Efficient Reconstruction of the Pattern Radiated by a ... - IEEE Xplore

Measurements Corner

Brian E. Fischer Integrity Applications Inc. 900 Victors Way, Suite 220 Ann Arbor, MI 48108 USA Tel: +1 (734) 997-7436 x4717 E-mail: bfi[email protected]

Ivan J. LaHaie Integrity Applications Inc. 900 Victors Way, Suite 220 Ann Arbor, MI 48108 USA Tel: +1 (734) 997-7436 x4724 E-mail: [email protected]

Measurements Corner Introduction A key concern in making near-field antenna measurements is measurement time, since most often the use of near-field processes entails stepping up to additional sampling. This month’s Measurements Corner article features an extension to previous work presented in this column by the authors – in this case, for an elongated antenna – realizing the promise of the proposed processing technique.

Efficient Reconstruction of the Pattern Radiated by a Long Antenna from Data Acquired via a Spherical-Spiral-Scanning Near-Field Facility Francesco D’Agostino, Flaminio Ferrara, Claudio Gennarelli, Rocco Guerriero, and Massimo Migliozzi D.I.In. Università di Salerno via Giovanni Paolo II, 132 - 84084 Fisciano (Salerno), Italy Tel: +39 089 964280 E-mail: [email protected]; fl[email protected]; [email protected]; [email protected]; [email protected]

Abstract This paper provides the experimental assessment of an effective near-field-to-far-field (NF-FF) transformation technique with spherical-spiral scanning, particularly suitable for long antennas. Such a technique allows a remarkable measurement-time saving, due to the use of continuous and synchronized movements of the positioning systems, and due to the reduced number of required near-field measurements. This is made possible by a non-redundant sampling representation of the voltage measured by the probe, obtained by using the unified theory of spiral scans for nonspherical antennas, and adopting a prolate-ellipsoidal source modeling. The near-field data needed by the classical spherical near-field-to-far-field transformation are then efficiently retrieved from those acquired along the spiral by an optimal sampling interpolation formula. Keywords: Antenna measurements; antenna radiation patterns; near-field – far-field transformations; spherical spiral scanning; non-redundant sampling representations of electromagnetic fields 146

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T

1. Introduction

he near-field-to-far-field (NF-FF) transformation techniques play a significant role in the antenna measurements area, since they allow overcoming of all drawbacks that make direct measurements in a conventional far-field range impractical and, indeed, represent the best choice, as long as complete pattern and polarization measurements are required. As a consequence, they have attracted considerable attention in the last four decades [1-5]. One of the hottest topics related to the near-field-to-far-field transformation techniques is the reduction of measurement time, since such a time is nowadays very much greater (even many orders of magnitude) than the computational time required to carry out the near-field-to-farfield transformation. As suggested in [6], an effective way to reduce the measurement time is the use of near-field-tofar-field transformations with innovative spiral scans [6-21], wherein the near-field data acquisition is performed on the fly, and continuous and synchronized movements of the positioning systems of the probe and antenna under test (AUT) are employed. Among these transformations, those [9-21] based on the non-redundant sampling representations of electromagnetic (EM) fields [22, 23] are even more effective from the measurement-time-reduction viewpoint, due to the reduced number of near-field data needed, and due to the lower number of spiral turns. In particular, those employing sphericalspiral scanning [14-20] are particularly attractive, since they retain the interesting feature of the spherical near-field-to-farfield transformations [24-32] to allow the full reconstruction of the antenna’s far field, and avoid errors related to the truncation of the scan surface. The two-dimensional non-redundant sampling representation for the voltage measured by the probe on a sphere, and the related optimal sampling interpolation (OSI) expansion, were obtained by developing a non-redundant representation on a spiral, the pitch of which is equal to the sample spacing required for the interpolation along a meridian. As a first step, the representation was obtained [14-16] by considering the AUT as enclosed in the smallest sphere able to contain it. To overcome the useless increase in the number of near-field data when dealing with elongated or quasi-planar antennas, near-field-to-far-field transformations with spherical spiral scanning for these kinds of antennas were then developed in [17-20] by applying the unified theory of spiral scans for non-spherical antennas [21]. In particular, a prolate [17] and an oblate ellipsoid [17, 18] were adopted for modeling an elongated and a quasi-planar antenna, respectively. Instead, a long AUT was assumed as enclosed in a cylinder capped in two half spheres in [19]. A surface formed by two circular bowls with the same aperture diameter but eventually different lateral bends was employed to shape a quasi-planar AUT in [19, 20]. The aim of this paper is to provide the experimental assessment of the near-field-to-far-field transformation with spherical-spiral scanning for non-spherical antennas [17] when dealing with elongated antennas (Figure 1). The validation in the case of quasi-planar antennas was already provided in a companion paper [18]. Very good agreement was also found in this case, both in the near-field reconstructions and in the far-field reconstructions, thus experimentally confirming the

