# An Effective Spherical NF/FF Transformation Suitable for Characterising an Antenna under Test in Presence of an Infinite Perfectly Conducting Ground Plane

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Sampling Representation over the Upper Hemisphere

## 3. Numerical Results

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Yaghjian, A.D. An overview of near-field antenna measurements. IEEE Trans. Antennas Propag.
**1986**, 34, 30–45. [Google Scholar] [CrossRef] - Appel-Hansen, J.; Dyson, J.D.; Gillespie, E.S.; Hickman, T.G. Antenna measurements. In The Handbook of Antenna Design; Rudge, A.W., Milne, K., Olver, A.D., Knight, P., Eds.; Peter Peregrinus: London, UK, 1986; pp. 584–694. [Google Scholar]
- Gillespie, E.S. (Ed.) Special Issue on near-field scanning techniques. IEEE Trans. Antennas Propag.
**1988**, 36, 727–901. [Google Scholar] - Francis, M.H.; Wittmann, R.W. Near-field scanning measurements: Theory and practice. In Modern Antenna Handbook; Balanis, C.A., Ed.; John Wiley & Sons Inc.: Hoboken, NJ, USA, 2008; pp. 929–976. [Google Scholar]
- IEEE Standard 1720-2012; IEEE Recommended Practice for Near-Field Antenna Measurements. Francis, M.H. (Ed.) IEEE: Piscataway, NJ, USA, 2012.
- Sierra Castañer, M.; Foged, L.J. Post-Processing Techniques in Antenna Measurement; SciTech Publishing IET: London, UK, 2019. [Google Scholar]
- Parini, C.; Gregson, S.; McCormick, J.; Van Rensburg, D.J. Theory and Practice of Modern Antenna Range Measurements; SciTech Publishing IET: London, UK, 2020. [Google Scholar]
- Gennarelli, C.; Ferrara, F.; Guerriero, G.; D’Agostino, F. Non-Redundant Near-Field to Far-Field Transformation Techniques; SciTech Publishing IET: London, UK, 2022. [Google Scholar]
- Jensen, F. On the probe compensation for near-field measurements on a sphere. Archiv Elektr. Übertr.
**1975**, 29, 306–308. [Google Scholar] - NBSIR 75-809; Non-Planar Near-Field Measurements: Spherical Scanning. Wacker, P.F. (Ed.) National Institute of Standards and Technology: Boulder, CO, USA, 1975.
- Larsen, F.H. Probe correction of spherical near-field measurements. Electr. Lett.
**1977**, 13, 393–395. [Google Scholar] [CrossRef] - Yaghjian, A.D. Simplified approach to probe-corrected spherical near-field scanning. Electr. Lett.
**1984**, 20, 195–196. [Google Scholar] - Yaghjian, A.D.; Wittmann, R.C. The receiving antenna as a linear differential operator: Application to spherical near-field measurements. IEEE Trans. Antennas Propag.
**1985**, 33, 1175–1185. [Google Scholar] [CrossRef] - Hansen, J.E.; Jensen, F. Spherical near-field scanning at the technical university of Denmark. IEEE Trans. Antennas Propag.
**1988**, 36, 734–739. [Google Scholar] [CrossRef] - Hald, J.; Hansen, J.E.; Jensen, F.; Larsen, F.H. Spherical Near-Field Antenna Measurements; Hansen, J.E., Ed.; Peregrinus: London, UK, 1998. [Google Scholar]
- Bucci, O.M.; Gennarelli, C.; Riccio, G.; Savarese, C. Data reduction in the NF-FF transformation technique with spherical scanning. J. Electromagn. Waves Appl.
**2001**, 15, 755–775. [Google Scholar] [CrossRef] - Laitinen, T.; Pivnenko, S.; Breinbjerg, O. Application of the iterative probe correction technique for a high-order probe in spherical near-field antenna measurements. IEEE Antennas Propag. Mag.
**2006**, 48, 179–185. [Google Scholar] [CrossRef] - Laitinen, T.; Pivnenko, S. Probe correction technique for symmetric odd-order probes for spherical near-field antenna measurements. IEEE Antennas Wirel. Propag. Lett.
**2007**, 6, 635–638. [Google Scholar] [CrossRef] - Laitinen, T.; Breinbjerg, O. A first/third-order probe correction technique for spherical near-field antenna measurements using three probe orientations. IEEE Trans. Antennas Propag.
**2008**, 56, 1259–1268. [Google Scholar] [CrossRef] - Laitinen, T. Double φ-step θ-scanning technique for spherical near-field antenna measurements. IEEE Trans. Antennas Propag.
**2008**, 56, 1633–1639. [Google Scholar] [CrossRef] - Laitinen, T. Modified θ-scanning technique for first/third-order probes for spherical near-field antenna measurements. IEEE Trans. Antennas Propag.
**2008**, 57, 1590–1596. [Google Scholar] [CrossRef] - Laitinen, T.; Pivnenko, S.; Nielsen, J.M.; Breinbjerg, O. Theory and practice of the FFT/matrix inversion technique for probe-corrected spherical near-field antenna measurements with high-order probes. IEEE Trans. Antennas Propag.
