## 2 Antenna Synthesis and Modeling
### Optimum Lattice Architecture  
The antenna lattice architectures must be devised as the best trade-off between performances and reduced number of elements, this latter constraint being motivated by the need to limit the number of active channels in the receiver and transmitter architectures. Whilst a regular circular lattice would have good scanning capabilities, the number of elements is high and the SLL is poor unless amplitude tapering is exploited. Amplitude tapering is generally undesired since, especially if there are substantial differences in the amplitudes, it reduces both aperture efficiency and power amplifier (PA) efficiency. Possible solutions for uneven yet semi-regular dispositions of elements like the non-uniform sunflower [10] would allow for SLL control without tapering, yet with somewhat inferior scanning capabilities and still a non-negligible amount of elements. The proposed approach is hence that of a thinned array [11; 12; 13; 14; 15; 16; 17]. In this kind of array, the optimum lattice is found by removing specific elements from the regular lattice. Such a removal, if optimized, leads to an intrinsic amplitude tapering (elements are generally more scattered at the outskirts of the array) and, if amplitude tapering proves necessary for SLL control anyway, the dynamic range, the ratio between maximum and minimum feed amplitude, proves to be generally much lower than for the full array, thanks to sparsity, and hence much less critical for PA efficiency. Furthermore, thinning is robust with respect to scanning since the thinned lattice is somewhat irregular and this is known to provide good scanning [17; 18; 19; 20; 21]. It is, anyway, important to notice that, backing the antenna, a beamforming network (BFN) section will be needed, and some regularity in the element deployment would be preferable to reduce costs, industrialization, testability, and installability [22; 23; 24]. This is again at a premium on a thinned array as compared to a random element distribution. Furthermore, in this paper, a modular approach has been implemented, by forcing the optimized solution to comprise a central disk and 12 identical sectors [25; 26; 27]. \begin{table}
\begin{tabular}{c c} \hline \hline
**Requirements** & **Value** \\ \hline Operation frequency range & 27.5–31 GHz \\ Coverage & \(\pm\)55\({}^{\circ}\) \\ SLL & \(<\)–25 dB \\ Grating lobes & Out of Earth \\ Directivity & \(>\)30 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Summary of requirements and constraints of the RX/TX antenna. The basic concept is to start from a regular array with elements deployed in concentric rings. Given the desired distance between elements \(d_{0}\), a central element is surrounded by other elements deployed on circles of radii \(nd_{0}\) with \(n=1,\ldots,M\). In the present case \(d_{0}=0.5\lambda_{0}\) is considered, \(\lambda_{0}\) being the free space wavelength at central frequency. \(M=13\) is necessary to comply with beam-width requirements, leading to a \(N_{E}=566\) radiating elements array (Figure 1a). The resulting antenna is then divided into 13 sectors: a circular central sector \(S_{0}\), with 7 elements, and 12 identical peripheral sectors, named \(S_{I}\) to \(S_{XII}\), each comprising 42 elements. The chosen number of sectors, 13, is a trade-off between a minimum number of different sectors and a minimum number of elements for each sector. This choice also imposes that in each ring the number of elements must be a multiple of 12; therefore, some elements are removed and rearranged so that no elements cross, or are too close to a sector border, leading to a starting point for thinning with \(N_{E}=511\) elements (Figure 1b). The sectored array is then the starting point for a multi-objective (MO) optimization based on the binary genetic algorithm (BGA) [28; 29; 30]. BGA is, of course, the best choice for an optimization problem where the variable space domain exhibits just two states (on and off) [29].