# An Effective Near-Field-Far-Field Transformation Technique for Elongated Antennas Using a Fast Helicoidal Scan [Measurements Corner]

**ABSTRACT** An experimental validation of a new near-field-far-field transformation technique with helicoidal scanning, tailored for elongated antennas, is provided in this paper. Such a transformation is based on non-redundant sampling representations of the electromagnetic fields. It makes use of a flexible source modeling, which allows one to very well fit many of these kinds of antennas by properly setting the geometric parameters. By employing such modeling instead of spherical modeling, it is possible to remarkably reduce the error related to the truncation of the scanning zone, since measurement cylinders with a diameter smaller than the antenna's height can be used. A comparison of the reconstructions recovered from the non-redundant measurements on the helix with those obtained from data directly measured on the classical cylindrical grid assesses the validity of this innovative scanning technique.

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- Brian E. Fischer, Ivan J. LaHaie, Francesco D'Agostino, Flaminio Ferrara, Claudio Gennarelli, Rocco Guerriero, Massimo Migliozzi[Show abstract] [Hide abstract]

**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 non-spherical 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.IEEE Antennas and Propagation Magazine 04/2014; 56(2):146-153. · 1.18 Impact Factor - SourceAvailable from: Rocco Guerriero
##### Conference Paper: A nonredundant spherical NF-FF transformation: Experimental tests @ UNISA antenna characterization lab

[Show abstract] [Hide abstract]

**ABSTRACT:**An experimental validation of a near-field - farfield transformation technique with spherical scanning for quasiplanar antennas requiring a minimum number of near-field data is provided in this work. Such a technique is based on the nonredundant sampling representations of the electromagnetic fields and on the optimal sampling interpolation expansions, and makes use of an oblate ellipsoid to model the antenna. It is so possible to remarkably lower the number of data to be acquired, thus reducing in a significant way the measurement time. The effectiveness of such a technique is experimentally assessed at the UNISA Antenna Characterization Lab by comparing the farfield patterns reconstructed from nonredundant measurements on the sphere with those obtained from the near-field data directly measured on the classical spherical grid.Electromagnetic Theory (EMTS), Proceedings of 2013 URSI International Symposium on; 01/2013 - SourceAvailable from: Rocco GuerrieroELECTROMAGNETIC WAVES. 01/2013; 140:719-732.

Page 1

Measurements Corner

Brian E. Fischer

General DynamicsAdvanced

Information Systems

Michigan Research and

Development Center

1200 Joe Hall Drive,

Ypsilanti,MI48197, USA

Tel: +1 (734) 480-5125

Fax: +1 (734)480-5328

Email: Brian.Fischer@gd-ais.com

Ivan J. laHaie

General DynamicsAdvanced

Information Systems

Michigan Research and

DevelopmentCenter

1200Joe Hall Drive,

Ypsilanti,MI48197, USA

Tel:+1 (734) 480-5207

Fax: +1 (734) 480-5328

Email: Ivan.LaHaie@gd-ais.com

Introduction

Accuracy and efficiency are two key drivers for any near-

field antenna-measurement technique. Traditional scanning tech-

niques are chosen for mathematical convenience, but in many cases

they suffer from data-collection inefficiencies, or errors due to

truncation effects. This issue's Measurements Column features a

paper by authors from the University of Salerno and MI Technolo-

gies, focused on experimental validation ofa near-field helicoidal-

scan technique that overcomes these limitations for a particular

class of antenna geometries.

