# Numerically Optimized Fourier Transform-Based Beamforming Accelerated by Neural Networks
## 8 Conclusions  
An inverse design method has been introduced, which combines FT-based beamforming, numerical optimization, and machine learning techniques. In our method, first, the aperture field distribution is obtained by applying Fourier transform to the desired far-field radiation pattern. Then, the aperture field amplitude information is discarded; however, the phase information is retained and is used as an input to the developed optimization algorithm that further improves beamforming accuracy. Next, this optimizing FT-based beamforming method is used to generate datasets of required aperture phase distributions vs. a number of single- and double-beam scenarios. The generated datasets are used to train a Beamforming Feed-forward Neural Network (BFNN), which is able to generate the necessary phase distribution data in a fraction of second. Moreover, to provide control of the PMS and generate the necessary controlling voltages for the unit cells of the PMS, we have proposed to use another trained Control Neural Network (CTRLNN), which, in practical applications, can also perform additional functions, such as compensating for the unit cell coupling in the PMS, etc. By employing the developed approach, the required phase distribution over the PMS and, consequently, the required bias voltages to control the PMS unit cells are calculated in real-time. The BFNN and CTRLNN networks have been trained on the generated datasets for a number of beamforming scenarios. The performance of the neural networks has been validated. The correlation between the BFNN output and the results of the optimizing FT-based beamforming method reaches 98% in the best cases, and is generally above 90% in other cases. The validated CTRLNN performance is also high, with the control voltage MSE below \(10^{-4}\) V\({}^{2}\). Overall, we have observed that the desired radiation patterns are recovered with high precision when either the optimizing FT-based method or the trained ANNs are used, despite ignoring the magnitude information at the Fourier transform stage. However, at high inclination angles, parasitic sidelobes may appear in the reconstructed patterns, a \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Ref.**} & **Frequency** & **Compon. 1** & **Type** & **Beam** & **Tunable** & **R.S. 2** & **C.C. 3** \\  & **[GHz]** & **per Cell** & **Type** & **Number** & & & \\ \hline This Work & 5 \& 10 & 4 Var. 4 & Analog & 1/2/M.B. 5 & Yes & Yes & Class I \\ [52] & 10 & – & – & 1 & No & No & Class I \\ [53] & 8-20 & – & Mech. 6 & 1 & Yes & Yes & Class I \\ [54] & 10–12 & 1 PIN & Digital & 1/2/4 & Yes & No & Class I \\ [55] & 5.8 & 2 Var. & Analog & 1 & Yes & Yes & Class II \\ [27] & 8–15 & 1 Bias & Digital & 1/2/4 & Yes & Yes & Class II \\  & & Diode & (1/2-bit) & 1/2/4 & Yes & Yes & Class II \\ [56] & 9.5 & 2 PIN & Digital & 2 & Yes & Yes & Class II \\  & & Diodes & (2-bit) & & & & \\ [57] & 3 SPDT 7 & Digital & & & & & \\  & & & (1-bit) & & & & \\ [58] & 8 & 2 PIN & Analog \& & & & \\  & & Diodes & Digital & & & & \\ [59] & 28 & 1 PIN & Digital & & & & \\  & & Diode & (1-bit) & & & & \\  & & 1 DFF 9 & & & & & \\  & & SR 10 \& & & & & \\  & & Dec. 11 & & & & & \\ \hline \hline \end{tabular}  
* Component; 2 Remembers State; 3 Complexity Class; 4 Varactor; 5 Multi-Beam; 6 Mechanical; 7 Single-Pole-Double-Throw switch chip; 8 Multi-Purpose; 9 D Flip-Flop; 10 Shift Register; 11 Decoder. 1 Under no circumstances this table should be considered as a complete list of representative works; it is only a selection based on our own preferences and limited knowledge.  
\end{table} 
Table 1: Comparison of different beamforming MS designs 4.  
phenomenon that is related to the rather small number of unit cell rows and columns in the PMS sample on which the study was performed. Conceptualization, S.M. ; methodology, S.M., K.K. and A.A.; software, K.K. and A.A.; validation, K.K., A.A. and S.M. ; formal analysis, S.M., K.K. and A.A.; investigation, K.K. and A.A.; resources, S.M. ; data curation, K.K.