Advanced Analysis and Synthesis Methods for the Design of Next Generation Reflectarrays

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International Do toral S hool in Information

and Communi ation Te hnology

DISI- University of Trento

Advan ed Analysis and Synthesis Methods

for the Design of Next Generation

Refle tarrays

Angelo Gelmini

Tutor:

PaoloRo a, Asso iate Professor

University of Trento

Advisor:

Gia omoOliveri, Asso iate Professor

University of Trento

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s ientist

I want to thank also all my familiy, my un les, ousins and Daniel for the

interest and the moral support.

An important thank-you goes to Giulia that has tolerate and support me

beyond every di ulties and also to her parents that have a ept me in the

family.

Spe ial thankstomy friendsfor thebeautiful momentsandfor havinguseful

and amusing onversations.

To my olleguesand friends of the ELEDIAgroupand toallthe people that

support me.

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The design of ree tarray surfa e urrents that satisfy both radiation and

user-dened antenna feasibility onstraints is addressed through a novel paradigm

whi htakesadvantageofthenon-uniquenessofinverse sour e(IS)problems. To

thisend, thesynthesis isformulatedinthe IS frameworkanditsnon-measurable

solutions are employed as a design DoF. Thanks to the adopted framework,

a losed-form expression for the design of ree tarray surfa e urrents is

de-rived whi h does not require any iterative lo al/global optimization pro edure

and whi h inherently satises both the radiation and the feasibility design

on-straints. The features and potentialities of the proposed strategy are assessed

through sele ted numeri alexperimentsdealing with dierent ree tarray

aper-ture types/sizesand forbiddenregion denitions.

Keywords

Ree tarray Synthesis, Non-RadiatingCurrents, Inverse Sour e Problems,

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[C1℄ G.Oliveri,F.Apolloni,A.Gelmini,E.Bekele,S.Ma i,andA.Massa,

Numeri alhomogenization andsynthesis ofwave polarizersthrough

the material-by-design paradigm, 9th European Conferen e on

An-tennas and Propagation (EUCAP 2015) (ISBN 978-8-8907-0185-6),

Lisbon, Portugal, pp. 1-4, April12-17, 2015.

[C2℄ N.Anselmi,M.Donelli,A.Gelmini,G.Gottardi,G.Oliveri,L.Poli,

P. Ro a, L.Tenuti, and A. Massa, Design and optimization of

ad-van edradarand ommuni ationssystemsandar hite tures

ELE-DIAResear h Center, Atti XXI Riunione Nazionale di

Elettromag-netismo(XXI RiNEm), Parma, pp. 164-167,12-14 Settembre 2016.

[C3℄ G.Oliveri, M. Salu i,A. Gelmini,L. Poli, P. Ro a, and A. Massa,

SARarraysynthesisfornextgenerationEarthobservationsystems,

11th European Conferen e on Antennas and Propagation (EUCAP

2017)(no. 978-88-907018-7-0/17/$31.00

2017IEEE),Paris,Fran e, pp. 2312-2314, Mar h 19-24, 2017.

[C4℄ G.Oliveri, A.Gelmini,M. Salu i,D. Bres iani,and A.Massa,

Ex-ploitingnon-radiating urrents in ree tarray antenna design, 11th

European Conferen e on Antennas and Propagation (EUCAP 2017)

(no. 978-88-907018-7-0/17/$31.00

2017 IEEE), Paris, Fran e, pp. 88-91,Mar h 19-24, 2017.

[C5℄ A.Gelmini,G. Gottardi,and T. Moriyama, A ompressive

sensing-based omputational method for the inversion of wide-band ground

penetrating radar data, 7th International Workshop on New

Com-putational Methods for Inverse Problems (NCMIP 2017), Ca han,

(8)

e tarray antennasimpli ationthrough non-radiating urrents

syn-thesis, 2017IEEE AP-S InternationalSymposium andUSNC-URSI

RadioS ien eMeeting (no. 978-1-5386-3284-0/17/$31.00

2017IEEE), San Diego,California,USA, pp. 1185-1186, July 9-15, 2017.

[C7℄ A. Gelmini, M. Salu i, G. Oliveri, and A. Massa, Robust

diagno-sis of planar antenna arrays through a Bayesian ompressive

sens-ing approa h, 6th Asia-Pa i Conferen e on Antennas and

Prop-agation (APCAP 2017), Xi'an, 2017, pp. 1-3. doi:

10.1109/AP-CAP.2017.8420936.

[C8℄ M.Salu i,A.Gelmini,G.Oliveri,andA.Massa,Exploitationof

fre-quen ydiversity inGPRimagingthrough aninnovative

onstrained-BCSmethod, 12thEuropean Conferen e onAntennasand

Propaga-tion (EUCAP 2018) , London, UnitedKingdom April9-13, 2018.

[C9℄ A. Gelmini, M. Salu i, G. Oliveri, and A. Massa, Innovative

syn-thesisof ree tarrays within the non-radiatinginverse sour e

frame-work,12thEuropeanConferen eonAntennasandPropagation

(EU-CAP 2018), London, United Kingdom, pp. 1-4, April 9-13, 2018

(DOI:10.1049/ p.2018.0864).

[C10℄ M. Bertolli,M. Donelli, A. Massa, G. Oliveri, A. Polo, F. Robol, L.

Poli,A.Gelmini,G.Gottardi,M.A.Hannan,L.T.P.Bui,P.Ro a,

C. Sa hi, F. Viani, T. Moriyama, T. Takenaka, and M. Salu i,

Computationalmethodsforwirelessstru tural healthmonitoringof

ultural heritages, 8th International Conferen e on New

Computa-tionalMethodsforInverseProblems(NCMIP2018),Ca han,Fran e,

May 25,2018.

[C11℄ M.Salu i,G.Oliveri,A.Gelmini,andA.Massa,Over oming

feasi-bility onstraints in ree rattary design by exploiting non-radiating

urrents, 2018 IEEE AP-S International Symposium and

USNC-URSIRadioS ien eMeeting Boston,Massa hussets,USA,July8-13,

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ationsELEDIAResear hCenter, Atti XXIIRiunioneNazionale

diElettromagnetismo(XXII RiNEm),Cagliari,pp. 325-328,3-6

Set-tembre 2018.

[C13℄ N. Anselmi, R. Azaro, P. Bui, A. Gelmini, G. Gottardi,A. Hannan,

G.Oliveri, L. Poli, A. Polo, F. Robol, P. Ro a, M. Salu i,and A.

Massa, Antenna Synthesis and Optimization ELEDIA Resear h

Center, Atti XXIIRiunione Nazionale di Elettromagnetismo (XXII

RiNEm),Cagliari, pp. 333-336,3-6Settembre 2018.

[C14℄ M. Salu i, A. Gelmini, G. Oliveri, and A. Massa, From

inverse-sour e problems to ree tarray design - An innovative approa h for

dealingwithmanufa turingandgeometri al onstraints, 13th

Euro-pean Conferen e on Antennas and Propagation (EUCAP 2019).

[C15℄ G.Oliveri,M.Salu i,A.Gelmini,and A.Massa,

Computationally-e ient synthesis of advan ed ree tarrays through a

system-by-designtool,13thEuropeanConferen eonAntennasandPropagation

(EUCAP 2019).

[C16℄ A. Polo, M. Salu i, A. Gelmini, G. Gottardi, G. Oliveri, P. Ro a,

and A. Massa, Advan ed tea hing in EM - Towards an integration

of theoreti al skills and appli ative/industrial skills, 13th European

Conferen e on Antennas and Propagation (EUCAP 2019).

[C17℄ G. Oliveri, M. Salu i, A. Gelmini, and A. Massa, Frontiers in

re-e tarray design, 2019 IEEE AP-S International Symposium and

USNC-URSI Radio S ien e Meeting, Atlanta,Georgia,USA.

