A PSO-Driven Spline-Based Shaping Approach for Ultra-Wideband (UWB) Antenna Synthesis

UNIVERSITY

OF TRENTO

DIPARTIMENTO DI INGEGNERIA E SCIENZA DELL’INFORMAZIONE

38123 Povo – Trento (Italy), Via Sommarive 14

http://www.disi.unitn.it

A PSO-DRIVEN SPLINE-BASED SHAPING APPROACH FOR

ULTRA-WIDEBAND (UWB) ANTENNA SYNTHESIS

L. Lizzi, F. Viani, R. Azaro, and A. Massa

January 2011

Wideband (UWB) Antenna Synthesis

Leonardo Lizzi, Federi oViani,Renzo Azaro, and Andrea Massa

Department of Information and Communi ation Te hnologies,

University of Trento,Via Sommarive 14,38050 Trento - Italy

Tel. +39 0461 882057,Fax +39 0461 882093

E-mail: andrea.massaing.unitn.it,

{leonardo.lizzi,federi o.viani, renzo.azaro}dit.unitn.it

Wideband (UWB) Antenna Synthesis

Leonardo Lizzi, Federi oViani,Renzo Azaro, and Andrea Massa

Abstra t

This paper deals with the synthesis of UWB antennas by means of a PSO-driven

spline-based shaping approa h. In order to devise a reliable and ee tive solution,

su h a topi is analyzed a ording to dierent perspe tives: 1) representation of

the antennashapewithasimple ande ient des ription,2) denitionofasuitable

des ription of the UWB Tx/Rx system, 3) formulation of thesynthesisproblem in

terms of an optimization one, 4) integration of the modelling of the UWB system

into a omputationally e ient minimization pro edure. As a result, aninnovative

synthesis te hnique based on the PSO-based iterative evolution of suitable shape

des riptorsis arefully detailed. Toassesstheee tiveness oftheproposedmethod,

asetofrepresentative numeri al simulationsareperformedandtheresultsare

om-pared with themeasurements from experimental prototypesbuilt a ording to the

designspe i ations omingfromtheoptimizationpro edure. Tofo usonthemain

advantages and features of the proposed approa h, omparisons with the results

obtained witha standard parametri approa h arereported, aswell.

Keywords: AntennaSynthesis,Ultra-Wide-Band,SplineRepresentation,Parti leSwarm

Ultra-wideband(UWB)isaterm ommonlyusedtodes ribeawirelesste hnology where

very short low-power time pulses, whi h o upy a very large frequen y bandwidth, are

transmitted/re eived. Su h a ommuni ation te hnique allows high transmission

data-rates, multipath immunity, low probability of inter ept, and low power onsumption.

Moreover, it enables the possibility for a single system to simultaneously operate in

dif-ferent ways (e.g., as a ommuni ation devi e, a lo ator or a radar) [1℄. Sin e the US

Federal Communi ation Commission (FCC)revision on UWB transmissions in 2002 [2℄

and the Ele troni Communi ationsCommittee (ECC)de ision inEuropeabout the use

of UWB te hnologies[3℄, thedesign ofUWB systems has drawn a onsiderableattention

be omingakey topi in the framework of wireless ommuni ations.

Asfarastheradiatingsub-systemis on erned,unlike onventionalnarrow-bandsystems

where modulatedsinusoidalwaveforms o upy smallportionsof the frequen y spe trum,

theassumptionofauniformbehavioroftheantennaisnolongerreliablewhendealingwith

UWB frequen y bandwidths and the distortion of the transmitted time-domain pulses

should be taken into a ount and arefully prevented. Consequently, UWB antennas

turn out to be riti al omponents of the whole system and their ele tri al parameters

need an a urate optimization in a wide range of frequen ies to minimize/redu e the

signal distortions. In order to properly address su h an issue, suitable te hniques are

ne essary sin e a standard synthesis pro ess aimed at determining lassi al

frequen y-domain parameters of both the transmitting antenna and the re eiving one (e.g., gain,

radiation patterns, ree tion oe ients, and polarization) is not enough to ensure the

distortionlessof the UWB systemand the orre ttransmission/re eption oftime-domain

pulses. More spe i ally, the design of an UWB antenna requires ustomized synthesis

te hniques to satisfy the UWB requirements [4℄ as well as proper analysis tools for an

a urate des riptionof the antennabehaviors inthe time domain[5℄.

