## 1 Introduction
This has been made possible since, according to [33], it is possible to adopt an effective non-redundant sampling representation of the voltage, which makes use of a minimum set of NF samples, whose number depends on the convex surface chosen to model the considered AUT. These voltage samples are then interpolated by an \"ad hoc\" two-dimensional optimal sampling interpolation (OSI) formula to reconstruct the voltage values that the measuring probe would have collected at the sampling points of the classical spherical grid, thus allowing one to get the FF pattern via [PERSON]'s NF/FF transformation technique. The choice of the modelling surface is crucial for obtaining a sampling representation that is really non-redundant and substantially depends on the given AUT, since the choice criterion is that the modelling surface is chosen such that it adheres as much as possible to its actual geometry. Thus, the modelling surface has been chosen in [16; 26; 27] coincident with that of an oblate spheroid [16; 27] or that obtained by two circular bowls sharing the apertures and with eventually different lateral bendings (double bowl modelling) [26] to characterise a quasi-planar AUT, while in [16; 26; 27] a prolate spheroidal surface [16; 27] or that bounding a cylinder terminated by two hemispherical caps (rounded cylinder) [26] has been adopted for a long AUT. A very adaptable surface, bounding a central cylindrical body ended by two circular bowls with the same apertures and different bases, has been recently proposed [34] to characterise 3-D modular AUTs. As a matter of fact, it is able to best shape the geometry of such a kind of antenna, as well as any volumetric AUTs, by properly setting the height and the radius of the central body and the radii of the lateral bendings, thus minimising the volumetric redundancy, as well as the amount of the required measurements to be collected, as compared to a spherical modelling of the source. Note that such a surface has four degrees of freedom in the choice of the geometrical parameters and contains the surfaces bounding a rounded cylinder, a double bowl, and a sphere as particular cases. The characterization of large and heavy AUTs in spherical NF setups requires the use of mechanically stable supports in order to ensure high repeatability of the measurements. In such a case, the size and weight of the considered AUT do not allow one to satisfy such a requirement. This occurs when testing one or more antennas mounted on a large platform or integrated with the body of a car or aircraft, and the interest is not just to test the single antennas but the whole structure (antenna plus platform/vehicle) in order to assess its capability to build up the wireless link with the network infrastructure (terrestrial or satellite network) for which it is intended in its operating conditions. To meet such a goal, the considered radiating structure can be characterised through a hemispherical NF setup, wherein such a structure is placed on a turning metallic ground, providing the required well-defined measurement environment to ensure a high degree of repeatability, and the NF measurements are collected by the measuring probe, guided along a circular arc or mounted on a rotating gantry, over the upper hemisphere. Obviously, the NF data at the points of the spherical sampling grid required by [PERSON]'s NF/FF transformation technique falling over the lower hemisphere must be set to zero, thus impairing the accuracy of FF reconstructions, which will be affected by a severe truncation error. This issue can be overcome, as proposed in [35; 36], by applying the principle of image [37] in the presence of an infinitely perfectly electric conducting (PEC) ground plane for synthesising the lacking NF data over the lower hemisphere. Such a suggestion has been followed in [38; 39] to develop non-redundant NF/FF transformations with a hemispherical scan for long AUTs.