[C18℄ G.Oliveri, A.Gelmini, G.Gottardi,and M. Salu i,

Metamaterial-by-design-A Paradigmfortheindustrialsynthesisof EM

manipula-tion devi es, 2019 IEEE International Conferen e on Mi rowaves,

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[R1℄ A.Gelmini,G. Gottardi,and T. Moriyama, A ompressive

sensing-based omputational method for the inversion of wide-band ground

penetratingradardata, JournalofPhysi s: Conferen eSeries,,vol.

904, pp. 1-7, 2017 (extended version submitted torevision by an

in-ternational omiteeof[C5℄)(DOI:10.1088/1742-6596/904/1/012002).

[R2℄ M. Salu i,A. Gelmini, L. Poli,G. Oliveri, and A. Massa,

Progres-sive ompressive sensing for exploiting frequen y-diversity in GPR

imaging, Journal of Ele tromagneti Waves and Appli ations, vol.

32,no. 9,pp. 1164-1193,2018(DOI:10.1080/09205071.2018.1425160).

[R3℄ M. Salu i, A. Gelmini, G. Oliveri, and A. Massa, Planar arrays

diagnosisbymeansofanadvan edBayesian ompressivepro essing,

IEEE Transa tions on Antennas and Propagation, vol. 66, no. 11,

pp. 5892-5906, November2018 (DOI: 10.1109/TAP.2018.2866534).

[R4℄ M.Salu i,A. Gelmini,G.Oliveri, N. Anselmi,and A. Massa,

Syn-thesisof shaped beamree tarrays with onstrainedgeometryby

ex-ploiting non-radiatingsurfa e urrents, IEEE Transa tions on

An-tennas and Propagation, vol. 66, no. 11, pp. 5805-5817, November

2018(DOI: 10.1109/TAP.2018.2869036).

[R5℄ M. Bertolli,M. Donelli, A. Massa, G. Oliveri, A. Polo, F. Robol,L.

Poli,A.Gelmini,G.Gottardi,M.A.Hannan,L.T.P.Bui,P.Ro a,

C. Sa hi, F. Viani, T. Moriyama, T. Takenaka, and M. Salu i,

Computationalmethodsforwirelessstru tural healthmonitoringof

ulturalheritages, Journal of Physi s: Conferen e Series,vol. 1131,

pp. 1-7,2018 (extended version submittedtorevision by an

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Instantaneous brain stroke lassi ation and lo alization from real

s attering data, Mi rowaveand Opti alTe hnologyLetters, vol. 61,

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1 Introdu tion 1

2 Problem Formulation 11

2.1 Radiation fromsurfa e urrent . . . 11

2.2 Dis retization of the ree tarray surfa e . . . 15

2.3 Inverse Sour e problemdenition . . . 18

3 Non-Measurable Currents-based Solution Method 19 3.1 Field dis retization . . . 20

3.2 Trun ated SingularValue De omposition . . . 22

3.3 Synthesis approa h . . . 26

3.4 Non-measurable/Non-radiatingdenition . . . 27

3.5 Forbidden region onstraint denition . . . 28

3.6 Final losed-formformulation . . . 30

4 Method Assessment 33 4.1 Error metri s . . . 33

4.2 Square ree tarray:

55 × 55

elements . . . 34

4.2.1 Step-by-step pro edure with lower dimensionality ase . . 35

4.2.2 Analysisvs. variousforbiddenregionshapeskeepingsame order dimension . . . 44

4.2.3 Changingthedimensionofthesametypeofforbiddenregion 48 4.2.4 Large dimension and omplex topology forbiddenregion . 52 4.3 Re tangular ree tarray:

81 × 69

elements . . . 56

4.3.1 Large dimension and omplex topology of the forbidden region . . . 59

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test ase . . . 65

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4.1 Square Aperture (

M × N = 55 × 55

,

∆x = ∆y = 3.73 × 10

−1

_λ

)

-Performan e Assessment - Varying the geometry. . . 45

M × N = 55 × 55

,

∆x = ∆y = 3.73 × 10

−1

_λ

)

-Performan e Assessment - Fixed geometryvarying the dimension. 51

M × N = 55 × 55

,

∆x = ∆y = 3.73 × 10

−1

_λ

)

-Performan e Assessment -Complex and large geometries. . . 55

4.4 Re tangularAperture (

M ×N = 81×69

,

∆x = ∆y = 3.07×10

−1

_λ

)

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1.1 Geometry omparisonof(a)PhasedArrayAntenna,(b)Ree tor

Antenna, ( )Ree tarray Antenna. . . 2

1.2 Ree tarray Antenna implemented by (a)[56℄ and by (b)Pozar

(http://www.e s.umass.edu/e e/pozar/ree t.jpg). . . 4

1.3 Geometry of the ree tarray antenna elements: (a)[15℄, (b)[64℄,

( )[65℄,(d)[66℄. . . 7

2.1 Geometry of the ree tarray antenna. . . 12

2.2 Co-polar and ross-polar unit ve tor following the Ludwig third

denition. . . 17

3.1 Example of singular values distribution

ψ

w

,

w = 1, ..., W

, taking intoa ount a trun ation order

H

and a trun ation threshold

τ

. 25 3.2 Geometry of the ree tarray antenna. . . 28

M × N = 55 × 55

,

∆x = ∆y = 3.73 × 10

−1

_λ

)

-Ree tarray geometry. . . 34

M × N = 55 × 55

,

∆x = ∆y = 3.73 × 10

−1

_λ

,

K = 11

) - Example of forbidden region

Φ

, E-Shape forbidden region with

K = 11

numberof elements. . . 35 4.3 Square Aperture (

M × N = 55 × 55

,

∆x = ∆y = 3.73 × 10

−1

_λ

)

- Plotof the referen e urrent (a) magnitude

J

ref

x

(x, y)

and (b) phase

∠J

ref

x

(x, y)

andradiatedeld( )magnitude

F

CO

ref

(u, v)

and (d) phase

∠F

ref

CO

(u, v)

. . . 36 4.4 Square Aperture (

M × N = 55 × 55

,

∆x = ∆y = 3.73 × 10

−1

_λ

)

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M × N = 55 × 55

,

∆x = ∆y = 3.73 × 10

−1

_λ

)

-Normalizederror

ξ

varying the SVD threshold

τ

M × N = 55 × 55

,

∆x = ∆y = 3.73 × 10

−1

_λ

,

K = 11

) - Plots of (a)( )(e)(g) the magnitude and (b)(d)(f)(h) thephaseof(a)(b)

J

ref

x

(r)

andsynthesized( )(d)

J

MN

x

(r)

,(e)(f)

F

_CO

ref

,and (g)(h)

F

MN

CO

M × N = 55 × 55

,

∆x = ∆y = 3.73 × 10

−1

_λ

,

K = 11

) - Plots of (a)( ) the magnitude and (b)(d) the phase of (a)(b)

J

N R

x

(r)

and synthesized ( )(d)

F

N R

CO

M × N = 55 × 55

,

∆x = ∆y = 3.73 × 10

−1

_λ

,

K = 11

) - Plots of (a)( ) the magnitude and (b)(d) the phase of the synthesized (a)(b)

J

x

(r)

and ( )(d)

F

CO

(r)

M × N = 55 × 55

,

∆x = ∆y = 3.73 × 10

−1

_λ

)

-Denitionof forbidden regions

Φ

M × N = 55 × 55

,

∆x = ∆y = 3.73 × 10

−1

_λ

)

-Plotsof

|J

x

(r)|

assuming(a)Cross-shaped(

K = 28

),(b) Ring-shaped (

K = 32

), ( ) Cir ular Ring-shaped (

K = 36

) and (d) Cir le-shaped (

K = 37

) forbiddenregions. . . 46 4.11 Square Aperture (

M × N = 55 × 55

,

∆x = ∆y = 3.73 × 10

−1

_λ

)

