# Numerically Optimized Fourier Transform-Based Beamforming Accelerated by Neural Networks
###### Abstract
Therefore, at the operating frequency of 5 GHz, each coupler may occupy an area of about 1 cm\({}^{2}\). Moreover, additional phase compensation lines are required to obtain the necessary phase shifts within the Butler matrix [18]. Thus, integrating the Butler network on the same PCB with antenna array elements may impose limits on the element placement density, especially when the number of input ports approaches the number of elements. On the other hand, among the more expensive actively controlled antenna systems, there are electronically/digitally controlled phased array antennas of various kinds, which can be assembled in linear, planar, or conformal configurations [19; 20; 21; 22; 23]. Such antennas can provide flexible electronic beam steering in less than milliseconds, eliminating any need for mechanical steering. They may produce multiple beams simultaneously and perform multifunction operations with independent beam control. Unlike passive phased array antennas, the active ones can also simultaneously emit and receive multiple radio signals  
Figure 1: Schematic of an 8 \(\times\) 8 Butler matrix comprising 8 phase shifters and 12 hybrid couplers [18]. The eight input ports are denoted as 1L–4L and 1R–4R. The eight output ports are marked as 1–8. at multiple frequencies in different directions, which is used for surveillance and tracking in radar applications. Although highly reconfigurable, these devices may also suffer from drawbacks such as limited coverage in both the azimuth and elevation planes (with a linear array, beam deflection is only possible in a single plane). Moreover, active phased array networks are very complex and require elements with sophisticated electronic controls that are available at a high cost, although their frequency agility can be limited [24; 25; 26]. During the last decade, an alternative concept that combines the advantages of the passive systems (e.g., low energy consumption and cost) and the active systems (e.g., flexible reconfigurability) has emerged: the concept of Programmable Metasurfaces (PMS) [27; 28; 29]. In recent years, PMS have been under active investigation. In particular, it has been demonstrated that PMS or Reflecting Intelligent Surfaces (RIS) can be used to adjust channel propagation conditions [30; 31]. In PMS-enhanced systems, the TX and RX antenna radiation patterns can be dynamically controlled to facilitate wireless communications in varying propagation environments. Indeed, it is known that microwave or optical metasurfaces (MS) can alter the propagation direction of the EM waves by creating phase gradients along their surface [32; 33]. In particular, the PMS may act as mirrors in reconfigurable reflector antennas. In such antennas, versatile beamforming can be achieved by controlling both the phase and amplitude of wavefronts reflected from an MS [34; 35; 36; 37]. However, in order to freely control the reflection amplitude at MS unit cells, active components (such as amplifiers or impedance inverters, etc.) may be needed. In this case, the effective surface impedance of the MS becomes complex. The sign of the real part of the surface impedance determines if there is attenuation or amplification of the signal locally reflected at that point. On the other hand, if the reflection amplitude control is not required, then the MS can be completely passive (reactive). Some form of amplitude control can be achieved even in purely reactive MS, e.g., by an approach that one may call \"surface mode engineering\", when the surface mode resonances supported by the MS are tuned in a way to redistribute the reactive power (i.e., power of local oscillations) along the MS.