Effective Antenna Modellings for NF-FF Transformations with Spherical Scanning Using the Minimum Number of Data

Abstract

Two efficient probe-compensated near-field-far-field transformations with spherical scanning for antennas having two dimensions very different from the third one are here developed. They rely on the nonredundant sampling representations of the electromagnetic fields and on the optimal sampling interpolation expansions, and use effective antenna modellings. In particular, an antenna with a predominant dimension is no longer considered as enclosed in a sphere but in a cylinder ended in two half spheres, whereas a surface formed by two circular “bowls” with the same aperture diameter but different lateral bends is adopted to shape an antenna with two predominant dimensions. These modellings are able to fit very well a lot of antennas by properly setting their geometric parameters. It is so possible to remarkably lower the number of data to be acquired, thus significantly reducing the measurement time. Numerical tests assessing the accuracy and the robustness of the techniques are reported.

1. Introduction

As is well known, near-field-far-field (NF-FF) transformation techniques play a significant role in modern antenna measurements [1–4]. As a matter of fact, the pattern evaluation from NF measurements allows one to overcome those drawbacks which, for electrically large radiating systems, make it unpractical to measure the pattern in a conventional FF range. Two main techniques for the FF pattern reconstruction from NF measurements can be found in literature. The former relies on the wave-mode expansion method and uses canonical scanning surfaces (sphere, cylinder, plane) [1–4]. The latter is based on the equivalence theorem [5, 6] and can be applied to arbitrary scanning surfaces although at a remarkably increased computational cost.

Among the NF-FF transformation techniques, the employing of the spherical scanning has attracted considerable attention [7–15], since it allows one to reconstruct, from a single set of NF measurements, the complete radiation pattern of the antenna under test (AUT). In the classical NF-FF transformation with spherical scanning [7–13], the choice of the highest spherical wave and the data spacings on the measurement sphere are determined according to a rule of thumb, related to the smallest sphere able to contain the AUT.

In [14], such an NF-FF transformation has been properly modified by taking into account the spatial band-limitation properties of the electromagnetic (EM) fields [16]. In particular, the choice of the highest spherical wave has been rigorously fixed by the band-limitation properties and the number of data to be acquired on the rings decreases towards the poles. Moreover, the nonredundant sampling representations of the EM field [17] have been applied in the same paper [14] to remarkably reduce the number of required NF data when antennas with one or two predominant dimensions are considered. These results have been obtained by considering the AUT as enclosed in a prolate or oblate ellipsoid, respectively. Then, an optimal sampling interpolation (OSI) formula has been developed to recover the data required by the aforementioned NF-FF transformation from the knowledge of a nonredundant number of NF samples. However, since the case of an ideal probe has been considered in [14], the effects of the measurement probe have not been taken into account in the evaluation of the antenna FF pattern. The ideal probe assumption has been removed in [15], wherein an efficient probe-compensated NF-FF transformation technique with spherical scanning tailored for elongated or quasiplanar antennas has been developed, by properly extending the nonredundant sampling representation to the voltage measured by a nondirective probe. Then, the nonredundant voltage samples are interpolated via an efficient OSI algorithm for reconstructing the NF data required by the probe-compensated NF-FF transformation technique with spherical scanning [7], again properly modified to account for the spatial bandlimitation properties.

It must be stressed that the reduction in the number of the NF data to be acquired reflects in a decrease of the measurement time, and this is a very important issue for the antenna measurement community, since such a time is very much greater than that needed to carry out the NF-FF transformation. A measurement time reduction can also be obtained by performing a fast electronic scanning via the modulated scattering technique [18], which employs arrays of scattering probes. However, antenna testing facilities based on such a technique are less flexible than those using mechanical scans. Accordingly, it appears more convenient the reduction of the needed NF data attainable by using the nonredundant sampling representations, which does not require to change the positioning systems, but only the software for controlling them.

The aim of this paper is to develop even more effective NF-FF transformations with spherical scanning tailored for antennas having one or two predominant dimensions. To this end, very flexible antenna modellings will be adopted. In particular, an electrically long antenna will be considered as enclosed in a rounded cylinder, namely, a cylinder which ends in two half spheres, whereas a surface formed by two circular “bowls” with the same aperture diameter but different lateral bends (double bowl) will be adopted to shape a quasiplanar antenna. The so-obtained NF-FF transformation techniques result to be more effective from the data reduction viewpoint than those employing the prolate or oblate ellipsoidal modelling. In fact, these flexible modellings allow one to fit the shape of a lot of actual antennas better by properly setting their geometric parameters. Moreover, they remain quite general and contain the spherical modelling as a particular case.

