## 2 Proposed EPP-Based Beam Scanning Approaches
### Proposed EPP-BSA
#### 2.1.1 Simulation Results of the Proposed EPP-BSA  
In this section, several simulations are carried out to verify the proposed approach and give a recommendation for the best case. Consider a broadside uniform linear antenna array (ULAA) consisting of \(N=8\) elements with uniform element spacing \(d=\lambda/2\) whose elements are arranged along the \(Z\)-axis. It is required to scan the array main beam to  
Figure 5: Geometry of a linear array of elements aligned along the \(Z\)-axis and has position perturbation along the \(Y\)-axis indicating the far field radiation from the second antenna element. the right and left of the broadside direction. This is achieved by performing EPP in the \(Y\)-direction following a known waveform pattern. In this case, the cosine waveform is utilized for the EPP where the cosine wave amplitude and period are expressed in terms of the operating wavelength \(\lambda\). The half period \(H\) of the cosine wave along the \(Z\)-axis is chosen related to the array length \(L_{h}=(N-1)\times d=3.5\ \lambda\). In this case, the element position in the Y-direction along the array line (\(Z\)-direction) is given by:  
\[d^{y_{n}}=\pm A_{c}\times cos\bigg{(}\frac{z\times\pi}{H}\bigg{)} \tag{20}\]  
where \(A_{c}\) is the amplitude of the cosine wave, \(z\) is the element position on the \(Z\)-axis, and \(H\) is the half period of cosine that is chosen to be \(H\geq L_{h}\). Both the amplitude of the cosine wave \(A_{c}\) and the half period \(H\) control the scanning angle of the array main beam. The beam scanning is performed at different compression ratios \(R\) that are defined as:  
\[R=\frac{L_{h}}{H},\ \ H\geq L_{h} \tag{21}\]  
It is noticed that as \(H\) increases more than \(L_{h}\), the array size in the Y-direction is reduced and becomes compact, so \(R\) is denoted as the compression ratio. Consequently, there are different test cases for beam scanning for different values of compression ratio. **Test case (1): \(\mathbf{R=1}\)**  
In this case, the cosine wave half period is chosen to be equal to the array size, \(H=L_{h}=3.5\ \lambda\), hence \(R=1\). 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 6a and 6b, respectively. When the cosine wave amplitude is changing from positive to negative and its magnitude is changing from \(A_{c}=0\) to \(A_{c}=0.19\ \lambda\), the array main beam is scanned from \(0^{\circ}\) to \(7.01^{\circ}\) with HPBW and the SLL is less than \(-10\) dB. However, as the cosine wave amplitude increases from than \(0.19\ \lambda\) to \(1.5\ \lambda\), the scan angle is increased from \(7.01^{\circ}\) to \(44.6^{\circ}\), the SLL is increased more than \(-10\) dB, and the HPBW is decreased by \(3^{\circ}\). However, as the cosine wave amplitude increases from than \(1.5\ \lambda\) to \(2\ \lambda\), the scan angle is increased from \(44.6^{\circ}\) to \(52.9^{\circ}\). But, the grating lobe appears, and the SLL is increased by more than \(-10\) dB as listed in Table 1. The synthesized radiation patterns using the EPP technique are shown in Figure 7.  
Figure 6: The EPPs of the array elements at compression ratio \(R=1\) for (**a**) \(A_{c}\) is changing from positive to negative and (**b**) \(A_{c}\) is changing from negative to positive.