AN EFFECTIVE NONREDUNDANT SAMPLING REPRESENTATION FOR PLANARNEAR-FIELD ANTENNA MEASUREMENTS

M. Ayyaz Qureshi, Carsten H. Schmidt, and Thomas F. Eibert

Lehrstuhl fur Hochfrequenztechnik, Technische Universitat Munchen80290 Munich, Germany, ayyaz.qureshi@mytum.de

ABSTRACT

A nonredundant sampling representation is described forplanar near-field antenna measurements. The minimumnumber of samples required is deduced from the num-ber of unknowns required to solve the linear system ofequations considering spherical expansion of radiated an-tenna fields. The antenna multipole order is computed inthe beginning in relation with the size of the antenna fol-lowing the angular spacing which is then mapped to theplanar surface by utilizing the distance between the AUTand the scan plane. The acquired near field on the pla-nar meshed grid is processed using the plane wave basedFast Irregular Antenna Field Transformation Algorithm(FIAFTA). Examples are presented afterwards on the ap-plicability of the proposed sampling representation.

Key words: Near-field measurements, planar, samplingtechniques.

1. INTRODUCTION

Near-field antenna measurements post-processed withnear-field far-field transformation algorithms are used todetermine the radiation pattern of an antenna under test(AUT) [1]. Planar, cylindrical, and spherical surfaces arecommonly used for collecting the near-field with appro-priate sampling, where modal expansion methods areused for determining the far field. An efficient sam-pling must be utilized in the measurements to reduceboth the cost and the computational effort. The classi-cal planar transformation algorithm employing two di-mensional Fast Fourier Transform (2D FFT) requireshalf-wavelength sample spacing to avoid aliasing er-rors [2][3]. However, transformation techniques based onintegral equations can process the near-field data col-lected on the standard surfaces using coarser irregularsampling. Therefore, several attempts have been made inthe past to decrease the measurement burden using op-timized sampling. A major contribution regarding nonre-dundant sampling representation on arbitrary surfaces hasbeen made by Bucci et al. in [4]. It has been shown thatthe electromagnetic (EM) fields radiated by bounded sur-

faces can be accurately represented over the arbitrary sur-faces with finite number of samples. The spatially ban-dlimited functions are used to approximate the EM fieldsafter extracting the phase propagation factor from thefield expression. Theoretical results from the nonredun-dant sampling representation are used in [5] to obtain aplanar mesh-grid for planar near-field measurement. Dueto variations in the sampling rate, a two dimensional opti-mal sampling interpolation (OSI) formula is presented forreconstructing the near field on a λ/2 grid required by theclassical transformation technique. Nonetheless, comput-ing additional near-field data using OSI introduces skep-ticism irrespective of its theoretical correctness and is notencouraged by the measurement industry.In our previous contribution, an adaptive samplingtechnique for planar near-field measurement togetherwith direct near-field far-field transformation was pro-posed [6]. The given technique applied higher samplingdensity at regions with strong variations and skipped datapoints from the smoother regions. A signal to noise ratio(SNR) based decision criterion was proposed for skip-ping the data point at run time. Significant reduction inthe number of measurement points was shown for anten-nas with smooth near-field distribution.In this contribution, we present a nonredundant planarsampling representation for the plane wave based FastIrregular Antenna Field Transformation Algorithm (FI-AFTA) [7]. FIAFTA directly makes use of reduced sam-ples for near-field far-field transformation without inter-polation of the intermediate data points unlike the pla-nar wide mesh scanning technique described in [5]. Inthe following we describe the fundamentals of FI-AFTA. The planar sampling representation is explainedin section 3. Section 4 summarizes the measurement re-sults using nonredundant sampling and its comparisonwith the results using standard sampling and section 5summarizes and concludes the paper.

2. FAST IRREGULAR ANTENNA FIELDTRANSFORMATION ALGORITHM

The classical technique employing modal expansion ofplane wave functions can only handle the data collectedon a regular grid. We utilize FIAFTA for efficient near-

field far-field transformation capable of handling near-field data collected on irregular grids. FIAFTA has lowcomputational complexity and offers full probe correc-tion without increase in the complexity. It has alreadybeen validated using simulations as well as experimen-tal results [7].

In near-field measurements, the field probe takes theweighted average of the field around the measurementpoint. Together with the receiving characteristics of theprobe, the output signal

U (rM) =y

Vprobe

wprobe (r) · E (r) dV (1)

is acquired at the measurement point rM. Vprobe is thevolume of the probe and wprobe contains the spatialweighting function of the probe. FIAFTA uses planewaves (I − kk) · J(k) as equivalent sources to recon-struct the radiated fields of the AUT. Unlike the classicalplane wave based approach, FIAFTA utilizes the com-plete Ewald sphere of propagating plane waves and re-lates the plane wave spectrum and near-field samples us-ing the diagonal translation operator TL(k, rM) (knownfrom the Fast Multipole Method [8]) according to

U(rM) = −jωµ

4π

{TL(k, rM)

P (k, rM) · (I − kk) · J(k)dk2(2)

where P(k, rM) contains the far-field pattern of the probefor probe correction. For optimum computational com-plexity, measurement points are grouped together to forma hierarchical structure similar to the Multilevel Fast Mul-tipole Method (MLFMM) [9].