2. Nonredundant Sampling Representation of the Voltage on a Sphere

Let us consider an AUT enclosed in a convex domain bounded by a rotational surface Σ and a nondirective probe scanning a spherical surface of radius 𝑑 in the antenna NF region. The spherical coordinate system (𝑟,𝜗,𝜑) will be adopted in the following to denote an observation point 𝑃 in the NF region. As shown in [19], the voltage measured by such a kind of probe has the same effective spatial bandwidth of the AUT field, and, accordingly, the nonredundant sampling representations of EM fields [17] can be applied to it. Moreover, since the scanning sphere can be represented by meridians and parallels, in the following we deal with the voltage representation on an observation curve 𝐶 described by an analytical parameterization 𝑟=𝑟(𝜉). According to [17], let us introduce the “reduced voltage”𝑉(𝜉)=𝑉(𝜉)e^{𝑗𝛾(𝜉)}, (1) where 𝛾(𝜉) is a proper phase function to be determined. The error, occurring when 𝑉 is approximated by a spatially bandlimited function, becomes negligible as the bandwidth exceeds a critical value 𝑊_𝜉 [17]. In fact, it exhibits a steplike behaviour whose transition occurs at 𝑊_𝜉 [16]. Accordingly, such an error can be effectively controlled [20] by choosing a bandwidth equal to 𝜒^′𝑊_𝜉, where 𝜒^′ is an excess bandwidth factor, slightly greater than unity for electrically large antennas.

When 𝐶 is a meridian, by adopting𝑊_𝜉=𝛽ℓ^′2𝜋 (2) (𝛽 being the wavenumber and ℓ^′ the length of the curve 𝐶^′ obtained as intersection between the meridian plane passing through the observation point 𝑃 and Σ), we get 𝛽𝛾=2𝑅₁+𝑅₂+𝑠^′₁−𝑠^′₂,𝜋𝜉=ℓ^′𝑅₁−𝑅₂+𝑠^′₁+𝑠^′₂, (3) where 𝑅_1,2 are the distances from 𝑃 to the tangency points 𝑃_1,2 on 𝐶^′, and 𝑠^′_1,2 are their arclength coordinates.

When the observation curve is a parallel, due to the involved symmetry, the phase function is constant and it is convenient to use the azimuthal angle 𝜑 as parameter. The corresponding bandwidth is given [17] by𝑊_𝜑=𝛽2max_𝑧′𝑅⁺−𝑅⁻=𝛽2max_𝑧′𝑧−𝑧^′2+𝜌+𝜌^′𝑧^′2−𝑧−𝑧^′2+𝜌−𝜌^′𝑧^′2, (4) wherein 𝜌=𝑑sin𝜗, 𝜌^′(𝑧^′) is the equation of Σ in cylindrical coordinates, and the maximum is attained [17] on that zone of the surface Σ lying on the same side of the observation circumference with respect to its maximum transverse circle.

In the light of these results, the reduced voltage at 𝑃(𝜉,𝜑) on the meridian fixed by 𝜑 can be evaluated via the OSI expansion𝑉𝜉,𝜑=^𝑚0+𝑞_{𝑚=𝑚0−𝑞+1}𝑉𝜉_𝑚,𝜑Ω_𝑀𝜉−𝜉_𝑚𝐷_𝑀′′𝜉−𝜉_𝑚, (5) where 𝑚₀=Int[𝜉/Δ𝜉] is the index of sample nearest (on the left) to 𝑃,2𝑞 is the number of the retained intermediate samples 𝑉(𝜉_𝑚,𝜑), that is, the reduced voltage values at the intersection points between the sampling parallels and the considered meridian, and𝜉_𝑚=𝑚Δ𝜉=2𝜋𝑚2𝑀^′′,𝑀+1_′′=Int𝜒𝑀^′+1,𝑀′=Int𝜒^′𝑊_𝜉+1, (6)𝜒 being a factor that controls the truncation error [17] and Int(𝑥) denoting the integer part of 𝑥. Moreover,𝐷_𝑀′′(𝜉)=sin2𝑀^′′+1𝜉/22𝑀^′′(),Ω+1sin𝜉/2_𝑀(𝜉)=𝑇_𝑀2cos²(𝜉/2)/cos²𝜉₀/2−1𝑇_𝑀2/cos²𝜉₀/2−1 (7) are the Dirichlet and Tschebyscheff Sampling functions, respectively, 𝑇_𝑀(⋅) being the Tschebyscheff polynomial of degree 𝑀=𝑀^′′−𝑀^′ and 𝜉₀=𝑞Δ𝜉.

