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An overview of the near-field-far-field (NF-FF) transformation techniques with innovative spiral scannings, useful to derive the radiation patterns of the antennas commonly employed in the modern wireless communication systems, is provided in this paper. The theoretical background and the development of a unified theory of the spiral scannings for quasi-spherical and nonspherical antennas are described, and an optimal sampling interpolation expansion to evaluate the probe response on a quite arbitrary rotational surface from a nonredundant number of its samples, collected along a proper spiral wrapping it, is presented. This unified theory can be applied to spirals wrapping the conventional scanning surfaces and makes it possible to accurately reconstruct the NF data required by the NF-FF transformation employing the corresponding classical scanning. A remarkable reduction of the measurement time is so achieved, due to the use of continuous and synchronized movements of the positioning systems and to the reduced number of needed NF measurements. Some numerical and experimental results relevant to the spherical spiral scanning case when dealing with quasi-planar and electrically long antennas are shown.

The design of the modern antenna systems, integrated on portable devices or employed in radio base stations, requires the experimental verification of the antenna performances in terms of frequency bands and radiation diagrams. Different types of antennas are employed in the abovementioned applications. In particular, printed or conformal antennas are generally used in portable devices (such as cellular phones, mobile pc, printers, scanners, DVD players, and digital projectors) [

As well known, the accurate measurement of the electromagnetic (EM) field radiated by an antenna can be performed only in an anechoic chamber, wherein the free-space propagation conditions are emulated by suppressing a large amount of the reflections from the lateral walls, ceiling, and floor. However, the far-field (FF) distance requirements cannot be practically satisfied in an anechoic chamber when dealing with antennas having large or even medium electrical dimensions, so that only near-field (NF) measurements can be carried out. This occurs specifically when characterizing antenna systems and arrays for space as well as radar applications [

Usually, the acquired NF data are transformed into FF patterns by using a suitable expansion of the field of the antenna under test (AUT) in terms of modes, that is, a complete set of solutions of the vector wave equation in the region outside the antenna. To this end, plane, cylindrical, or spherical waves are generally employed [

In this framework, the application of the spatial bandlimitation properties of radiated EM fields [

The time needed for the acquisition of the NF data can be drastically reduced by using the modulated scattering technique, wherein arrays of scattering probes, which allow a very fast electronic scanning, are employed [

A more convenient way for reducing the measurement time is, as suggested in [

by choosing the spiral in such a way that its step, specified by two consecutive intersections with the considered meridian curve (generatrix, radial line, and meridian), be equal to the sample spacing needed for the interpolation along this curve;

by developing the nonredundant sampling representation along the spiral.

Helicoidal scanning.

Planar spiral scanning.

Spherical spiral scanning.

It must be stressed that the use of effective AUT modellings when dealing with elongated or quasi-planar antennas offers, besides the reduction of the number of required NF data, another great advantage in the helicoidal and planar spiral scanning case, respectively. Specifically, it makes possible the consideration of measurement cylinders (planes) with a radius (distance) smaller than one-half of the antenna maximum size, thus reducing the error related to the truncation of the scanning surface.

A unified theory of the spiral scannings for spherical and nonspherical antennas has been also developed in [

Moreover, direct NF-FF transformations with helicoidal scanning, which allow the evaluation of the antenna far field in any cut plane directly from a minimum set of NF data without interpolating them, have been recently proposed in [

Other NF-FF transformation techniques with spiral scannings have been also proposed [

The aim of this paper is just to provide an overview of the NF-FF transformation techniques with spiral scannings, useful to characterize the antennas commonly used in the modern wireless communication systems. The necessary theoretical background on the nonredundant sampling representations of EM fields is summarized in Section

Let us consider an antenna enclosed in a convex domain

To obtain a nonredundant representation, that is, a representation requiring a minimum number of samples, first of all, we must minimize the “local” bandwidth

It can be easily recognized that, by choosing

As regards the parameter

It can be easily verified that when

Relevant to a meridian observation curve.

Note that the angular-like parameter

It can be shown [

When

Accordingly, since

It must be stressed that, according to [

As a consequence, an effective source modelling when dealing with an electrically long antenna is got, f.i., by choosing

Ellipsoidal source modelling: prolate case.

Relation (

It can be easily shown [

Let us now consider a nondirective probe scanning a proper spiral wrapping an arbitrary rotational surface

In order to obtain a nonredundant sampling representation of the probe voltage on the surface

to choose the spiral in such a way that its pitch, specified by two consecutive intersections with a meridian curve, be equal to the sample spacing needed for the interpolation along this curve,

to develop a nonredundant sampling representation along the spiral.

