## 2 Proposed EPP-Based Beam Scanning Approaches
### Proposed SLL/EPP-BSA
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Cosine Wave Amplitude \(A_{c}\)**} & \multicolumn{3}{c}{**CST Results**} & \multicolumn{3}{c}{**MATLAB Results**} \\ \cline{2-7}  & **Scan Angle** & **HPBW** & **SLL** & **Scan Angle** & **HPBW** & **SLL** \\ \hline \(A_{c}=0\) & \(0^{\circ}\) & \(12.3^{\circ}\) & \(-13\) dB & \(0^{\circ}\) & \(12.8^{\circ}\) & \(-12.8\) dB \\ \hline \(0.25\,\lambda\) & \(8^{\circ}\) & \(12.2^{\circ}\) & \(-12.2\) dB & \(8.56^{\circ}\) & \(12.6^{\circ}\) & \(-10.09\) dB \\ \hline \(0.5\,\lambda\) & \(17^{\circ}\) & \(12^{\circ}\) & \(-7.5\) dB & \(16.81^{\circ}\) & \(12.4^{\circ}\) & \(-7.88\) dB \\ \hline \(0.75\,\lambda\) & \(24^{\circ}\) & \(11.6^{\circ}\) & \(-5.6\) dB & \(24.54^{\circ}\) & \(12^{\circ}\) & \(-6.181\) dB \\ \hline \(\lambda\) & \(31^{\circ}\) & \(11.2^{\circ}\) & \(-4.1\) dB & \(31.76^{\circ}\) & \(11.2^{\circ}\) & \(-4.917\) dB \\ \hline \(1.25\,\lambda\) & \(37^{\circ}\) & \(10.7^{\circ}\) & \(-3.2\) dB & \(37.95^{\circ}\) & \(10.8^{\circ}\) & \(-4.009\) dB \\ \hline \(1.5\,\lambda\) & \(43^{\circ}\) & \(10.2^{\circ}\) & \(-2.1\) dB & \(43.14^{\circ}\) & \(10.5^{\circ}\) & \(-3.386\) dB \\ \hline \(1.75\,\lambda\) & \(47^{\circ}\) & \(9.7^{\circ}\) & \(-1.3\) dB & \(47.75^{\circ}\) & \(9.7^{\circ}\) & \(-2.949\) dB \\ \hline \(2\,\lambda\) & \(51^{\circ}\) & \(9.1^{\circ}\) & \(-2.1\) dB & \(51.36^{\circ}\) & \(9.2^{\circ}\) & \(-2.668\) dB \\ \hline \hline \end{tabular}
\end{table} 
Table 7: The CST simulation results for \(A_{c}\) changing from 0 to 2 \(\lambda\) compared to the MATLAB simulation results indicating the resultant scan angle, SLL, and HPBW. The even excitation vector can be determined as follows:  
\[\mathbb{C}_{(N-1)\times 1_{E}}=\left.\mathbb{C}_{I\times 1}\right|_{even} \tag{25}\]  
where the elements \(\mathbb{C}_{E}(n,1)\) of the vector \(\mathbb{C}_{(N-1)\times 1_{E}}\) can be obtained from the elements \(\mathbb{C}(i,1)\) of the vector \(\mathbb{C}_{I\times 1}\) such that:  
\[\mathbb{C}_{E}(n,1)=\mathbb{C}(2i,1) \tag{26}\]  
where \(1\leq(i=n)\leq\left(\frac{I+1}{2}\right)-1\)  
Accordingly, the synthesized array factor using odd excitation coefficients \(AF_{SO}(\theta)\) is given by:  
\[AF_{SO}(\theta)=\sum_{n=1}^{\lfloor\frac{I+1}{2}\rceil}\mathbb{C}_{O}(n,1)e^{ (jk(n-1)d_{s}\cos\theta)} \tag{27}\]  
While the synthesized array factor using even excitation coefficients \(AF_{SE}(\theta)\) is given by:  
\[AF_{SE}(\theta)=\sum_{n=1}^{(\frac{I+1}{2})-1}\mathbb{C}_{E}(n,1)e^{(jk(n-1)d_ {s}\cos\theta)} \tag{28}\]  
where \(d_{s}\) is the synthesized element spacing determined by the GA [13]. The GA optimizes the element spacing such that the synthesized array factors \(AF_{SO}(\theta)\) and \(AF_{SE}(\theta)\) provide the same HPBW as the original array factor \(AF(\theta)\). It is worth mentioning that both synthesized \(AF_{SO}(\theta)\) and \(AF_{SE}(\theta)\) provide a twofold decrease in the SLL compared to the original array factor \(AF(\theta)\). After SLL reduction, the synthesized excitation coefficients \(\mathbb{C}_{(N\times 1)_{O}}\) and \(\mathbb{C}_{(N-1)\times 1_{E}}\) are kept constant and the EPP is performed for perturbing the elements' positions in the azimuth plane to steer the main beam to the desired direction. Accordingly, the synthesized array factor using odd excitation coefficients \(AF_{SO}(\theta)\) and EPP can be expressed as:  
\[AF_{SO-EPP}(\theta)=\sum_{n=1}^{\lfloor\frac{I+1}{2}\rceil}\mathbb{C}_{O}(n,1 )e^{(jk(n-1)d_{s}\cos\theta+jkd_{p_{n}}\sin\theta)} \tag{29}\]  
While the synthesized array factor using even excitation coefficients \(AF_{SE}(\theta)\) and EPP can be expressed as:  
\[AF_{SE-EPP}(\theta)=\sum_{n=1}^{(\frac{I+1}{2})-1}\mathbb{C}_{E}(n,1)e^{(jk(n -1)d_{s}\cos\theta+jkd_{p_{n}}\sin\theta)} \tag{30}\]  
where \(d_{y_{n}}\) is related to the synthesized element spacing \(d_{s}\) according to Figure 26 shown below.