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Figure 1. Spherical-spiral scanning for an elongated antenna.

effectiveness of such a new near-field-to-far-field transformation. Fast and accurate near-field-to-far-field transformations with spherical-spiral scanning for antennas having one or two predominant dimensions and allowing a remarkable measurement-time saving are thus available.

2. Non-Redundant Sampling Representation Let us consider a long AUT and a nondirective probe that scans a spiral wrapping a sphere of radius d in the antenna’s near-field region. Let us adopt the spherical coordinate system (r , ϑ , ϕ ) to denote an observation point P (Figure 1). Because the voltage, V, measured by such a probe has the same spatial bandwidth as the field, the non-redundant sampling representations of EM fields [22] can be applied to it. Consider the AUT to be enclosed in a convex domain bounded by a proper rotational surface, Σ . Let the observation curve, C, be described by means of an optimal analytical parameterization, r = r (η ) . Then, the “reduced voltage,” V (η ) = V (η ) e jψ (η ) ,

(1)

where V is the voltage V1 or V2 measured by the probe or by the rotated probe and ψ (η ) is a proper phase function, can be closely approximated by a bandlimited function. The resulting approximation error is negligible as the bandwidth exceeds a critical value, Wη [22], and can be effectively controlled by choosing the bandwidth of the approximating function equal to χ' Wη , χ ′ > 1 being a bandwidth-enlargement factor. As shown 147

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in [22], r (η ) and ψ (η ) depend on the modeling of the AUT. The choice of a modeling that fits the AUT’s geometry well is mandatory to reduce the number of samples needed. Since the considered AUT is long, it is convenient to choose the surface Σ coincident with a prolate ellipsoid, having major and minor semi-axes equal to a and b (Figure 1). According to the unified theory of spiral scanning for nonspherical antennas [21], a two-dimensional optimal-sampling interpolation expansion to reconstruct the voltage from a nonredundant number of samples lying on a spherical spiral can be obtained by developing a non-redundant sampling representation on a spiral having a pitch such that it intersects any meridian at points the spacing of which is equal to that needed for the interpolation. The bandwidth, Wη , the parameterization,

η , relevant to a meridian, and the corresponding phase function, ψ , are therefore [17, 21] Wη =

(

)

4a E π 2 ε2 , λ

(

(2)

)

−1 2  π  E sin u ε  , = η 1+ 2  2 E 2 π ε  

(

)

(3)

   1− ε 2 v2 − 1 = ψ β a v − E cos −1 ε 2 , (4) 2 2 2 2   v −ε  v − ε   wherein λ is the wavelength, E (   ) is the elliptic integral of the second kind, = u (r1 − r2 ) 2 f and = v ( r1 + r2 ) 2a are the elliptic coordinates, with r1,2 being the distances from the point P to the foci of the ellipse C ′ , intersection between a meridian plane and Σ , and 2 f is its focal distance. Moreover, β is the wavenumber, and ε = f a is the eccentricity of C ′ . The expression in Equation (3) is valid for ϑ belonging to the range [ 0, π 2] . For ϑ from π 2 to π , the result is η= π − η ′ , where η ′ is the optimal parameter value corresponding to the point at π − ϑ . Note that in any meridian plane, the curves ψ = const and η = const are [22] ellipses and hyperbolas confocal to C ′ (see Figure 2). The parametric equations of the spiral, obtained as a projection of a proper spiral wrapping the ellipsoid Σ on the spherical surface by means of the hyperbolas at η = const [17, 21], are  x = d sin θ (η ) cosφ   y = d sin θ (η ) sinφ ,   z = d cos θ (η )

(5)

wherein φ is the angular parameter describing the spherical spiral, and η = k φ . The parameter k is such that the spiral step = ∆η 2π ( 2 N ′′ + 1) required for is equal to the sample spacing 148

