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An effective near-field - far-field (NF - FF) transformation with spherical scanning for quasi-planar antennas from irregularly spaced data is developed in this paper. Two efficient approaches for evaluating the regularly spaced spherical samples from the nonuniformly distributed ones are proposed and numerically compared. Both the approaches rely on a nonredundant sampling representation of the voltage measured by the probe, based on an oblate ellipsoidal modelling of the antenna under test. The former employs the singular value decomposition method to reconstruct the NF data at the points fixed by the nonredundant sampling representation and can be applied when the irregularly acquired samples lie on nonuniform parallels. The latter is based on an iterative technique and can be used also when such a hypothesis does not hold, but requires the existence of a biunique correspondence between the uniform and nonuniform samples, associ- ating at each uniform sampling point the nearest irregular one. Once the regularly spaced spherical samples have been recovered, the NF data needed by a probe compensated NF - FF transformation with spherical scanning are efficiently evaluated by using an optimal sampling interpolation algorithm. It is so possible to accurately compensate known posi- tioning errors in the NF - FF transformation with spherical scanning for quasi-planar antennas. Some numerical tests assessing the accuracy and the robustness of the proposed approaches are reported.

Near-field - far-field (NF - FF) transformation techniques [1-5] have been widely investigated in the last four decades and used for applications ranging from cellular phone antennas to large phased arrays and complex multi-beam communication satellite antennas. They allow one to overcome the drawbacks which, for electrically large radiating systems, make unpractical the measurement of the antenna pattern in a conventional FF range and represent the better choice when complete pattern and polarization measurements are required. Moreover, they provide the necessary information to determine the radiating field on the surface of the antenna. Such an information can be properly used for the diagnostics of surface errors in a reflector antenna or of faulty elements in an array (microwave holographic diagnostics [

It must be stressed that the inaccurate control of the positioning systems and their finite resolution do not allow one to acquire the NF data at the points fixed by the sampling representation. On the other hand, their position can be accurately determined by using optical devices. Accordingly, the development of an accurate and stable reconstruction process from irregularly spaced data appears indispensable. In this framework, a procedure based on the conjugate gradient iteration method and using the unequally spaced fast Fourier transform (FFT) [

The aim of this paper is to develop and compare numerically analogous algorithms to compensate the probe positioning errors in the NF - FF transformation with spherical scanning for quasi-planar antennas, which will be now assumed as enclosed in an oblate ellipsoid (see

The paper is organized in six sections. Section 2 briefly describes the classical probe compensated NF - FF transformation with spherical scanning as modified in [14,15]. Section 3 is devoted to the nonredundant sampling representation of the probe voltage over a sphere, based on an oblate ellipsoidal modelling of the AUT. Section 4 describes the techniques for reconstructing the nonredundant samples from the irregularly spaced acquired ones. Section 5 is devoted to discuss the numerical results assessing the accuracy and the robustness of the proposed approaches. Finally, conclusions are drawn in Section 6.

The key steps of the classical probe compensated NF - FF transformation with spherical scanning as modified in [14,15] are reported for reader’s convenience.

Let us consider a probe scanning a sphere of radius d in the antenna NF region, and adopt the spherical coordinate system to denote an observation point

both in the NF and in the FF region (

wherein the index of the highest spherical wave to be considered is rigorously fixed by the bandlimitation properties of the EM field and [13-15] is given by:

where a is the radius of the smallest sphere enclosing the AUT, b is the wavenumber, is the bandwidth enlargement factor, and Int(x) denotes the integer part of x. The vector wave functions are given by:

being the normalized associated Legendre functions as defined in [

Let us consider a non directive probe scanning a spherecal surface of radius d in the NF region of a quasi-planar antenna enclosed in an oblate ellipsoid Σ having major and minor semi-axes equal to a and b (see

Since the voltage V measured by such a kind of probe has the same effective spatial bandwidth of the field [