-Plotsof

∆F

CO

(u, v)

whenassuming(a)Cross-shaped(

K = 28

), (b)Ring-shaped(

K = 32

),( )Cir ularRing-shaped(

K = 36

) and (d) Cir le-shaped (

K = 37

)forbidden regions. . . 47 4.12 Square Aperture (

M × N = 55 × 55

,

∆x = ∆y = 3.73 × 10

−1

_λ

)

- Denition of forbidden regions

Φ

keeping the same shape but varying the dimension: (a)

K = 4

, (b)

K = 25

, ( )

K = 49

and (d)

K = 100

M × N = 55 × 55

,

∆x = ∆y = 3.73 × 10

−1

_λ

)

-Plots of

|J

x

(r)|

assuming dierent dimension of a Square-shape (a)

K = 4

, (b)

K = 25

, ( )

K = 49

and (d)

K = 100

forbidden regions. . . 49

M × N = 55 × 55

,

∆x = ∆y = 3.73 × 10

−1

_λ

)

- Plots of

∆F

CO

(u, v)

when assuming dierent dimension of a Square-shape (a)

K = 4

, (b)

K = 25

, ( )

K = 49

and (d)

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M × N = 55 × 55

,

∆x = ∆y = 3.73 × 10

−1

_λ

,

Square-shaped forbiddenregion)-Behaviourof

ξ

and

∆t

versus

K

M × N = 55 × 55

,

∆x = ∆y = 3.73 × 10

−1

_λ

)

Φ

with omplex shape and large dimension: (a) Triangle-shaped

K = 55

nearer to the orner, (b)Triangle-shaped

K = 55

,( )ELEDIA-shaped

K = 54

and (d) Diamond-shaped

K = 115

M × N = 55 × 55

,

∆x = ∆y = 3.73 × 10

−1

_λ

)

-Plots of

|J

x

(r)|

assuming dierent dimension of a Square-shape (a)

K = 4

, (b)

K = 25

, ( )

K = 49

and (d)

K = 100

forbidden regions. . . 53

M × N = 55 × 55

,

∆x = ∆y = 3.73 × 10

−1

_λ

)

- Plots of

∆F

CO

(u, v)

when assuming dierent dimension of a Square-shape (a)

K = 4

, (b)

K = 25

, ( )

K = 49

and (d)

K = 100

forbiddenregions.. . . 54 4.19 Re tangularAperture (

M ×N = 81×69

,

∆x = ∆y = 3.07×10

−1

_λ

)

- Ree tarray geometry. . . 56

4.20 Re tangularAperture (

M ×N = 81×69

,

∆x = ∆y = 3.07×10

−1

_λ

)

- Plotof the referen e urrent (a) magnitude

J

ref

x

(x, y)

and (b) phase

∠J

ref

x

(x, y)

and radiated eld ( ) magnitude

F

CO

ref

(u, v)

and (d) phase

∠F

ref

CO

(u, v)

and the minimum-norm solution (e) magnitude

J

MN

x

(x, y)

and (f) phase

∠J

MN

x

(x, y)

and radiated eld (g) magnitude

F

MN

CO

(u, v)

and (h) phase

∠F

MN

CO

(u, v)

. . . . 58 4.21 Square Aperture (

M × N = 55 × 55

,

∆x = ∆y = 3.73 × 10

−1

_λ

)

Φ

with omplex shape and large dimension: (a) ELEDIA-shaped

K = 54

and (b) Diamond-shaped

K = 115

. . . 59 4.22 Re tangularAperture (

M ×N = 81×69

,

∆x = ∆y = 3.07×10

−1

_λ

)

-Plots of(a)( )the magnitudeandand (b)(d)thephase of

J

x

(r)

when assuming (a)(b) ELEDIA-shaped (

K = 54

) and ( )(d) Diamond-shaped (

K = 115

) forbiddenregions. . . 60 4.23 Re tangularAperture (

M ×N = 81×69

,

∆x = ∆y = 3.07×10

−1

_λ

)

-Plotsof

∆F

CO

(u, v)

whenassuming(a)ELEDIA-shaped(

K =

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M × N = 55 × 55

,

∆x = ∆y = 3.73 × 10

−1

_λ

)

-Denition of forbiddenregions

Φ

with dierent shapes: (a) E-shaped

K = 11

, (b) Cross-shaped

K = 28

, ( ) Ring-shaped

K = 32

, (d) Cir ular Ring-shaped

K = 36

and (e) Cir le-shaped

K = 37

M × N = 55 × 55

,

∆x = ∆y = 3.73 × 10

−1

_λ

)

-Plots of

|J

x

(r)|

assuming an(a) E-shaped

K = 11

, (b) Cross-shaped

K = 28

, ( ) Ring-shaped

K = 32

K = 36

K = 37

forbidden. . . 63 4.26 Square Aperture (

M × N = 55 × 55

,

∆x = ∆y = 3.73 × 10

−1

_λ

)

-Plotsof

∆F

CO

(u, v)

whenassumingan(a)E-shaped

K = 11

,(b) Cross-shaped

K = 28

, ( ) Ring-shaped

K = 32

K = 36

K = 37

forbidden region. . . 64

4.27 Re tangularAperture(

M ×N = 81×69

,

∆x = ∆y = 3.07×10

−1

_λ

)

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Introdu tion

Antennas abletoexhibithigh gainsand arefullyshaped patternsare of

fun-damental importan e in radar, satellite remote sensing and

long-distan e/high- apa ity ommuni ationsystems [1℄-[6℄.

In order to meet su h ambitiousperforman e requirements,the te hnologies

traditionallyemployed areree tor antennas[6℄andphasedarrays[7℄,[8 ℄. Infa t

both te hnologies an a hieve a very high gain and are able togenerate shaped

patterns. In ree tor antenna the high gain apability is a hieved thanks to the

abilityto on entratetheeldthatis omingfromthefeedertoasingledire tion

inspa e(duetothe paraboli ree tor),whilethe shaped beam an beobtained

by shaped prole ree tors (adding some bumps into the paraboloid ree tor

surfa e) [9℄.

In [9℄itisproposed touse anoptimizationpro edurethattakesininput the

typeof the ree tor surfa e ( ir ular, ellipsoid, square, re tangular, oni et ..),

the feed onguration (horn antenna, array antennas, et ..) and position (at

the enter or shifted),the ree tor onguration(single ree tor, dual ree tor,

Cassegrain,et ..) andthe radiation hara teristi s. The optimizer omputes the

radiatedpattern applying the theory of generalized dira tion, i.e. the physi al

theory of dira tionis used toanalyse the antennaand produ ea ost fun tion

to quantify the mat hing of the radiation hara teristi s.If the ost fun tion is

not minimized, the optimizer reates anew trialsolution.

In phased arrays, high gain and a properly shaped beam are given by the

position of the elements (regular latti e, sparse latti e, random latti e, et ..),

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Figure 1.1: Geometry omparison of (a) Phased Array Antenna, (b) Ree tor

Antenna, ( ) Ree tarray Antenna.

tapering,thinning,phasesynthesis,timemodulatedarray, lusteringte hniques,

et ..)[7 ℄[8℄.

Though able toa hievethe desiredrequirements,bothsolutionshave

signi- ant drawba ks. Ree tor antennas exhibit high manufa turing omplexity, are

di ulttobeimplemented as re ongurable antennas(unless me hani al

steer-ing is onsidered, whi h is typi allyavoided in spa e appli ations) and are also

hara terizedby non- onformalshapes[6℄. Moreover, spa eappli ation ree tor

antennas suer manufa ture toleran e and deformation problems [10℄ that an

severely ae t the antenna operational. Phased antennaarrays are expensive in

terms of fabri ation and power onsumption (and, onsequently, need

tempera-ture ontrol, not suitablefor spa eappli ations). In addition,su hantennas are

also heavy due to support and feeding network, and their design is not trivial

[7℄[8℄.