An Effective Near-Field-Far-Field

Transformation Technique for Elongated

Antennas Using a Fast Helicoidal Scan

F. D'Agostino1, F. Ferrara1, J. A. Fordham2, c. Gennarelli1, R. Guerriero1, M. Migliozz;1,

G. Riccio1, and C. Rizzo3

lDipartimento di Ingegneria dell'lnformazione ed Ingegneria Elettrica

University of Salerno. via Ponte Don Melillo. 1-84084 Fisciano (Salerno), Italy

2MI Technologies

1125 Satellite Blvd, Suite 100 Suwanee, Georgia 30024-4629, USA

Tel: +1 (678) 475-8365; E-mail: jfordham@mi-technologies.com

3MITechnologies Europe

3 Hither Green Southbourne Emsworth, P010 8JA, UK

Abstract

An experimental validation of a new near-field-far-field transformation technique with helicoidal scanning, tailored for

elongated antennas, is provided in this paper. Such a transformation is based on non-redundant sampling representations of

the electromagnetic fields. It makes use of a flexible source modeling, which allows one to very well fit many of these kinds of

134

IEEEAntennasandPropagation MagaZine, Vol. 51, No.4, August 2009

Page 2

.antennas by properly setting the geometric parameters. By employing such.modeling i ~ s t e a d

possible to remarkably reduce the error related to the truncation of t h ~ scanning zone, since .measurement cylinders with a

diameter smaller than the antenna's height can be used. A comparison of the reconstructions recovered from the non-·

redundant measurements on the helix with those obtained from data directly measured on the classical cylindrical grid

assesses the validity of this innovative scanning technique.

of spherical m.odeling,.it is

Keywords: Near-field-far-field (NF-FF) transformation techniques; n o n - r e d u n d ~ n t sampling; h e l i c o i d ~ 1

measurements; antenna radiation patterns; cylindrical antennas; electromagnetic measurements; optimization methods

~ c a ~ n i n g ; antenna

1. Introduction

T

attractive alternatives to conventional far-field (FF) and compact-

range measurements. Accordingly, they have been widely investi-

gated, and used for applications ranging from cellular-phone

antennas to large phased arrays, and complex multi-beam commu-

nication-satellite antennas [1-4]. In addition, near-field measure-

ments can be performed in a controlled environment - such as in

an indoor anechoic chamber - thus overcoming those drawbacks

due to weather conditions, electromagnetic (EM) interference, and

such cannot be eliminated in far-field measurements.

he techniques for reconstructing an antenna's far field from

near-field (NF) measurements have proven to be efficient and

The antenna-measurement community can profit today from

about fifty years of research activity on near-field data acquisition

and related near-field-far-field transformations. Over these years,

many solutions have been proposed to meet the demands of the

various applications. In this framework, significant improvements

in the performance ofnear-field measurements have been recently

obtained. They are based on the spatial band-limitation properties

of radiated EM fields [5], on their non-redundant sampling repre-

sentations [6], and on the optimal-sampling-interpolation (OS1)

expansions of the central type [7]. In particular, a significant

reduction in the number ofrequired near-field data (and, as a con-

sequence, in the corresponding measurement time) has been

achieved for all the conventional scanning methods (see [8-11] for

planar scanning, [12, 13] for cylindrical scanning, and [14] for

spherical scanning).

Moreover, innovative scanning techniques have been pro-

posed to further reduce the time needed for data acquisition. As

suggested in [15], they are implemented by means of a continuous

movement ofthe positioning systems ofthe probe and the antenna

under test (AUT). In particular, the helicoidal scan [16, 17], the

planar scan [17, 18], and spherical [17, 19] spiral scans have been

achieved. In all of these cases, by assuming that the AUT is

enclosed in the smallest spherical surface able to contain it, a non-

redundant sampling representation of the voltage data acquired by

the measurement probe on the curve considered (a helix or spiral)

has been developed by applying the theoretical results to the non-

redundant representations of the EM fields [6]. The choice of the

spiral angular step equal to the sample spacing needed for interpo-

lating the data along the corresponding meridian curve (the gen-

eratrix, radial line, or meridian) has then allowed one to get the

desired two-dimensional optimal-sampling-interpolation formula.

It has been so possible to recover the near-field data required by the

near-field-far-field transformation technique using the related con-

ventional scanning [3].

