## 4 Results and Discussion
### Second Application: Control of Tilt, Sidelobes and Nulls  
#### 4.3.1 Operation Conditions  
A classic application of PAs radiation pattern control is the direction of arrival (DoA) problem, which aims at separating multiple signals on the same frequency channel through spatial filtering algorithms and adaptive antennas. A DoA problem occurs when interfering signals of high power hinder or even prevent the reception of the desired signal, so that the signal-to-noise ratio is degraded or the input level of the receptor is saturated [13]. It is possible to reject such interference, positioning nulls of the radiation pattern in the directions of arrival of the interfering signals. As in the case of tilt angle control (first application), positioning these nulls requires control of the PAR elements excitation. This changes the radiation pattern, increasing the beamwidth and reducing gain and RSLL [13]. Then, in the second application of GA-MMC, there were three optimization objectives. The first two objectives are the same as the first application, which is to determine the main lobe tilt angle and reduce the sidelobes level. The third objective is to direct nulls of the radiation pattern to reject two interfering \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c c c c c c c} \hline \hline \(\theta_{0}\) & \multicolumn{3}{c}{\([A_{1}A_{2}A_{3}A_{4}A_{5}A_{6}A_{7}A_{8}A_{9}A_{10}A_{11}A_{12}A_{13}A_{14}A_{15}A_{16}A_{17}A_{18}A_{19}A_{20}]\)} & \multicolumn{3}{c}{\(\beta\)} \\ \hline \(90\)\({}^{\circ}\) & _[6_ & 5 & 6 & 4 & 6 & 4 & 4 & 5 & 5 & 0 & 6 & 3 & 0 & 4 & 2 & 0 & 2 & 1 & 0 & 1] \\ \(60\)\({}^{\circ}\) & _[7_ & 7 & 7 & 7 & 6 & 6 & 5 & 6 & 4 & 5 & 3 & 4 & 3 & 2 & 3 & 1 & 3 & 0 & 2] \\ \(45\)\({}^{\circ}\) & _[6_ & 6 & 7 & 5 & 7 & 4 & 6 & 4 & 5 & 4 & 3 & 4 & 2 & 3 & 1 & 3 & 1 & 2 & 0 & 1] \\ \(30\)\({}^{\circ}\) & _[6_ & 6 & 6 & 6 & 6 & 6 & 5 & 5 & 4 & 4 & 3 & 4 & 2 & 3 & 2 & 2 & 1 & 1 & 1 & 1]_ & \(-39.00\)\({}^{\circ}\) \\ \(10\)\({}^{\circ}\) & _[6_ & 6 & 6 & 6 & 6 & 6 & 5 & 5 & 4 & 4 & 4 & 3 & 3 & 2 & 2 & 2 & 1 & 1 & 1 & 1] \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Individuals with higher fitness in tests of the GA-MMC first application [13]. Figure 13: Radiation pattern for \(\theta_{0}=45^{\circ}\) in Table 2. (**a**) Polar form. (**b**) Cartesian form. signals (e.g., jammers), \(A_{\text{11}}\) and \(A_{\text{12}}\), with known directions of arrival, according to the objective function given in Equation (6). The null is defined as the place of the radiation pattern with negligible transmission or reception capacity. In this paper, the adopted null value was 50 dB below the main lobe. GA-MCC uses the exact same operating conditions (\(A_{n}\), \(\beta\), \(N\) e \(d\)) established for the first application. Furthermore, there are known directions of arrival, \(\theta_{\text{11}}\) and \(\theta_{\text{12}}\), for two simultaneous interfering signals. To illustrate, Figure 14 contains an operating condition in which the PAR tracks a target in \(\theta_{\text{0}}=45\,^{\circ}\) and suffers interference from two jammers in \(\theta_{\text{11}}=10\,^{\circ}\) and \(\theta_{\text{12}}=135\,^{\circ}\), as well as details the associated linear PA, which was used in the second GA-MMC application tests [13]. Due to the large number of possible operational situations, obtained by combining the values of \(\theta_{\text{0}}\), \(\theta_{\text{11}}\) and \(\theta_{\text{12}}\), it was decided to test only the region of interest to a PAR of four faces (Figure 7b), that is, \(45\,^{\circ}\leq\theta_{\text{0}}\leq 90\,^{\circ}\). Despite this simplification, it is possible to use GA-MMC in any operational situation even to check impossible situations in practice (as \(\theta_{\text{0}}\cong\theta_{\text{11}}\), for instance).