## 2 Sampling Representation over the Upper Hemisphere  
Let us characterise a large and heavy AUT through a spherical NF setup. The scanning is realised by combining the azimuthal movement of a metallic turntable, which can be conveniently modelled as a PEC plane with infinite extension, with the elevation motion of a small first-order measurement probe, guided along a circular arc or mounted on a rotating gantry. Accordingly, the considered probe, which possesses the same bandwidth as the AUT one [40], collects the NF measurements over the upper hemisphere of radius \(R_{hs}\) in presence of an infinite PEC ground plane (Figures 1 and 2). Let us introduce a spherical coordinate system \((r,\ \theta,\ \varphi)\) having its origin \(O\) on the infinite PEC ground and coincident with the centre of the spherical measurement system to denote any observation point \(Q\). Moreover, let us suppose that the considered AUT is \(d_{gg}\) away from the ground plane. The non-redundant sampling representation over the scanning hemisphere is developed by reformulating the problem according to the principle of image [37], which provides a convenient way to study a radiating structure in the presence of an infinite PEC ground plane. As a matter of fact, it enables us to consider, instead of the original problem (Figure 2a), an equivalent one (Figure 2b), wherein the infinite PEC ground plane is substituted by the image of the AUT above it. The AUT image is below the ground plane at the same distance as the AUT is above it, in order that the boundary condition of the equivalent problem is equal to the PEC boundary condition of the original problem. Accordingly, the AUT and its image can be considered together in the equivalent problem as a single combined AUT. To fulfil the requirements of the non-redundant representations of EM fields [33], (i) the combined AUT (the considered AUT plus its image) has to be modelled by a convex surface, \(\Sigma_{M}\), which fits as much as possible its actual geometry; (ii) each meridian and parallel describing the spherical observation surface must be represented in terms of a convenient parameter \(\tau\); and (iii) a suitable phase factor \(\mathrm{e}^{\mathrm{j}\tau(\tau)}\) has to be determined in order to introduce the so-called \"reduced voltage\"  
\[\widetilde{V}(\tau)=\widetilde{V}_{p,r}(\tau)=V_{p,r}(\tau)\mathrm{e}^{\mathrm{ j}\tau(\tau)} \tag{1}\]  
with \(V_{p}\), \(V_{r}\) denoting the voltage at the terminals of the probe in two of its orientations (in the second one, the probe is rotated by \(90^{\circ}\) around its axis). Such a reduced voltage is particularly convenient since, despite the mere voltage, it exhibits the features of a bandlimited function [33]. Indeed, it is only quasi-bandlimited, but it can be shown [33] that a very small error is made if \(\widetilde{V}(\tau)\) is approximated by a rigorously bandlimited function. To this end, its bandwidth \(W_{\tau}\) is increased by a factor \(\chi^{\prime}\), named excess bandwidth factor,  
Figure 1: Spherical scan of heavy volumetric AUTs over a PEC ground plane. Figure 2: View in the meridian plane at \(\varphi=0\): (**a**) original formulation of the problem; (**b**) equivalent formulation of the problem. which is able to control the aliasing error [33]. A \(\chi^{\prime}\)-factor slightly greater than unity is enough when considering AUTs of large size, as those here considered. The combined AUT of the equivalent problem can be considered a volumetric antenna, whose encumbrance has one or two predominant dimensions as compared to the other one(s), depending on the shape of the original antenna and its distance from the PEC ground. In order to choose the modelling surface \(\Sigma_{M}\) able to best fit the real geometry of the combined AUT, whatever its encumbrance, the very adaptable surface proposed in [34] can be conveniently employed.