The intermediate samples 𝑉(𝜉_𝑚,𝜑) can be evaluated via the OSI formula𝑉𝜉_𝑚,𝜑=^𝑛0+𝑝_{𝑛=𝑛0−𝑝+1}𝑉𝜉_𝑚,𝜑_𝑛,𝑚Ω_𝑁𝑚𝜑−𝜑_𝑛,𝑚×𝐷_{𝑁𝑚′′}𝜑−𝜑_𝑛,𝑚, (8) where 𝑉(𝜉_𝑚,𝜑_𝑛,𝑚) are the reduced samples on the parallel fixed by 𝜉_𝑚,2𝑝 is the retained samples number, 𝑛₀=Int(𝜑/Δ𝜑_𝑚), and𝜑_𝑛,𝑚=𝑛Δ𝜑_𝑚=2𝜋𝑛2𝑁_𝑚^′′+1,𝑁_𝑚^′′=Int𝜒𝑁^′_𝑚𝑁+1,_′𝑚=Int𝜒^∗𝑊_𝜑𝜉_𝑚+1,𝑁_𝑚=𝑁_𝑚^′′−𝑁^′_𝑚,𝜒^∗=1+𝜒^′−1sin𝜗𝜉_𝑚^−2/3. (9)

The variation of 𝜒^∗ with 𝜉 is required to ensure a band-limitation error constant with respect to 𝜉.

By properly matching (5) and (8), the following two-dimensional OSI expansion results: 𝑉=𝜉,𝜑_{𝑚0+𝑞𝑚=𝑚0−𝑞+1}Ω_𝑀𝜉−𝜉_𝑚𝐷_𝑀′′𝜉−𝜉_𝑚^𝑛0+𝑝_{𝑛=𝑛0−𝑝+1}×𝑉𝜉_𝑚,𝜑_𝑛,𝑚Ω_𝑁𝑚𝜑−𝜑_𝑛,𝑚𝐷_{𝑁𝑚′′}𝜑−𝜑_𝑛,𝑚, (10) which makes it possible to recover the NF data needed to carry out the spherical NF-FF transformation [7] as modified in [15].

2.1. Double Bowl Modelling

An effective and flexible modelling for an AUT having a quasiplanar geometry is obtained by choosing Σ coincident with the smallest surface formed by two circular “bowls” with the same aperture diameter 2𝑎 (see Figures 1 and 2). Note that their lateral surfaces have not the same bend because they are generally determined by rotating two different circular arcs, each equal to a quarter of circumference (with radius 𝑐 and 𝑐^′). This surface allows one to fit very well a lot of actual antennas by properly choosing the values of the parameters 𝑐, 𝑐^′, and 𝑎. For instance, Σ coincides with a sphere if 𝑐=𝑐^′=𝑎, and it becomes a half sphere if 𝑐=0 and 𝑐^′=𝑎, and it reduces to a circular dish for 𝑐=𝑐^′=0. It can be easily verified that, for such a modelling, the length of the curve 𝐶^′ is ℓ^′=2[𝑏+𝑏^′+(𝑐+𝑐^′)𝜋/2], wherein 𝑏=𝑎−𝑐 and 𝑏^′=𝑎−𝑐^′. The phase function and the optimal parameter relevant to a meridian can be determined by substituting in relations (3) the appropriate values of 𝑅_1,2 and 𝑠^′_1,2, whose expressions change depending on the location of the tangency points 𝑃_1,2. For such a modelling, five cases must be considered for 𝜗 ranging in [0,𝜋] (see Figure 2).

Figure 1 

Spherical scanning for a quasiplanar antenna.

Figure 2 

Relevant to the double bowl modelling.

When 0≤𝜗≤𝜗_𝐴=sin⁻¹(𝑎/𝑑), it results that 𝑅₁=𝑑²+𝑏²+2𝑏𝑑sin𝜗−𝑐²,𝑠^′₁=−𝑏+𝑐𝛼₁,𝛼₁=tan⁻¹𝑅₁𝑐−tan⁻¹(𝑏+𝑑sin𝜗),𝑅𝑑cos𝜗₂=𝑑²+𝑏²−2𝑏𝑑sin𝜗−𝑐²,𝑠^′₂=𝑏+𝑐𝛼₂,𝛼₂=tan⁻¹𝑅₂𝑐−tan⁻¹(𝑏−𝑑sin𝜗).𝑑cos𝜗 (11)

