## 3 Numerical Results  
In this section, some representative results numerically assessing the feasability of the proposed spherical NF/FF transformation for characterising volumetric AUTs in the presence of an infinitely perfectly conducting ground plane are presented. In order to show the capability of the proposed AUT modelling to reduce volumetric redundancy and, as a consequence, the number of required NF samples, two different kinds of antennas have been considered. The former (Case A) is a volumetric AUT whose geometry is characterised by two predominant dimensions as, f.i., a parallelepiped having its width (\(W\)) and length (\(L\)) greater than the height (\(H\)), while the latter (Case B) is a volumetric AUT having a predominant dimension as, f.i., a parallelepiped for which \(H>W,L\). The AUT of Case A has been properly arranged to simulate an automotive body. To this end, a rectangular parallelepiped surmounted by a rectangle has been considered. The parallelepiped, whose geometric centre lies on the \(z\)-axis of the adopted \((x,y,z)\) reference system, has width \(W_{x}=14\,\lambda\), length \(L_{y}=35\,\lambda\), and height \(H_{z}=6\,\lambda\) and is located at \(2\,\lambda\) from the PEC ground plane. The rectangle is \(4.5\,\lambda\) above the parallelepiped and is \(14\,\lambda\) wide and \(25\,\lambda\) long. A planar array of elementary Huygens sources, spaced by \(\lambda/2\) and polarized along the \(z\)-axis, is located on each of the faces of the parallelepiped at \(y=17.5\,\lambda\) and \(y=-17.5\,\lambda\). Moreover, the rectangle surmounting the parallelepiped is covered by another planar array of Huygens sources spaced by \(\lambda/2\) and polarized along the \(y\)-axis. An open-ended WR430 rectangular waveguide, operating at \(2.45\) GHz, scans a hemisphere of radius \(R_{hs}=30\,\lambda\). To account for the presence of the infinite ground plane and properly develop the non-redundant representation over the entire sphere, the proposed technique exploits the principle of image, according to which the voltage at any sampling point is due to the contributions from the AUT and its image. To properly simulate the NF samples, the image of the AUT has been obtained by mirroring each electric and magnetic dipole, realising an elementary Huygens source of the AUT. Then, the obtained combined AUT has been modelled by an adaptable surface having parameters \(a=18.85\,\lambda\), \(h_{c}=16\,\lambda\), and \(c=4.5\,\lambda\). Moreover, rule (7) has been used to synthesise the NF samples, which should have been measured by the probe and rotated probe on the lower hemisphere. The non-redundant sampling representation has been obtained by adopting \(\chi^{\prime}=\chi=1.10\). To test the precision of the proposed sampling strategy and of OSI algorithms (8) and (13) from a qualitative standpoint, Figures 3 and 4 show the comparisons of the exact and reconstructed normalised amplitudes and phases of the voltages \(V_{p}\) and \(V_{r}\) on the meridians at \(\varphi=0^{\circ}\) and \(\varphi=90^{\circ}\). As can be seen, the recovered patterns over the upper hemisphere are in very good agreement with the exact ones and do not suffer from any truncation errors. For the sake of completeness, the recoveries of the amplitudes of \(V_{p}\) and \(V_{r}\) relevant to the meridian at \(\varphi=60^{\circ}\) are also reported in Figure 5. The effectiveness of the OSI algorithm is confirmed by the normalised mean-square reconstruction errors. Such errors have been obtained by comparing the exact and recovered values of \(V_{p}\) and \(V_{r}\) on a dense grid of the scanning hemisphere and normalizing them to the maximum voltage value over it. As shown in Figure 6, these errors, computed for \(\chi^{\prime}=1.10\), \(\chi=1.05,1.10,1.15\), and \(p\), \(q\) ranging from 2 to 11, decrease as \(\chi\), \(p\), and \(q\) increase. This provides a tool for properly choosing the sampling/OSI parameters for a given reconstruction error.