## 2 Proposed EPP-Based Beam Scanning Approaches
### Test case (2): \(R=0.875\)  
In this case, the cosine wave half period is chosen to be greater than the array size, \(H>L_{h}\) and equals \(H=4\ \lambda\), hence \(R=3.5\ \lambda/4\ \lambda=0.875\). Accordingly, the EPPs of the array elements for \(A_{c}\) changing from positive to negative and \(A_{c}\) changing from negative to positive are shown in Figures 9a and 9b, respectively.  
When the cosine wave amplitude is changing from positive to negative and its magnitude is changing from \(A_{c}=0\) to \(A_{c}=0.25\ \lambda\), the array main beam is scanned from \(0^{\circ}\) to \(8.56^{\circ}\) with a constant HPBW and the SLL is less than \(-10\) dB. However, as the cosine wave amplitude increases from \(0.25\ \lambda\) to \(2\ \lambda\), the scan angle is increased from \(8.56^{\circ}\) to \(51.36^{\circ}\),  
Figure 8: Polar plot of the synthesized radiation pattern using the EPP-BSA approach for 8-element UAAA at compression ratio \(R=1\) for \(A_{c}=2\ \lambda\) and changing from negative to positive.  
Figure 9: The EPPs of the array elements at compression ratio \(R=0.875\) for (**a**) \(A_{c}\) is changing from positive to negative and (**b**) \(A_{c}\) is changing from negative to positive.  
the SLL is increased by more than \(-10\) dB, and the HPBW is decreased by \(3.6^{\circ}\) as listed in Table 2. The synthesized radiation patterns using the EPP technique are shown in Figure 10. It is clear that as the amplitude of the cosine waveform increases, the scanning angle increases away from the broadside direction, the HPBW decreases, and the SLL increases as shown in Figure 11.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Figure** & **Cosine Wave Amplitude \(A_{c}\)** & **Main Beam Direction \(\theta_{\sigma}\), in \({}^{\circ}\)** & **SLL (dB)** & **HPBW in \({}^{\circ}\)** & **Maximum Scanning Range** \\ \hline Figure 10a & 0 & 90\({}^{\circ}\) & \(-12.8\) dB & \(12.8^{\circ}\) \\ \hline Figure 10b & \(0.25\,\lambda\) & \(81.44^{\circ}\) & \(-10.09\) dB & \(12.6^{\circ}\) \\ \hline Figure 10c & \(0.5\,\lambda\) & \(73.19^{\circ}\) & \(-7.88\) dB & \(12.4^{\circ}\) \\ \hline Figure 10d & \(0.75\,\lambda\) & \(65.46^{\circ}\) & \(-6.181\) dB & \(12^{\circ}\) \\ \hline Figure 10e & \(\lambda\) & \(58.24^{\circ}\) & \(-4.917\) dB & \(11.2^{\circ}\) \\ \hline Figure 10f & \(1.25\,\lambda\) & \(52.05^{\circ}\) & \(-4.009\) dB & \(10.8^{\circ}\) \\ \hline Figure 10g & \(1.5\,\lambda\) & \(46.86^{\circ}\) & \(-3.386\) dB & \(10.5^{\circ}\) \\ \hline Figure 10h & \(1.75\,\lambda\) & \(42.25^{\circ}\) & \(-2.949\) dB & \(9.7^{\circ}\) \\ \hline Figure 10i & \(2\,\lambda\) & \(38.64^{\circ}\) & \(-2.668\) dB & \(9.2^{\circ}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: The relation between the amplitude changes of the cosine waveform and the scanning angle, HPBW, SLL, and maximum scanning range \(\theta_{\sigma}\) for 8-element ULAA at compression ratio \(R=0.875\).  
On the other hand, when the cosine wave amplitude is changing from negative to positive and its magnitude is changing from \(A_{c}=0\) to \(A_{c}=0.25\,\lambda\), the array main beam is scanned from \(0^{\circ}\)_to_\(-8.56^{\circ}\) with a constant HPBW and the SLL is less than \(-10\) dB. However, as the cosine wave amplitude increases from \(0.25\,\lambda\) to \(2\,\lambda\), the scan angle is increased from \(-8.56^{\circ}\) to \(-51.36^{\circ}\), the SLL is increased by more than \(-10\) dB, and the HPBW is decreased by \(3.6^{\circ}\) as shown in Figure 12. The achieved maximum scanning range is \(-51.36^{\circ}\)\(\leq\theta_{r}\leq 51.36^{\circ}\) around the broadside direction.