## 1 Introduction  
With the deployment of 5G networks [1], there is an increased need for the electromagnetic characterization of devices for applications that range from the Internet of Things [2; 3], smart cities [4], Industry 4.0 [5] and self-driving cars [6], satellite-terrestrial relay networks [7; 8] and beam-forming for mm-wave networks [9], among others [10]. Of particular interest is the mm-wave frequency range (FR) 2, from 24.25 GHz to 71 GHz, since it provides a readily available wide spectrum, unlike FR1 in the sub-6 GHz spectrum, which is already very crowded with services that range from wireless local area networks to aerial communications [11]. Of all the FR2 bands, the lower bands around 28 GHz are especially appealing since they have been allocated by the most countries, including, but not limited to, the European Union, the United States, Japan, China and South Korea [11], but also because they present relatively low attenuation losses compared to other mm-wave frequencies [12]. Thus, antenna measurement systems must evolve to fulfil the characterization needs of the devices developed for these new applications. To this end, a number of alternatives have been proposed for the measurement of radiation patterns at mm-wave bands [12; 13; 14; 15]. However, most of these techniques involve a last stage of data post-processing to obtain the radiation pattern. Another alternative consists in directly acquiring the radiation pattern, for instance, with far-field (FF) ranges. However, since the antenna under test (AUT) needs to be placed in the FF region of the probe, FF ranges usually result in bulky systems. By placing the AUT in the near-field (NF) region of the probe a near-field range is obtained, but, again, data post-processing is required. In this regard, compact antenna test ranges (CATR) [16] solve these issues by simulating open range conditions in the Fresnel region of the probe so the far field may be readily acquired in a close range. Typically, parabolic antennas have been employed, due to their geometrical properties, for the collimation of spherical waves into planar waves in a region known as the quiet zone (QZ) [17]. To avoid diffraction effects due to the low illumination taper in the reflector surface, serrated edges are employed [18]. However, CATR systems based on single or multiple parabolic reflectors have various drawbacks: they are bulky and expensive to manufacture, especially at high frequencies at which the required surface error is very low [16]. A more interesting approach is the use of plane-wave generators (PWG). They consist of array antennas that are optimized to generate a plane wave in their NF region. Although the concept is not new [19; 20], it has gained traction in the past few years for over-the-air measurements of 5G devices in the mm-wave ranges [21; 22]. The use of conventional arrays generates an issue of requiring complex feeding networks to control the excitation of the array elements [21]. This, in turn, may make the in-band synthesis of these arrays more challenging since the feeding network would, ideally, have to be considered in the simulations to obtain more accurate results [23]. Using spatially fed arrays [24] overcomes this limitation, since the feed may be readily included in the in-band synthesis procedure simply by obtaining the incident field at the array aperture at a number of frequencies, either by means of measurements or use of full-wave simulations. Then, the array element may be easily simulated by assuming local periodicity according to the Floquet theorem [25; 26]. Both reflectarrays [27; 28; 29] and transmitarrays [30; 31] have been proposed for use as probes in CATR systems. However, due to the strong illumination taper imposed by the feed, an NF synthesis technique is necessary to shape the NF field in the QZ such that it complies with the requirements of a plane wave. Recently, several synthesis techniques were proposed for the NF-shaping of spatially fed arrays [28; 32; 33; 34].