# Numerically Optimized Fourier Transform-Based Beamforming Accelerated by Neural Networks
## 2 FT-Based Beamforming Methodology for Reflecting PMS  
We consider a PMS-based beamforming antenna with separately controllable \(N\times M\) reflecting elements, which are the unit cells of the PMS. Here, \(N\) and \(M\) are the numbers of rows and columns of the PMS elements, respectively. Moreover, we suppose that the antenna operates at either \(f_{1}=5\) GHz or \(f_{2}=10\) GHz. These two frequencies will be used to conduct simulations and numerical studies of the proposed PMS and the beamforming methodology. The period of the PMS structure, which is also the distance between the element centers along \(x\) and \(y\) directions, is \(d=6.8\) mm, which is about \(\lambda_{1}/9=\lambda_{2}/4.5\), where \(\lambda_{1}\) and \(\lambda_{2}\) are the free-space wavelengths at the two considered frequencies. A feed antenna with low directivity illuminates the PMS, Figure 3a, and the far-field radiation pattern is formed after the PMS reflects the illuminating field, Figure 3b. In Figure 3a, \(R_{f}\) is the distance from the feed antenna to the PMS plane (the focal distance) and \(R_{mn}\) (denoted as \(R(m,n)\) in the figure) is the radial distance from the same antenna to the \((m,n)\)-th element of the PMS, Equation (1):  
\[R_{mn}=\sqrt{R_{f}^{2}+\bigg{(}\Big{(}m-\frac{M-1}{2}\Big{)}^{2}+\Big{(}n-\frac {N-1}{2}\Big{)}^{2}\bigg{)}d^{2}}. \tag{1}\]  
Radiation patterns are typically expressed in the spherical coordinate system. However, for us it will be more convenient to specify the antenna patterns in the \((u,v)\)-space with the orthogonal coordinates \(u\) and \(v\) that are related to the spherical angles \(\theta\) and \(\varphi\)  
Figure 3: Reconfigurable MS-based reflect array antenna. (**a**) Illuminating a reconfigurable MS-based reflect array antenna by a source antenna. (**b**) Reflected beam at the prescribed direction produced by a reconfigurable MS-based reflect array antenna. Figure 2: Conceptual diagram of numerically optimized FT-based beamforming accelerated by FFNN. as follows: \(u=k_{0}d\sin\theta\cos\varphi\), \(v=k_{0}d\sin\theta\sin\varphi\), where \(k_{0}\) is the free-space wavenumber, which equals either \(2\pi/\lambda_{1}\) or \(2\pi/\lambda_{2}\), according to the selected operating wavelength. It makes sense to consider a set of aperture patterns, \(F_{\alpha}(u,v)\), represented by superpositions of sinc-like terms, where \(\text{sinc}\ x=\frac{\sin\pi x}{\pi x}\), Equation (2),  
\[F_{\alpha}(u,v)=\sum_{i=1}^{L}a_{i}\,\text{sinc}\,\frac{s_{x}(u-u_{0,i})}{2\pi }\,\text{sinc}\,\frac{s_{y}(v-v_{0,i})}{2\pi}, \tag{2}\]  
as these are good approximations for the elementary patterns of a rectangular aperture of \(s_{x}d\) by \(s_{y}d\) size. Here, \(s_{x}\) and \(s_{y}\) are aperture dimensions (along the \(x\)- and \(y\)-axes) normalized to the PMS period \(d\). In this study, \(s_{x}=M\) and \(s_{y}=N\). Increasing or decreasing the aperture sizes makes the beams narrower or wider, respectively. The amplitude factors \(a_{i}\) are set to unity in this study for simplicity. However, as is explained in Section 3.2, each beam can have a separate scale factor assigned to it during the aperture phase optimization process. Physically, \(F_{\alpha}(u,v)\) represents an approximation of the radiation pattern of a dense array of isotropic emitters. The pattern has \(L\) main beams pointing at the directions in the \((u,v)\)-space defined by vectors \((u_{0,i},v_{0,i})=(k_{0}d\sin\theta_{i}\cos\varphi_{i},k_{0}d\sin\theta_{i} \sin\varphi_{i})\), where \(\theta_{i}\) and \(\varphi_{i}\) are the spherical angles corresponding to the \(i\)-th beam direction. Due to vectorial nature of the EM fields, the radiation pattern of a reflecting PMS must include a correction that takes into account the direction of the surface electric currents induced on the PMS. This is determined by the incident electric field polarization produced by the feed antenna.