Unfortunately, the use of spherical AUT modeling, even if

quite general, prevents the possibility of considering measurement

cylinders with a radius smaller than one-half of the AUT's maxi-

mum size. This drawback occurs in helicoidal scanning when con-

IEEEAntennasandPropagation Magazine, Vol. 51, No.4, August 2009

sidering elongated antennas. Obviously, this results in an increase

ofthe error related to the truncation ofthe measurement surface. In

fact, for a given size of the scanning zone, such an error increases

with increasing radius. Moreover, the "volumetrical" redundancy

ofthe spherical modeling gives rise to an increase in the near-field

data requirement when the AUT geometry departs from a spherical

geometry. To overcome these drawbacks, two effective near-field-

far-field transformations with cylindrical scanning, tailored for

elongated antennas, were developed in [20-22]. The former [20]

made use ofa very flexible source modeling, named in the follow-

ing "rounded cylinder," wherein the AUT is enclosed in the small-

est surface, L, formed by a cylinder capped by two half-spheres

(see Figure 1). The latter [21, 22] employed a prolate ellipsoidal

model ofthe AUT.

The aim ofthis paper is to provide experimental validation of

the near-field-far-field transformation using rounded-cylinder

modeling. This is more effective from a data-reduction viewpoint

than is prolate-ellipsoidal modeling. As a matter of fact, such

flexible modeling allows one to better fit the shape ofmany actual

antennas by properly setting its geometric parameters. The experi-

mental validation was out at the laboratory of antenna characteri-

zation of the University of Salerno, where an advanced cylindrical

near-field measurement facility, supplied by MI Technologies, was

available.

2. Helicoidal Scanning for Elongated

Antennas: Rounded-Cylinder Modeling

The results [20] concerning the reconstruction of the voltage

from the knowledge of a non-redundant number of its samples

acquired along a helix are summarized here.

Let us consider an AUT and a non-directive probe scanning a

proper helix, lying on a cylinder of radius d (see Figure 1). Let us

adopt a spherical coordinate reference system (r,9,qJ) to denote

the observation point, P. Since the voltage, V,measured by such a

probe has the same effective spatial bandwidth as the field, the

theoretical results on the non-redundant representation of EM

fields [6] can be applied to the voltage. Accordingly, ifthe AUT is

enclosed in a convex domain bounded by a surface L with rota-

tional symmetry, and the observation curve is described by a proper

analytical parameterization, 1:.. = 1:..(1]),it is convenient to introduce

the probe "reduced voltage:"

(1)

with If/ (1]) being a phase function to be determined. The band-

limitation error, occurring when the reduced voltage is approxi-

mated by a bandlimited function, becomes negligible as the band-

135

Page 3

width exceeds a critical value, WT/ [6]. Accordingly, it can be

effectively controlled by choosing a bandwidth equal to X'WT/'

where X' is an excess-bandwidth factor slightly greater than unity

for an electrically large AUT.

Whenconsidering elongatedantennas,suchas those

employed in radio base stations, it is convenient to choose as the

surface :E a rounded cylinder, namely a cylinder of height h'

capped at the ends with two half-spheres of radius a' (see Fig-

ures 1 and 2).

The two-dimensional optimal-sampling-interpolation algo-

rithm for reconstructing the voltage from a non-redundant number

of its samples collected by the probe along a proper helix was

obtained in [20]. This was done by developing a non-redundant

sampling representation of the voltage on a helix, the step size of

which must be chosen equal to the sample spacing required to

interpolate the data along a generatrix. This target was achieved by

heuristically extending the rigorous approach in [17], valid when

adopting the spherical AUT modeling, to the case of elongated

antennas modeled as enclosed in a rounded cylinder. By consider-

ing a rounded cylinder as surface :E,and by adopting WT/ =Pf.'/2;r

(with P being the wavenumber, and f.' being the length of the

intersection curve, C', between the meridian plane and :E), the

phase function, If!, and the parameterization, T/, to be used for

describing a generatrix become [6, 20]

Figure 3. The curves If!= constant and T/= constant.