In order to deal with the aforementioned issues, ree tarray antennas have

emerged as a possible solution to yield high-gain shaped beam antennas with

lowrealization osts, at/ onformalshapes, andlow- onsumptionfeednetworks

[6℄,[11℄. Thanks to their potentials and exibility [6℄,[12℄-[36℄, the design of

shaped-beam ree tarray antennas has be ome a very a tive resear h eld and

several methodologieshave been proposed tothis end [37℄-[52℄.

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apa-bilityof ombiningthe positivefeatures of both lassi alree tor antennas(i.e.,

highgain, low ost andeasyfabri ation)[6℄and phasedarrays (i.e.,

re ongura-bility and lowprole)[7℄. Typi ally, they onsist of a planar array of mi rostrip

pat hes printed on a ground-ba ked diele tri substrate and illuminated by a

feeder (e.g., a horn antenna, or also a phased array). Size, shape and

arrange-mentofthemetalli pat hesareproperlydesignedsu hthattheeldree tedby

thepassive/a tivesurfa emeetsthedesiredpatternfeatures(e.g.,steeringangle,

sidelobe level, bandwidth,et ..)[6 ℄. As a main onsequen e, ree tarrays do not

requirethe use ofabulkyparaboli dis , whilethe tuningofthe radiatedeldis

obtained without the need for expensive beam-formingnetworks or me hani al

steering[14℄.

The rst exampleof aree tarray antennawasproposed atthe beginningof

60's by Berry [11℄, who proposes to use trun ated waveguides as ree ting

ele-ments. Thesewaveguideshavedierentlengthsthat areable toimposeaproper

phase shift to obtain a desired ree ted pattern. The ree tarrays produ ed

with this te hnology an a hieve good performan es and an handle very high

power (no diele tri substrate) at the ost of using a heavy stru ture. For this

reason, only with the development of the mi rostrip te hnology in the late 80's

the ree tarray arose asa leadingte hnology.

Before mi rostrip te hnology, another kind of ree ting stru ture was

ana-lyzed: the Spiralphase ree tarray [53℄. In this work four arms of spirals are

onne ted with swit hing diodes that a tivate a dierent pair of arms and thus

permit to ontrol the s an angle of the ir ular polarized ree tarray. However,

due to the diodes ir uitand the spiral avity (

λ/4

), the stru ture be omes too bulky topermit ane ient implementation.

The simplestdesign of a ree tarray is proposed in [54℄ and [55℄ and

imple-mented by [56℄,[57℄ [Fig. 1.2(a)℄, and onsists in mi rostrip pat hes with xed

shapeanddierentadaptingstubs. Sin ethesestubshavedierentlengths,they

anprovideadierentimpedan e, and onsequently adierentphase shift. The

major problem with the stub te hnology is that this method is inherently

nar-rowband,sin ethe stubstru turemustbedimensionedforaspe i wavelength,

and parasiti oupling withadja ent elements ouldbea possible issue.

Pozar et al. in [58℄,[59℄ and Chaharmir et al. [60℄ propose to introdu e, in

the same planar stru ture, pat hes with dierent dimension, rotation, or even

(24)

(a) (b)

Figure 1.2: Ree tarray Antenna implemented by (a)[56℄ and by (b)Pozar

(http://www.e s.umass.edu/e e/pozar/ree t.jpg).

Thiskindofdesignsolvestheproblemsofthestubs,improvesthebandwidthand

allowsthe designertohaveabetter ontrolonpolarization. However, atrade-o

must be taken into a ount when designing ree tarray antennas that radiated

shaped beams. Ifaparti ularshapedbeamisdesiredinorderto overonlysome

regionsof theEarth (e.g. abeam that an over northernEuropewithout

send-ing power on sea areas) the phase distribution on the ree tarray aperture has

a non-smooth behavior. This meansthat adja ent elements ouldhave a

signif-i antly dierent phase shift and this implies very dierent and omplex shapes.

As a onsequen e, manufa turing osts are high (also due to manufa ture

toler-an es),and there may alsobe problems involvingin orre t shape denition and

oupling. This kind of design is improved by En inar et al. in [15℄,[16℄,[37℄,[61℄

that propose to design shaped beams by using more layers (2 or 3) of dierent

shaped pat hes and exploiting an optimizationte hnique in order to dene the

bestphasedistributionondierentlayers. Thiskindofdesign,basedonmultiple

layers, animprovetheperforman eandde reasethe omplexityofea hsingular

layer, although the overall stru ture is still omplex, expensive to manufa ture,

and it ould behard to inserta ontrolnetwork for beam-steeringappli ations.

In general, ree tarray layouts are usually synthesized by a two-step

pro e-dure inwhi h:

(25)

omputed;

(b) the feed and ree ting elements (e.g., mi rostrip pat hes) able to

approxi-mately generate su h urrentsare dedu ed/designed.

Several methods have been developed in the literature to solve (b) for various

unit ell geometries and ar hite tures [2℄,[6 ℄,[15℄,[59℄,[62℄-[66℄. On the ontrary,

veryfew approa hes have been proposed toaddress (a) [37℄-[39℄.

Oneexampleofsolutioninliteraturethatdealswithstep (b)is[59℄,inwhi h

theauthorsdes ribeamethodof omputingthephaseresponse hara teristi sfor

asquarepat hmi rostripand thensynthesizingthe pat h distributiontoobtain

apen ilbeamin dierent ree tarray ongurations: squared ree tarray oset

beam having the feeder in broadside, ir ular diameter ree tarray with both

feeder and far-eld maximum inbroadside, square ree tarray withprime fo us

re tangularhorn and square ree tarray in Cassegrian onguration.

Instead [15℄ [Fig. 1.3(a)℄, in order to enlarge the operational bandwidth of

the system, a multi-layer stru ture is employed. In parti ular the number of

layer isset to 2 and a simple square pat h is sele ted to have the desired phase

shift. More in detail, the size of the side of the square pat h an vary the phase

response of the spe i ell and by xing the ratio between the ell in the two

layers(theupperlayerpat hesare

0.7

timesthelowerlayerones)theree tarray issynthesizedusingthesimplephasedelay ompensation(thephaseofthepat h

has to ompensatethe same travelling time that shouldbe o urred in ase ofa

ree tor) and good performan e are obtained within

16.67%

of the operational bandwidth.

When ree tarray pat hes are designed, a problem that an o ur is that

their phase response doesnot over the full

360 ◦

phase range. Toover ome this

problem[64℄ [Fig. 1.3(b)℄propose to use a kind of stru ture that is y li . This

kind of element omes ba k to the original geometry shape when a full phase

rangeis overed. Infa tthe proposed elementisaphoenixelement (i.e. alled

phoenix for its rebirthing apabilities) that is omposed by a entered square

pat h of xed dimension, an external ring of xed dimension that delimits the

element with the adja ent ones and a variable ring that an move from the

inner tothe outer. Furthermore,this elementis designed to be metal-only,thus

(26)

thee ien yoftheree tarraysin eit an overthefullphaserangeanditdoes

not require any diele tri substrate. Nonetheless, the onne ting strips exhibits

some drawba ks in ontrolling the ross-polarization.

Tobetter ontrolthe ross polarization,butmaintainingthe fullrange over,

it is proposed by [65℄ [Fig. 1.3( )℄ to use two dierent y les to dene the

element. Firstly the element is made by dipole rossed with same arms (to

ensuredual-polarization),whosewidth ishalfof their length. To implementthe

y le, the length is in reased until the element tou hes the adja ent ones, then

the elementgeometry hangesand be omes agrid. The se ondstep of the y le

is done keeping xed the length and vary only the width of the arms unless the

metalizationdisappear, then the y le restart asa rossed dipole. The designed

ree tarray using this elements an handle both polarizations and demonstrate

tohave anoperationalbandwidth of

11.1%

.