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

When C is a meridian, by choosing, being the length of the ellipse (intersection between the meridian plane through the observation point P and Σ), we get the following expressions [

where denotes the elliptic integral of second kind and and are the elliptic coordinates, being the distances from P to the foci of the ellipse. Moreover, is its eccentricity and 2f its focal distance. The expression of the parameter in (6) is valid when the angle belongs to the range [0,]. For ranging from to, it results, where is the parameterization value corresponding to the point specified by the angle. It is worthy to note that the curves γ = const and = const are ellipses and hyperbolas confocal to [

When the curve C is a parallel, the phase function is constant and it is convenient to choose the angle as parameter. The corresponding bandwidth [13,18] is

wherein is the polar angle of the asymptote to the hyperbola through P.

According to these results, the voltage at P on the meridian fixed by can be evaluated via the OSI expansion

where is the index of the sample nearest (on the left) to P, 2q is the number of retained intermediate samples,

is an oversampling factor required to control the truncation error [

Moreover,

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

The intermediate samples on the meridian through P can be determined by means of a similar OSI expansion along. The two-dimensional OSI expansion for reconstructing the data at any point P on the sphere can be obtained [

where in, 2p is the retained samples number along,

and the other symbols have the same meanings as in (9). The variation of with is required to ensure a bandlimitation error constant with respect to.

By using expansion (15), it is possible to evaluate the NF data needed by the classical NF - FF transformation with spherical scanning [

Two different techniques to retrieve the uniformly distributed samples from the acquired irregularly spaced ones will be presented in this section and numerically compared in the subsequent one.

The SVD-based approach can been applied when the starting two-dimensional problem of the uniform samples reconstruction can be reduced to find the solution of two independent one-dimensional problems. Accordingly, let us now suppose that, apart from the sample at the north pole, the irregularly distributed samples lie on nonuniformly spaced parallels. This assumption can really represent the spatial distribution of the measured data when the acquisition is carried out by moving along parallels, as required to exploit the possibility of reducing the number of NF data on noncentral parallels, offered by the described nonredundant representation.

Let us first consider the recovery of the uniformly spaced samples on each nonuniform parallel. Given a sequence of known nonuniform sampling points on the nonuniform parallel at (where is the number of the corresponding uniform sampling points ), the known reduced voltage at each nonuniform sampling point can be expressed in terms of the unknown uniform samples via the OSI expansion along, thus getting the linear system:

This last can be rewritten in the matrix form, where is the sequence of the known nonuniform samples, x is that of the unknown uniform ones, and is a matrix, whose elements are given by the weight functions in the considered OSI expansion:

and, for any fixed row j, are equal to zero when the index m is out of the range. The best approximated solution in the least squares sense of the system is obtained by means of SVD.

Once the uniform samples on the nonuniform parallels have been so retrieved, the OSI expansion along is used to determine the intermediate samples at the intersection points between the nonuniform parallels and the meridian passing through P. Obviously, these samples are again irregularly spaced and, accordingly, the voltage at P can be evaluated by first reconstructing the uniformly spaced intermediate samples via SVD and then interpolating them by using the OSI formula (9). It must be stressed that it is convenient to determine the same number of samples on each of the uniform parallels to minimize the computational effort. This number is that corresponding to the equator. In such a way, although the so retrieved NF data are slightly redundant in, the number of SVD relevant to the meridians is minimized. Once the uniform samples have been reconstructed, the NF data needed by the classical spherical NF - FF transformation [

Note that, to avoid a strong ill-conditioning of the related linear system [

When the hypothesis that the irregularly distributed samples lie on nonuniformly spaced parallels does not hold, the SVD technique could be still used, but the dimension of the involved matrix would become very large, thus requiring a massive computational effort. In fact, in such a hypothesis, the starting two-dimensional problem can no longer be tackled as two independent one-dimensional ones and it is more convenient to resort to the iterative technique [22,23]. Accordingly, let us assume in the following that, as required for the convergence of the iterative technique, the nonuniformly distributed samples are such that it is possible to build a biunique corresponddence, which associates at each uniform sampling point the “nearest” nonuniform one. In such a case, by expressing the reduced voltage at each nonuniform sampling point as a function of the unknown values at the nearest uniform ones via the two-dimensional OSI expansion (15), we get:

This last can be again rewritten as, where is a sparse banded matrix, whose elements are given by the weight functions in the considered OSI expansion (Q being the overall number of the nonuniform/uniform samples), is the sequence of the known irregularly distributed samples, and x is that of the unknown uniform ones.