If!=(P/2)[R1+R2+sj - S2],

(2)

(3)

where (see Figure 2)

z

f.'=2(h'+;ra'),

(4)

When imposing its passage through a fixed point, Po , ofthe

generatrix at qJ=0, the equations ofthe helix are

are the distances from P to the tangency points, 1\.2' on C' , and

sb are their arc-length coordinates, given by

(5)

(6)

(8)

(7)

,_ , . _1!a'd+R1[(h'/2)-ZJ}

sl-asIn

2,2

R1+a

'

j

x=d cos(¢-¢i)

y =d sin(¢-¢i) '

Z =d cot[.9(T/)]

,,'[. _,!a'd+R2[(h'/2)+ZJ}]

;r - SIn

s2=h +a

22.

R2+a'

It

If!= constant and T/ = constant become those displayed in Fig-

ure 3, instead ofthe circumferences and radial lines ofthe spherical

modeling.

muststressed that inany meridian plane,the curves

h'

x

Figure 2. Relevant to a cylinder generatrix.

wherein ¢ is the angular parameter describing the helix, ¢i is the

value of ¢ at Po , and T/ = k¢ . Such a helix is obtained by project-

136

IEEEAntennasandPropagation Magazine, Vol. 51, No.4, August 2009

Page 4

h

Figure 1. Helicoidal scanning for an elongated AUT: rounded-

cylinder modeling.

135

135

'"

Amphlude(dBj

: :~ .. .. ,

I)

~

'"

45

4!

.,

AZMJTH 0

AZM./TH 0

-<5

-<5

.,

.,

.$0

.$0

·'35

·'35

. 4

Amplllude(dBj

1 : ~ I :..

I :.,.' .,

.':,.,

. 4

VERTICAl

Figure 8. Contour level plots of the measured near-field data.

IEEEAntennasandPropagation Magazine, Vol. 51, No.4, August 2009

VERTICAl

Figure 9. Contour level plots of the interpolated near-field

data.

137

Page 5

ing a proper spiral wrapping the surface L modeling the AUT on

the scanning cylinder via the curves at 17 = constant [20]. In order

to allow the two-dimensional interpolation, the step size ofthe spi-

ral must be equal to the sample spacing needed for interpolating

the voltage along the generatrix. The parameter k is then such that

the spiral step, determined by two consecutive intersections (at ¢

and ¢+2n ) with a given generatrix, is chosen equal to the sample

spacing

where

~ 1 7 = 2 1 r / ( 2 N " + I ) ,

N"=Int(XN')+I,

with

N' = Int (X'W1] )+1. Accordingly, since 11.17 = 21rk, it follows that

k =1/(2N" +1). The function Int(x) gives the integer part ofx,

and X > 1 is an over-sampling factor needed to control the trunca-

tion error.

namely, the voltage values at the intersection points between the

helix and the generatrix passing through a given point, P. Once

these samples have been evaluated, the near-field data required by

the probe-compensated near-field-far-field transformation tech-

nique with cylindrical scanning [23] can be reconstructed via the

following optimal-sampling-interpolation formula:

no+q

V[17(.9),lp] == L

V(17n) QN (17 -17n)DNW

( 17-17nI- (13)

n=no-q+l

where no = Int[(17 -170)/ ~

voltage samples,

7]J, the V(17n)

are the intermediate

with M" == Int(XM') +1 and M' = Int ( X ' W ~ )+1. Moreover,

mo+p

V(q)= L

V ( q m ) Q M ( ~ - ~ m ) D M w ( q - ~ m ) '

(9)

m=mo-p+l

and all the other symbols have the same or analogous meanings as

in Equation (9).

3. Experimental Validation

The above-described scanning technique was experimentally

validated in the anechoic chamber of the antenna characterization

laboratory of the University of Salerno, where an advanced cylin-

-- drical near-field measurement facility supplied by MI Technologies

wasavailable.Thedimensions

8 m x 5 m x 4 m. Pyramidal absorbers were positioned in order to

minimize the reflections, thus ensuring a background noise lower

than -40dB. The chamber was equipped with a vertical scanner

and a rotating table, so that by properly matching their movements,

the near-field data could be acquired at any point on a cylindrical

surface surrounding the AUT. The vertical scanner had a height of

240 em, and was characterized by a linear precision of ±0.005 em.