Ree tarrayare used alsofornon-mi rowave appli ationmovingtothe

tera-hertzdomain[66℄[Fig. 1.3(d)℄. Inthisdomainstubsormany ells withdierent

shapes an not be manufa tured (or are too expensive). Thus, it is proposed

touse metalblo ks with dierent height inorder to ompensate the phase with

respe t toareferen e plane. Inthis way itisthesameasifthe physi albehavior

of a ree tor is obtained by sampling and then applying a modulus operation

with respe t to the wavelength atthe heights of the blo ks. It is demonstrated

thatwith thisapproa hagoodgain an bea hieved and thepatternbehavioris

quitestabletothefrequen yband(

30%

)obtainingalsoaverygoodperforman e inantenna e ien y due tothe absen e of diele tri s.

Consideringstep(a),theexploitationoflo aloptimizationstrategies(su has

theInterse tionApproa h [37℄,[39℄)has been proposed asarst stepofashaped

beam ree tarray synthesis [37℄,[39℄. However, su h methodologies an be

om-putationallyexpensive(espe iallyif wideapertures are athand)and their

ee -tiveness and onvergen e rate stronglydepend onthe hoi e of theinitialization

point [37℄. Alternatively, ray-tra ing te hniques have been proposed to dedu e

theree tarray surfa e urrentsstartingfromtheknowledgeofapreviously

syn-thesizedshaped ree toraordingthedesiredbeam pattern[38℄. Unfortunately,

su ha strategy doesnot allowthe designer to spe ify any feasibility onstraints

onthe solution (e.g., presen e of forbiddenregions inthe array aperture) and

(27)

"Square" Patch Dual Layer

g

(1)

g

(2)

"Square" Patch Dual Layer

g

(1)

g

(2)

(a) (b)

( ) (d)

Figure 1.3: Geometry of the ree tarray antenna elements: (a)[15℄, (b)[64℄,

( )[65℄, (d)[66℄.

In the Interse tion Approa h [67℄ two sets are onsidered: the rst set is

omposedbyalltheradiationpatternthatrespe tstherequiredspe i ationand

the se ond ontains all the radiation pattern that the ree tarray an radiate.

Roughly,thesynthesispro eduremakes ontinueproje tionofthepatternsinthe

twosetsfromtherstsettothese ond,untilthemismat hbetweentheproje ted

patterns is almost null. Thus, as the dimension of the ree tarray in reases the

dimension of the sets in reases as well, and this is one of the drawba ks of

te hniques explained in [37℄,[39℄. While [37℄ has a ree tarray made with three

layerof squared pat hes and an a hieve verygoodperforman e in overing the

SouthAmeri aregionwithabandwidthof

10%

,and[39℄(thathas alsoused the FFT toin reasee ien y oftheapproa h) ana hievegoodperforman ewitha

ree tarray made ofbla k boxes(itdoes not takeintoa ountthe real element,

onlyitsree tion oe ient)synthesizinganisoux patternand ashaped-beam

for the Europe overing with aDire t Broad astSatellite (DBS).

Froma dierentperspe tive,itisknown thatthe relationbetween the

ree -tarray urrentsand theirradiatedpatterns an beee tivelymodeledexploiting

(28)

and the unknowns are the surfa e urrents.

In [69℄ and [70℄ the problem addressed is toretrieve the urrent distribution

that radiates a measured eld. In the rst one it is minimized the distan e of

re onstru ted Equivalent Magneti Current (EMC) by the near-eld measured

ina ylindri alway usingtheMarquardtalgorithm,and inthe se onditsalmost

the same but taking into a ount a near-eld measured on a spheri al surfa e

(hen ethree omponents of the eld, instead of onlytwo).

In [71℄ the problem is to re onstru t equivalent urrents distribution using

integral equationalgorithm. Usingthe integralequation the authorsare ableto

re onstru tthe urrentoveruser-dened surfa es, notonly ylindri alor

spheri- alsurfa e (thatare easierto omputeusingthe tangentialeldsandthe

Equiv-alen e Prin iple)but also, for example,on the surfa e of a horn antenna.

In[72℄ metalli bodies are re onstru tedasequivalent urrents. Inparti ular

the Sour e Re onstru tion method is applied to the retrievalof metal obje t in

an investigation domain and use a minimization (using a Conjugate Gradient

method)ofa ostfun tionthat,takingintoa ounttheTikhonov regularization

and the normalization of the equations terms, of the

L

2

-norm of the measured and re onstru ted eld (by the radiationof the equivalent urrent).

Intheframeworkofinverses atteringandantennadiagnosis/ hara terization

[69℄-[72℄,su haproblemisknowntobeill-posedbe auseofthenon-uniquenessof

theradiationoperator[73℄,whi hisrelatedtotheexisten eofnon-me

asurable/non-radiating urrents [74℄-[76℄. While this feature an be an issue in traditional

inverse problems requiring suitable ountermeasures [74℄-[76℄, it a tually

repre-sents a degree-of-freedom (DoF) in the framework of onstrained ree tarray

design. In fa t,bysuperimposingasuitably designednon-measurable urrentto

anavailable(e.g., minimum-norm[74℄-[77℄)solutionoftheIS problem,a urrent

ould be synthesized whi h radiates the desired far-eld pattern, and omplies

with the user-dened onstraints.

A ordingtosu h onsiderations,aninnovativeparadigmtosynthesize

ree -tarray surfa e urrents[i.e.,toaddressstep (a)℄isproposedwhi h,byleveraging

on the non-uniqueness of the IS problem as a design DoF, enables to dedu e

solutionsalsosatisfyinguser-dened antenna feasibility onstraints (e.g.,on the

presen e and shape of forbiddenregions in the aperture). To this end, the

(29)

the IS problemisrstly derived, and then(ii)asuitablenon-measurable sour e

is omputedsothattheresultingsurfa e urrent[i.e.,thesuperpositionofthe

so-lutions(i) and (ii)℄ omplieswith the user-dened requirements. Thanks tothe

features of the proposed formulation, a losed-formexpression is nally derived

for both the minimum-norm and the non-measurable urrents whi h does not

require any iterative lo al/global optimization pro edure and whi h inherently

satisesboth the radiationand the feasibilitydesign onstraints.

Inparti ular,itisproposedtoapplytheSingularValueDe omposition(SVD)

to a dened Green's operator. The out ome of this pro ess are two set of

or-thonormal bases and a matrix of singular values. This output has to be

ana-lyzed inorder to nd agoodtrade-o between, onone side, the pre isionof the

minimum-norm urrent able to radiate the desired eld; on the other side, the

possibility to have the greatest number of non-measurable bases. This analysis

it is done by dening a variable threshold on the value of the singular values

and olle tingdierent ombinationof orthonormalbases that are linked to the

singularvaluesabove or belowthe threshold.

The innovative methodologi al ontributions of the paper therefore in lude

theintrodu tion,forthersttimetothebestoftheauthorknowledge,ofa

ree -tarray surfa e urrentsynthesisparadigmwhi hleveragesonthenon-uniqueness

of the IS problemand the existen e of non-measurable urrents to improve the

features of the obtained solution (e.g., in terms of feasibility), and the

intro-du tion of expli it losed-form expressions for the omputation of ree tarray

surfa e urrentsaordingadesiredfar-eldpatternand omplyingwith

(30)

The thesis isorganized asfollows. After the formulationof the shaped-beam

onstrained ree tarray urrents synthesis problem (Chapter 2), the proposed

designmethodisillustratedanditsnal losed-formsolutionisderived (Chapter

3). A set of numeri al examplesbased on realisti ree tarray ar hite tures are

then illustrated to assess the ee tiveness and potentialities of the onsidered

design paradigm (Chapter 4). At the end are presented the on lusion and

(31)

Problem Formulation

In this hapter the problem formulation is explained throgh the understanding

of the radiation problem in a ree tarray antenna and how to formulate it as

an Inverse Sour e problem. In parti ular, rstly it is des ribed the radiation

problemformulationanditsdis retizationontheree tarraysurfa e(thatisnot

ontinuous) and then the problem to nd the urrent that generate a spe ied

radiatedeld is formulated asanInverse Sour e problem.

