NF – FF Transformation with Helicoidal Scan for Long Antennas: Positioning Errors Compensation F. D’Agostino (1) , F. Ferrara (1) , C. Gennarelli (1) , R. Guerriero (1) , M. Migliozzi (1) (1) D.I.I.I.E. - University of Salerno via Ponte Don Melillo, 84084 Fisciano (SA), Italy gennar@diiie.unisa.it Abstract— An iterative technique is proposed in this communica- tion for compensating the probe positioning errors in a near-field – far-field transformation with helicoidal scanning for long antennas. It relies on a very effective source modelling wherein the surface enclosing the antenna under test is a cylinder ended in two half-spheres and uses an optimal sampling interpolation algorithm for reconstructing the data needed by a standard near- field – far-field transformation technique with cylindrical scan- ning from the acquired irregularly spaced ones. Numerical tests confirm the effectiveness and the robustness of the approach. I. INTRODUCTION As well-known, the near-field – far-field (NF–FF) trans- formation using the cylindrical scanning is particularly attrac- tive when considering antennas that concentrate the electro- magnetic (EM) radiation in an angular region centred on the horizontal plane. In this framework, a remarkable measure- ment time reduction is obtained by employing a helicoidal scan which exploits continuous and synchronized movements of the positioning systems of the probe and antenna under test (AUT), as suggested in [1]. To this end, two efficient NF–FF transformations with helicoidal scans for long antennas have been recently developed in [2, 3], by heuristically extending [4] the rigorous approach [5, 6]. They rely on the nonredun- dant sampling representations of EM fields [7] and use opti- mal sampling interpolation (OSI) formulas to reconstruct the NF data required by the standard NF–FF transformation with cylindrical scanning [8]. In particular, a very effective model- ling wherein the surface enclosing the AUT is a cylinder ended in two half-spheres (Fig. 1) has been adopted in [3]. Unfortunately, it may be unpractical to get regularly spaced NF measurements due to an inaccurate control of the position- ing systems, but their position can be accurately read by opti- cal devices. Moreover, the finite resolution of the positioning systems and their imprecise synchronization do not allow one to exactly locate the probe at the points fixed by the sampling representation. Accordingly, the development of an accurate and stable reconstruction process from nonuniformly distri- buted data becomes relevant. The formulas available in litera- ture for the direct reconstruction from nonuniform samples are not user friendly, unstable, and valid only for particular sam- pling points arrangements [9]. A convenient strategy is to re- cover cover the uniform samples from those irregularly spaced and then determine the value at any point of the scanning sur- Fig. 1 Geometry of the problem face by an accurate and stable OSI formula. Two different ap- proaches have been proposed [9, 10]. The former [9] is based on an iterative technique which has been found convergent only if there exists a biunique correspondence, associating at each uniform sampling point the nearest nonuniform one. The latter [10] relies on the singular value decomposition (SVD) method and has been applied when the two-dimensional prob- lem can be tackled as two independent one-dimensional ones. This occurs, for instance, in the cylindrical scanning, wherein the nonuniformly distributed data can be really assumed to lie on not regularly spaced rings [10]. Such a hypothesis is no longer valid in the helicoidal scanning and, therefore, the it- erative technique will be here applied to recover the uniformly spaced helicoidal samples from the acquired irregularly distri- buted data. Obviously, the SVD-based approach could be gen- eralized to such a two-dimensional problem, but the involved matrix dimension would become computationally too large. II. NONREDUNDANT REPRESENTATION ON A CYLINDER Let us consider an elongated AUT, enclosed in a convex domain bounded by a surface  with rotational symmetry, and a non directive probe scanning a proper helix lying on a cylin- der of radius d (Fig. 1). The spherical coordinate system (r,  ,  ) will be adopted to denote an observation point P both in the NF and FF region. A shape suitable to fit such an AUT is obtained by choosing  coincident with a cylinder of height x y d h' 2a'  z h 978-1-4244-7306-9/10/$26.00 ©2010 IEEE 2010 Loughborough Antennas & Propagation Conference 8-9 November 2010, Loughborough, UK 173   978-1-4244-7307-6/10/$26.00 ©2010 IEEE 