The rotating table, MI-6111B, mounted with its rotary axis parallel

to the vertical scanner, ensured an angular precision of ±O.05°.

The controller, MI-4190, was used to control the motion of the

positioners, and was completed by the MI-4193 option, so that it

was able to simultaneously drive both the positioners. Moreover, it

was connected to a host computer by means ofanIEEE-488 inter-

face. The amplitude and phase measurements were performed by

means of a computer-controlled vector network analyzer, Anritsu

37247C, characterized by a wide dynamic range, and high sensi-

tivity and linearity over the range from 40 MHz to 20 GHz. The

probe was an open-ended rectangular waveguide, MI-6970-WR90,

the end of which was tapered to minimize the diffraction effects.

The AUT, located in the x == 0 plane, was properly designed and

optimized by using the commercial Ansoft HFSS software. It was a

resonant slotted waveguide array, 37.7 cm long, fed at the center of

the bottom broad wall by a coaxial line, operating at 10 GHz (see

Figure 4). It was obtained from a WR-90 waveguide by cutting in it

two rows each of 10 round-ended slots. These rows were at the

same distance from the centerline of the waveguide's broad wall.

The slots were longitudinally directed and uniformly spaced by

Ag/2, where Agis the guide wavelength.

ofthechamberwere

(11)

(12)

sin[(2M" +1 ) ~ /2]

DM" (.;)= (2M" +l)sin(.;/2),

The phase function, r , and the parameterization,

used for obtaining a non-redundant sampling representation along

the helix can be determined according to the heuristic approach

proposed in [20]. In particular, by generalizing the corresponding

relations valid when adopting the spherical AUT modeling, the

phase function r coincides with If/ defined in Equation (2). Fur-

thermore, the parameter ~

is f3/ W ~

of the projecting point that lies on the spiral wrapping the surface

L. Moreover, as suggested in [20],

f3/n times the length of the spiral wrapping the surface L from

pole to pole. Namely, the spiral, r, and ~

cide with those relevant to the spherical modeling, when the sur-

face L leads to a sphere.

~ ,

to be

times the curvilinear abscissa

W ~

must be chosen equal to

are such that they coin-

where rno = Int {[q- ~ (¢i)J/I1.q} is the index of the sample near-

est (on the left) to the point Q, 2p is the number ofretained sam-

ples, V ( ~ m ) ' and

According to the above results, the voltage at any point, Q, of

the helix can be recovered via the optimal-sampling-interpolation

expansion [20]:

are the Dirichlet and Tschebyscheff sampling functions. TM(q) is

the Tschebyscheff polynomial of degree

~= p ~ q .

M = M" - M', and

Theoptimal-sampling-interpolation

tion (9) can be used to evaluate the "intermediate samples,"

expansioninEqua-

According to the sampling representation described in Sec-

tion 2, the AUT was modeled as enclosed in the surface L formed

by a cylinder ofheight h' =37.5 cm, capped in two half-spheres of

radius a' = 2.5 em. The probe output voltages were acquired on a

helix lying on a cylinder with h =237 em and d =18em. In order

to assess the effectiveness of the two-dimensional optimal-sam-

pIing-interpolation expansion, the amplitudes of the recovered

138

IEEEAntennasandPropagation Magazine, Vol. 51, No.4, August 2009

Page 6

q=p= 6

X=1.20

x'= 1.35

q=p =6

X = 1.20

x'= 1.35

2040 6080100

z (em)

00-20

2-

.g -30

3

}-4O

til

~- s o

g

"0

;;. -60

'5

0-g-70

c:.>

.~

-80

til

Q)

0:: _90

ULJ-l..-l....L...1...J.....L.J....J-.L...LL.J.....LLJ...J.....L....L...1....L....L.J....J-.1-L.L....l...JL...J....l...J.....L....L...1.............J....J

