## 3 Numerical Results
Note that a very low reconstruction error of about \(-60\) dB occurs for the adopted sampling/OSI parameters \(\chi^{\prime}=\chi=1.10\) and \(p=q=8\). Such an error can be considered negligible if compared with the measurement one. Moreover, to investigate the algorithm stability, the NF samples have been simulated as corrupted with random errors. In particular, a background noise (having an amplitude bound to \(\Delta\xi\) and an arbitrary phase) and an uncertainty of \(\pm\Delta\xi_{r}\) in amplitude and \(\pm\Delta\psi\) in phase have been added to each exact NF sample over the upper hemisphere. Figure 7 shows the reconstructions of amplitude and phase on the meridian at \(\varphi=90^{\circ}\) obtained from these error-affected samples, which assess the algorithm's stability even in a real measurement environment. Figure 4: \(V_{r}\) on the meridian at \(\varphi=90^{\circ}\). Blue solid line: exact. Red dots: interpolated from the non-redundant samples: (**a**) Amplitude; (**b**) Phase. Figure 3: \(V_{p}\) on the meridian at \(\varphi=0^{\circ}\). Blue solid line: exact. Red dots: interpolated from the non-redundant samples: (**a**) Amplitude; (**b**) Phase. Figure 5: Voltage amplitude on the meridian at \(\varphi=60^{\circ}\). Blue solid line: exact. Red dots: interpolated from the non-redundant samples: (**a**) \(V_{p}\); (**b**) \(V_{r}\). Figure 6: Normalised mean-square errors in the reconstruction of the probe voltage: (**a**) \(V_{p}\); (**b**) \(V_{r}\). Figure 7: \(V_{r}\) on the meridian at \(\varphi=90^{\circ}\). Blue solid line: exact. Red dots: recovered from the error affected NF samples: (**a**) Amplitude; (**b**) Phase. respectively, set by the developed sampling strategy. Figure 8 shows the comparisons between the recovered FF patterns in the cut planes at \(\varphi=0^{\circ}\), \(60^{\circ}\), and \(90^{\circ}\) and the corresponding exact ones over the upper hemisphere. As can be seen, the exact and recovered FF patterns agree very well, thus proving the efficiency of the devised NF/FF transformation with hemispherical scanning in the presence of an infinite PEC ground plane for volumetric AUT exhibiting two predominant dimensions. The AUT of Case B has been properly arranged to simulate an antenna having a predominant dimension. To this end, a rectangular parallelepiped, with dimensions equal to \((W_{x}\times L_{y}\times H_{z})=(15\lambda\times 15\lambda\times 23\lambda)\) and having its centre on the \(z\)-axis of the used \((x,y,z)\) reference system, has been considered. It is placed \(2\,\lambda\) away from the PEC ground plane, and its lateral faces are covered by planar arrays of elementary Huygens sources, spaced by \(\lambda/2\) and polarized along the \(z\)-axis. To simulate the combined AUT, the image of such an antenna with respect to the \(xy\) plane has been obtained as described in the previous example. Then, the combined AUT has been considered enclosed in the adaptable surface with parameters \(a=10.6\,\lambda\), \(h_{c}=50\,\lambda\), and \(c=2.5\,\lambda\). The non-redundant sampling representation on the entire sphere has been obtained by adopting \(\chi^{\prime}=\chi=1.10\). The NF samples at the corresponding sampling points have been simulated as acquired (or synthesised according rule (7)) on a sphere of radius \(R_{\mathrm{bs}}=35\,\lambda\). To qualitatively appreciate the effectiveness of OSI algorithms (8) and (13), the reconstruction of the voltage \(V_{r}\) on the meridian at \(\varphi=90^{\circ}\) is shown in Figure 9. As can be seen, the reconstruction over the upper hemisphere is very accurate. The same accuracy is also experienced for the voltage reconstructions relevant to other meridians. These reconstructions are not shown here to save space. In any case, the accuracy of the devised strategy and of the related OSI algorithm can be appreciated by the normalised mean-square reconstruction  
Figure 8: Far-field pattern. Blue solid line: exact.