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Figure 2. Prolate-ellipsoidal modeling: curves ψ = const and η = const .

the interpolation along a meridian, where = N ′′  χ N ′′ + 1 and = N ′  χ ′Wη  + 1 . Accordingly, since ∆η = 2π k , it follows = that k 1 (2 N ′′ + 1) . The function  x  gives the integer part of x, and χ > 1 is an over-sampling factor controlling the truncation error [22]. It is worth noting that the spiral angle, θ  , unlike the zenithal angle, ϑ , can assume negative values. The unified theory of spiral scans [21] also allows the determination of the non-redundant representation on the spiral. The optimal parameter, ξ , for describing the spiral is equal to β Wξ times the arc length of the projecting point that lies on the spiral wrapping, Σ , and the related phase function, γ , coincides with that ψ relevant to a meridian. The bandwidth, Wξ , is chosen equal to β π times the length of the spiral wrapping, Σ , from pole to pole [17, 21], so that ξ covers a 2π range when the point moving on the scanning spiral encircles the AUT once. Note that if a = b , the ellipsoid leads to a sphere, and the results of the spherical modeling case are reproduced [21]. According to the above results, the voltage in P on the meridian at ϕ can be recovered via the optimal sampling interpolation expansion [17]: V η (ϑ ) , ϕ  = e

− jψ (η )

n0 + q

∑

V (ηn ) G (η ,ηnη , N , N ′′ )

n = n0 − q +1

(6)

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wherein the V (η n ) are the intermediate samples, i.e., the reduced voltage values at the intersection points between the spiral and the meridian through P; 2q is the number of retained samples; n0 = (η − η0 ) ∆η  , η = q ∆η , = N N ′′ − N ′ , and

ηn = η n (ϕ ) = kϕ + n ∆η = η0 + n ∆η .

(7)

Moreover, G (η ,ηn ,η , N, N ′′= ) DN ′′(η −η n ) Ω N ( η −η n ,η ) (8) wherein DN ′′ (η ) =

sin ( 2 N ′′ + 1)η 2  , ( 2 N ′′ + 1) sin (η 2 )

(9)

TN  2cos 2 (η 2 ) cos 2 (η 2 ) − 1 ΩN (η ,η ) = , (10) TN 2 cos 2 (η 2 ) − 1 are the Dirichlet and Tschebyscheff sampling functions [17, 18, 22], with TN (  ) being the Tschebyscheff polynomial of degree N.

The intermediate samples, V (ηn ) are reconstructed [17] by means of a similar optimal sampling interpolation expansion along the spiral: V  ξ (η n ) =

m0 + p

∑

3. Experimental Assessment The near-field-to-far-field transformation with spherical spiral scan described was experimentally validated in the anechoic chamber of the UNISA Antenna Characterization Lab, which is equipped with a roll ( ϕ axis) over azimuth ( ϑ axis) spherical near-field facility. The amplitude and phase measurements were accomplished by a vector network analyzer and an open-ended WR90 rectangular waveguide. This waveguide, which scanned a spiral wrapping a sphere with radius d = 45.2 cm, was employed as probe. The results reported refer to the field radiated at 10.4 GHz by an X-band resonant slotted waveguide array made by PROCOM A/S. This was realized by cutting 12 rounded-ended slots on both the broad walls of a WR90 rectangular waveguide, and soldering two cylinders onto its narrow walls (see Figure 3). Such an antenna was mounted in such a way that the broad walls were parallel to the plane y = 0 , and its axis was coincident with the z axis (Figure 1). According to the described representation, it was considered as enclosed in a prolate ellipsoid with a = 18.17 cm and b = 3.75 cm. Figures 4 and 5 show the reconstruction of the amplitude and phase of the voltage, V1 , on the meridian at ϕ= 90° . As could be seen, there was good agreement between the reconstructed voltage and the directly measured voltage, thus assessing the effectiveness of the optimal sampling interpolation expansion. Note that an enlargement bandwidth factor χ ′ such that the sample spacing was reduced by a factor of five was used in the zones of the spiral specified by the 20 samples around each pole. The far-field patterns in the E and H principal planes reconstructed from the near-field samples acquired along

V (ξ m ) G  ξ (η n ) , ξ m , ξ , M , M ′′

m = m0 − p +1

(11)

, M ′′  χ M ′ + 1 , 2p is the ξ ∆ξ  , ξ = p ∆ξ= = number of retained samples, M ′  χ ′Wξ  + 1 , = M M ′′ − M ′ , and where m = 0

ξ m = m∆ξ = 2π m ( 2M ′′ + 1) .