2.1 Radiation from surfa e urrent

We onsider a ree tarray antenna, oriented like in Fig. 2.1, with both ground

plane and pat hes made by a Perfe t Ele tri Condu tor separated by a layer

of substrate withstandard omplex permittivityvalue

ε = ε

0 ε

r

(1 − j tan δ)

and illuminatedby a feeder positionedin

r

f

_{= (x}

f

, y

f

, z

f

)

that infar-eld generates a plane-wave that has a relative angular position

(θ

inc

, φ

inc

)

(see Fig. 2.1). The in ident ve tor for ea h ell of the ree tarray is

ν

inc

(r) =− (sin θ

inc

cos φ

inc

,

sin θ

inc

sin φ

inc

, cos θ

inc

)

.

The in ident plane wave ona ree tarray element an bemodelas:

"

E

θ

inc

E

_inc

φ

#

=

"

E

θ

0 E

₀

φ

#

e

−jk

(

ν

inc

(r)·r

)

(2.1) where

E

0

isthe ve torthat des ribes amplitudeand polarizationofthe in ident plane-wave,

r

is the position of the ree tarray element,

k = 2πf √µε

,

µ

,

ε

are

(32)

x

z

y

U

T

I

y

'

x

'

:

5HIOHFWDUUD\

6XUIDFH

)HHG

ܚ

ଵଵ

ܚ

௠௡

_ܚ

ெே

Figure2.1: Geometry of the ree tarray antenna.

the free-spa e wave number, permeability, and permittivity, respe tively, and

f

isthe frequen y.

The presen e of the groundeddiele tri slab and ofthe layerprinted pat hes

generates dierent kind of eld that are ba k-radiated. The total eld that is

present in the region of the spa e in front of the ree tarray antennas an be

des ribed asthe sum of these ontributes:

E

_tot

= E

_inc

+ E

_RGDS

+ E

_{RP P}

(2.2)

The term

E

RGDS

indi atesthe ree tedeld by the innitegrounded diele -tri slab without any kindof pat hprinted on, and an be deniteas:

E

RGDS

= RE

0 e

jk(x sin θ

inc

cos φ

inc

+y sin θ

inc

sin φ

inc

−z cos θ

inc

)

(2.3)

where matrix

R

is the diagonal ree tion matrix, and its non-null entries

R

θθ

and

R

φφ

are dened asin [79℄.

The other term

E

RP P

represents the ree ted eld when the mi rostrip pat hes are present. On this pat hes, made of PEC, the

E

inc

indu es a

(33)

sur-fa e urrentthat radiates aeld dened as:

E

_{RP P}

= SE

₀

e

jk(x sin θ

inc

cos φ

inc

+y sin θ

inc

sin φ

inc

−z cos θ

inc

)

(2.4)

where

S

isthe s attering matrix and its oe ients hara terize the ree tion:

S

=

"

S

θθ

S

θφ

S

φθ

S

θθ

#

(2.5)

Ea h s attering oe ient is dened as the ratio between the s attered and

in ident eld of the mi rostrip surfa e for ea hpolarization:

S

ji

=

E

_{RP P}

j

(z = 0)

E

i

inc

(z = 0)

j, i = {θ, φ}

(2.6)

These oe ients an be omputed for ea h mi rostrip pat h and then used

to obtain the surfa e urrent on the ree tarray aperture. Sin e we want to

dened the urrenton the mi rostrip surfa e

J

s

:

J

s

= ˆ

n × H

(2.7)

(where

n

ˆ

isthe normaltothe surfa e)and wehavedened allthe termsin(2.2) we an express alsothe total magneti eld as:

H =

1 η

ν

inc

× E

inc

+

1 η

ν

ref l

× E

RGDS

+

1 η

ν

ref l

× E

RP P

(2.8) where

ν

ref l

is the spe ular ree tion dire tion (Snell's law on a PEC) of the

in iden e dire tion

ν

inc

, and

η

is the free-spa e impedan e.

In ase of far-eld, following [80℄, the radiatedeld by an ele tri urrent

J

s

an be approximates as:

E

RAD

(r) ≈ −jη

exp (−jkr)

2λr

(N

θ

(θ, φ) + N

φ

(θ, φ))

(2.9) where the radiationve torin arthesian oordinates

N

an be expressed as

N (θ, φ) =

R R

_Ω

J

x

(r) ˆ

x + J

y

(r) ˆ

y

×

exp j

2π

_λ

(x sin θ cos φ + y sin θ sin φ)

dx dy

(34)

and

λ

is the wavelength.

In order to be easier omputed the radiation ve tors an be expressed in

spheri al oordinates form:

(

N

θ

(θ, φ) = N

x

(θ, φ) cos θ cos φ + N

y

(θ, φ) cos θ sin φ

N

φ

(θ, φ) = −N

x

(θ, φ) sin φ + N

y

(θ, φ) cos φ

(2.11)

In this way, bu substituting(2.11) in (2.9) we an obtain:

(

E

RAD,θ

(r) =

−jη

exp(−jkr)

2λr

(N

x

(θ, φ) cos θ cos φ + N

y

(θ, φ) cos θ sin φ)

E

RAD,φ

(r) =

−jη

exp(−jkr)

2λr

(−N

x

(θ, φ) sin φ + N

y

(θ, φ) cos φ)

(2.12)

The equations(2.12)and (2.10) ompletelydes ribethe far-eldradiationof

an indu ed urrent from a feeder on the ree tarray surfa e. In order to make

possible the utilizationof these equation and the problemdenition we need to

(35)

2.2 Dis retization of the ree tarray surfa e

Lets assume to dis retize the ree tarray surfa e in a regular latti e, and

ea h ell of the latti e in ludes only one ree tarray element. Thus, the

ree -tarray onsists of agrid of

M × N

elements withunit ells of size

∆x × ∆y

and in order to dis retize (2.10) we will apply a pixel-basis fun tion

P

entered at

r

mn

_,

_{m −}

M

2 ∆x, n −

N

2 ∆y, 0

,

m = 1, ..., M

,

n = 1, ..., N

tothe urrent distributionand we obtain:

J

q

(r)

,

M

X

m=1

N

X

n=1

J

mn

q

P

mn

_(r)

q ∈ {x, y}

(2.13)

In parti ular, the x- omponent of the urrent an beexpressed as:

J

x

|

mn

= −

ν

inc

x

(r

mn

) sin(θ)

η

(1 + S

θθ

(r

mn

)) E

inc

θ

(r

mn

) +

+S

θφ

(r

mn

) E

inc

φ

(r

mn

)

i

+

−

ν

inc

z

(r

mn

) cos(θ) cos(φ)

η

(1 − S

θθ

(r

mn

)) E

inc

θ

(r

mn

) +

−S

θφ

(r

mn

) E

inc

φ

(r

mn

)

i

+

ν

inc

z

(r

mn

) sin(φ)

η

(1 − S

φθ

(r

mn

)) E

inc

θ

(r

mn

) +

−S

φφ

(r

mn

) E

inc

φ

(r

mn

)

i

(2.14)

and the y- omponent as:

J

y

|

_mn

= −

ν

inc

y

(r

mn

) sin(θ)

η

(1 + S

θθ

(r

mn

)) E

inc

θ

(r

mn

) +

+S

θφ

(r

mn

) E

inc

φ

(r

mn

)

i

+

ν

z

inc

(r

mn

) cos(θ) sin(φ)

η

(1 − S

θθ

(r

mn

)) E

inc

θ

(r

mn

) +

−S

θφ

(r

mn

) E

φ

inc

(r

mn

₎

i

₊

+

ν

z

inc

(r

mn

) cos(φ)

η

(1 − S

φθ

(r

mn

)) E

inc

θ

(r

mn

) +

−S

φφ

(r

mn

) E

inc

φ

(r

mn

)

i

(2.15)

where the ree tion matrix of the ground plane is substituted by

1

sin e it is perfe tly ree ting, while the omponent z is obviously null (

J

z

|

mn

= 0

,

m =

(36)

