# Numerically Optimized Fourier Transform-Based Beamforming Accelerated by Neural Networks
###### Abstract  
Conventional beamforming methods for reconfigurable reflector antennas assume full control over the amplitude and phase of the reflected field. Here, we develop a novel beamforming methodology for reflecting Programmable Metasurfaces (PMS) with capacitive memory. Although utilizing such fully reactive PMS simplifies antenna design and reduces energy consumption, the PMS reflection magnitude is unity and thus a global optimization of the reflection phases over the PMS unit cells is required in each beamforming scenario. We propose an implementation of such an optimization method rooted in the traditional Fourier transform-based beamforming and evaluate its performance. Additionally, we show that a pair of trained feed-forward neural networks (FFNN) with one input, one hidden, and one output layer can replace time-consuming global optimizations in the case of a PMS comprising \(3\times 10\) unit cells. We train the FFNNs on a dataset obtained for typical single- and dual-beam beamforming scenarios. After training, the FFNNs perform requested beamforming tasks within a fraction of second and with about the same accuracy as the original optimization algorithm. The proposed methodology may find applications in future mobile telecommunication systems that require real-time beamforming on low-end hardware. The same beamforming methodology can be also employed in short-range wireless power transfer systems. metasurface; Fourier transform-based beamforming; optimization; neural networks +
Footnote †: journal: _Applied_  
0000-0002-4000-0000-0000-0000-0000-0000-0000-0000-0000-0000-0000-0000-00000-00000-00000-00000-00000-00000-00000-00000-00000-  
channel diversity [1; 2; 3; 4; 5; 6]. Usually, elementary antennas in such MIMO systems (or antenna arrays) can be realized with inexpensive Printed Circuit Board (PCB)-based techniques (e.g., microstrip or coplanar) and can be used for transmitting and receiving all kinds of information [7; 8; 9; 10; 11; 12; 13; 14]. Several methods exist to create and steer antenna array beams, ranging from the classical Butler matrix approach to the actively developed smart and reconfigurable antenna concepts. In 1961, [PERSON] and [PERSON] proposed their methodology for passive beamforming [15; 16]; this was a development of an earlier work by [PERSON] [17]. Butler matrices are commonly used for feeding various antenna arrays with wide angular coverage [18]. A typical system may comprise an \(N\times N\) matrix of fixed-value phase shifters and hybrid couplers, where \(N\) is a power of 2. An antenna array equipped with such a matrix may have \(N\) input ports (the beam ports) to which power is applied and \(N\) output ports (the element ports) to which \(N\) antennas are connected. The Butler matrix provides power to the elements with a progressive phase difference between the elements such that the antenna main beam points in a selected direction. The beam direction is controlled by switching the input power between the beam ports. More than one beam or even all \(N\) of them can be activated simultaneously. Since [PERSON]'s matrix can be realized with microstrip lines, it is a cost-effective solution for achieving directional beam control without the need to deploy an array of expensive active phase shifters. Hence, its fabrication cost can be just a small fraction of the cost of a full-fledged antenna array feeding network. However, its relative hardware simplicity comes at the cost of having to deal with a bulky structure and a fixed set of predefined beam directions. Indeed, Figure 1 depicts a Butler matrix-based beamforming system [18] comprising eight input ports and eight antenna ports. This system is able to produce up to eight fixed beams with inclinations in the range of about \(\pm 55^{\circ}\). Per antenna element, this design requires one input feed, one phase shifter, and one and a half hybrid couplers. In typical designs, the 90-degree hybrid couplers are formed by four segments of quarter-wavelength microstrip lines.