## 4 Discussion  
The optimization results demonstrate the impact of parameters \(\alpha\) and \(\beta\) on the clustering process, respectively. The modified K-means clustering introduces a tradeoff between the uniformity of the beam occupation and the uniformity of the beam width. On the one hand, the reduction in \(\alpha\) helps in tuning the determination matrix from sorting the distances to the object from each centroid to sorting the distances to the centroid from each object. On the other hand, the growth of \(\beta\) keeps increasing the obligatory minimum number of objects in a single beam. Figure 12 illustrates the standard deviation of objects per beam in terms of \(\alpha\) and \(\beta\). The minimum value equals 6.95 when \(\alpha\) is 0.2 and \(\beta\) is 0.8. This indicates that a larger value of \(\beta\) with a relative smaller value of \(\alpha\) tends to obtain uniformity of the beam occupation. On the contrary, if \(\beta\) is smaller than 0.5 and \(\alpha\) is close to 1, the standard deviation is raised to _c.a._ 16, which means a greater variation in the object number per beam. The uniformity of the beam occupation is a key specification that determines the performance. However, to make the whole proposal feasible, other factors should also be taken into consideration, such as the key characteristics of the antenna system. Figure 13a illustrates the standard deviation of the half-cone angles in terms of \(\alpha\) and \(\beta\). Figure 13b illustrates the maximum of the half-cone angles in terms of \(\alpha\) and \(\beta\). As mentioned above, the uniformity of the beam occupation is maximized when \(\alpha\) is 0.2 and \(\beta\) is 0.8. In this case, the maximum of the half-cone angles is 3.94\({}^{\circ}\) and the standard deviation is 0.79\({}^{\circ}\). The growth of \(\alpha\) and reduction of \(\beta\) mitigates the large variation of the beam widths. The variance is the key characteristic that challenges the performance of the antenna and the feasibility of the proposal. The uniformity of the beam width can be obtained when \(\alpha\) is 1 and \(\beta\) is 0.4, which suggests that the determination matrix is degenerated to the conventional K-means with a moderate obligatory minimum number \begin{table}
\begin{tabular}{c c c c c} \hline \multirow{2}{*}{\(A\)} & \multirow{2}{*}{\(\beta\)} & **Number of Objects in Each Beam** & **Half-Cone Angle of Beam** \\ \cline{3-5}  & & **Avg.** & **Std.** & **Avg.** & **Std.** \\ \hline
1 & 0 & & 16.55 & 1.96\({}^{\circ}\) & 0.45\({}^{\circ}\) \\
0 & 0 & & 12.92 & 1.77\({}^{\circ}\) & 0.95\({}^{\circ}\) \\
0.5 & 0.2 & 40.95 & 12.55 & 1.95\({}^{\circ}\) & 0.46\({}^{\circ}\) \\
0.5 & 0.5 & & 12.03 & 1.96\({}^{\circ}\) & 0.49\({}^{\circ}\) \\
0.5 & 0.8 & & 9.87 & 1.98\({}^{\circ}\) & 0.56\({}^{\circ}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Statistic of the impact of \(\alpha\) and \(\beta\) on the results. Figure 12: Standard deviation of objects per beam in terms of \(\alpha\) and \(\beta\). of objects per beam. In this case, the maximum of the half-cone angles is \(2.57^{\circ}\) and the standard deviation is \(0.33^{\circ}\). It worth noting that there is a plateau when \(\alpha\) is larger than \(0.6\) and \(\beta\) is smaller than \(0.4\) as shown in Figures 11 and 12. The moderate obligatory minimum number of objects per beam and the modified determination matrix tune the results to be similar with conventional K-means clustering, which aims at the uniformity of the beam sizes. However, the uniformity of the beam occupation can only be obtained from the plateau area with a larger value of \(\beta\) and smaller value of \(\alpha\). Finally, the computational complexity of the proposed algorithm is studied. The calculating time of Equation (1) to generate the Euclidean distance matrix is \(O(nkd)\), where \(n\) is the number of objects, \(k\) is the number of clusters and \(d\) is the dimension. This step has the same implementation cost with the conventional K-means algorithm [32].