Generallyspeaking, twomain methodologiesfortheantennadesign anbeusually

re og-nized. The former, indi atedas parametri approa h, onsiders areferen e shapedened

by a xed numberof geometri des riptors to be optimized to satisfy the proje t

gular stru ture [9℄, diamond [10℄[11℄, and bow-tie [12℄. On the other hand, the so- alled

building-blo kapproa h synthesizesthe antennageometrythroughasuitable ombination

of elementarybuilding-blo ks asshown in [13℄ and [14℄.

In this paper, by exploiting the best features of both approa hes, an innovative UWB

antenna synthesis method,whosepreliminaryresultshavebeen presented in[15℄,is

are-fullydes ribed andanalyzed. InordertodesignUWB antennasthatmeetthe userneeds,

both the simpli ity in des ribing some geometri al hara teristi s (e.g., the feedline

ex-tension, the groundplane,and substrate dimensions) of the parametri approa hand the

exibility of a motherstru ture-based method are ne essary toallowa fastand ee tive

synthesispro edure. Moreover, thereisthe needtointegratethesynthesiste hniquewith

an analysis method aimed at evaluating not only the impedan e bandwidth and

radia-tion propertiesbut alsotaking intoa ount the ee ts ofthe propagation hannelonthe

UWB system. Towards this end, both the transmitting antenna and the re eiving one

are onsideredto simulate the whole system.

The outline of the paper is as follows. Se tion 2 gives a des ription of the proposed

approa h fo using on the representation of the antenna shape as well as on the analysis

method. As for the assessment, a set of numeri aland experimentalresults are reported

and dis ussed (Se t. 3). Final omments and on lusions follow inSe t. 4.

2 Mathemati al Formulation

TheUWBantennasynthesisproblemisformulatedintermsofanoptimizationonewhere

a set ofunknown representativedes riptors are tunedthrough aniterativepro ess aimed

at ttingsuitablerequirements/ onstraints onthe ele tri albehavior inthe UWB

band-width [2℄[3℄. Towards this end, a spline-based[16℄ shapegenerator and a PSO pro edure

[17℄are usedtodene the onvergent su essionof trialsolutions. By usingaset of

 on-trolpointsandgeometri aldes riptorsto ode theantennashaperepresentation, aset of

trialgeometriesisevaluatedatea hiterationandupdatedthroughthePSO strategyuntil

a suitablemat hing between estimated and required spe i ations is obtained. As far as

University of Trento and implemented following the guidelines in [19℄ and [20℄, is used

to simulate both the transmitting antenna and the re eiving one (assumed of identi al

shapes) as well asthe propagation hannel.

Forthesakeof larity,thekey-pointsoftheproposedapproa haredetailedinthefollowing

sub-se tions.

2.1 UWB Antenna Shape Representation

The rst issue, addressed in dealing with UWB antenna synthesis, is on erned with a

suitable and exible representation of the antenna shape. To this end, there is the need

to provide an antenna geometry generator (

AGG

) able to model a wide set of planar pat h stru tures as a fun tion of a smallnumber of optimization variables. A ordingly,

a ombination of parametri  des riptors and spline urves [16℄ (assumed as building

blo ks)seems to be a good solution. In parti ular, some antenna features are expressed

intermsofthevaluesofgeometri alparametersinxedranges. Inparti ular,thefeedline

isassumedofre tangularshapeaswellasboththegroundplaneandthesubstrate(Fig. 1).

Thosepartsarefullydeterminedbyxingtheirextensions(i.e.,theirlengthsandwidths).

The des ription of the antenna geometry is ompleted by a spline-based representation

aimed at oding omplex ontours by means of a limitedset of ontrol points.