(12)

Note that when reconstructing the intermediate samples nearby the poles, the factor χ ′ must be properly increased to avoid a significant growth of the band-limitation error [17]. By applying the optimal sampling interpolation expansions of Equations (6) and (11), it is thus possible to reconstruct the near-field data required to perform the spherical near-fieldto-far-field transformation from the data acquired along the spiral.

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Figure 3. A photo of the X-band resonant slotted waveguide array.

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Figure 4. The amplitude of V1 on the meridian at ϕ= 90° : the solid line is the measured values; the crosses were recovered from near-field data acquired via the sphericalspiral scanning.

Figure 6. The E-plane pattern: the solid line is the reference; the crosses were recovered from near-field data acquired via the spherical-spiral scanning.

Figure 5. The phase of V1 on the meridian at ϕ= 90° : the solid line is the measured values; the crosses were recovered from near-field data acquired via the spherical-spiral scanning.

Figure 7. The H-plane pattern: The solid line is the reference; the crosses were recovered from near-field data acquired via the spherical-spiral scanning; the dashed line is the reconstruction error.

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4. Conclusions The near-field-to-far-field transformation with spherical spiral scan for elongated antennas using prolate-ellipsoidal source modeling was experimentally assessed. As stressed in the last comment of the previous section, it was even more effective from the data-reduction viewpoint, and, as a consequence, from the measurement-time-saving viewpoint than that for quasi-planar antennas [18]. Effective near-field-to-far-field transformation techniques with spherical-spiral scan tailored to elongated or quasi-planar AUTs and allowing a remarkable measurement-time saving have thus been made available.

5. References Figure 8. The far-field pattern in the cut plane at ϕ= 90° . The solid line is the reference; the crosses were recovered from near-field data acquired via the spherical-spiral scanning; the dashed line is the reconstruction error.

1. J. Appel-Hansen, J. D. Dyson, E. S. Gillespie, and T. G. Hickman, “Antenna Measurements,” in A. W. Rudge, K. Milne, A. D. Olver, and P. Knight (eds.), The Handbook of Antenna Design, London, UK, Peter Peregrinus, 1986, Chapter 8. 2. A. D. Yaghjian, “An Overview of Near-Field Antenna Measurements,” IEEE Transactions on Antennas and Propagation, AP-34, 1, January 1986, pp. 30-45.

the spiral are compared in Figures 6 and 7 with those (references) obtained from the near-field data directly acquired on the classical spherical grid. In both cases, the far-field reconstructions were obtained by using the MI-3000 software package, implementing the standard spherical near-field-to-far-field transformation [26]. At last, the reconstruction of the far-field pattern in the cut plane at ϕ= 90° is shown in Figure 8. Note that in Figures 7 and 8, the reconstruction errors are also plotted, to allow the reader to better appreciate their levels. As could be seen, all reconstructions were very accurate, save for the zones characterized by very low field levels, thus confirming the effectiveness of the approach.

3. E. S. Gillespie (ed.), “Special Issue on Near-Field Scanning Techniques,” IEEE Transactions on Antennas and Propagation, AP-36, 6, June 1988, pp. 727-901.

It is interesting to compare the number of samples employed (1179, including the 160 extra samples) with the number (5100) needed by the MI software package, and with the number (3622) that would be required by the spiral scanning technique [14-16], based on the spherical AUT modeling. Note that in the above example, the reduction rates of the needed near-field data with respect to those required in the spherical spiral scan [14-16] using the spherical modeling and in the classical case were much greater than the corresponding rates obtained in the companion paper [18]. This is a general validity result: for elongated antennas, the time savings achievable using the spiral scan is usually remarkably greater than that for quasi-planar antennas. This is since the number of spiral turns is related to the length of C ′ , whereas the average number of samples on a turn depends on the maximum transverse radius of Σ .