**2010**, 58, 2623–2631. [Google Scholar] [CrossRef] - Hansen, T.B. Spherical near-field scanning with higher-order probes. IEEE Trans. Antennas Propag.
**2011**, 59, 4049–4059. [Google Scholar] - Cornelius, R.; Heberling, D. Spherical wave expansion with arbitrary origin for near-field antenna measurements. IEEE Trans. Antennas Propag.
**2017**, 65, 4385–4388. [Google Scholar] [CrossRef] - Cornelius, R.; Heberling, D. Spherical near-field scanning with pointwise probe correction. IEEE Trans. Antennas Propag.
**2017**, 65, 995–996. [Google Scholar] [CrossRef] - D’Agostino, F.; Ferrara, F.; Gennarelli, C.; Guerriero, R.; Migliozzi, M. Effective antenna modellings for NF-FF transformations with spherical scanning using the minimum number of data. Int. J. Antennas Propag.
**2011**, 2011, 936781. [Google Scholar] [CrossRef] - D’Agostino, F.; Ferrara, F.; Gennarelli, C.; Guerriero, R.; Migliozzi, M. Non-redundant spherical NF—FF transformations using ellipsoidal antenna modeling: Experimental assessments. IEEE Antennas Propag. Mag.
**2013**, 55, 166–175. [Google Scholar] [CrossRef] - D’Agostino, F.; Ferrara, F.; Gennarelli, C.; Guerriero, R.; Migliozzi, M. A spherical near-to-far-field transformation using a non-redundant voltage representation optimized for non-centered mounted quasi-planar antennas. Electronics
**2020**, 9, 944. [Google Scholar] [CrossRef] - Varela, F.R.; Iragüen, B.G.; Sierra Castañer, M. Fast spherical near-field to far-field transformation for offset-mounted antenna measurements. IEEE Antennas Wirel. Propag. Lett.
**2020**, 19, 2255–2259. [Google Scholar] [CrossRef] - Hansen, T.B. Numerical investigation of the system-matrix method for higher-order probe correction in spherical near-field antenna measurements. Int. J. Antennas Propag.
**2012**, 2012, 493705. [Google Scholar] [CrossRef] - Qureshi, M.A.; Schmidt, C.H.; Eibert, T.F. Adaptive sampling in spherical and cylindrical near-field antenna measurements’. IEEE Antennas Propag. Mag.
**2013**, 55, 243–249. [Google Scholar] [CrossRef] - Saccardi, F.; Rossi, F.; Mioc, F.; Foged, L.J.; Iversen, P.O. Application of the translated-SWE algorithm for the characterization of antennas installed on cars using a minimum number of samples. In Proceedings of the Antenna Measurement Techniques Association Symposium, Atlanta, GE, USA, 15–20 October 2017; pp. 1–6. [Google Scholar]
- Bucci, O.M.; Gennarelli, C.; Savarese, C. Representation of electromagnetic fields over arbitrary surfaces by a finite and nonredundant number of samples. IEEE Trans. Antennas Propag.
**1998**, 46, 351–359. [Google Scholar] - D’Agostino, F.; Ferrara, F.; Gennarelli, C.; Guerriero, R.; Riccio, G. Pattern reconstruction of 3-D modular antennas by means of a non-redundant near-field spherical scan. Electronics
**2022**, 11, 2060. [Google Scholar] [CrossRef] - Camacho-Perez, J.R.; Moreno, P. Initial considerations towards hemispherical near-field antenna measurements. IEEE Antennas Wirel. Propag. Lett.
**2014**, 13, 1441–1444. [Google Scholar] [CrossRef] - Mauermayer, R.A.M.; Eibert, T.F. Spherical field transformation above perfectly electrically conducting ground planes. IEEE Trans. Antennas Prop.
**2018**, 66, 1465–1478. [Google Scholar] - Harrington, R.F. Time-Harmonic Electromagnetic Fields; John Wiley & Sons, Inc.: New York, NY, USA, 2001. [Google Scholar]
- D’Agostino, F.; Ferrara, F.; Gennarelli, C.; Guerriero, R.; Migliozzi, M. A non-redundant spherical NF/FF transformation for an AUT measured over an infinite perfectly conducting ground plane. In Proceedings of the Antennas Propagation Conference, Birmingham, UK, 11–12 November 2019; pp. 1–5. [Google Scholar]
- D’Agostino, F.; Ferrara, F.; Gennarelli, C.; Guerriero, R.; Migliozzi, M. AUT far-field pattern reconstruction from a reduced set of spherical near-field data collected in presence of an infinite perfectly conducting ground plane. In Proceedings of the European Conference on Antennas and Propagation, Copenhagen, Denmark, 15–20 March 2020; pp. 1–5. [Google Scholar]
- Bucci, O.M.; D’Elia, G.; Migliore, M.D. Advanced field interpolation from plane-polar samples: Experimental verification. IEEE Trans. Antennas Propag.
**1998**, 46, 204–210. [Google Scholar] [CrossRef]

