## 3 Results  
Simulations, performance assessments, and the implementation of the optimization algorithms were carried out within the MATLAB R2023a environment on a workstation system with an AMD Ryzen 5 5600X processor, 32 GB DDR5 RAM, and an NVIDIA RTX 2060 GPU. The parameter calibration process for the MO-BDE and MO-PSO algorithms is detailed in Table 2; specific configurations were refined to enhance optimization efficacy. The multi-objective implementation incorporating the three defined objective functions is visualized in Figure 5. In Figure 5a, candidate solutions satisfying \(N=N_{T}\) are in blue, while non-candidate solutions are colored along a gradient to red, indicating the deviation from the desired target. The evaluation of the initial solution is depicted in black. In this graph, the first Pareto front of the three-dimensional solution space is projected onto a 2D plane, where the thinning objective is represented through a color mapping. The figure illustrates how non-candidate solutions with \(N\
eq N_{T}\) can still contribute to the exploration process, enhancing the overall search by promoting diversity in the solution space. Figure 5b presents the solutions for the excitation tuning phase using the MO-PSO algorithm. This plot shows the obtained first Pareto front in the \(OB_{1}\)-\(OB_{2}\) space, where the dense clustering of optimized solutions toward the lower-left corner reflects successful convergence, where both objectives are minimized. \begin{table}
\begin{tabular}{l c c c} \hline \hline \multicolumn{5}{c}{SIMULATION PARAMETERS} \\ \hline Population size & & 2000 & \\ Number of generations & & 2000 & \\ Evaluation angle samples (\(\theta\)) & & 1440 & \\ Fractal structure stages & \(p=2\) & \(p=3\) & \(p=4\) & \(p=5\) \\ Individual gene size & \(N=16\) & \(N=64\) & \(N=256\) & \(N=1024\) \\ \hline \multicolumn{5}{c}{MO-BDE} \\ \hline Thinning factor & & 0.625 & \\ Differential weight parameter (\(F\)) & & 2 & \\ Crossover probability (\(C_{R}\)) & & 0.75 & \\ Mutation probability (\(M_{R}\)) & & \(\dfrac{1}{2\cdot N}\) & \\ Average duration at stages & 30 m & 85 m & 160 m & 760 m \\ \hline \multicolumn{5}{c}{MO-PSO} \\ \hline Amplitude weight values & & \(x\in\mathbb{Z}\ |\ 1\leq x\leq 10\) \\ Initial inertia weight & & \(\omega_{0}=2\) & \\ Initial acceleration coefficient & & \(c_{1}=1.6\) & \(c_{2}=1.4\) \\ Average duration at stages & 6 m & 35 m & 82 m & 260 m \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Simulation and algorithm parameters. The implementation of the mask function is illustrated in Figure 6. The plots are presented at frequency boundaries, (\(f_{l}\)) and (\(f_{u}\)). Corresponding mask functions are applied to minimize SLL while maintaining the integrity of the main lobe. The resulting array factor (AF) is designed to mitigate secondary-lobe interference and promote a more uniform distribution of radiated energy across the intended coverage region. The proposed optimization framework is applied to two design scenarios to evaluate the capabilities of the Eisenstein fractal geometry and the wideband optimization approach. Figure 5: Solution space and convergence of algorithms; (a) MO-BDE for array thinning and (b) MO-PSO for amplitude weight tuning  
Figure 6: Array factor performance of optimized pattern and original array using mask function objectives at stage 4. By leveraging the first front of the multi-objective solution set, optimal configurations can be selected to meet distinct design objectives. Optimization at \(f_{u}\) results in balanced performance across the entire bandwidth, while targeting \(f_{l}\) leads to enhanced performance at the initial operating frequency, though with potential trade-offs in effectiveness at higher frequencies:  
1. Design A: initial frequency optimization (\(f_{l}\));
2. Design B: wideband compromise optimization (\(f_{l}\) to \(f_{u}\)).