# Numerically Optimized Fourier Transform-Based Beamforming Accelerated by Neural Networks
## 4 Optimization Results for Single-Beam and Dual-Beam Cases  
To suppress sidelobes and adjust the main beams to realize the target pattern with a higher accuracy, an optimization algorithm, fmincom, which finds a minimum of constrained nonlinear multivariable function, was employed. When evaluating the cost function from Section 3.2, the parameters \(C_{1}\), \(C_{2}\), and \(t\) were set to 10, 1, and 1.3, respectively, in order to increase the convergence rate and enhance the optimization results. These values have been obtained by a \"trial and error\" method by running several test optimizations and evaluating their outputs. The coefficients \(\alpha_{i}\) have been set to \(1/\max|F(u,v)|_{PA}\) within each PA, since our objective is to recover all beams with the same magnitude. The coefficient \(\beta\) was determined based on the estimated attainable maximum of \(|\mathbf{E}(u,v)|\) within the VA: \(\beta\approx\max|\mathbf{E}(u,v)|_{VA}\). This value can be determined only after a few trial optimizations. As an initial guess, one can use the value obtained with unoptimized electric field pattern. The optimal value of \(\beta\) is larger than that. In some cases that we have considered, the optimal value was found to be about five times greater than the initial guess value. To test the proposed optimization approach, we have performed a number of numerical simulations. In Figure 8, we show a couple of example phase distributions over the PMS before and after applying the optimization process for single- and dual-beam scenarios for a PMS composed of \(10\times 3\) unit cells. Despite the fact that the initial phase distributions [Figure 8a,c] show irregular patterns, the optimization process recovers the expected phase gradients over the PMS surface in the two considered scenarios [Figure 8b,d]. Figure 8: Phase distributions over a \(10\times 3\) PMS in the single-beam and dual-beam scenarios. (**a**) Single-beam scenario before applying optimization. (**b**) Single-beam scenario after applying optimization. (**c**) Dual-beam scenario before applying optimization. (**d**) Dual-beam scenario after applying optimization. Figure 7: Evaluation of optimization process performance for single-beam and dual-beam patterns generated by a \(10\times 3\) PMS. The plots show evolution of the cost function values, step size, and optimality vs. number of iterations. (**a**) Single-beam scenario. (**b**) Dual-beam scenario. Figure 9 shows the optimized normalized electric field pattern in the same two scenarios, depicted in the \((u,v)\)-space [Figure 9a,c] and in the spherical coordinates in 3D space corresponding to the VA [Figure 9b,d]. In the single-beam scenario, tilting of the main beam in the \(xz\)-plane with respect to the normal to the PMS (the \(z\)-axis) is clearly visible. Inclination of the beam in \(yz\)-plane is also recognizable; however, because the PMS size along the \(y\) direction is smaller, the beam directivity in this plane is poor. In the realized dual-beam scenario shown in Figure 9b,d, the two beams have symmetric inclinations with respect to the PMS normal. The optimization process was able to recover the two main beams, despite the fact that the initial aperture phase data, if used directly, would result in a merged beam like that shown in Figure 5b. Finally, Figure 10 shows the project pattern, the electric field pattern before optimization, and the electric field pattern after optimization in \(xz\)- and \(yz\)-planes depicted in polar coordinates. As is evident from Figure 10a-e, omitting the amplitude information produces suboptimal radiation patterns (pink dotted curves). However, by applying the proposed optimization strategy, the project patterns (black dashed curves) are recovered with a high precision (blue solid curves). From these examples, one can also notice that in single-beam scenarios with strong inclinations of the main beam, besides the main beam, there are parasitic sidelobes in the VA.