When 𝜗_𝐴<𝜗≤𝜗_𝐵=cos⁻¹(𝑐/𝑑), 𝑅₁, 𝑠^′₁ and 𝛼₁ are again given by relations (11), whereas it results that𝑅₂=𝑑²+𝑏^′2−2𝑏^′𝑑sin𝜗−𝑐^′2,𝑠^′₂𝜋=𝑏+𝑐2+𝑐^′𝛼₂,𝛼₂=tan⁻¹𝑅₂𝑐^′−tan⁻¹𝑑cos𝜗𝑑sin𝜗−𝑏^′. (12)

When 𝜗_𝐵<𝜗≤𝜗_𝑐=𝜋−cos⁻¹(𝑐^′/𝑑), 𝑅₂, 𝑠^′₂, and 𝛼₂ are again given by relations (12), whereas it results that𝑅₁=𝑑²+𝑏²−2𝑏𝑑sin𝜗−𝑐²,𝑠^′₁=𝑏+𝑐𝛼₁+𝜋2,𝛼₁=−tan⁻¹𝑅₁𝑐−tan⁻¹𝑑cos𝜗().𝑑sin𝜗−𝑏 (13)

When 𝜗_𝐶<𝜗≤𝜗_𝐷=𝜋−sin⁻¹(𝑎/𝑑), 𝑅₁, 𝑠^′₁, and 𝛼₁ are again given by relations (13), whereas it results that𝑅₂=𝑑²+𝑏^′2+2𝑏^′𝑑sin𝜗−𝑐^′2,𝑠^′₂=𝑏+2𝑏^′+𝑐+𝑐^′𝜋2+𝑐^′𝛼₂,𝛼₂=tan⁻¹𝑅₂𝑐^′−tan⁻¹𝑑sin𝜗+𝑏^′||.𝑑cos𝜗 (14)

When 𝜗_𝐷<𝜗≤𝜋, 𝑅₂, 𝑠^′₂, and 𝛼₂ are again given by relations (14), whereas it results that𝑅₁=𝑑²+𝑏^′2−2𝑏^′𝑑sin𝜗−𝑐^′2,𝑠^′₁𝜋=𝑏+𝑐2+𝑐^′𝜋2−𝛼₁,𝛼₁=tan⁻¹𝑅₁𝑐^′−tan⁻¹𝑏^′−𝑑sin𝜗||.𝑑cos𝜗 (15)

For what concerns the evaluation of the maximum in (4) which allows the determination of the azimuthal bandwidth 𝑊_𝜑, it is convenient (when 𝜗≤𝜋/2) to introduce the angular coordinate 𝜂 (Figure 2) such that 𝑧^′=𝑐cos𝜂 and 𝜌^′=𝑏+𝑐sin𝜂. It can be easily verified through simple geometrical considerations or analytical manipulations that the sought maximum corresponds to the 𝜂 value that zeroes the derivative of 𝑅⁺−𝑅⁻ with respect to 𝜂. A quite analogous result holds when 𝜋/2<𝜗≤𝜋.

2.2. Rounded Cylinder Modelling

An effective modelling for an elongated AUT is got by considering it as enclosed in a cylinder of height ℎ^′ ended in two half spheres of radius 𝑎^′ (see Figures 3 and 4). In such a case, it results that ℓ^′=2(ℎ^′+𝜋𝑎^′). The phase function and the optimal parameter relevant to a meridian can be again obtained by substituting the appropriate values of 𝑅_1,2 and 𝑠^′_1,2 in relations (3). It can be easily verified that their expressions change depending on the location of the tangency points 𝑃_1,2. In particular, three cases must be considered for 𝜗 ranging in [0,𝜋] (see Figure 4).

Figure 3 

Spherical scanning for an elongated antenna.

Figure 4 

Relevant to the rounded cylinder modelling.

When 0≤𝜗≤sin⁻¹(𝑎^′/𝑑), it results that𝑅₁=(𝑑sin𝜗)²+ℎ𝑑cos𝜗−^′2²−𝑎^′2,𝑠^′₁=𝑎^′sin⁻¹𝑎^′𝑑sin𝜗+𝑅₁ℎ^′/2−𝑑cos𝜗𝑅²₁+𝑎^′2,𝑅₂=𝑅₁,𝑠^′₂=𝑎^′sin⁻¹𝑎^′𝑑sin𝜗−𝑅₂ℎ^′/2−𝑑cos𝜗𝑅²₂+𝑎^′2. (16)

When sin⁻¹(𝑎^′/𝑑)<𝜗≤𝜋−sin⁻¹(𝑎^′/𝑑), the expressions of 𝑅₁ and 𝑠^′₁ are again given by relations (16), whereas it results that𝑅₂=(𝑑sin𝜗)²+ℎ𝑑cos𝜗+^′2²−𝑎^′2,𝑠^′₂=ℎ^′+𝑎^′𝜋−sin⁻¹𝑎^′𝑑sin𝜗+𝑅₂ℎ^′/2+𝑑cos𝜗𝑅²₂+𝑎^′2. (17)

