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In this paper, an effective technique to compensate the positioning errors in a near-field—far-field (NF-FF) transformation with helicoidal scanning for elongated antennas is presented and validated both numerically and experimentally. It relies on a nonredundant sampling representation of the voltage measured by the probe, obtained by considering the antenna as enclosed in a cylinder ended in two half-spheres. An iterative scheme is used to reconstruct the helicoidal NF data at the points fixed by the representation from the acquired irregularly spaced ones. Once the helicoidal data have been retrieved, those needed by a classical NF-FF transformation with cylindrical scanning are efficiently evaluated by using an optimal sampling interpolation algorithm. Some numerical tests, assessing the accuracy of the approach and its stability with respect to random errors affecting the data, are reported. Experimental tests performed at the Antenna Characterization Lab of the University of Salerno further confirm the validity of the proposed technique.

Nowadays, the reduction of the time required to acquire the near-field data is assuming an ever growing relevance for the antenna measurement community. In fact, such a time is currently very much greater than that needed to perform the near-field - far-field (NF-FF) transformation. As suggested in [

It is worth noting that other spiral scanning techniques have been proposed [14,15], but since these last approaches do not exploit the nonredundant representations of EM fields, they need a useless large amount of NF measurements.

From a practical viewpoint, it may be impossible to get regularly distributed NF measurements due to an inaccurate control of the positioning systems. On the other hand, their position can be accurately read by optical devices. In addition, the finite resolution of the positioning systems does not allow one to exactly locate the probe at the points fixed by the sampling representation. Therefore, the development of an accurate and stable reconstruction process from irregularly spaced data becomes relevant. A procedure relying on the conjugate gradient iteration method and employing the unequally spaced fast Fourier transform (FFT) [

The aim of this paper is to present and validate, both numerically and experimentally, the iterative scheme for compensating the positioning errors in the NF-FF transformation with helicoidal scanning for long antennas [

For reader’s convenience the key steps of the classical probe compensated NF-FF transformation with cylindrical scanning [

Let us consider a probe scanning a cylinder of radius d in the antenna NF region, and adopt the spherical coordinate system (r, J, j) to denote an observation point both in the NF and in the FF region (see

According to the classical probe compensated NF-FF transformation with cylindrical scanning, the modal coefficients and of the cylindrical wave expansion of the field radiated by the AUT are related to: 1) the two-dimensional Fourier transforms and of the output voltages of the probe for two independent sets of measurements (the probe is rotated 90˚ about its longitudinal axis in the second set); 2) the coefficients () and () of the cylindrical wave expansion of the field radiated by the probe and the rotated probe, respectively, when used as transmitting antennas. The key relations [

wherein, is the Hankel function of second kind and order n, b is the wavenumber, and, are the output voltages of the probe and the rotated probe at the point of cylindrical coordinates.

In the classical approach [

where r' is the radius of the smallest cylinder enclosing the AUT and l is the wavelength.

Once the modal coefficients have been determined, the FF components of the electric field can be evaluated by means of the following relations:

which can be efficiently computed by performing the summations via the FFT algorithm.

In the next subsection, it is shown how the NF data at the grid spacings (5) can be recovered from a nonredundant (i.e. minimum) number of samples collected by the probe and rotated probe along a helix wrapping the measurement cylinder [

Let us consider an elongated AUT, enclosed in a convex domain bounded by a surface S with rotational symmetry, and a nondirective probe scanning a proper helix lying on a cylinder of radius d (

where is the voltage measured by the probe or by

the rotated probe, and is a proper phase function. The error, occurring when is approximated by a bandlimited function, becomes negligible as the bandwidth exceeds a critical value [

The unified theory of spiral scans for nonspherical antennas [

where is the length of (intersection curve between the meridian plane and S), are the arclength coordinates of the tangency points between the cone of vertex at the observation point P and, and

are the distances from P to (see

The helix is obtained by projecting onto the scanning cylinder a proper spiral wrapping S. The projection is obtained via the curves at h = const [7,11]. The helix step, determined by two consecutive intersections with a given generatrix, is equal to the sample spacing required to interpolate the voltage along a generatrix. Note that, where Int(x) denotes the integer part of x, , and is an oversampling factor that controls the truncation error [

wherein f is the parameter describing the helix, is the value of f at, and, k being a parameter related to the helix step by. A nonredundant representation along the helix is then obtained by enforcing the optimal parameter x for describing it equal to times the arclength of the projecting point on the spiral wrapping S and by choosing the related phase function g coincident with that y relevant to a generatrix. Moreover, the bandwidth is chosen equal to times the length of the spiral wrapping S from pole to pole [

By exploiting the above results, the reduced voltage at any point of the helix can be recovered [

where is the index of the sample nearest to the output point, 2p the number of retained samples, and

with and. Moreover,

wherein

are the Dirichlet and Tschebyscheff sampling functions, respectively, being the Tschebyscheff polynomial of degree and.