According to condition

The development of a nonredundant sampling representation of the voltage along the spiral is a more difficult task, which has been heuristically tackled in [

Accordingly, the main results of the unified theory of spiral scannings for antennas enclosed in a spherical surface [

Geometry of the problem in the plane

By substituting (

On the other hand [

By taking into account such a relation and substituting (

According to such a relation, when adopting a spherical modelling, the optimal parameter

It is worth noting that the scanning spiral can be viewed as obtained by radially projecting the corresponding one wrapping, with the same pitch, the modelling sphere. Since such a spiral is a closed curve, it is convenient to choose the bandwidth

Let us now turn to the more general case wherein the AUT is no longer modelled as enclosed in a sphere. The parameterization

According to these results, the reduced voltage at any point

It is worth noting that small variations of

The OSI expansion (

The effectiveness of the OSI algorithms depends on the choice of

Some representative numerical and experimental results relevant to the spherical spiral scanning case when considering quasi-planar and electrically long antennas, respectively, are reported in the following for reader’s convenience. The interested reader can find other numerical and experimental results relevant to the spherical spiral scanning case as well as to the helicoidal and the planar spiral one in the quoted references. The reason for the choice of spherical spiral scanning case, notwithstanding its less effectiveness from the data reduction and measurement time saving viewpoints with respect to the helicoidal and planar spiral scan, is due to the fact that such a scanning, as the spherical one, allows the reconstruction of the whole AUT radiation pattern. Moreover, since it allows the characterization of planar, quasi-planar and elongated antennas, commonly adopted in modern wireless communication systems, the flexible AUT modellings adopted in the following examples have been employed also in the planar spiral scanning and in the helicoidal one, respectively. On the other hand, these two AUT modellings together with the already described ellipsoidal ones provide a whole survey on how to effectively model nonspherical antennas.

The following numerical simulations are relevant to the spherical spiral scanning and to a quasi-planar antenna modelled as enclosed in a double bowl, that is, a surface

Spherical spiral scanning for a quasi-planar antenna.

The reconstructions of the amplitude and phase of the probe voltage

Amplitude of the probe voltage

Phase of the probe voltage

Normalized maximum errors in the reconstruction of

Normalized mean-square errors in the reconstruction of

H-plane pattern. Solid line: exact. Crosses: reconstructed from NF measurements.

It is worthy to note that the number of samples on the spiral is 28483, significantly less than the one (43664) needed by the approach in [

For reader’s convenience, we report in the following some experimental results already published in [

Spherical spiral scanning for an elongated antenna.

Photo of the X-band resonant slotted waveguide array.

The effectiveness of the two-dimensional OSI expansion is assessed by comparing the amplitude and phase of the recovered voltage

Amplitude of

Phase of

At last, the E-plane and H-plane FF patterns reconstructed from the NF set of measurements acquired by means of the spherical spiral scan are compared in Figures

E-plane pattern. Solid line: reference. Crosses: recovered from NF data acquired via the spherical spiral scanning.

H-plane pattern. Solid line: reference. Crosses: recovered from NF data acquired via the spherical spiral scan. Dashed line: reconstruction error.

Note that the number of used samples is 1024, significantly less than those (3622 and 5100) required by the NF-FF transformation with spiral scanning [

It is worth noting that, in this example, the reduction rates of the needed NF data with respect to those required in the spherical spiral scan [

This paper provides a comprehensive overview of the nonredundant NF-FF transformations using fast spiral scanning techniques, which allow a drastic time saving, since they are realized through continuous and synchronized movements of the positioning systems and require a reduced number of NF measurements. The authors have first recalled the results on the nonredundant representations of EM fields, which constitute the theoretical background of the spiral scanning techniques. Then, they have described the reasoning involved in the heuristic approach that allowed the development of the unified theory concerning the spiral scannings for nonspherical antennas from that for the quasi-spherical ones. The OSI algorithm, used to determine the voltage measured by a nondirective probe on a quite arbitrary rotational surface from a nonredundant number of its samples collected along a proper spiral, has been also presented. Although this theory is valid for spirals lying on quite arbitrary rotational surfaces, its application to spirals wrapping the conventional scanning surfaces has made possible the development of accurate NF-FF transformations which use a nonredundant number of NF spiral data and allow remarkably reducing the measurement time, since the NF data needed by the corresponding classical NF-FF transformation are accurately reconstructed from those collected on the spiral path. NF-FF transformation techniques allowing a remarkable time saving without losing the efficiency of the classical ones are so made available to the antenna designers and to the measurement community involved in the development of the modern wireless communication systems.

The authors declare that there is no conflict of interests regarding the publication of this paper.