## 3 Results and Verification
### Radiation Pattern of the Thinned Array  
With the array structure in Figure 2a, the 2D normalized array factors of the large-scale thinned array with 2400 elements for the \(\varphi\) = 0\({}^{\circ}\), \(\varphi\) = 45\({}^{\circ}\), \(\varphi\) = 90\({}^{\circ}\), and \(\varphi\) = 135\({}^{\circ}\) cut-planes are shown in Figure 5. From Figure 5, it can be seen that in the \(\varphi\) = 0\({}^{\circ}\), \(\varphi\) = 45\({}^{\circ}\), \(\varphi\) = 90\({}^{\circ}\), and  
Figure 4: Simulated radiation patterns of the array element in the cases of single element (Figure 2b) and in the array (Figure 2c) at 29 GHz. (**a**) E-plane (_xz_-plane). (**b**) H-plane (_yz_-plane).  
Figure 3: Simulated S-parameters of the array element in the cases of single element (Figure 2b) and in the array (Figure 2c).  
\(\varphi=135^{\circ}\) cut-planes, the radiation patterns in the sidelobe level regions meet the requirements. Figure 5a,c show that the first sidelobe levels are lower than \(-21.8\) dB at the \(\varphi=0^{\circ}\) and \(\varphi=90^{\circ}\) cut-planes, while Figure 5b,d show that the first sidelobe levels are lower than \(-22.6\) dB at the \(\varphi=45^{\circ}\) and \(\varphi=135^{\circ}\) cut-planes. To better illustrate the array factor, Figure 6 shows the 3D normalized array factor of the large-scale thinned array with 2400 elements. From Figure 6, it is apparent that, for each cut-plane with different azimuth angles, the gains at the sidelobe level regions are still small.  
To verify the array factor, the radiation patterns of the array model in Figure 2c and the radiation patterns of the array prototype in Figure 2d are simulated and measured, respectively. The simulated 3D radiation pattern is shown in Figure 7. Figure 7 shows that the proposed thinned array can form a sharp beam with a high gain. The simulated and measured normalized radiation patterns in the \(\varphi=0^{\circ}\) and \(\varphi=45^{\circ}\) cut-planes are shown in Figure 8. From Figure 8, it is evident that the simulated radiation patterns match the array factor very well. The high sidelobe level at the large elevation angle might come from the radiation of the fringing field. Also, Figure 8 shows that the measured radiation patterns agree with the simulated ones. The slightly higher sidelobe level might come from the metal screws and the measured error.  
Figure 5: Two-dimensional normalized array factor of the large-scale sparse array with 2400 elements. (**a**) \(\varphi=0^{\circ}\) cut-plane. (**b**) \(\varphi=45^{\circ}\) cut-plane. (**c**) \(\varphi=90^{\circ}\) cut-plane. (**d**) \(\varphi=135^{\circ}\) cut-plane.  
Figure 6: Three-dimensional normalized array factor of the large-scale thinned array with 2400 elements.  
In addition, because the array elements are large and the desired sidelobe level is complex, other methods, like nonlinear optimization methods (DE, PS, SA, IWO, or GA), decomposition, Fourier transform, etc., cannot easily converge. The present branch and bound technique with convex optimization is an efficient method for the synthesis of the large-scale thinned array with specific radiation pattern restrictions.