclusions  
Based on previous studies, we discussed a universal approach for calculating array factors in the form of 3D random arrays, including conformal arrays. We propose an amplitude relational method based on Bernstein polynomial optimization and phase compensation. The proposed phase calculation method is not limited to specific grid structures (e.g., rectangular and triangular grids), but it is widely applicable to any 3D phased array. Unlike previous studies on phase compensation in conformal arrays, the method does not require different expressions for different structures; rather, one expression can be used for any 3D structure. The proposed phase calculation method was verified by simulating uniform amplitude feeding using an \(8\times 8\) random array of a cavity-backed patch antenna operating at 10 GHz. Next, we developed a relationship between amplitude tapering and phase compensation by analyzing the results obtained using GLPSO in the Bernstein polynomial. Unlike previous studies on conformal array amplitudes, optimization is not performed for different element shapes, array shapes, and radiation patterns; rather, an expression is developed based on the optimization results obtained in a case with which the amplitude for the radiation pattern is quickly derived. The proposed method needs to be improved as optimization is not performed each time to obtain the best results in each case, as in previous studies. However, universal equations can be derived, and prediction and control of the performance of patterns for arbitrary 3D arrays can be achieved in a short time. To verify the proposed method, we applied phase compensation and amplitude tapering to an array and verified the performance through simulations. Future work will involve the fabrication of the proposed cavity-backed patch antennas and its implementation in an \(8\times 8\) random grid array system for experimental verification. The \(8\times 8\) random grid array shown in this paper can be measured by manufacturing a metal jig that matches the shape of the array. To date, studies are ongoing to develop and commercialize conformal arrays. Based on our results herein and those of previous studies, a universal array factor can be obtained for all array types through phase compensation and amplitude  
Figure 14: Comparison of \(8\times 8\) random-grid amplitude-tapering 2D radiation patterns: (**a**) array factor of conformal array calculated with MATLAB, (**b**) comparison of simulated radiation patterns for uniform and amplitude tapering feeding.  
tapering for conformal arrays. For all phased arrays (such as planar and conformal arrays), mathematical and beam synthesis methods can be developed to control SLL without the need for optimization for each case.  
Conceptualization, J.P., H.J.L. and K.C.H.; methodology, J.P. and H.J.L.; software, J.P. and S.T.-V.; validation, J.P., H.J.L., D.P., Y.K.J. and K.C.H.; formal analysis, J.P. and S.T.-V.; investigation, J.P., H.J.L. and D.L.; resources, J.P. and D.P.; data curation, J.P., H.J.L., S.T.-V. and D.P.; writing--original draft preparation, J.P.; writing--review and editing, J.P., H.J.L. and K.C.H.; visualization, J.P.; supervision, K.C.H.; project administration, K.C.H. All authors have read and agreed to the published version of the manuscript.  
This work was supported by a grant-in-aid of HANWHA SYSTEMS.  
Not applicable.  
All Data has been included in study.  
The authors declare no conflict of interest.