## 2 Materials and Methods
### Scaling Transformation  
We assume a radiating aperture of surface \(S\) on the \(xy\) plane, having an excitation distribution \(I\left(x/\sqrt{S},y/\sqrt{S}\right)\). The respective far-field pattern reads:  
\[AF(\phi,\theta)=\iint\limits_{Aperture}I\left(x/\sqrt{S},y/\sqrt{S}\right) \exp\left(-j\frac{2\pi}{\lambda}(x\cos\phi\sin\theta+y\sin\phi\sin\theta) \right)dxdy. \tag{1}\]  
If we substitute in (1):  
\[\zeta=x/\sqrt{S};\ \zeta=y/\sqrt{S}, \tag{2}\]  
we have:  
\[AF(\phi,\theta)=\iint\limits_{Aperture}I(\zeta,\xi)\exp\left(-j\frac{2\pi}{ \lambda}(\zeta\cos\phi+\zeta\sin\phi)\sqrt{S}\sin\theta\right)d\zeta d\zeta. \tag{3}\]  
From (1) and (3), it is obvious that the relation between real space and the observation angle \(\theta\) is solely included in the term \((2\pi/\lambda)\sqrt{S}\sin\theta=\beta\sqrt{S}\sin\theta\):  
\[AF(\phi,\theta)=f\left(\beta\sqrt{S}\sin\theta\cos\phi,\beta\sqrt{S}\sin \theta\sin\phi\right). \tag{4}\]  
As a result, if we apply a scaling procedure, mapping the geometry \((x,y)\) to \((x^{\prime},y^{\prime})\), then the observed field in the angle \(\theta\) is moved to angle \(\theta^{\prime}\). The equivalence relation is given by the following:  
\[r\sin\theta=r^{\prime}\sin\theta^{\prime}\Leftrightarrow\theta^{\prime}=\sin^ {-1}\bigl{(}r\sin\theta/r^{\prime}\bigr{)}, \tag{5}\]  
where \(r\) and \(r^{\prime}\) are the distances of a point from the center of the aperture in the initial and scaled cases. Equivalently, one can reach a relation:  
\[ \begin{array}{l}xu=x^{\prime}u^{\prime}\Leftrightarrow x\cos(\phi)\sin( \theta)=x^{\prime}\cos(\phi)\sin(\theta^{\prime})\\ yv=y^{\prime}v^{\prime}\Leftrightarrow y\sin(\phi)\sin(\theta)=y^{\prime}\sin( \phi)\sin(\theta^{\prime})\end{array} . \tag{6}\]  
The previous equation suggests that an up-scaling in real space (up-scaling the continuous or discrete aperture) results in shrinking (following a reciprocal scaling law) of the \(u\)-\(v\) space in such a way that previously invisible (having imaginary \(u\) and \(v\)) regions are entering the visible unit circle. It is noticed that the above given formulation holds not only for planar apertures, but also for line sources (see Figure 1). For a line source of length \(L\), instead of (1) and (3), now it is:  
\[ \begin{array}{l}AF(\phi,\theta)=\int\limits_{x=-L/2}^{x=+L/2}I(x/(L/2))\exp(- j2\pi(x\sin\theta))dx=(L/2)\int\limits_{-1/2}^{+1/2}I(\zeta)\exp(-j2\pi(L/2) \sin\theta\zeta)d\zeta\\ \zeta=x/(L/2)\end{array} . \tag{7}\]  
The first pictorial example for comprehension reasons is given next for a linear array, which is a case of line source. Indeed, we use a line source of length \(L=3\lambda\) with the following excitation:  
\[I(x/(L/2))=\sum_{v=-3}^{+3}\delta(x/(L/2)-v/3), \tag{8}\]  
where \(\delta\) is the Dirac Delta function. Equation (8) represents a uniformly excited linear array of 7 point sources in equidistance \(d=0.5\lambda\). Using a scaling factor of \(\mu=2\), we reach a new array with \(d=1\lambda\), where the field-of-view range is limited to \(\theta_{0}=\sin^{-1}(\sin(\pi/2)/\mu)=\sin^{-1}(0.5)=30^{\circ}\). It is obvious that the initial pattern is compressed, as the [0, 90\({}^{\circ}\)] visible region is shrunk to the new one [0, 30\({}^{\circ}\)]. Additionally, the half-power beamwidth, (_HPBW_) decreases from ~7.3\({}^{\circ}\) degrees to ~3.6\({}^{\circ}\). Based on Relation (5), one can formulate a mapping between FFoV and LFoV systems. This mapping, which is derived next, apart from the transform in Equation (5), includes the following:  
* Scaling factor equation (Equation (9));
* HPBW equation (Equation (10));
* Directivity equation (Equation (18));
* Sidelobe level (SLL). To construct the mapping, first, the scaling factor, \(\mu\), must be computed.