## 2 Antenna Synthesis and Modeling
### Optimum Lattice Architecture
On the other hand, MO allows us to obtain a Pareto set--an approximation of the problem's Pareto front--containing the non-dominated solutions with respect to a set of conflicting requirements. In our case, the requirements were the lowest number of elements and the best SLL over the entire scan range. Several MO runs were carried out and lattices were evaluated by analyzing Pareto sets. The chosen case for further investigations (Figure 1a) exhibits \(N_{E}=319\), that is a substantial 44% reduction in the number of elements, and a 20 dB SLL, 7 dB better with respect to the full array of Figure 1a. While the number of elements is fine, SLL could still be improved; hence, an amplitude tapering is considered. This is implemented as a function of the distance from the center. The number of rings, starting from the outer one, with variable amplitude values, is a design parameter. In this case, due to the continuous nature of the amplitude tapering, an Invasive Weed Optimization (IWO) is exploited [31; 30; 32; 33] in a single objective framework, optimizing only for the SLL. Fixed amplitudes are considered unitary; variable amplitudes of outer rings span the range \([0,2]\), to also allow for elements with higher feed amplitudes, if necessary. A parametric study is carried out with the number of outer rings \(N_{r}\) whose amplitude feed is to be optimized in the set \(5,8,10,12\). Figure 1: The full regular circular array composed of 13 rings and \(N_{E}=566\) elements (**a**); and the sectored array of \(N_{E}=511\) elements. (**a**) Circular regular array; (**b**) circular sectored array; red shows central sector \(S_{0}\), blue shows \(S_{I}\) sector, green ones are duplicate sectors. Table 3 summarizes the results. The number of concentric rings, from outer to inner, is given in the first column; the second column reports the number of elements with a tapered amplitude; the third column shows the attained SLL; the fourth column shows the dynamic range of tapering, the ratio between maximum and minimum amplitude. The best configuration is the one with 12 rings since it attains almost 2 dB better SLL and exhibits a limited amplitude dynamic, slightly above 3. Full results are shown in Table 4 and Figures 2 and 3. The array geometry is shown with feed amplitudes in a color scale from lower (blue) to higher (red). The 3D broadside pattern and the two principal cut planes are also reported. In addition SLL attained in scanning is reported in Figure 4. Scanning has been done for an extended \(\theta\) range (up to 57\({}^{\circ}\)) range and in one-eighth of the azimuthal range, spanning more than a single sector and is hence representative of the complete azimuthal range. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline
**Ring** & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 \\ \hline
**Amp.** & 1 & 1 & 0.93 & 0.64 & 1.96 & 1.45 & 1.52 & 1.41 & 1.15 & 1.10 & 0.83 & 0.70 & 0.73 & 0.69 \\ \hline \multicolumn{11}{c}{Fixed} & & & & & & & & & & & & & \\ \hline \end{tabular}
\end{table} 
Table 4: Amplitude tapering for the selected case of Table 3. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Concentric Rings** & **Tapered Elements** & **SLL (dB)** & **Dynamic \(A_{max}/A_{min}\)** \\ \hline
5 & 168 (53\%) & \(-20.85\) & 2.55 \\
8 & 240 (75\%) & \(-20.94\) & 2.12 \\
10 & 288 (90\%) & \(-21.61\) & 4.21 \\
**12** & **312 (98\%)** & \(-21.79\) & **3.06** \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Antenna optimization results. All cases are considered for 319 “on” elements. Bold line highlights the case selected for further analysis. SLL value is the worst SLL case computed while the main beam scans the whole range. Figure 2: Optimization results. **(a)** A Pareto set derived from one of the many optimization runs with highlighted configuration 1, chosen for subsequent SO optimization; **(b)** geometry, with color-highlighted tapering (red: higher amplitude, blue: lower amplitude).