## 2 Proposed EPP-Based Beam Scanning Approaches
### Test case (3): \(R=0.7778\)
#### Test case (4): \(R=0.7\)  
In this case, the cosine wave half period is chosen to be equal to \(H=5\)\(\lambda\), and the array length equals \(L_{h}=3.5\)\(\lambda\), hence \(R=0.7\). 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 16a and 16b, respectively. When the cosine wave amplitude is changing from positive to negative and its magnitude is changing from \(A_{c}=0\) to \(A_{c}=0.39\)\(\lambda\), the array main beam is scanned from \(0^{\circ}\) to \(10.68^{\circ}\) with a constant HPBW and the SLL is less than \(-10\) dB. However, as the cosine wave amplitude increases from \(0.39\)\(\lambda\) to \(2\)\(\lambda\), the scan angle is increased from \(10.68^{\circ}\) to \(45.69^{\circ}\), the SLL is increased by more than \(-10\) dB, and the HPBW is decreased by \(1.1^{\circ}\) as listed in Table 4. The synthesized radiation patterns using the EPP technique are shown in Figure 17.  
Figure 14: Polar plots of the synthesized radiation patterns using the EPP-BSA approach for an 8-element ULAA at compression ratio \(R=0.7778\) at (**a**) \(A_{c}=0\), (**b**) \(A_{c}=0.25\)\(\lambda\), (**c**) \(A_{c}=0.27\)\(\lambda\), (**d**) \(A_{c}=0.3\)\(\lambda\) (**e**) \(A_{c}=0.5\)\(\lambda\), (**f**) \(A_{c}=0.75\)\(\lambda\), (**g**) \(A_{c}=\lambda\), (**h**) \(A_{c}=1.25\)\(\lambda\), (**i**) \(A_{c}=1.5\)\(\lambda\), (**j**) \(A_{c}=1.75\)\(\lambda\), and (**k**) \(A_{c}=2\)\(\lambda\) and changing from positive to negative. Figure 15: Polar plot of the synthesized radiation pattern using the EPP-BSA approach for 8-element ULAA at compression ratio \(R=0.7778\) for \(A_{c}=2\)\(\lambda\) and changing from negative to positive. Figure 16: The EPPs of the array elements at compression ratio \(R=0.7\) for **(a)**\(A_{c}\) is changing from positive to negative and **(b)**\(A_{c}\) is changing from negative to positive. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Figure** & **Cosine Wave Amplitude \(A_{c}\)** & **Main Beam Direction \(\theta_{\
u}\), in \(\lx@math@degree\)** & **SLL (dB)** & **HPBW in \(\lx@math@degree\)** & **Maximum Scanning Range** \\ \hline Figure 17a & \(0\) & \(90\lx@math@degree\) & \(-12.8\) dB & \(12.8\lx@math@degree\) & \\ \hline Figure 17b & \(0.25\,\lambda\) & \(82.99\lx@math@degree\) & \(-11.2\) dB & \(12.74\lx@math@degree\) & \\ \hline Figure 17c & \(0.35\,\lambda\) & \(80.41\lx@math@degree\) & \(-10.36\) dB & \(12.66\lx@math@degree\) & \\ \hline Figure 17d & \(0.39\,\lambda\) & \(79.38\lx@math@degree\) & \(-10.05\) dB & \(12.66\lx@math@degree\) & \\ \hline Figure 17e & \(0.5\,\lambda\) & \(76.28\lx@math@degree\) & \(-9.198\) dB & \(12.65\lx@math@degree\) & \\ \hline Figure 17f & \(0.75\,\lambda\) & \(70.1\lx@math@degree\) & \(-7.39\) dB & \(12.5\lx@math@degree\) & \\ \hline Figure 17g & \(\lambda\) & \(63.91\lx@math@degree\) & \(-5.869\) dB & \(12.42\lx@math@degree\) & \\ \hline Figure 17h & \(1.25\,\lambda\) & \(58.24\lx@math@degree\) & \(-4.68\) dB & \(12.4\lx@math@degree\) & \\ \hline Figure 17i & \(1.5\,\lambda\) & \(53.08\lx@math@degree\) & \(-3.791\) dB & \(12.7\lx@math@degree\) & \\ \hline Figure 17j & \(1.75\,\lambda\) & \(48.44\lx@math@degree\) & \(-3.185\) dB & \(13.4\lx@math@degree\) & \\ \hline Figure 17k & \(2\,\lambda\) & \(44.31\lx@math@degree\) & \(-2.819\) dB & \(13.9\lx@math@degree\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The relation between the amplitude changes of the cosine waveform and the scanning angle, HPBW, SLL, and maximum scanning range \(\theta_{\tau}\) for 8-element ULAA at compression ratio \(R=0.7\). Figure 17: _Cont._  
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 \(A_{c}=0.39\)\(\lambda\), the array main beam is scanned from \(0^{\circ}\)\(to-10.68^{\circ}\) with a constant HPBW and the SLL is less than \(-10\) dB.