When 𝜋−sin⁻¹(𝑎^′/𝑑)<𝜗≤𝜋, the expressions of 𝑅₂ and 𝑠^′₂ are again given by relations (17), whereas it results that𝑅₁=(𝑑sin𝜗)²+ℎ𝑑cos𝜗+^′2²−𝑎^′2,𝑠^′₁=ℎ^′+𝑎^′𝜋2−sin⁻¹𝑅₁𝑑sin𝜗+𝑎^′ℎ^′/2+𝑑cos𝜗𝑅²₁+𝑎^′2. (18)

As regards the azimuthal bandwidth 𝑊_𝜑, it can be shown that the maximum in (4) is attained at𝑧^′=|𝑧|≤ℎ𝑧,^′2,ℎ^′2+|𝑧|−ℎ^′/2𝑎^′2(𝑑sin𝜗)²+|𝑧|−ℎ^′/2²(𝑧),|𝑧|>ℎsgn^′2, (19) where sgn(⋅) is the sign function.

3. NF-FF Transformation with Spherical Scanning

For reader’s convenience, the key steps of the probe-compensated NF-FF transformation with spherical scanning [7] as modified in [15] by properly taking into account the spatial band-limitation properties of the EM field [16] are reported in this section.

According to [7], the electric field in the FF region can be expressed in terms of the following truncated spherical wave expansion:𝐸()=e𝑅,Θ,Φ^{−𝑗𝛽𝑅}𝑅^𝑁max_𝑛=1×^𝑛_𝑚=−𝑛𝑗^𝑛+1𝑏¹_𝑛𝑚𝑓_1𝑛𝑚(Θ)+𝑗^𝑛𝑏²_𝑛𝑚𝑓_2𝑛𝑚(Θ)×e^𝑗𝑚Φ, (20) where the spherical coordinate system (𝑅,Θ,Φ) has been adopted to specify the FF observation point.

The choice of the highest spherical wave to be considered is determined in the classical approach according to the following rule of thumb:𝑁_max=Int𝛽𝑎+10, (21) where 𝑎 is the radius of the smallest sphere enclosing the AUT. Whereas in the approach [15], the highest spherical harmonic is rigorously fixed by the aforementioned band-limitation properties of the radiated EM fields [16]. Accordingly,𝑁_max=Int𝜒^′𝛽𝑎+1. (22)

The vectorial functions 𝑓_1,2𝑛𝑚(Θ) are given by𝑓_1𝑛𝑚(Θ)=−𝑚|𝑚|^𝑚1()×2𝜋𝑛𝑛+1𝑗𝑚sinΘ𝑃_𝑛^|𝑚|()𝑑cosΘΘ−𝑑Θ𝑃_𝑛^|𝑚|()Φ,𝑓cosΘ_2𝑛𝑚(Θ)=−𝑚|𝑚|^𝑚1()×d2𝜋𝑛𝑛+1dΘ𝑃_𝑛^|𝑚|(cosΘ)Θ+𝑗𝑚sinΘ𝑃_𝑛^|𝑚|(cosΘ)Φ, (23)𝑃_𝑛^|𝑚|(cosΘ) being the normalized associated Legendre functions as defined in [21].