By assuming a symmetry with respe t to the

y

− z

plane, only one half of the physi al stru ture of the antennahas tobemodeled. Let

{ϕ

l

; l = 1, ..., L; L = 4}

(1)

bethesetofparametri des riptors. Morespe i ally,

ϕ

1

isthesubstratelength,

ϕ

2

isone half of the substrate width,

ϕ

3

is one half of the feed-line width, and

ϕ

4

is the length of the groundplane. Moreover,

ϕ

(min)

l

≤ ϕ

l

≤ ϕ

(max)

l

,

ϕ

(min)

l

and

ϕ

(max)

l

beingxed onstants

that denethe range of admissiblevariation of the

l

-th des riptor. As for the remaining part of the antenna geometry, let

∆ (t) = {x (t) , y (t)}

be the mathemati al des ription ofthe B-splinepat hperimeterasafun tionofthe urvilinear oordinate

t

. Su ha urve turns out to be the linear ombinationof

M

-degree polynomials

δ

n,M

asfollows

∆ (t) =

N

X

n=1

M

X

m=0

δ

n,M

(P

n−M+m,t

) ,

M

= 3

(2)

where

P

n

= (x

n

, y

n

)

,

n

= 1, ..., N

,denotes the

n

-th ontrolpointwhose Cartesian oordi-nates are

(x

n

, y

n

)

. Inorder toavoidthe generationof unrealisti stru tures, thefollowing onstraints are imposed, aswell:

x

1

= ϕ

3

and

x

N

= 0

.

A ordingtosu harepresentation,atrialantennashape

A

istheresultoftheappli ation of the operator

AGG

A

= AGG ξ

where

ξ

= {(x

n

, y

n

) , n = 1, ..., N; ϕ

l

; l = 1, ..., L} = {ξ

j

, j

= 1, ..., J; J = 2 × N + L}

(3)

odes the setofvariables,tobeoptimizedtofullltheUWB proje trequirements,whi h

unequivo ally identies the orresponding geometri almodelof

A

.

2.2 UWB Antenna System Chara terization

These ondsteptobefa ed onsistsinidentifyingafastandreliablenumeri alpro edure

to evaluate the ele tri behaviorof ea h trialshape. Be ause of the very large frequen y

band of operationof UWB systems and the transmission/re eption of short time pulses,

a natural approa h would onsider a time-domain des ription. However, from an

exper-imental point-of-view, a frequen y domain analysis ould be preferable be ause of the

higher a hievable measurement a ura y[21℄.

Takingintoa ountthese onsiderations,atimedomainrepresentationisinitiallyadopted

to dene the requirementsof the transmitting-re eiving antennasystem. Then,these

re-quirementsaretranslated inthe frequen y domainusing the s atteringparameters

rep-resentationtodealwithphysi alquantitiesexperimentally-dete tableinastraightforward

and a urate way.

In order to have a distortionless behavior, the frequen y domain transfer fun tion (i.e.

responsewithalineardependen eonthe frequen yor,analogously, onstantgroupdelay.

Undergoodimpedan emat hing onditions[22℄[23℄,thetransferfun tion anbe

approx-imated with the s attering oe ient

s

21

. Notwithstanding su h a omputation usually needsthe numeri almodellingof thewholetransmitting/re eivingantennasystem witha

very expensive and time- onsuming omputationalpro edure. Tosimplify the numeri al

model and to redu e the omputationalburden, let us onsider a simple method

analo-gous to the so- alled Pur ell antenna gain measurement method [24℄. More spe i ally,

the two-antenna system [Fig. 2(a)℄ is substituted by an antenna and a ree ting surfa e

lo atedjustinfrontofthetransmittingdevi eatadistan eequaltoonehalfoftheTx/Rx

distan e [Fig. 2(b)℄. Although su h an operation annot always be performed (sin e an

image antenna is not generally identi al to the original one even though a perfe tly

re-e tingsurfa e ofinnitearea is onsidered[25℄),howeverthe antennasystem undertest

fullysatisesthemethodhypothesesbe auseofthe axialsymmetryofthereferen eshape

(Fig. 1). Consequently, after simple manipulations,the mutual s attering oe ient

s

21

an beexpressed as follows

s

21

= s

12

= s

11

− s

M P

11

(4) where

s

11

=

b

1

a

1







a

2

=0

is the inputree tion oe ient of the originalantenna system [Fig.