-100 -80-60-40-200

00-20

2-

~

-30

.~}-4O

til

§b-so

s

"0

;;. -60

'5

0-g-70

c:.>

.~

-80

til

Q)

0:: _90

CJ.J-L...1....L..J....J...L.L....l...Jc...LL....L..J.....l...L....L..J....CL.L...L....L...l.....L..J....L.LL..L.J...L.J......L..J....L.1ll.J1..L..J

-100 -80-60-40-200

Figure 6. The amplitude of the probe voltage on the generatrix

at

rp = 45°: the solid line is measured data; the crosses are

interpolated values.

drical grid. The same software was to get the far-field reconstruc-

tions from the helicoidal near-field data. To this end, the two-

dimensional optimal-sampling-interpolation

employed for recovering the cylindrical data required to carry out

the near-field-far-field transformation. As can be seen, in both

planes there was a very good agreement, thus confirming the

validity of the proposed near-field-far-field transformation with

helicoidal scanning.

algorithm was

204060 80100

z (em)

Figure 5. The amplitude ofthe probe voltage on the generatrix

at rp =0° : the solid line is measured data; the crosses are inter-

polated values.

It is interesting to compare the number of data (788) needed

by such a near-field-far-field transformation with the number

(5120) required by the traditional cylindrical near-field scanning to

cover the same scanning zone. Note that this number is comparable

with that (880) needed by the non-redundant near-field-far-field

transformation with cylindrical scanning [12). It is significantly

less than that needed by the helicoidal scanning technique [24],

which requires the same number as the classical cylindrical

approach. As a conclusion, the proposed near-field-far-field trans-

formation with helicoidal scan retains the accuracy of the classical

cylindrical approach, and allows one to remarkably reduce the time

required for the data acquisition.

Figure 4. A photo of the slotted antenna.

probe voltage relevant to the generatrices at rp =0° and rp =45°

are compared with those directly measured on the same gen-

eratrices, in Figures 5 and 6. As can be seen, the reconstructions

were everywhere very good, save for the peripheral zone (below

about -60dB), where the error was caused both by the truncation

of the scanning zone and the residual environmental reflections.

Note that due to the filtering properties of the interpolation func-

tions, the spatial harmonics relevant to the noise sources outside

the AUT's spatial bandwidth were cut away. This was reflected in

a smoother behavior of the reconstructed voltage with respect to

the measured voltage. The comparison between the phase of the

recovered voltage and the measured voltage on the generatrix at

rp =0° is also reported in Figure 7. In order to improve the

readability of the plot, the comparison is shown only in the range

[0 em, 100 em]. It is worth noting that all the reported reconstruc-

tions were obtained by using X'=1.35, X=1.20, and p =q=6 .

Finally, the contour-level plots in the z-sp plane for the measured

and interpolated near-field data are reported in Figures 8 and 9,

respectively. Note that these plots are shown only in the range of rp

from [-135°,135°], to improve their readability.

In order to assess the overall effectiveness of the proposed

near-field-far-field transformation technique, the far-field pattern

in the principal planes, H and E, reconstructed from the acquired

helicoidal near-field data, is compared in Figures 10 and 11 with

the pattern obtained by using the software package MI-3000 from

the knowledge of the data directly measured on the classical cylin-

IEEEAntennas andPropagation Magazine, Vol. 51, No.4, August2009

139

Page 7

Figure 7. The phase of the probe voltage on the generatrix at

cp=0°: the solid line is measured data; the crosses are interpo-

lated values.

180

~

135

'"'"'"

OIl

....

