# Numerically Optimized Fourier Transform-Based Beamforming Accelerated by Neural Networks
###### Abstract
When comparing the PMS-based systems with the Butler matrix-based ones, one may notice that, per unit cell, the PMS-based systems typically need just a few Surface-Mounted (SMT) components with dimensions on the order of 1 mm or less, whilst Butler matrices require a network of hybrid couplers and phase shifters realized with microstrip lines whose dimensions are mandated by the operation wavelength and the power handling requirements. Therefore, in PMS-based systems, the unit cell size is not as restricted by the size of the auxiliary (i.e., control) components. At microwave frequencies, this enables realizations with unit cell dimensions significantly smaller than the wavelength (e.g., from 1/10 to 1/5 of the wavelength). Having smaller unit cells is important in complex beamforming scenarios as it improves accuracy of the aperture field representation. For example, a single-layer passive MS is proposed in [38; 39] to control the phase and amplitude of the EM field in an antenna aperture plane. The MS is intended to transform a cylindrical incident wave into a directive far-field pattern with the main beam inclined by 30\({}^{\circ}\) with respect to the MS normal. A fully reactive MS has been attained by applying an optimization algorithm to generate the desired near and far fields. In their method, the MS surface impedance as a function of location is initially obtained from the desired aperture field distribution. However, the impedance found in this way is complex with the real part changing sign at different locations on the MS. Next, the real part is made negligible by applying a global optimization to the surface impedance distribution with a goal to obtain a merely reactive MS, while keeping the same far-field radiation pattern. In a nutshell, such optimization adjusts the MS surface reactance at different locations in order to generate a proper evanescent wave spectrum near the MS and maintain the surface waves needed to redistribute the reactive power on the MS and produce the desired far-field pattern. Metaheuristic optimization and Artificial Intelligence (AI) algorithms such as Particle Swarm Optimization (PSO), Genetic Algorithms (GA), and various Artificial Neural Network (ANN) algorithms have been employed in the design of metasurfaces for various applications [40; 41; 42]. In particular, Machine Learning (ML) and ANN approaches have gained massive attention in recent years. Indeed, ML concepts have already helped to solve many practical problems and demonstrated promising results in optimization, identification, and prediction problems in systems that exhibit nonlinear, dynamic, and complex behaviors. We believe that Feed-Forward Neural Networks (FFNN) can open new research venues in beamforming and highly efficient wave manipulation [43; 44; 42], because FFNN are able to model complex nonlinear systems with only a limited set of input parameters after suitable training is performed. In this article, we supersede the traditional Fourier transform-based (FT-based) beamforming methods that rely on the well-known link between the far-field pattern and the antenna aperture field by proposing a Fourier transform-based beamforming enhanced with a pretrained FFNN. Indeed, in many works on reconfigurable beamforming MS, it is assumed that the required phase distribution on the MS aperture is determined by the Fourier transform of the far-field radiation pattern. However, additional steps are necessary in order to increase beamforming accuracy. Preliminary concepts of how FFNN could be used to implement these steps were introduced in [45; 46]. As was discussed earlier, although simultaneous amplitude and phase control of the aperture field is, in principle, possible with active PMS, it leads to increased complexity and fabrication costs. As a measure of complexity, one can use the number of controllable elements and other electronic components per unit cell in a given design. For example, the RIS described in [47] uses only two pin diodes per unit cell to adjust the reflection phase.