2(a)℄, while

s

M P

11

=

b

′

₁

a

′

₁

isthe same quantity, but in the image onguration[Fig. 2( )℄.

Moreover, underthe assumptionsofantennaswith negligiblestru tural s attering

ontri-butions[21℄, the evaluationof

s

11

an be arriedoutby onsideringa singleantennathat radiates infree spa e [Fig. 2(d)℄

s

11

⋍ s

0

11

=

b

0

a

0

.

(5)

A ordingly, itappears that

s

21

= s

0

11

− s

M P

11

.

(6)

In su h away, the whole system an be modeledbymeans of a singleantenna instead of

from the value of

s

11

with and withoutthe presen e of aninnite metalli plate.

2.3 PSO-Based Optimization of the UWB Antenna Shape

On e an ee tivemethodto hara terize atrialshape of the

U W B

antenna is available, there is the need to dene an evolution strategy able toguide the synthesis pro ess to a

nal design that fully satisesthe proje tguidelines and onstraints dened by the user.

To this end, let us reformulate the synthesis problem as an optimization one where a

suitable ost fun tion

Ω (A)

is minimizeda ording toa

P SO

-based strategy

b

A

= arg min

A

{Ω (A)}

(7)

to determine the nal shape

A

b

= AGG



bξ



of the

U W B

antenna.

In general, evolutionary algorithms use the on ept of tness to represent how well a

parti ularsolution omplywiththedesignobje tive. Thedegreeoftnesstotheproblem

athandofea htrialsolutionisequaltothe orrespondingvalueofthe ostfun tion. Sin e

the design obje tive is to synthesize a

U W B

system hara terized by good impedan e mat hing onditions and by distortionless properties, the ost fun tion

Ω

is a ordingly dened.

Let

f

1

and

f

2

be the lowest and highest frequen y of the band of interest, respe tively. As far as the impedan e mat hing is on erned, the following onstraint is imposed

|s

11

(f )| ≤





s

(d)

11

(f )







f

(min)

≤ f ≤ f

(max)

(8)

wherethe supers ript

(d)

 denotes the targetvalueof the designspe i ation. Moreover,

by assuming that (8) holds true and that

H

≃ s

21

, the onditions for a distortionless system anbereformulatedintermsof: 1)a onstraintonthemagnitudeof

s

21

[Condition (a) - Flat amplitude response℄

△ |s

21

| ≤ △

(d)

|s

21

|

(9)

where

△ |s

21

| , max

an-△τ ≤ △

(d)

τ

(10)

where

△τ = max

f

(min)

_{≤f ≤f}

(max)

{τ (f )}−min

_f

(min)

_{≤f ≤f}

(max)

{τ (f )}

,

τ

(f ) = −

1

2π

d

df

{∠s

21

(f )}

. Starting from these onditions on the s attering parameters, the ost fun tion, whi h

maps the designspe i ations into atness index, isdened asfollows

Ω (A) =

Z

f

(max)

f

(min)

Ω

11

(A) H {Ω

11

(A)} df + Ω

21

(A) H {Ω

21

(A)} + Ω

GD

(A) H {Ω

GD

(A)}

(11) where

Ω

11

(A) =

|s

11

(f )| −





s

(d)

11

(f )











s

(d)

11

(f )







(12)

Ω

21

(A) =

△ |s

21

| − △

(d)

|s

21

|

△

(d)

_|s

21

|

(13)

Ω

GD

(A) =

△τ − △

(d)

_τ

△

(d)

_τ

(14)

H

being the Heaviside fun tion

H {Ω} =











1 Ω ≥ 0

0 Ω < 0

.