90

'" 2-

'"

~..c

0-

'"

.::l

'0>:;

0-

:;

0 -135

45

0

OIl

-45 f-

-90 f-

f-

-180

0

I

p= q=6

X = 1.20

X'= 1.35

20

I

40

I

60

I

80

100

z (em)

4. Conclusions

An experimental validation of an efficient probe-compen-

sated near-field-far-field transformation technique with fast heli-

coidal scan, which makes use ofthe rounded-cylinder modeling of

the AUT, has been provided in this paper. Such a modeling allows

one to efficiently fit elongated antennas by properly setting their

geometric parameters. It is also possible to consider measurement

cylinders having a diameter less than the antenna's maximum size,

thus remarkably reducing the error related to the truncation ofthe

scanning zone in the case of elongated antennas. This is a very

important feature of the described technique, which overcoming

the main and serious drawback ofthe previous approach based on

spherical AUT modeling, makes the helicoidal scanning more and

more appealing from a practical viewpoint. Moreover, the adopted

source modeling also allows a significant reduction in the required

number ofnear-field measurements.

5. References

10. O. M. Bucci, F. D'Agostino, C. Gennarelli, G. Riccio, and C.

Savarese, ''NF-FF Transformation with Plane-Polar Scanning:

5. O. M. Bucci and G. Franceschetti, "On the Spatial Bandwidth of

Scattered Fields," IEEE Transactions on Antennas and Propaga-

tion, AP-35, 12, December 1987, pp. 1445-1455.

1. A. D. Yaghjian, "An Overview of Near-Field Antenna Meas-

urements," IEEE Transactions on Antennas and Propagation, AP-

34,1, January 1986, pp. 30-45.

3. C. Gennarelli, G. Riccio, F. D'Agostino, and F. Ferrara, Near-

Field-Far-Field Transformation Techniques, 1, Salerno, Italy,

CUES,2004.

7. O. M. Bucci, C. Gennarelli, and C. Savarese, "Optimal Interpo-

lation of Radiated Fields over a Sphere," IEEE Transactions on

Antennas and Propagation, AP-39, 11, November 1991, pp. 1633-

1643.

2. "Special Issue on Near-Field Scanning Techniques," IEEE

Transactions on Antennas and Propagation, AP-36, 6, June 1988,

pp. 727-901.

9. O. M. Bucci, C. Gennarelli, G. Riccio, and C. Savarese, "Near-

Field - Far-Field Transformation from Nonredundant Plane-Polar

Data: Effective Modellings of the Source," lEE Proceedings

Microwaves, Antennas and Propagation, 145, February 1998, pp.

33-38.

4. C. Gennarelli, G. Riccio, F. D'Agostino, F. Ferrara, and R.

Guerriero, Near-Field-Far-Field Transformation Techniques, 2,

Salerno, Italy, CUES, 2006.

8. F. Ferrara, C. Gennarelli, R. Guerriero, G. Riccio, and C.

Savarese, "An Efficient Near-Field to Far-Field Transformation

Using the Planar Wide-Mesh Scanning," Journal ofElectromag-

netic Wavesand Applications, 21, 3, 2007, pp. 341-357.

6. O. M. Bucci, C. Gennarelli, and C. Savarese, "Representation of

Electromagnetic Fields over Arbitrary Surfaces by a Finite and

Nonredundant Number ofSamples," IEEE Transactions on Anten-

nas and Propagation, AP-46, 3, March 1998,pp. 351-359.

45

100

o

+

X'=1.35

X= 1.20

p=q= 6

7550

p =q =6

X =1.20

X' = 1.35

-30

l.....-'- -<-...L......J'--'--'-.L.......L---'--'--L......1.-<-...L......J'--'--'-.L.....J.--'--'--'--'--'-.J.......J

25

20

~IO

co

2-

'" ]0

~E

C':S -10

-0

<i

u:

-20

90135180

<p (degrees)

Figure 11. The E-plane pattern: the solid line is the reference;

the crosses are values reconstructed from near-field data

acquired via helicoidal scanning.

25

20

~co

2-

'"

.g

i5.

E

os

-0

<i

u:

15

-0

10

5

0

-5

-10

-180-135-90-45

125150

t} (degrees)

Figure 10. The H-plane pattern: the solid line is the reference;

the crosses are values reconstructed from near-field data

acquired via helicoidal scanning.

140

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