**Figure 2.**View in the meridian plane at $\phi $ = 0: (

**a**) original formulation of the problem; (

**b**) equivalent formulation of the problem.

**Figure 3.**${V}_{p}$ on the meridian at $\phi $ = 0°. Blue solid line: exact. Red dots: interpolated from the non-redundant samples: (

**a**) Amplitude; (

**b**) Phase.

**Figure 4.**${V}_{r}$ on the meridian at $\phi $ = 90°. Blue solid line: exact. Red dots: interpolated from the non-redundant samples: (

**a**) Amplitude; (

**b**) Phase.

**Figure 5.**Voltage amplitude on the meridian at $\phi $ = 60°. Blue solid line: exact. Red dots: interpolated from the non-redundant samples: (

**a**) ${V}_{p}$; (

**b**) ${V}_{r}$.

**Figure 6.**Normalised mean-square errors in the reconstruction of the probe voltage: (

**a**) ${V}_{p}$; (

**b**) ${V}_{r}$.

**Figure 7.**${V}_{r}$ on the meridian at $\phi $ = 90°. Blue solid line: exact. Red dots: recovered from the error affected NF samples: (

**a**) Amplitude; (

**b**) Phase.

**Figure 8.**Far-field pattern. Blue solid line: exact. Red dots: obtained from the non-redundant samples: (

**a**) $\phi $—component on the cut plane at $\phi $ = 0°; (

**b**) ϑ—component on the cut plane at $\phi $ = 60°; (

**c**) $\phi $—component on the cut plane at $\phi $ = 60°; (

**d**) ϑ—component on the cut plane at $\phi $ = 90°.

**Figure 9.**${V}_{r}$ on the meridian at $\phi $ = 90°. Blue solid line: exact. Red dots: interpolated from the non-redundant samples: (

**a**) Amplitude; (

**b**) Phase.

**Figure 10.**Relevant to the reconstruction of ${V}_{r}$: (

**a**) Normalised mean-square errors; (

**b**) Amplitude on the meridian at $\phi $ = 90°. Blue solid line: exact. Red dots: recovered from the error-affected NF samples.

**Figure 11.**Far-field pattern. Blue solid line: exact. Red dots: interpolated from the non-redundant samples: (

**a**) ϑ—component on the cut plane at $\phi $ = 90°; (

**b**) ϑ—component on the cut plane at ϑ = 90°.

AUT Case | Here Proposed | NF/FF Transformation [15] in Presence of PEC Ground Plane |
---|---|---|

Case A | 14,521 | 20,736 |

Case B | 13,297 | 36,864 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ferrara, F.; Gennarelli, C.; Guerriero, R.; Riccio, G.
An Effective Spherical NF/FF Transformation Suitable for Characterising an Antenna under Test in Presence of an Infinite Perfectly Conducting Ground Plane. *Electronics* **2024**, *13*, 397.
https://doi.org/10.3390/electronics13020397

**AMA Style**

Ferrara F, Gennarelli C, Guerriero R, Riccio G.
An Effective Spherical NF/FF Transformation Suitable for Characterising an Antenna under Test in Presence of an Infinite Perfectly Conducting Ground Plane. *Electronics*. 2024; 13(2):397.
https://doi.org/10.3390/electronics13020397

**Chicago/Turabian Style**

Ferrara, Flaminio, Claudio Gennarelli, Rocco Guerriero, and Giovanni Riccio.
2024. "An Effective Spherical NF/FF Transformation Suitable for Characterising an Antenna under Test in Presence of an Infinite Perfectly Conducting Ground Plane" *Electronics* 13, no. 2: 397.
https://doi.org/10.3390/electronics13020397