(15)

As far as the

P SO

-based sampling of the solution spa e is on erned, ea h parti le of the swarm of dimension

R

odes a andidate solution

ξ

r

(

r

= 1, ..., R

) of the optimiza-tionproblem. Atea hiteration

k

(

k

= 1, ..., K

)ofthe minimizationpro ess,ea hparti le moves losertoitsownbestpositionfoundsofar

ζ

(k)

r

= arg min

k=1,...,K

n

Ω

h

AGG



ξ

(k)

_r

io

(indi ated aspersonalbest position)aswellastowards thepositionofthe best parti leof

the swarm,

ς

(k)

_{= arg min}

r=1,...,R

n

Ω

h

AGG



ζ

(k)

_r

io

(referred to as global best position) [17℄ a ording to the rules of evolution of the PSO [26℄. Starting from a swarm of

par-ti les randomly hosen (i.e., a random hoi e of the ontrol points of the initial shapes

when

Ω



AGG ς

(k)



_{≤ η}

conv

,

η

conv

being a user-dened threshold. Then,

bξ = ς

(k)

and

b

A

= AGG ς

(k)

. 3 Numeri al Results

In this se tion,asele ted set ofnumeri alresults fromseveral experimentsis reported to

showthe behaviorof thesynthesis methodaswellastoassess itsreliabilityand e ien y

in ttingthe designrequirements.

Whatever the test ase or experiment, if it is not spe ied, the proje t onstraints have

been xed to





s

(d)

11

(f )





 = −10 dB

,

△

(d)

_|s

21

(f )| = 6 dB

, and

△

(d)

_τ

_{= 1 nsec}

to guarantee

suitable performan es of the synthesized

U W B

antenna systems. As for the sto hasti optimizer, a population of

R

= 7

parti leshas been used and the onvergen e threshold has been xedto

η

conv

= 10

−3

withthe maximumnumberof iterationsequalto

K

= 400

. Moreover, the values of the

P SO

ontrol parameters have been xed to

C

1

= C

2

=

2.0

and

w

= 0.4

. They have been sele ted a ording to the suggestions given in the referen eliteraturewherethey havebeenfound toprovidegoodperforman e. Foramore

detailed study of onvergen e hara teristi s in orresponden e with dierent values of

these parameters, please referto [27℄.

The rst test ase is about the synthesis of an

U W B

antenna operating in the range

f

(min)

_{= 6 GHz}

-

f

(max)

_{= 9 GHz}

and ompliant with the guidelines of the Ele troni

Communi ations Committee (ECC) [3℄. Su h an experiment is aimed at analyzing the

behavior of the optimization pro ess both in terms of onvergen e and evolution of the

antenna shape. Startingfromarandomlygenerated perimeter[

k

= 0

-Fig. 3(a)℄ hara -terizedby

N

= 4

ontrolpoints,whi hdoesnottatallthedesign obje tives(Fig. 4)as onrmedbythe valueof the ost fun tion

Ω

(i.e.,

Ω



AGG ς

(0)

 ∼

_{= 5}

),the trialsolution

improves until the onvergen e onguration [Fig. 3(d)℄ is found. In order to point out

the mainadvantagesofthe proposedspline-basedmethodoveraparametri optimization

approa h, the same problem(in terms of user requirements) has been addressed by

on-sidering the referen e ir ular geometry shown in Fig. 5. The same PSO algorithmhas

been used to modify the unknown parameters

ξ

= {(x

1

, y

1

) ; r; ϕ

l

, l

= 1, ..., 4}

. Figure 6 and Figure 7 show the evolution of the trial geometry and the orresponding s attering

maximum number of iterations (

k

= K

) has been performed, the onvergen e solution does not t the whole set of proje t requirements (

△ |s

21

(f )| > 6 dB

). Con erning the omputationalissues,the plotof the ost fun tionin orresponden e withthe parametri

approa his omparedinFig. 8withthatofthe proposedapproa h. Besides thewiderset

ofpossiblesolutions,thespline-basedte hniqueturnsouttobemoree ientinsampling

the solutionspa e sin e abetter solutionin rea hed in just

k

conv

= 28

iterations.