6. R. G. Yaccarino, L. I. Williams, and Y. Rahmat-Samii, “Linear Spiral Sampling for the Bipolar Planar Antenna Measurement Technique,” IEEE Transactions on Antennas and Propagation, AP-44, 7, July 1996, pp. 1049-1051.

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4. M. H. Francis and R. W. Wittmann, “Near-Field Scanning Measurements: Theory and Practice,” in C. A. Balanis (ed.), Modern Antenna Handbook, Hoboken, NJ, John Wiley & Sons, Inc., 2008, Chapter 19. 5. M. H. Francis (ed.), “IEEE Recommended Practice for NearField Antenna Measurements,” IEEE Std 1720-2012, New York, NY, USA, IEEE, 2012.

7. S. Costanzo and G. Di Massa, “Far-Field Reconstruction from Phaseless Near-Field Data on a Cylindrical Helix,” Journal of Electromagnetic Waves and Applications, 18, 8, 2004, pp. 1057-1071. 8. S. Costanzo and G. Di Massa, “Near-Field to Far-Field Transformation with Planar Spiral Scanning,” Progress in Electromagnetics Research (PIER), 73, 2007, pp. 49-59. 9. O. M. Bucci, F. D’Agostino, C. Gennarelli, G. Riccio, and C. Savarese, “Probe Compensated Far-Field Reconstruction by Near-Field Planar Spiral Scanning,” IEE Proceedings Microwaves, Antennas and Propagation, 149, 2, April 2002, pp. 119-123.

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10. F. D’Agostino, F. Ferrara, C. Gennarelli, R. Guerriero, and M. Migliozzi, “An Effective NF-FF Transformation Technique with Planar Spiral Scanning Tailored for Quasi-Planar Antennas,” IEEE Transactions on Antennas and Propagation, AP-56, 9, September 2008, pp. 2981-2987.

20. F. D’Agostino, F. Ferrara, C. Gennarelli, R. Guerriero, and M. Migliozzi, “Far-Field Reconstruction from Near-Field Data Acquired via a Fast Spherical Spiral Scan: Experimental Evidences,” Progress in Electromagnetics Research (PIER), 140, 2013, pp. 719-732.

11. F. D’Agostino, F. Ferrara, J. A. Fordham, C. Gennarelli, R. Guerriero, M. Migliozzi, G. Riccio, and C. Rizzo, “An Effective Near-Field–Far-Field Transformation Technique for Elongated Antennas Using a Fast Helicoidal Scan,” IEEE Antennas and Propagation Magazine, 51, 4, August 2009, pp. 134-141.

21. F. D’Agostino, F. Ferrara, C. Gennarelli, R. Guerriero, and M. Migliozzi, “The Unified Theory of Near-Field – FarField Transformations with Spiral Scannings for Nonspherical Antennas,” Progress in Electromagnetics Research B (PIER B), 14, 2009, pp. 449-477.

12. F. D’Agostino, F. Ferrara, C. Gennarelli, R. Guerriero, and M. Migliozzi, “Experimental Results Validating the Near-Field to Far-Field Transformation Technique with Helicoidal Scan,” The Open Electrical and Electronic Engineering Journal, 4, 2010, pp. 10-15.

22. O. M. Bucci, C. Gennarelli, and C. Savarese, “Representation of Electromagnetic Fields over Arbitrary Surfaces by a Finite and Nonredundant Number of Samples,” IEEE Transactions on Antennas and Propagation, AP-46, 3, March 1998, pp. 351-359.

13. F. D’Agostino, F. Ferrara, C. Gennarelli, R. Guerriero, and M. Migliozzi, “Laboratory Tests Assessing the Effectiveness of the NF – FF Transformation with Helicoidal Scanning for Electrically Long Antennas,” Progress in Electromagnetics Research (PIER), 98, 2009, pp. 375-388.

23. O. M. Bucci and C. Gennarelli, “Application of Nonredundant Sampling Representations of Electromagnetic Fields to NF-FF Transformation Techniques,” International Journal of Antennas and Propagation, 2012, ID 319856, 2012, 14 pages.

