## 2 Materials and Methods
### Scaling Transformation
Assuming that the field-of-view range is scaled down from \(\pi/2\) (the full-field-of-view range) to \(\theta_{0}\) (the limited-field-of-view range), then using (5) we reach the following equation (scaling factor equation):  
\[\mu=\frac{r_{L}}{r_{F}}=\sin\frac{\pi}{2}/\sin\theta_{0}\geq 1, \tag{9}\]  
where subscript F stands for FFoV while subscript L stands for LFoV systems. Figure 1: The initial full-field-of-view (red) and the resulting limited-field-of-view (green) array patterns after the scaling procedure. Then, given the \(HPBW_{\rm L}\) equation (Equation (5)), we can determine the \(HPBW_{\rm F}\). In the current work, we focus on broadside beams where \(sin(\theta)\cong\theta\) practically holds. As a result, the \(HPBW\) equation reads:  
\[HPBW_{L}=\frac{r_{F}}{r_{L}}\cdot HPBW_{F}=\frac{1}{\mu}\cdot HPBW_{F}. \tag{10}\]  
Apart from the previous equations, a relation between the directivities of the two designs exists. This is made possible if the radiating elements of the LFoV case have the following radiation pattern:  
\[EP(\theta^{\prime})= \begin{cases}1&\theta^{\prime}\leq\theta_{0}\\ 0&\theta_{0}<\theta^{\prime}\end{cases} . \tag{11}\]  
We start from the directivity of the FFoV array:  
\[D_{F}=4\pi\left(\int\limits_{0}^{2\pi}\int\limits_{0}^{\overline{\theta}=\pi/ 2}\left|AF\left(\cos\phi\sqrt{S}\sin\theta,\sin\phi\sqrt{S}\sin\theta\right) \right|^{2}\sin\theta d\phi d\theta\right)^{-1}. \tag{12}\]  
Now, enforcing a transformation from variable \(\theta\) to \(\theta^{\prime}\), using (5) we have:  
\[\theta=\sin^{-1}\left(\frac{r^{\prime}\sin\theta^{\prime}}{r}\right)=\sin^{-1 }\left(u\sin\theta^{\prime}\right);\ d\theta=d\theta^{\prime}\mu\frac{\cos \theta^{\prime}}{\cos\theta}. \tag{13}\]  
The directivity index after some algebra is given below:  
\[D_{F}=\frac{4\pi}{\mu^{2}}\cdot\left(\int\limits_{0}^{2\pi}\int\limits_{0}^{ 2\pi}\int\limits_{0}^{\infty}\left|AF\left(\cos\phi\sqrt{S^{\prime}}\sin\theta ^{\prime},\sin\phi\sqrt{S^{\prime}}\sin\theta^{\prime}\right)\right|^{2}\sin \theta^{\prime}d\phi d\theta^{\prime}\left(\frac{\cos\theta^{\prime}}{\cos \theta}\right)\right)^{-1}. \tag{14}\]  
Figure 2 depicts \(\cos\theta/\cos\theta^{\prime}=\cos\theta_{F}/\cos\theta_{L}\) (blue curves) for \(\mu\) = (5, 6, 7, 8, 9, 10). The vertical red lines depict the upper border of the LFoV range, \(\left[\sin^{-1}\left(\frac{1}{\mu}\sin(\pi/2)\right)\right]\). Increasing \(\mu\), we move the border to the left. Note that for scaling factors at least higher than 5 in the LFoV range, the following approximation holds:  
\[\cos\theta^{\prime}/\cos\theta\approx 1. \tag{15}\]  
So, (14) results in:\[.D_{\rm F}\approx\frac{4\pi}{\mu^{2}}\cdot\left(\int\limits_{0}^{2\pi}\int\limits_{ 0}^{\overline{\theta}=\sin^{-1}}\left(\frac{1}{\mu}\sin\left(\pi/2\right)\right) \right. \tag{16}\]  
If we assume that we use radiating elements with the pattern given by (11) for the LFoV case, the corresponding directivity takes the following form:  
\[.D_{L}=4\pi\left(\int\limits_{0}^{2\pi}\int\limits_{0}^{\overline{\theta^{ \prime}}}\left|AF\left(\cos\phi\sqrt{S^{\prime}}\sin\theta^{\prime},\sin\phi \sqrt{S^{\prime}}\sin\theta^{\prime}\right)\right|^{2}\sin\theta^{\prime}d\phi d \theta^{\prime}\right)^{-1} \tag{17}\]  
Equations (16) and (17) result in the directivity equation:  
\[D_{\rm F}=\frac{1}{\mu^{2}}D_{L}\Rightarrow \begin{array}{l}D\\ LFoV\ with\ Brick\ Wall\ Element\\ Pattern\end{array} \quad \begin{array}{l}(dBi)=10\log_{10}\mu^{2}+D\\ Pattern\end{array} \quad \begin{array}{l}FFoV\ with\ Omni\ Element\\ Pattern\end{array} \quad \begin{array}{l}(dBi).\end{array}  \tag{18}\]  
Note that, for example, for a scaling factor of \(\mu=7\), the previous expression gives a boost of approximately 16.9 dB in the directivity of the LFoV design. Last, as far as the \(SLL\) is concerned, it remains constant between the two designs, since the scaling transformation varies the angle of view but not the amplitude of the field. The transformation results are gathered in Table 1.