Letusnowanalyzethedependen eoftheiterativepro essonthedimensionofthesolution

spa e. Sin e the generation of the spline geometry stri tly depends on the number of

ontrol points that also denes the dimension of the solution spa e, some simulations

have been performed by varying

N

. As a general rule, a smallnumber of ontrol points wouldde reasethedimensionofthesolutionspa eallowingafastersear h,butatthe ost

ofaredu ed apa ityofrepresenting omplex ontours. Onthe ontrary,alargernumber

of ontrol points would allow the des ription of a wider set of geometries and of more

omplex antenna shapes, even though with a higher omputationalburden and the need

to enlargethe dimension of the swarm tofully exploit the additionaldegrees of freedom.

In order to better understand the behavior of the optimization in orresponden e with

dierent values of

N

, letus onsider the followingquality indexes

F

11

=

1

∆f

Z

f

(max)

f

(min)

|bs

11

(f )| df

(16)

F

21

=

1

∆f

Z

f

(max)

f

(min)

(

|bs

21

(f )| − |s

21

(f )|

|bs

21

(f )|

)

df

(17)

F

GD

=

1

∆f

Z

f

(max)

f

(min)

(

|b

τ

(f )| − |τ (f )|

|b

τ

(f )|

)

df

(18)

wherethesupers ript

¯

 indi atesthemeanvalue,overthefrequen yband

∆f = f

(max)

₋

f

(min)

, of the s attering parameters at the onvergen e [

b

s

ij

= s

ij



b

A



℄. Figure 9 shows the values of the quality indexes versus

N

as well as the plot of

k

conv

. As expe ted, the number of iterations needed to nd a onvergen e solution in reases with

N

sin e the solution spa e be omes larger. However, the quality indexes do not proportionally

turns out to be at

N

= 7

ontrol points. In the following, this value will be assumed as referen e.

After the analysis on the optimization pro ess and on the sensitivity of the method to

the antenna shape des riptors, the se ond part of this se tion is devoted to present the

resultsfromthenumeri alandexperimentalassessment. Fortheexperimentalvalidation,

a set of antenna prototypes has been built a ording to the results from the numeri al

simulations by meansof a photo-lithographi printing ir uitte hnology and onsidering

an Arlon substrate (

ε

r

= 3.38

) of thi kness

0.78 mm

as diele tri support. Con erning the measurements,ea hprototypehas been fedwitha oaxialline,equipped withaSMA

onne tor, onne ted at the input port (Fig. 1). Moreover, an Anritsu Ve tor Network

Analyzer (37397D Lightning) has been used to olle t the data in a non- ontrolled

envi-ronmentand the measurementsof the parameter

s

21

havebeenperformedby onsidering a distan e of

d

= 15 cm

between two identi al (Tx/Rx) antennaprototypes[28℄. Be ause of the operatingfrequen ies of UWB systems and the dimensions of the prototypes, the

antennas an be reasonably onsidered inthe far eld region of ea h other.

The se ond test ase deals with the synthesis of a

U W B

antenna that operates in the frequen y range

4 GHz 6 f 6 9 GHz

and omplieswith the FCC standard. The values of the antennades riptors obtained by the

P SO

-based optimizationpro edureare given in Tab. I and the prototype of the antenna is shown in Fig. 10. As for the ele tri

performan es,Figure11showsthesimulatedandmeasuredbehaviorsofthe

s

ij

parameters and ofthegroupdelay

τ

. Asit an beobserved, theantennats theproje trequirements from

f

= 4 GHz

up to

f

= 9 GHz

(i.e., within the FCC mask [2℄) and the bandwidth turnsouttobeof

△f = 5 GHz

(thefra tionalbandwidthbeingequalto

76.9

%a ording to [2℄) from simulated as well as measured data. In parti ular,

|bs

11

(f )| < −10 dB

[Fig. 11(a)℄, the widest variation of the magnitude of

s

b

21

is less than

5 dB

(i.e.,

△ |bs

21

(f )| <

△

(d)

