# Numerically Optimized Fourier Transform-Based Beamforming Accelerated by Neural Networks
## 2 FT-Based Beamforming Methodology for Reflecting PMS
In what follows, we assume that the incident electric field is linearly polarized along the \(x\)-axis, and, therefore, the induced currents on the PMS are also dominantly \(x\)-polarized. It can be shown that in this case we must define a set of polarization-corrected project patterns, \(F(u,v)\), as  
\[F(u,v)=\sqrt{1-\left(\frac{u}{k_{0}d}\right)^{2}}F_{\alpha}(u,v). \tag{3}\]  
Note that by using the terminology of phased array antennas, \(F_{\alpha}(u,v)\) can be regarded as the array factor and the square root term in front of it can be regarded as the radiation pattern of a single element of the array. Instead of using \(F(u,v)\) as a continuous function of two variables, let us take discrete samples of it, \(F_{kl}\), obtained as follows:  
\[F_{kl}=F\left(\frac{2\pi(k-\frac{M-1}{2})}{M-1},\frac{2\pi(l-\frac{N-1}{2})}{N -1}\right). \tag{4}\]  
Here, the indices \(k\in[0;M-1]\) and \(l\in[0;N-1]\) label the \((k,l)\)-th sample of the radiation pattern in the \((u,v)\)-space. Consequently, the necessary amplitude and phase distribution on the PMS aperture can be found by applying a discrete Fourier transform to the discretized project pattern, since, as is well known, the far-field radiation pattern of a rectangular aperture in such coordinates is equivalent to the Fourier transform of the aperture wave field. As was mentioned in the introduction, we neglect the amplitude of the Fourier-transformed pattern and just consider its phase. In addition to the aperture phase distribution dictated by the Fourier-transformed project pattern, in order to find the actual aperture field phase, \(\Phi_{mn}\), at a given PMS element, we must compensate for the varying delay acquired by the incident field when propagating from the feed antenna to the given element. Hence, by using the two-dimensional discrete Fourier transform, the aperture phase can be expressed as in Equation (5):  
\[\Phi_{mn}=k_{0}R_{mn}+\angle\sum_{k=0}^{M-1}\sum_{l=0}^{N-1}F_{kl}e^{-2\pi i \left(\frac{(m-\frac{M-1}{2})(k-\frac{M-1}{2})}{M}+\frac{(m-\frac{N-1}{2})(l- \frac{N-1}{2})}{N}\right)}, \tag{5}\]where we have used the notation \(\angle z\equiv\arg z\). Next, the phase \(\Phi_{mn}\) is used to obtain the complex phasor \(\Omega_{mn}=a_{mn}e^{i\Phi_{mn}}\), with the amplitude factor \(a_{mn}\) set to unity for all unit cells. Finally, the actual far-field electric field pattern (here we are interested in unnormalized magnitude pattern) produced by the PMS is expressed as follows:  
\[|\mathbf{E}(u,v)|\propto\sqrt{1-\left(\frac{u}{k_{0}d}\right)^{2}}\sum_{m=0}^ {M-1}\sum_{n=0}^{N-1}\frac{\Omega_{mn}e^{i\left(m-\frac{M-1}{2}\right)u+i\left( n-\frac{N-1}{2}\right)v-ik_{0}R_{mn}}}{R_{mn}}. \tag{6}\]  
This expression includes the spherical wave amplitude factor \(1/R_{mn}\) to account for the incident field propagation decay. When the actual pattern given by Equation (6) is compared with the project pattern, Equation (3), we have noticed that, despite fixing the magnitude to unity and keeping only the aperture phase information, the project pattern is accurately regenerated with acceptable sidelobe levels for some single-beam cases. However, in many other cases, the reconstructed pattern has multiple sidelobes with unacceptably high levels. Moreover, a shift may occur in the direction of the main beam. For instance, the project pattern and the reconstructed electric field pattern in the \((u,v)\)-space created by a \(10\times 3\) PMS with 50 cm focal distance that operates at 5 GHz and produces a single-beam directed at \((u_{0,1},v_{0,1})=(-0.47,-0.41)\) are depicted in Figure 4a,b. In these figures, the green circle represents the beam visibility range \(\sqrt{u^{2}+v^{2}}\leq k_{0}d\). The red oval depicts the protected area range, the meaning of which will be explained in Section 3.1. One can see that in this case the reconstructed main beam is wider than the project pattern main beam. In addition, the main beam is displaced from the original position and there are strong secondary maxima outside the visibility range.