The OSI expansion (15) can be applied to evaluate the “intermediate samples”, namely, the reduced voltage values at the intersection points between the helix and the generatrix passing through the observation point P. Once these samples have been determined, the reduced voltage at P can be evaluated by using a quite similar formula [

where, 2q is the number of the retained intermediate samples, ,

and the other symbols have the same meaning as in (15). Expansion (20) can be used to evaluate the voltage at any point P on the cylinder and, in particular, at those needed to perform the classical NF-FF transformation with cylindrical scanning [

Let us now turn to the case of irregularly spaced samples (

where Q is the overall number of sampling points. Such a linear system can be rewritten in the matrix form

, where is the sparse matrix whose elements are

b is the vector of the collected nonuniform data, and x is the vector of the unknown uniform helicoidal samples.

By splitting into its diagonal part and nondiagonal one, it results

multiplying both members of (24) by and rearranging the terms, we get

The following iterative procedure thus results

where is the vector of the uniform helicoidal samples estimated at the step. Necessary conditions for the convergence [

wherein

being the index of the intermediate sampling point nearest to the uniform one.

The numerical simulations refer to a uniform planar array of 0.6l spaced elementary Huygens sources, polarized along the z axis and covering a zone in the plane y = 0, formed by a rectangle ended in two half-circles (see

A reconstruction example of the voltage V^{1} (the most significant one) on the generatrix at j = 90˚, obtained by 0 and 6 iterations, is shown in Figures 4 and 5, respectively. As can be seen, 6 iterations are enough to get a very good recovery. The evaluation of the mean-square errors (normalized to the voltage maximum on the cylinder) in the reconstruction of the uniform samples assesses more quantitatively the effectiveness of the technique. They have been determined by comparing the reconstructed helicoidal samples and the exact ones in the central zone of the scanning surface, to assure the existence of the guard samples. As can be seen (

principal planes are shown in Figures 8 and 9. As can be seen, the exact and recovered fields are indistinguishable, thus providing an overall assessment of the technique.

Note that the number of used samples for reconstructing the NF data over the considered cylinder is 29,260, about one half than that (46,080) needed by the standard cylindrical scanning and by the helicoidal scanning technique [

The described technique has been experimentally validated in the anechoic chamber available at the UNISA Antenna Characterization Lab. The chamber, whose dimensions are 8 m × 5 m × 4 m, is equipped with an advanced cylindrical NF facility supplied by MI Technologies. An open-ended WR90 rectangular waveguide is used as probe. The AUT, located in the plane x = 0, is a very simple H-plane monopulse antenna, operating at 10 GHz in the sum mode. It has been realized by using two pyramidal horns (8.9 × 6.8 cm) made by Lectronic Research Labs at a distance of 26 cm (between centers) and a hybrid Tee (

a' = 4.2 cm. The helix lies on a cylinder with d = 17.5 cm and h = 240 cm. To assess the effectiveness of the technique in severe conditions as in the case of measurements performed by using bad positioning systems, we have enforced the acquisition of the NF data in such a way that the distances in x and h between the position of each nonuniform sample and the associate uniform helicoidal one are random variables uniformly distributed in and. It is worth noting that no optical device has been used to read the actual sampling positions, since those given by the employed positioners were more than safe.

The amplitude and phase of the probe voltage relevant to the generatrix at j = 0˚ reconstructed by using 10 iterations are compared in Figures 11 and 12 with those directly measured. According to

At last, the FF patterns in the principal planes E and H recovered from the irregularly spaced helicoidal NF data are compared in Figures 13 and 14 with those (references) obtained from the data directly measured on the classical cylindrical grid. In both the cases, the software package MI-3000 has been used to get the FF reconstructions. Obviously, once the uniform data have been retrieved, the OSI algorithm has been used to recover the cylindrical data needed to carry out the NF-FF transformation.

Note that all the reported reconstructions have been obtained by using, , and p = q = 6.

It is interesting to compare the number of acquired nonuniform NF data (1512) with that (5760) required by the traditional cylindrical and helicoidal scanning techniques [

An effective iterative technique for compensating the probe positioning errors in the NF-FF transformation with helicoidal scanning using a cylinder ended in two

half-spheres to model an electrically long antenna has been here proposed. In particular, the helicoidal NF data at the points fixed by the sampling representation are efficiently reconstructed from the acquired irregularly spaced ones, whose positions are known. Once the helicoidal data have been retrieved, those needed by a classical NF-FF transformation with cylindrical scanning are efficiently evaluated via the two-dimensional OSI algorithm. The effectiveness of the approach has been assessed both numerically and experimentally.