## 5 Conclusions  
In this work, a spline-based UWB monopole antenna with an adjustable number of parameters has been presented. The structure has been optimized in a bi-objective setup using an algorithm that enables sequential generation of the Pareto designs between the extreme solutions. Each step of the design process involves a trust region-based optimization of the structure in a nested loop. The latter gradually increases the dimensionality of the antenna. The trade-off solutions are generated through the optimization of candidate designs, determined from bisections between the already existing solutions. The proposed method was used to obtain a total of nine antenna designs, representing compromises between a minimization of reflection within 3.1 GHz to 10.6 GHz band and size reduction. The geometries of the optimized radiators were represented using 30 to 46 independent parameters. The performance and sizes of the final designs along the Pareto front varied from \(-\)10.3 dB to \(-\)20.5 dB and from 230 mm\({}^{2}\) to 757 mm\({}^{2}\), respectively. The method has been favorably validated against a genuine multi-objective optimization algorithm in terms of computational cost and the quality of the obtained solutions. Furthermore, the smallest \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Design**} & \multicolumn{6}{c}{**Number of Model Evaluations [R]**} & \multirow{2}{*}{**Total**} & \multirow{2}{*}{**Total**} \\ \cline{2-2} \cline{6-9}  & \(n=1\) & \(n=2\) & \(n=3\) & \(n=4\) & \(n=5\) & \(n=6\) & [R] & [\(h\)] \\ \hline \(x_{1-dn}\)\({}^{*}\) & 64 & 92 & 94 & 156 & 326 & 284 & 1016 & 36.97 \\ \(x_{2-dn}\) & N/A & N/A & N/A & 187 & 2 & 192 & 381 & 13.86 \\ \(x_{3-dn}\) & N/A & N/A & N/A & 158 & 41 & 145 & 344 & 12.52 \\ \(x_{4-dn}\)\({}^{*}\) & N/A & N/A & N/A & 193 & 119 & 98 & 410 & 14.92 \\ \(x_{5-dn}\) & N/A & N/A & N/A & 128 & 198 & 146 & 472 & 17.18 \\ \(x_{6-dn}\) & N/A & N/A & N/A & 127 & 119 & 284 & 530 & 19.28 \\ \(x_{7-dn}\) & N/A & N/A & N/A & 158 & 41 & 238 & 437 & 15.90 \\ \(x_{8-dn}\)\({}^{*}\) & N/A & N/A & N/A & 156 & 81 & 96 & 333 & 12.12 \\ \(x_{9-dn}\) & 56 & 46 & 118 & 156 & 40 & 50 & 466 & 16.96 \\ \hline Total & 120 & 138 & 212 & 1419 & 967 & 1533 & 4389 & 159.71 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Cost breakdown of the multi-objective optimization of the spline-parameterized antenna. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Antenna**} & \(f_{L}\) & **Bandwidth** & **Dimensions** & **Size** & **Dimensions** & **Footprint \({}^{\bullet}\)** \\  & **[GHz]** & **[GHz]** & **[mm \(\times\) mm]** & **[mm\({}^{2}\)]** & **[\(\lambda_{g}\times\lambda_{g}\)]** & **[\(\lambda_{g}^{2}\)]** \\ \hline
[55] & 2.9 & 6.6 & 32.0 \(\times\) 36.0 & 1152 & 0.56 \(\times\) 0.64 & 0.359 \\
[56] & 3.0 & 8.0 & 28.0 \(\times\) 38.0 & 1064 & 0.51 \(\times\) 0.69 & 0.352 \\
[57] & 3.1 & 8.4 & 14.5 \(\times\) 28.0 & 406 & 0.27 \(\times\) 0.52 & 0.140 \\
[11] & 2.9 & 19.3 & 25.0 \(\times\) 17.0 & 425 & 0.44 \(\times\) 0.30 & 0.132 \\
[18] & 3.0 & 7.8 & 10.0 \(\times\) 32.5 & 325 & 0.18 \(\times\) 0.59 & 0.106 \\
[54] & 3.0 & 7.6 & 10.0 \(\times\) 25.0 & 250 & 0.17 \(\times\) 0.43 & 0.073 \\
[58] & 3.1 & 8.3 & 7.00 \(\times\) 25.0 & 175 & 0.13 \(\times\) 0.47 & 0.061 \\ \hline This work & 3.1 & 8.9 & 12.5 \(\times\) 18.4 & 230 & 0.32 \(\times\) 0.22 & 0.069 \\ \hline \hline \end{tabular} \({}^{\bullet}\) For fair comparison, the antenna size is expressed in terms of the guided wavelength corresponding to the substrate properties the design is implemented on.  
\end{table} 
Table 2: Spline-parameterized antenna (\(x_{1.20}\)) in comparison with the benchmark designs. Pareto-optimal antenna was compared against the state-of-the-art monopoles from the literature. The optimized radiator is characterized not only by competitive performance, but also a small footprint, which makes it of potential use for UWB sensors, mobile localization services, or body area networks.