# Numerically Optimized Fourier Transform-Based Beamforming Accelerated by Neural Networks
## 6 Prototyping and Preliminary Experimental Results  
In this section, we report preliminary experimental results obtained with a PMS prototype with \(3\times 10\) unit cells. The prototype was designed using the analytical-numerical model developed in [29]. To control direction of the reflected beam, the phase gradient along the PMS was generated by applying varying dc voltages to the varactors that belong to neighboring unit cells. The phase gradient is approximately proportional to the bias  
Figure 14: The optimized electric field patterns and the ANN-predicted electric field patterns in polar coordinates, as produced by a \(3\times 10\) PMS. (**a**) Single-beam scenario in polar coordinates at 5 GHz, \(xz\)-plane. (**b**) Same scenario, \(yz\)-plane. (**c**) Same scenario, \(xy\)-plane. (**d**) Single-beam scenario in polar coordinates at 5 GHz, \(xz\)-plane. (**e**) Same scenario, \(yz\)-plane. (**f**) Same scenario, \(xy\)-plane. (**g**) Single-beam scenario in polar coordinates at 5 GHz, \(xz\)-plane. (**h**) Same scenario, \(yz\)-plane. (**i**) Same scenario, \(xy\)-plane. (**j**) Dual-beam scenario in polar coordinates at 10 GHz, \(xz\)-plane. (**k**) Same scenario, \(yz\)-plane. (**l**) Same scenario, \(xy\)-plane. Figure 15: The corresponding MSE as a function of the epoch number and reflection phase versus applied bias voltage. (**a**) MSE as a function of the epoch number. (**b**) Reflection phase versus applied bias voltage. voltage gradient. In this way, we achieved beam steering in the direction of the applied voltage gradient. A view of the prototype PMS installed in a plastic holder is shown in Figure 16a. The PMS radiation pattern measurements were performed using the anechoic chamber facilities of the Department of Electronics, Telecommunications and Informatics of University of Aveiro. The measurement setup is shown in Figure 16b. In the experimental setup, the TX antenna (a circular horn) and the PMS prototype are fixed on the same rotating support so that the incidence is always close to normal. The incident wave is linearly polarized with the electric field vector parallel to the elongated side of the PMS. The RX antenna (a Vivaldi antenna, not seen in the figure) is fixed and is placed a few meters away from the rotating support. When the support rotates, the RX antenna measures the signal scattered by the PMS sample at varying angles. Figure 16c depicts the results of the radiation pattern measurements in the \(xz\)-plane (\(\theta=0^{\circ}\) corresponds to the normal to the PMS) alongside the results of the analytical beamforming model and full-wave simulations in SIMULIA CST Studio Suite, for the cases with and without the bias voltage. The applied voltage gradient corresponds to the reflection phase variation from \(-50^{\circ}\) to \(50^{\circ}\) (varactors' capacitance values vary from 0.25 to 0.2 pF) in the direction of the long side of the PMS, which should produce the main beam pointing at \((\theta_{0},\,\varphi_{0})\approx(19^{\circ},0^{\circ})\). As can be seen in this figure, the maximum of the PMS radiation pattern without the applied voltage points at \(\theta_{0}\approx 0^{\circ}\), as expected (green dashed curve). When the voltage gradient is applied, the maximum of the radiation pattern shifts to \(\theta_{0}\approx 17^{\circ}\) (black dash-dotted curve). The radiation pattern generated using the beamforming methodology developed in this article (blue solid curve with a maximum at \(\theta=19^{\circ}\)) and the result of full-wave simulations in SIMULIA CST Studio Suite (red dash-dotted curve with the maximum at \(\theta_{0}\approx 18^{\circ}\)) are also depicted in the same figure. We may conclude that beam inclinations in all these cases are in good agreement. However, the beam shape of the prototype PMS is distorted, and sidelobes with relatively high amplitude are clearly visible.