Substituting (2.13)in (2.10):

N

q

(θ, ϕ) =

P

M

m=1

P

N

n=1

exp [jkˆ

rr

mn

] J

q

mn

×

R

Ω

mn

P

mn

_{(r) exp}

_jkˆ

_rr

Ω

mn

dΩ

mn

q ∈ {x, y}

(2.16)

where

Ω

mn

is the area of the

mn

-th re tangular pixel. Due to the presen e of the pixel-basisfun tion the integralin the radiationve tor formulabe omes:

Z

x

n

+

∆x

2 x

n

−

∆x

2 Z

y

m

+

∆y

2 y

m

−

∆y

2 exp

j

2π

λ

x sin θ cos φ

exp

j

2π

λ

y sin θ sin φ

dx dy

(2.17)

where

(x

n

, y

m

)

isthe enter of the

mn

-three tarray element/re tangularpixel. Moreover, with some simple step an be proven that this integral an be solved

as:

exp

j

2π

_λ

x sin θ cos φ

exp

j

2π

_λ

y sin θ sin φ

_×

4 ∆x∆y

sinc

k∆x

2 sin θ cos φ

sinc

k∆y

₂

sin θ sin φ

(2.18)

Nowsubstituting the integral solution(2.18) in (2.16)we nallyhave:

N

q

(θ, ϕ) =

_∆x∆y

4 sinc

k∆x

₂

sin θ cos φ

sinc

k∆y

₂

sin θ sin φ

_×

P

M

m=1

P

N

n=1

J

q

mn

exp [jk (x

n

sin θ cos φ + y

m

sin θ sin φ)]

(2.19)

Thisnalequationisimportantbe auseitdes ribes, inadis retizedway,the

radiationve torsand thuswe an ompute thefar-eldhaving anon ontinuous

urrent denition.

Theusualway todes ribethe far-eldpatterninree tarrayantennasystem

isusing the thirdLudwig denition [78℄[6℄[81℄ of the oordinatesystem.

The far eld radiated by a ree tarray displa ed on a surfa e

Ω

(Fig. 2.1) an be modeled as[68℄

E

_{(r) ≈}

jµf

2 exp (−jkr)

r

[F

CO

(θ, ϕ)

p

b

CO

+ F

CX

(θ, ϕ)

b

p

CX

]

(2.20) where

r = |r|

,

r

= (r sin θ cos ϕ, r sin θ sin ϕ, r cos θ)

is the position ve tor, and the o-polar

b

p

CO

and ross-polar

p

b

CX

unitve torsagreewiththethirddenition of Ludwig[78℄[6℄[81℄(Fig. 2.2)

(

b

p

CO

= cos (ϕ) b

θ

− sin (ϕ) b

ϕ

b

p

CX

= sin (ϕ) b

θ

+ cos (ϕ)

ϕ

b

(2.21)

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Figure 2.2: Co-polar and ross-polar unit ve tor following the Ludwig third

denition.

with some manipulation the o-polar and ross-polar pattern an be retrieved

from(2.20) and (2.12):

F

CO

(θ, ϕ) =

{1 + cos

2 (ϕ) [1 − cos (θ)]} N

x

(θ, ϕ) +

+ [cos (θ) − 1] sin (ϕ) cos (ϕ) N

y

(θ, ϕ) +

− sin (θ) cos (ϕ) N

z

(θ, ϕ)

(2.22)

F

CX

(θ, ϕ) =

[cos (θ) − 1] sin (ϕ) cos (ϕ) N

x

(θ, ϕ) +

+

1 + sin

2 _{(ϕ) [1 − cos (θ)]}

N

y

(θ, ϕ) +

− sin (θ) sin (ϕ) N

z

(θ, ϕ)

(2.23)

At this point we have dened how to ompute the referen e pattern in the

standard omponents we an move to the synthesis problem denition. Sin e

therearemanyworksintheliteraturethatdealwiththeproblemtondaproper

te hnology (e.g. printed pat hes type, number of layers, all-metal stru tures,

et ...)[2℄,[6℄,[15℄,[59℄,[62℄-[66℄ toobtainthe wanted value ofthe s attering matrix

S

for a given urrent distribution, now we do not take this step into a ount (step (b))and wewill ontinue handlingonlythe problemrelated tothe surfa e

(38)

2.3 Inverse Sour e problem denition

A ording to the previous formulation, the synthesis of the onstrained

sur-fa e urrents of a ree tarray withdesired far-eld shaped-beam pattern

E

ref

(r)

,

jµf

2 exp (−jkr)

r

h

F

CO

ref

(θ, ϕ)

p

b

CO

+ F

CX

ref

(θ, ϕ)

p

b

CX

i

(2.24)

an be formulated asan inverse sour e problem.

The inverse sour e problemis dened as:

Constraint-Geometry Ree tarray-Currents Synthesis problem

(CG-RCS)and its denition is:

Find the surfa e urrent

J (r)

(or its numeri al ounterpart

J

q

,

J

mn

q

; m = 1, ..., M

,

n = 1, ..., N}

,

q ∈ {x, y}

), whose radiatea far-eld,

E

,whoseasso iated o-polarand ross-polar omponent,tthe following referen epattern mat hing ondition:

(

F

_CO

ref

(θ, ϕ) = F

CO

(θ, ϕ)

F

_CX

ref

(θ, ϕ) = F

CX

(θ, ϕ)

(2.25)

having that

J

q

∈ S

q

,

q ∈ {x, y}

.Where

S

q

,

q ∈ {x, y}

, are the fea-sibility sets, whi h a ount for the onstraints provided by the

end-user/designer(i.e.,thepresen eofforbidden regions intheaperture).

For example, if

Φ

identies the arbitrary-shaped user-dened 2-D forbidden re-gion (withintheree tarray

Φ ∈ Ω

),thefeasibility onditionstatethat

J (r) = 0

if

r ∈ Φ

(i.e.: in numeri al form:

J

mn

q

= 0

,

q = {x, y}

, if

r

mn

_{∈ Φ}

). It is worth

remarking that many te hniques an be adopted for the synthesis of feed and

asso iated ree tarray elements (step (b)) (depending on the sele ted unit- ell

geometry[2℄,[6℄,[15℄,[59℄,[62℄-[66℄)on e

J

q

,

q ∈ {x, y}

, has been found by solving the above problem.

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Non-Measurable Currents-based

Solution Method

In this hapter the solution method, that solve the Inverse Sour e problem, is

explained. Inparti ular,aftersomemathemati al omputationneeded toobtain

a matrix formulation of the problem, it is applied a Trun ated Singular Value

De omposition (T-SVD) in order to obtain the minimum-norm solution. This

solution an radiate the desired eld but an not deal withfeasibility onstraint

(e.g. forbidden region). In order to over ome this problem, it is proposed to

superimpose the non-radiating/non-measurable urrents, that are derived from

theT-SVD,totheminimum-norm solution. Moreover,giventhedenitionofthe

handledfeasibility onstraint (i.e. forbidden region),the losed-formformulation

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3.1 Field dis retization

In order to address the CG-RCS problem, taking in onsideration that the

z- omponentofthe urrentisnotpresent,itsdis retizedversionisrstly omputed

bysubstituting(2.22)and(2.23),in(2.25),samplingitinasetof

L

angles

(θ

l

, ϕ

l

)

,

l = 1, ..., L

as follows











F

_CO

ref

(θ

l

, ϕ

l

) = {1 + cos

2 (ϕ

l

) [1 − cos (θ

l

)]} N

x

(θ

l

, ϕ

l

) +

+ [cos (θ

l

) − 1] sin (ϕ

l

) cos (ϕ

l

) N

y

(θ

l

, ϕ

l

)