By subdividing into its diagonal part and nondiagonal one, multiplying both members of the matrix relation by and rearranging the terms, we get:

The following iterative scheme is so obtained:

where is the sequence of the uniform samples estimated at the step. Necessary conditions for the convergence of the above scheme are that the modulus of each element on the principal diagonal of be not zero and greater than those of the other elements on the same row and column [22,23]. These conditions are certainly verified in the assumed hypothesis of one-to-one correspondence between each uniform sampling point and the nearest nonuniform one.

By making (23) in explicit form, we finally get:

The OSI expansion (15) is then used to interpolate the so recovered uniform NF samples for reconstructing the NF data needed to carry out the NF - FF transformation.

The numerical tests are relevant to a uniform planar circular array (see

The first set of simulations (from Figures 2-9) refers to the case of irregularly spaced samples lying on nonuniformly distributed parallels, so that the reconstruction of the uniform samples can be reduced to the solution of two independent one-dimensional problems. The nonuniform samples (whose positions are assumed

known) have been generated by imposing that the distance between the position of each of these parallels and the associated uniform one is a random variable uniformly distributed in. Similarly, the dis-

placements between the irregularly spaced sampling points and the corresponding regularly spaced ones on the nonuniform parallels are random variables uniformly distributed in. The reconstructed am-

plitude and phase of the rotated probe voltage (the most significant one) on the meridian at 90˚ are shown in Figures 2 and 3. As can be seen, the exact and reconstructed curves are indistinguishable in spite of the considered large values of the probe positioning errors, very pessimistic in an actual scanning procedure. The performances of the SVD algorithm for compensating the positioning errors have been assessed in a more quantitative way by evaluating the maximum and mean-square errors in the reconstruction of the uniform samples. They are normalized to the voltage maximum value on the sphere and have been obtained by comparing the reconstructed and the exact uniform samples. As can be seen from

The second set of simulations (from Figures 10-14) refers to the case of irregularly spaced samples that do not lie on parallels. In such a situation, it is more convenient, from the computational viewpoint, to apply the iterative approach, that requires the existence of a one-to-one correspondence between the uniform and nonuniform samples, associating at each uniform sampling point the nearest irregular one. Accordingly, the irregularly distributed samples have been generated in such a way that the displacements in and between each nonuniform sampling point and the corresponding uniform one are random variables uniformly distributed in and. The reconstructions of the amplitude and phase of the probe voltage obtained after 5 iterations are shown in Figures 10 and 11. The normalized maximum and mean-square errors in the reconstruction of the uniform

samples are reported in

For sake of completeness, we stress that the reported results have been obtained by using 15,986 NF samples, which are remarkably lower than those (32,514) required by the classical NF - FF transformation with spherical scanning [

The compensation of known positioning errors in the NF - FF transformation with spherical scanning for quasiplanar antennas has been tackled in this paper. To this end, two different techniques to evaluate the uniformly distributed spherical samples from the nonuniform ones have been developed and numerically compared. The former uses the SVD method and can be conveniently employed when the nonuniform sampling points lie on parallels. The latter employs an iterative algorithm and can be applied also when the nonuniform sampling points do not lie on parallels, but requires the existence of a biunique correspondence that associates at each uniform sampling point the nearest irregular one. The reported numerical results assess the accuracy and robustness of both approaches in spite of the considered large values of the probe positioning errors, very pessimistic in an actual scanning procedure.