## 1 Introduction  
The design and manufacturing processes of an antenna require, as a crucial step, its characterization. Indeed, it is a very challenging task since the characteristics of the antenna (frequency, electric dimension, far-field angular coverage, environmental conditions, etc.) and the intended application determine the choice of the measurement technique to be used that guarantees the greatest accuracy [1]. The pros and cons of far-field versus near-field test ranges are well known to the antenna measurement community. The far-field test ranges are usually employed when dealing with antennas working at low frequencies, and only the far-field (FF) pattern in the principal planes is required. Near-field (NF) test setups are instead used when dealing with antennas working at higher frequencies, and complete FF pattern evaluations and polarisation measurements are required. Note that, unlike NF test setups, the FF test range is usually an outdoor setup and has the disadvantage that the weather conditions and surroundings (buildings, hills, objects, electromagnetic interference from other wireless networks) can impair the accuracy of the measurements. Nevertheless, the use of the NF test setups requires a greater number of measurements than the FF ones, since the collected measurements have to be properly post-processed through a NF/FF transformation to obtain the FF pattern in any desired cut plane. The choice of the NFtest setup to be used is always a trade-off between the type of antenna, the measurement requests, and the involved analytical and mechanical complexities. Accordingly, the planar NF test setup is usually chosen to characterise high-directivity antennas under test (AUTs); the cylindrical NF test setup is well-suited for AUTs mainly radiating in an angular zone centred on the horizontal plane; the spherical NF test setup is particularly recommended for AUTs without preferential radiation directions. Obviously, depending on the NF test setup adopted to characterise the antenna under test (AUT), the NF/FF transformation employed to evaluate the FF pattern can be planar, cylindrical, or spherical [1; 2; 3; 4; 5; 6; 7; 8]. It must be stressed that the spherical NF test setup shows the greatest mechanical complexity, and the related NF/FF transformation is analytically the most complex. In any case, it can be adopted to characterise any kind of antennas, allowing a FF characterization over the whole FF sphere without any truncation error, and, accordingly, can be considered the purest and most complete NF antenna measurement solution. This is the reason why, over the years, it has attracted the attention of the antenna measurements community (see [9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32] as a non-exhaustive list of references). In particular, many efforts have been devoted to developing NF/FF transformations with spherical scan using sampling techniques more efficient from a data reduction viewpoint than the classical NF/FF transformation developed by [PERSON] in [15]. As a matter of fact, this last makes use of a spherical sampling grid, in which the number of parallels and that of the samples along each parallel are related, according to \"the minimum sphere rule,\" to the radius of the minimum sphere containing the AUT. Obviously, the larger the AUT size, the greater the number of measurements and, as a consequence, the longer the acquisition time. The theoretical results concerning the non-redundant sampling representations of the radiated or scattered electromagnetic (EM) fields [33] have been exploited to develop NF/FF transformations [16; 26; 27; 28], which make use of a smaller number of measurement points as compared to that utilised by classical [PERSON]'s technique [15]. These techniques have proven to be effective from a numerical and experimental standpoint without impairing the accuracy of the FF reconstruction process.