## 2 Proposed EPP-Based Beam Scanning Approaches
### Proposed EPP-BSA  
To derive a closed form expression of the synthesized array factor applying EPP, we considered a linear antenna array (LAA) whose elements distributions are shown in Figure 1 before and after EPP. Consider a uniform LAA consisting of \(N\) antenna elements with uniform element spacing \(d\) as shown in Figure 2, its array factor \(AF(\theta)\) is given by [18]:  
\[AF(\theta)=\sum_{n=1}^{N}a_{n}\ \mathrm{e}^{ik(n-1)d\cos\theta} \tag{1}\]  
where \(a_{n}\) is the excitation coefficient of the \(n^{th}\) antenna elements, \(k=2\pi/\lambda\) is the wave number, and \(\lambda\) is the wavelength. If the positions of the \(N\) antenna elements of the linear antenna array are perturbed along the \(Y\)-axis as shown in Figure 2, the array factor of Equation (1) should be modified. The total electric field \(E_{t}\) at the far field point is the superposition of the individual electric fields arising from each antenna element such that:  
\[E_{t}=E_{1+}E_{2}+\cdots\cdots\cdots+E_{N} \tag{2}\]  
where  
\[E_{1}=a_{1}\frac{E_{0}}{r_{1}}e^{-jkr_{1}} \tag{3}\]  
\[E_{2}=a_{2}\frac{E_{0}}{r_{2}}e^{-jkr_{2}} \tag{4}\]  
\[E_{N}=a_{N}\frac{E_{0}}{r_{N-1}}e^{-jkr_{N-1}} \tag{5}\]  
Then, the total field can be expressed as:  
\[E_{t}=\left(\frac{E_{0}}{r_{1}}a_{1}e^{-jkr_{1}}+\frac{E_{0}}{r_{2}}a_{2}e^{-jkr _{2}}+\cdots+a_{N}\frac{E_{0}}{r_{N-1}}e^{-jkr_{N-1}}\right) \tag{6}\]  
where \(r_{n}\), \(n=1,2,\ 3,\ \ldots,\ N\) is the distances from the \(n^{th}\) antenna element to the far field point. For far field approximation we use the following assumptions:  
\[\frac{1}{r_{1}}=\frac{1}{r_{2}}=\frac{1}{r_{3}}=\cdots=\frac{1}{r_{N}}=\frac{ 1}{r} \tag{7}\]  
\[r_{1}\parallel r_{2}\parallel r_{3}\parallel\cdots\parallel r_{N} \tag{8}\]  
Figure 1: Linear antenna array (**a**) without EPP and (**b**) with EPP. Figure 2: Geometry of a linear array of elements aligned along the \(Z\)-axis and has position perturbation along the \(Y\)-axis. where \(r\) is the distance from the origin or reference point to the far field point and \(\parallel\) is the parallel operator. And, \(\theta\) is the angle between the \(r\) vector and the \(Z\)-axis. \(d\) is the distance between successive elements on the \(Z\)-axis, \(d_{ym}\) is the distance of the \(n^{th}\) element in the Y-direction, \(d_{n}\) is distance between the \(n^{th}\) element and origin, and \(\theta_{n}\) is the angle between \(d_{n}\) and the \(Z\)-axis as shown in \[E_{t}=E_{1+}E_{2}+\cdots\cdots\cdots+E_{N} \tag{2}\]0  
Based on Figure 3, the angle \(\theta_{n}\) can be calculated as follows:  
\[E_{t}=E_{1+}E_{2}+\cdots\cdots\cdots+E_{N} \tag{2}\]1  
\[E_{t}=E_{1+}E_{2}+\cdots\cdots\cdots+E_{N} \tag{2}\]2  
To determine \(r_{1}\) for the first antenna element consider the array geometry shown in \[E_{t}=E_{1+}E_{2}+\cdots\cdots\cdots+E_{N} \tag{2}\]3 In this case, \(r_{1}\) can be expressed as:  
\[E_{t}=E_{1+}E_{2}+\cdots\cdots\cdots+E_{N} \tag{2}\]4  
On the other hand, \(r_{2}\) of the second antenna element can be determined as shown in Figure 5 such that: \(r_{2}=r-d_{2}\text{cos}(\theta_{2}-\theta)\)  
\[E_{t}=E_{1+}E_{2}+\cdots\cdots\cdots+E_{N} \tag{2}\]5  
\[E_{t}=E_{1+}E_{2}+\cdots\cdots\cdots+E_{N} \tag{2}\]6 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 first antenna element. \[E_{t}=E_{1+}E_{2}+\cdots\cdots\cdots+E_{N} \tag{2}\]7 The triangular relation between the element position in the \(Y\)-direction and the distance from the reference antenna element.