# Ultra-Wideband Antenna Design for 5G NR Using the Bezier Search Differential Evolution Algorithm
## 3 Optimization Results and Discussion  
The optimization process was utterly successful by producing a result that included 5G NR in its operational bandwidth. This Figure 2 shows the original antenna from [1]. The resulting antenna, as depicted in Figure 3, was much more simplified compared to the initial one, shown in Figure 2. During the optimization process, after a number of iterations, some of the variables were set to such values by the algorithm, which deemed them non-essential for the optimal result, i.e., the width of the initial slots is equal to 0. This \begin{table}
\begin{tabular}{c c c c c c} \hline  & **BeSD** & **GA** & **BBO** & **DE** & **CMA-ES** \\ \hline \(f1\) & \(\mathbf{0.000\times 10^{00}}\) & \(1.170\times 10^{-33}\) & \(1.606\times 10^{-02}\) & \(4.216\times 10^{-13}\) & \(5.819\times 10^{-25}\) \\ \(f2\) & \(\mathbf{0.000\times 10^{00}}\) & \(13.867\times 10^{00}\) & \(3.349\times 10^{00}\) & \(5.980\times 10^{01}\) & \(1.593\times 10^{02}\) \\ \(f3\) & \(\mathbf{0.000\times 10^{00}}\) & \(1.563\times 10^{-12}\) & \(0.491\times 10^{-01}\) & \(7.846\times 10^{-09}\) & \(9.731\times 10^{-10}\) \\ \(f4\) & \(\mathbf{0.000\times 10^{00}}\) & \(9.356\times 10^{-33}\) & \(2.149\times 10^{-02}\) & \(7.462\times 10^{-16}\) & \(9.874\times 10^{-30}\) \\ \(f5\) & \(\mathbf{0.000\times 10^{00}}\) & \(6.950\times 10^{-02}\) & \(4.350\times 10^{-02}\) & \(1.429\times 10^{02}\) & \(9.343\times 10^{03}\) \\ \(f6\) & \(\mathbf{0.000\times 10^{00}}\) & \(\mathbf{0.000\times 10^{00}}\) & \(5.343\times 10^{-06}\) & \(2.625\times 10^{-73}\) & \(3.118\times 10^{-08}\) \\ \(f7\) & \(5.580\times 10^{-08}\) & \(\mathbf{0.000\times 10^{00}}\) & \(3.592\times 10^{-15}\) & \(1.987\times 10^{-21}\) & \(8.680\times 10^{-07}\) \\ \(f8\) & \(\mathbf{0.000\times 10^{00}}\) & \(2.478\times 10^{-34}\) & \(2.027\times 10^{-02}\) & \(3.771\times 10^{-1}\) & \(1.800\times 10^{-26}\) \\ \(f9\) & \(\mathbf{0.000\times 10^{00}}\) & \(\mathbf{0.000\times 10^{00}}\) & \(1.665\times 10^{-15}\) & \(\mathbf{0.000\times 10^{00}}\) & \(1.296\times 10^{-06}\) \\ \(f10\) & \(\mathbf{0.000\times 10^{00}}\) & \(4.005\times 10^{-18}\) & \(1.131\times 10^{-15}\) & \(1.976\times 10^{-38}\) & \(4.778\times 10^{-12}\) \\ \hline \end{tabular}
\end{table} 
Table 2: Tested algorithms’ performance with benchmark functions based on the optimal result. The bold font shows lower values. \begin{table}
\begin{tabular}{c c c c c c} \hline  & **BeSD** & **GA** & **BBO** & **DE** & **CMA-ES** \\ \hline Friedman & 1.3 & 2.1 & 3.6 & 3.0 & 3.7 \\ Normalized Ranking & 1 & 2 & 4 & 3 & 5 \\ \hline \end{tabular}
\end{table} 
Table 3: Friedman non-parametric statistical testing of algorithms’ performance was either due to their inefficiency in changing the response of the antenna design or due to their non-essentiality, as the result could be achieved with fewer degrees of freedom. After discarding those variables from the design, a simplified version of the antenna was optimized from scratch using the BeSD algorithm, and the optimal result consisted of the main circular patch with partial grounding and a stub on it. That is, the substrate was Isola FR408 (\(\epsilon_{r}=3.75\) and \(tan\delta=0.012\)), with a thickness equal to 1.6 mm. This stub was the main reason for the widening of the bandwidth to include 5G NR, as well as including Wi-Fi in the harvested technologies, as proven by comparing the response graph of the  
The cost function is a value that is returned to the algorithm and is how the design is evaluated (Equation (1)). In this minimization problem, the cost function of each agent was defined by two factors, firstly by whether the antenna operated in the whole target bandwidth (from 1.4 GHz to 3.9 GHz), or in a fraction of it, so that the condition \(S_{11}<-10\) dB was met for all frequency points, and by the response \(S_{11}\) of the center frequency of the simulation process, \(f_{c}=3.55\) GHz. More specifically, the input arguments of BeSD, as depicted in Algorithm 1, were set as follows:  
* Independent Trials: 5.