As shown in [7], when using a probe with a first-order azimuthal dependence pattern (e.g., an open ended circular waveguide), the modal expansion coefficients in (20) can be determined from the knowledge of the voltages 𝑉 and 𝑉^′ measured by the probe and rotated probe, respectively, and are given by𝑏¹_𝑛𝑚=2𝑛+1∫16𝜋^𝜋0∫₀^2𝜋𝐼₁(𝜗)𝑉−𝑗𝐼₂(𝜗)𝑉^′e^{−𝑗𝑚𝜑}sin𝜗d𝜑d𝜗∑^𝜐max_𝜐=1𝑏^′1_𝜐1𝐴^𝑛_𝜐1𝛽𝑑−𝑏^′2_𝜐1𝐵^𝑛_𝜐1,𝑏𝛽𝑑_2𝑛𝑚=2𝑛+1∫16𝜋^𝜋0∫₀^2𝜋𝐼₂(𝜗)𝑉−𝑗𝐼₁(𝜗)𝑉^′e^{−𝑗𝑚𝜑}sin𝜗𝑑𝜑𝑑𝜗∑^𝜐max_𝜐=1𝑏^′1_𝜐1𝐵^𝑛_𝜐1𝛽𝑑−𝑏^′2_𝜐1𝐴^𝑛_𝜐1,𝛽𝑑 (24) where 𝜐_max is the highest spherical wave of the probe,𝐼₁(𝜗)=𝑑^𝑛_1𝑚(𝜗)−𝑑^𝑛_−1𝑚(𝜗,𝐼₂)(𝜗)=𝑑^𝑛_1𝑚(𝜗)+𝑑^𝑛_−1𝑚(𝜗,𝑑_𝑛𝜇𝑚)(𝜗)=𝑛+𝜇!𝑛−𝜇!(𝑛+𝑚)!(𝑛−𝑚)!_𝜎×𝜎()𝑛+𝑚𝑛−𝜇−𝜎𝑛−𝑚−1^{𝑛−𝜇−𝜎}×𝜗cos2^{2𝜎+𝜇+𝑚}𝜗sin2^{2𝑛−2𝜎−𝜇−𝑚} (25) are the rotation coefficients [7, 22], the summation over 𝜎 involving all terms in which the binomial coefficients do not lead to negative arguments for the factorials. Moreover,𝐴^𝑛_𝜐𝜇𝛽𝑑=(2𝑛+1)(2𝜐+1)𝑛(𝑛+1)𝜐(𝜐+1)𝜐+𝜇!𝑛−𝜇!!!×()𝜐−𝜇𝑛+𝜇−1^𝜇12𝑗^{𝑛−𝜐𝑛+𝜐}_{||𝑝=𝑛−𝜐}×𝑗^−𝑝𝑛()()ℎ𝑛+1+𝜐𝜐+1−𝑝𝑝+1×𝜏𝜇,𝑛,−𝜇,𝜐,𝑝_𝑝⁽²⁾,𝐵𝛽𝑑_𝑛𝜐𝜇𝛽𝑑=(2𝑛+1)(2𝜐+1)𝑛(𝑛+1)𝜐(𝜐+1)𝜐+𝜇!𝑛−𝜇!!!×()𝜐−𝜇𝑛+𝜇−1^𝜇12𝑗^{𝑛−𝜐𝑛+𝜐}_{||𝑝=𝑛−𝜐}×𝑗^−𝑝2𝑗𝜇𝛽𝑑𝜏𝜇,𝑛,−𝜇,𝜐,𝑝ℎ_𝑝⁽²⁾𝛽𝑑 (26) are the translation coefficients [7, 11, 23], wherein ℎ_𝑝⁽²⁾(𝑥) is the spherical Hankel function of second kind, and 𝜏(𝜇,𝑛,−𝜇,𝜐,𝑝) are the linearization coefficients defined by the expansion of the product of two unnormalized associated Legendre functions𝑃^𝜇_𝑛(𝑥)𝑃_𝜐^−𝜇(𝑥)=^𝑛+𝜐_{||𝑝=𝑛−𝜐}𝜏𝜇,𝑛,−𝜇,𝜐,𝑝𝑃_𝑝(𝑥). (27)

The probe’s expansion coefficients 𝑏^′1,2_𝜐1 in relations (24) can be evaluated from the knowledge of its tangential electric field on a scanning sphere when it is used as transmitting antenna𝑏^′1,2_𝜐1=1𝛽𝑔_1,2𝜐𝛽𝑑^𝜋₀^𝜋_−𝜋𝐸^′_td,𝜗,𝜑⋅𝑓^∗_1,2𝜐1(𝜗)×e^−𝑗𝜑sin𝜗d𝜑d𝜗, (28)(⋅) and (∗) denoting the inner product and the complex conjugate, respectively, and𝑔_1,𝜐(𝑥)=ℎ_𝜐⁽²⁾(𝑥),𝑔_2,𝜐(𝑥)=1𝑥𝑑d𝑥𝑥ℎ_𝜐⁽²⁾(𝑥). (29)

According to (28), the probe’s expansion coefficients can be determined from the simulated measurements of the field radiated by an open-ended circular waveguide by using the uncompensated NF-FF transformation [14].