14. O. M. Bucci, F. D’Agostino, C. Gennarelli, G. Riccio, and C. Savarese, “NF–FF Transformation with Spherical Spiral Scanning,” IEEE Antennas and Wireless Propagation Letters, 2, 2003, pp. 263-266. 15. F. D’Agostino, C. Gennarelli, G. Riccio, and C. Savarese, “Theoretical Foundations of Near-Field–Far-Field Transformations with Spiral Scannings,” Progress in Electromagnetics Research (PIER), 61, 2006, pp. 193-214. 16. F. D’Agostino, F. Ferrara, J. A. Fordham, C. Gennarelli, R. Guerriero, and M. Migliozzi, “An Experimental Validation of the Near-Field – Far-Field Transformation with Spherical Spiral Scan,” IEEE Antennas and Propagation Magazine, 55, 3, June 2013, pp. 228-235. 17. F. D’Agostino, F. Ferrara, C. Gennarelli, R. Guerriero, M. Migliozzi, and G. Riccio, “A Nonredundant Near-Field to Far-Field Transformation with Spherical Spiral Scanning for Nonspherical Antennas,” The Open Electrical and Electronic Engineering Journal, 3, 2009, pp. 4-11. 18. F. D’Agostino, F. Ferrara, C. Gennarelli, R. Guerriero, and M. Migliozzi, “Experimental Assessment of an Effective Near-Field – Far-Field Transformation with Spherical Spiral Scanning for Quasi-Planar Antennas,” IEEE Antennas and Wireless Propagation Letters, 12, 2013, pp. 670-673. 19. F. D’Agostino, F. Ferrara, C. Gennarelli, R. Guerriero, and M. Migliozzi, “Far–Field Reconstruction from a Minimum Number of Spherical Spiral Data Using Effective Antenna Modellings,” Progress in Electromagnetics Research B (PIER B), 37, 2012, pp. 43-58.

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24. P. F. Wacker, “Non-Planar Near-Field Measurements: Spherical Scanning,” NBSIR 75-809, Boulder, CO, USA, 1975. 25. A. D. Yaghjian and R. C. Wittmann, “The Receiving Antenna as a Linear Differential Operator: Application to Spherical Near-Field Measurements,” IEEE Transactions on Antennas and Propagation, AP-33, 11, November 1985, pp. 1175-1185. 26. J. Hald, J. E. Hansen, F. Jensen, and F. H. Larsen, Spherical Near-Field Antenna Measurements, London, UK, Peter Peregrinus, 1988. 27. O. M. Bucci, F. D’Agostino, C. Gennarelli, G. Riccio, and C. Savarese, “Data Reduction in the NF–FF Transformation Technique with Spherical Scanning,” Journal of Electromagnetic Waves and Applications, 15, 2001, pp. 755-775. 28. T. B. Hansen, “Spherical Near-Field Scanning with Higher-Order Probes,” IEEE Transactions on Antennas and Propagation, AP-59, 11, November 2011, pp. 4049-4059. 29. F. D’Agostino, F. Ferrara, C. Gennarelli, R. Guerriero, and M. Migliozzi, “Effective Antenna Modellings for NF–FF Transformations with Spherical Scanning Using the Minimum Number of Data,” International Journal of Antennas and Propagation, 2011, ID 936781, 2011, 11 pages. 30. M. A. Qureshi, C. H. Schmidt, and T. F. Eibert, “Adaptive Sampling in Spherical and Cylindrical Near-Field Antenna Measurements,” IEEE Antennas and Propagation Magazine, 55, 1, February 2013, pp. 243-249.

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31. F. D’Agostino, F. Ferrara, C. Gennarelli, R. Guerriero, and M. Migliozzi, “Non-Redundant Spherical NF – FF Transformations Using Ellipsoidal Antenna Modeling: Experimental Assessments,” IEEE Antennas and Propagation Magazine, 55, 4, August 2013, pp. 166-175. 32. F. D’Agostino, F. Ferrara, C. Gennarelli, R. Guerriero, and M. Migliozzi, “Experimental Testing of Nonredundant NearField to Far-Field Transformations with Spherical Scanning Using Flexible Modellings for Nonvolumetric Antennas,” International Journal of Antennas and Propagation, 2013, ID 517934, 2013, 10 pages.

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Solicitation for Measurements Corner We welcome contributions for future installments of the Measurements Corner. Please send them to Brian Fischer and Ivan LaHaie, and they will be considered for publication as quickly as possible. Contributions can range from short notes to full-length papers on all topics related to RF measurement technology and its applications, including antennas, propagation, materials, scattering, and radar cross section. New or unique measurement techniques are of particular interest.

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