F

CX

ref

(θ

l

, ϕ

l

) = [cos (θ

l

) − 1] sin (ϕ

l

) cos (ϕ

l

) N

x

(θ

l

, ϕ

l

) +

+

1 + sin

2 (ϕ

l

) [1 − cos (θ

l

)]

N

y

(θ

l

, ϕ

l

)

l = 1, ..., L

(3.1)

whi h, by exploiting (2.19), an berewritten as











F

_CO

ref

(θ

l

, ϕ

l

) = Γ (θ

l

, ϕ

l

) ({1 + cos

2 (ϕ

l

) [1 − cos (θ

l

)]}

×

P

M

m=1

P

N

n=1

J

mn

x

e

mn

(θ

l

, ϕ

l

) +

+ [cos (θ

l

) − 1] sin (ϕ

l

) cos (ϕ

l

)

×

P

M

m=1

P

N

n=1

J

y

mn

e

mn

(θ

l

, ϕ

l

)

F

_CO

ref

(θ

l

, ϕ

l

) = Γ (θ

l

, ϕ

l

) ([cos (θ

l

) − 1] sin (ϕ

l

) cos (ϕ

l

)

×

P

M

m=1

P

N

n=1

J

mn

x

e

mn

(θ

l

, ϕ

l

) +

+

1 + sin

2 (ϕ

l

) [1 − cos (θ

l

)]

×

P

M

m=1

P

N

n=1

J

y

mn

e

mn

(θ

l

, ϕ

l

)

l = 1, ..., L

(3.2)

where,for easy of ompa tness:

e

mn

(θ

l

, ϕ

l

)

, exp [jk

0 (m∆x sin θ

l

cos ϕ

l

+ n∆y sin θ

l

sin ϕ

l

)]

(3.3)

and

Γ (θ

l

, ϕ

l

) =

4 ∆x∆y

sinc

k∆x

2 sin θ cos φ

sinc

k∆y

2 sin θ sin φ

(3.4)

In order to further simplify the notation and to better handle the problem

we need toexpress the equation in matrix form.

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following matrix equation

F

ref

_{= GJ}

(3.5) where

F

ref

_,

h

_F

ref

CO

, F

ref

CX

i

T

,

F

ref

t

=

n

F

t

ref

(θ

l

, ϕ

l

) , l = 1, ..., L

o

,

t ∈ {CO, CX}

,

J

, [J

x

, J

y

]

T

, and:

G ,

"

G

CO,x

G

CO,y

G

CX,x

G

CX,y

#

(3.6)

is the

(2 × L) × (2 × P )

overall Green matrix (

· T

being the transpose operator)

featuringthe sub-matri es:

G

CO,x

, {Γ (θ

l

, ϕ

l

) e

mn

(θ

l

, ϕ

l

) {1 + cos

2 (ϕ

l

) [1 − cos (θ

l

)]}

G

CO,y

= G

CX,x

, {Γ (θ

l

, ϕ

l

) e

mn

(θ

l

, ϕ

l

) [cos (θ

l

) − 1] sin (ϕ

l

) cos (ϕ

l

)

G

CX,y

, {Γ (θ

l

, ϕ

l

) e

mn

(θ

l

, ϕ

l

)

1 + sin

2 (ϕ

l

) [1 − cos (θ

l

)]

m = 1, ..., M, n = 1, ..., N, l = 1, ..., L

(3.7)

where

Γ

was dened in (3.4) and

e

mn

in(3.3).

Now we have all the formulation ready for start ta king the Inverse Sour e

problem.

The problem to retrieve a urrent distribution from a eld is well-known to

beill-posed. Thismeansthat multiple urrent distribution anradiatethe same

eld. Inthe literature, oneof the mostused tooltoobtaina minimum-norm(or

generalized)solution of the system is the regularizationand inversion te hnique

Trun ated Singular Value De omposition. Using this algorithm the obtained

solution is the ones that best represents the radiated eld with the smallest

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3.2 Trun ated Singular Value De omposition

Inorderto omputetheminimumnormsolutioniswell-knownthe pro edure

based onthe trun ated version of the SingularValue De omposition

(SVD)[73℄-[77℄.

Weassumethatthenumberofthe ree tarrayelements(

2 ×P = 2×M ×N

) is less than the number of eld samples (

2 × L

). Given that

ψ

2

1 , ψ

2

2 , ..., ψ

2 W

(

W

, min {2 × L, 2 × P }

)arethepositiveeigenvalues ofsymmetri matrix

G

∗

_G

(where

∗

indi atethe onjugatetranspose)and

c

1 , c

2 , ..., c

2×P

the orresponding orthonormaleigenve tors:

G

∗

_Gc

j

= ψ

j

2 c

j

c

∗

j

c

k

= δ

jk

j, k = 1, ..., 2 × L

(3.8) being:

ψ

j

b

j

= Gc

j

j = 1, ..., 2 × L

(3.9)

substituting (3.8)in (3.9) it an be obtained:

G

∗

b

j

= ψ

j

c

j

j = 1, ..., 2 × L

(3.10)

multiplying left and right side of (3.10) for

G

and thanks to (3.9) immediately follow that:

GG

∗

b

j

= ψ

j

2 b

∗

j

b

j

= δ

jk

j, k = 1, ..., 2 × L

(3.11)

Equations(3.10)and(3.11)showntheorthonormalpropertiesofthetwomatri es

B

and

C

.

In matrix notation(3.9) an bewritten as:

(43)

an be noti edthat

Ψ

isa diagonalmatrix in the form:

Ψ =







ψ

1

0

. . . . . .

0 ψ

W







W

×W

(3.13)

another note is that the singular values are ordered in des ending order (i.e.,

ψ

w

≥ ψ

w+1

,

w = 1, ..., W − 1

).

Writing nowthe minimum-norm urrent as aweighted sum as:

J

MN

= Cγ

with

γ

=







γ

1

. . . . . .

γ

W







(3.14)

from(3.9) we an obtain that:

γ

j

ψ

j

b

j

= γ

j

Gc

j

j = 1, . . . , W

(3.15)

and using the ration expressed in(3.5):

F

ref

=

W

X

j=1

γ

j

ψ

j

b

j

(3.16)

Sin e

ψ

i

b

j

= 0

for

j > W

, itis simply to derive that only the rst

W c

w

bases that are used to des ribe

J

MN

are measurable.

For these bases the oe ients

γ

j

are given using:

γ

j

= ψ

−1

j

b

∗

_j

F

ref

j = 1, ..., W

(3.17) and substituting (3.17)in (3.14):

J

MN

=

W

X

j=1

ψ

−1

j

b

∗

_j

F

ref

c

j

(3.18)

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Dually inmatrix form (3.18) be ame:

J

MN

= CΨ

−1

B

∗

F

ref

(3.19)

This resulting problem is well-known [74℄-[77℄ to be not well-posed due to the

ill- onditioningof the

G

matrix.

The solution instability o urs due to the fa t that some singular values

ψ

are mu h lower in magnitude with respe t to the rst one (

ψ

1

). This problem an be measured by using the ondition number that is dened as:

d =

ψ

1 ψ

W

(3.20)

This value measures the instability of the problem. In fa t as higher is the

value, as higher is the instability, and this means that the a small variation in

the

F

ref

generate a greatvariationin

J

MN

.

In literature this problem iswell-known [74℄-[77℄ and the solution isto use a

trun ated version of the SVD.

It is dened

H

as the trun ation order, and is omputed as:

H

, arg

min

w

ψ

w

ψ

1 − τ

s.t.

ψ

w

ψ

1 ≥ τ

(3.21)

where

τ

beingtheasso iateduser-denedSVD trun ationthreshold. The thresh-old

τ

, the trun ation order

H

and an example of singular value behavior

ψ

are shown in Fig. 3.1. By sele ting the value of the SVD trun ation threshold

τ

theuser impli itlydenes the pre isiononthe reprodu tionthe far-eldand the

instability of the urrent, thus lower value of the threshold means better

repro-du tion of the far-eld but also higher variation in the solution (e.g. urrent