The integration over 𝜑 in relations (24) can be carried out [14] by expanding the voltage measured by the probe and rotated probe on each parallel in Fourier series with respect to 𝜑, and this can be efficiently performed by employing the Fast Fourier Transform (FFT) algorithm. In a quite similar way, the remaining integration over 𝜗 can be efficiently performed through the following steps: (i) the aforementioned Fourier series coefficients are extended [7] from the range [0,𝜋] to [0,2𝜋] and then expanded in Fourier series via FFT, (ii) the rotation coefficients are expanded in Fourier series too [7, 22]. The double integrals are so transformed in double summations, of indexes ℓ and 𝑖, involving the Fourier series coefficients (in 𝜗) and the integrals^𝜋₀e^{𝑗(ℓ−𝑖)𝜗}sin𝜗d𝜗. (30)

It must be stressed that, to take advantage of the numerical efficiency of the FFT, the number of NF parallels to be considered in the NF-FF transformation and the number of samples on them must be the first power of two greater or equal to 𝑁_max and 2𝑀_𝜑, respectively, where 𝑀_𝜑=Int(𝜒^∗𝛽𝑎sin𝜗)+1.

It is worth noting that, by inverting the summations order, the spherical wave expansion (20) can be rewritten as𝐸()=e𝑅,Θ,Φ^{−𝑗𝛽𝑅}𝑅^𝑀Φ_{𝑚=−𝑀Φ}^𝑁max_{|𝑚|𝑛=()𝑛≠0}×𝑗^𝑛+1𝑏¹_𝑛𝑚𝑓_1𝑛𝑚(Θ)+𝑗^𝑛𝑏²_𝑛𝑚𝑓_2𝑛𝑚(Θ)e^𝑗𝑚Φ=e^{−𝑗𝛽𝑅}𝑅^𝑀Φ_{𝑚=−𝑀Φ}𝑐_𝑚(Θ)e^𝑗𝑚Φ, (31) where 𝑀_Φ=Int(𝜒^∗𝛽𝑎sinΘ)+1. Relation (31) allows an efficient evaluation of the antenna far field at the considered elevation angle Θ by carrying out the summation again via the FFT algorithm.

From a numerical viewpoint, it is convenient to apply the described FF reconstruction process to evaluate only the FF samples required by the OSI expansion in [24], properly modified to deal with an even number of samples on the FF parallels: 𝐹()=Θ,Φ_{𝑚0+𝑞𝑚=𝑚0−𝑞+1}Ω_𝑁Θ−Θ_𝑚𝐷_𝑁′′Θ−Θ_𝑚×2𝐿_Φ^′′−12𝐿_Φ^{𝑛′′0+𝑝}_{𝑛=𝑛0−𝑝+1}𝐹Θ_𝑚,Φ_𝑛,𝑚×Ω_𝐿ΦΦ−Φ_𝑛,𝑚𝐷_{𝐿Φ′′−1}Φ−Φ_𝑛,𝑚 (32) wherein Θ_𝑚=𝑚ΔΘ=2𝜋𝑚2𝑁^′′,𝑁+1_′′=Int𝜒𝑁_max+1,𝑁=𝑁^′′−𝑁_max,Φ_𝑛,𝑚=𝑛ΔΦ_𝑚𝜋=𝑛𝐿_Φ^′′,𝐿_Φ^′′=2^𝑖≥Int𝜒𝑀_Φ+1,𝐿_Φ=𝐿_Φ^′′−𝑀_Φ, (33)𝐹(Θ,Φ)=𝑅e^j𝛽𝑅𝐸(𝑅,Θ,Φ), and the other symbols have the same or analogous meanings as in (10).

It is useful to note that the expansion (32) is based on a spherical modelling of the antenna. OSI expansions requiring a lower number of FF samples can be obviously derived when using more effective modellings tailored to the actual antenna geometry.

4. Numerical Tests

Many numerical simulations have been performed in order to assess the effectiveness and robustness of the proposed NF-FF transformation techniques with spherical scanning for antennas having two dimensions very different from the third one. Two sets of simulations are reported in the following. The former (from Figure 5 to 11) refers to a quasiplanar antenna, whereas the latter (from Figure 12 to 16) is relevant to the case of an elongated one. In both the cases, an open-ended circular waveguide having radius 0.338𝜆, 𝜆 being the wavelength is chosen as measurement probe.

Figure 5 

Amplitude of the output voltage 𝑉′ on the meridian at 𝜑=90°. Solid line: exact. Crosses: interpolated.