3. PLANAR SAMPLING REPRESENTATION

In a practical measurement setup, two orthogonalpolarizations of radiated AUT field are usually re-quired. Therefore, two complex voltages are defined andthe discrete representation of spectral integral over Ewaldsphere by using numerical quadrature [8] is given as

U1/2 (φm, θn, rM) = −jwµ

4π

∑kφ

∑kθ

TL

(k, rM

)W (kθ)

P1/2 (kφ, kθ, φm, θn) ·(

I− kk)· J (kθ, kφ)

(3)

whereW (kθ) is a weighting factor andm = 1, ...,M andn = 1, ..., N denote the number of observation pointsin φ- and θ-direction, respectively. The radiation patternof the AUT using equivalent plane wave sources can bereconstructed accurately by using an efficient samplingcriterion in the discrete representation. Although over-sampling tends to reduce the effect of measurement er-rors but an efficient sample spacing is desirable provid-ing sufficient accuracy in the transformed field. In this

section, we determine a nonredundant sampling repre-sentation for planar scanning surfaces. A linear systemof equations can be developed from the discrete repre-sentation for many measurement points as

U′ = −jwµ

4π‖C‖ · J′ (4)

where U′ is the measured voltage vector containingall measurement points with both orthogonal polariza-tions and J

′contains the plane wave amplitudes of

the AUT. The diagonal translation operator TL(k, rM),weighting factor W (kθ) and probe correction coeffi-cient P(k, rM) are combined to form a coupling matrix‖C‖. More details about the arrangement of each elementin the matrix can be found in [7].

The number of φ and θ samples required to solve thegiven system of equations should ideally at least equal thenumber of unknown plane wave coefficients which de-pends on the AUT multipole order LAUT. Consequently,the spacing between φ and θ samples considering uni-form sampling distribution on a complete sphere is givenas

4φ = π/(α1LAUT) (5)

4θ = π/(α2LAUT). (6)

The empirical values of the proportionality constants α1

and α2 vary from 1 to slightly higher values depending onthe noise conditions of the given measurement environ-ment. LAUT is commonly approximated as (kd/2 + 10)with d as the diameter of minimum sphere enclosing theantenna [10]. Since FIAFTA can handle the measure-ments on arbitrary grids, the derived spherical samplespacing criterion remains valid when projected to otherarbitrary surfaces. We map the spherical sampling distri-bution to a planar surface according to the desired validangle requirements. It is observed that unlike classicalplanar sampling, the projected sample spacing on a pla-nar surface is not uniform but increases in outward direc-tion away from the center. A meshed grid thus obtainedis shown in Fig. 1.

Figure 1. 2D representation of near-field sampling pointson a planar surface.

4. MEASUREMENT RESULTS

The proposed sampling procedure for planar near-fieldmeasurements is verified using real measurements. Thenear field of a medium gain antenna operating at11.95 GHz is collected on a planar surface xz placed atdp = 1.8288m. The length of the square shaped pla-nar surface is 2.26 m. The antenna is looking in −y di-rection and the near field distribution using the regularλ/2 sampling and the nonredundant sampling is shownin Fig. 2. The samples are mapped from spherical to pla-nar grid in a way that sample spacing increases in threesteps from λ/2 to 3λ/2 causing remarkable reduction inthe number of measurement points as evident from thewhite spaces in Fig. 2b. The white spaces represent theskipped samples in reference to the regular λ/2 spacing.

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

x [m]

z [m

]

(a)

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

x [m]

z [m

]

(b)

Figure 2. Near-field distribution [dB] of medium gainantenna on a planar surface with regular λ/2 (a) and theproposed nonredundant sampling (b).

The near-field data collected using both regular andnonredundant sampling is processed by FIAFTA. Fig. 3shows the transformedE- andH-plane pattern cuts alongwith the error. The error level is computed by taking thelogarithm of the difference between the normalized pat-tern magnitude using regular sampling and normalizedpattern magnitude using nonredundant sampling. As ob-vious from the results, good agreement in the valid regionof both patterns is observed even with 62% decrease inthe number of sampling points.

255 260 265 270 275 280 285−60

−50

−40

−30

−20

−10

0

Phi [°]

FF

[dB

]

Reg. sampl.

Nonredun. sampl.

Error

(a) E-plane pattern cut

255 260 265 270 275 280 285−60

−50

−40

−30

−20

−10

0

Theta [°]F

F [

dB

]

Reg. sampl.

Nonredun. sampl.

Error

(b) H-plane pattern cut

Figure 3. Transformed E- and H-plane pattern cuts formedium gain antenna.

5. CONCLUSION

A nonredundant sampling representation for plane wavebased near-field far-field transformed planar near-fieldmeasurements was proposed. The sampling representa-tion was found dependent on the size of the antenna andthe separation between the AUT and the measurementplane. The proposed sampling strategy was assessed us-ing practical measurements and a remarkable decreasesin the number of measurement points was observed ascompared to traditional technique employing λ/2 samplespacing. Also, good agreement in the transformed pat-tern using traditional and nonredundant sampling was ob-served.

ACKNOWLEDGEMENTS

The authors are grateful to D. J. van Rensburg (NSI) forproviding the near-field measurement data of the mediumgain antenna presented in section 4.

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