In the first set, the numerical tests refer to a scanning sphere having radius 𝑑=25𝜆 and to three uniform planar circular arrays placed at 𝑧=−2𝜆,1𝜆,4𝜆, having radius equal to 18𝜆,19.5𝜆, and 16𝜆, respectively. Their elements, symmetrically placed with respect to the plane 𝑦=0, are elementary Huygens sources linearly polarized along 𝑦 and spaced of 0.5𝜆 along the radial and azimuthal lines. Such an AUT has been fitted by the double bowl modelling with 𝑎=20𝜆,𝑐=4.5𝜆, and 𝑐^′=2.5𝜆. Figures 5 and 6 show representative reconstruction examples of the amplitude and phase of the rotated probe voltage 𝑉^′ on the meridian at 𝜑=90°. As can be seen, the exact and recovered voltages are practically indistinguishable. To assess in a more quantitative way the performances of the algorithm, the maximum and mean-square reconstruction errors (normalized to the voltage maximum value over the sphere) have been evaluated for 𝜒^′=1.20, 𝜒=1.10,1.15,1.20,1.25, and 𝑝=𝑞 ranging from 3 to 10 (see Figures. 7 and 8). They have been computed by comparing the reconstructed and exact voltage 𝑉^′ on a spherical grid with Δ𝜗=Δ𝜑=1°. As can be seen, the errors decrease up to very low values on increasing the oversampling factor and/or the number of retained samples, thus assessing the effectiveness of the interpolation procedure. The algorithm robustness has been verified by adding random errors to the exact samples. These errors simulate a background noise (bounded to Δ𝑎 in amplitude and with arbitrary phase) and an uncertainty on the voltage samples of ±Δ𝑎_𝑟 in amplitude and ±Δ𝛼 in phase. As shown in Figures 9 and 10, the algorithm works well also when dealing with error-affected data. At last, the developed interpolation algorithm has been applied to recover the NF data required to carry out the spherical NF-FF transformation [7], as modified in [15]. The reconstruction of the antenna FF pattern in the principal plane E is shown in Figure 11. As can be seen, the FF reconstruction, in spite of the significantly reduced number of used NF samples (see Table 1), is everywhere very accurate thus validating the effectiveness of the proposed approach.

Table 1 

Figure 6 

Phase of the output voltage 𝑉′ on the meridian at 𝜑=90°. Solid line: exact. Crosses: interpolated.

Figure 7 

Normalized maximum errors in the reconstruction of the voltage 𝑉′.

Figure 8 

Normalized mean-square errors in the reconstruction of the voltage 𝑉′.

Figure 9 

Amplitude of the output voltage 𝑉′ on the meridian at 𝜑=90°. Solid line: exact. Crosses: interpolated from error-affected data.

Figure 10 

Phase of the output voltage 𝑉′ on the meridian at 𝜑=90°. Solid line: exact. Crosses: interpolated from error-affected data.

Figure 11 

E-plane pattern. Solid line: exact. Crosses: reconstructed from NF data.

Figure 12 

Amplitude of the output voltage 𝑉′ on the meridian at 𝜑=90°. Solid line: exact. Crosses: interpolated.

The second set of figures refers to a scanning sphere having radius 𝑑=35𝜆 and to a uniform planar array of elementary Huygens sources polarized along the 𝑧 axis, spaced by 0.5𝜆. These sources cover a zone in the plane 𝑦=0 formed by a 16𝜆×44𝜆 rectangle ended in two half-circles. Accordingly, such an antenna has been modelled by a rounded cylinder with ℎ^′=44𝜆 and 𝑎^′=8𝜆. An example of voltage reconstruction relevant to the meridian at 𝜑=90° is reported in Figure 12. Again, a very accurate reconstruction results. Figure 13 confirms the algorithm accuracy by showing the normalized maximum and mean-square errors for 𝑝=𝑞 ranging from 3 to 10 and 𝜒^′=𝜒=1.20. Also in such a case, the algorithm is resulted to be very stable with respect to random errors affecting the exact samples as shown in Figure 14. At last, the reconstructions of the antenna FF pattern in the E and H planes are shown in Figures 15 and 16, respectively. As can be seen, the exact and recovered fields are practically indistinguishable, thus assessing the effectiveness of the developed NF-FF transformation. Also in such a case, it can be interesting to compare the number of the employed NF data with those required by the other techniques (see Table 2).

Table 2 

Figure 13 

Normalized errors in the reconstruction of the voltage 𝑉′.

Figure 14 

Amplitude of the output voltage 𝑉′ on the meridian at 𝜑=90°. Solid line: exact. Crosses: interpolated from error-affected data.

Figure 15 

E-plane pattern. Solid line: exact. Crosses: reconstructed from NF data.

Figure 16 

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

5. Conclusions

Fast and accurate probe-compensated NF-FF transformations with spherical scanning for nonspherical antennas have been here developed by employing flexible AUT modellings, which allow one to fit very well the shape of a lot of actual antennas. In particular, an electrically long antenna has been considered as enclosed in a rounded cylinder, whereas a double bowl has been adopted to shape a quasiplanar antenna. The use of these modellings makes possible to remarkably lower the number of data to be acquired, thus significantly reducing the measurement time. The developed techniques work very well as widely assessed by the numerical simulations.