_|s

21

(f )|

) [Fig. 11(b) and Tab. I℄, and

△b

τ

= 0.5 nsec < △

(d)

_τ

[Fig. 11(d)℄. Given

the separation

d

= 15 cm

between theantennas,thea hievedvalues for

s

b

21

arereasonable and in agreement with the values obtained from the Friis formula. As regards to the

dependen e of

|bs

21

(f )|

onthe distan e between the twoantenna prototypes, the in rease of the separation distan e turns out in a shift of the plot towards lower values. As

a onsequen e, the value of

△ |bs

21

(f )|

does not present signi ant variations with the distan e.

As far as the group delay is on erned, although in a reasonable a ordan e with

simu-lations, measured data present larger u tuationsprobably ausedby the non- ontrolled

environmentwhere the experimentalmeasurementshave been arried out. Nevertheless,

theatbehaviorsofthemagnitudeof

s

21

andofthegroupdelay,orequivalentlythelinear trend of the phase of

s

21

[Fig. 11( )℄,assure that the synthesized antenna is suitable for UWB ommuni ations where atransmitted signal should not be distorted.

The lastexample is on erned with amore hallenging problemwhere awider frequen y

band is required (i.e.,

∆f = 6 GHz

,

f

(min)

_{= 4 GHZ f}

(max)

_{= 10 GHZ}

) and a further

onstraint on the dimension of the antenna is imposed for a more easy integration in

ommer ial produ ts (i.e.,

ϕ

(max)

1

= 35 mm

and

ϕ

(max)

2

= 15 mm

), even though a smaller threshold on the amplitude of

s

21

is xed (

△

(d)

_|s

21

(f )| = 10 dB

).

The des riptive parameters and the main ele tri al indexes of the synthesized antenna

are summarized in Tab. II. Moreover, the plots of the ele tri al parameters over the

whole frequen y range are reported in Fig. 12. As expe ted, the antenna performan es

generally degrade ompared to that of the previous example due to the in reased band.

As amatteroffa t, thedesigned antenna has afra tionalbandwidthof

85.7

%. However, the maximum variationof

τ

isof just about

0.15 ns

(simulateddata) and the dimensions ofthe antennaare verymoderate(

ϕ

b

1

= 33.3 mm

and

ϕ

b

2

= 12.2 mm

). On theotherhand,

△ |bs

21

(f )|

turns out tobe slightly greaterthan the desiredvalue (

△ |bs

21

(f )| = 12 dB

vs.

△

(d)

_|s

21

(f )| = 10 dB

)sin ethemaximumnumberofiterations

K

= 400

hasbeenrea hed andthe onvergen e riterionhas notbeensatised. Probably,agreaterdimensionofthe

swarm and alarger number of iterationswould be enoughto arefullymat halsosu h a

onstraint, but su h asetup would require very expensive omputations.

The omparisonbetween thesimulationresultandthedata measuredfromtheprototype

(Fig. 13) further onrm the reliability of the synthesis pro edure. As a matter of fa t,

there is ana eptable agreement between the two plots (Fig. 12) despitesome ree tion

In this paper, an innovative synthesis approa h based on the use of a spline-based

rep-resentation for

U W B

antennas has been presented. The nal shape of the antenna is determined by means of a

P SO

-based optimization pro edure that exploits a suitable and omputationally e ient ele tromagneti representation of the whole Tx/Rx UWB

system.

Theassessmenthasbeen ondu tedondierenttest ases. Firstly,theproposedte hnique

has been testedby onsideringnumeri alexamplestoshowthefeatures andthe behavior

of the iterative pro edure. In order to point out some of the main advantages of the

proposed method, the same test ase has been addressed with a parametri approa h,

as well. Se ondly, we have analyzed the dependen e of the synthesis method on the

des riptors of the antenna shape. Finally, the synthesis results have been veried with

some omparisons with the experimentaldata measured from the antenna prototypes.

The A on lusion, whi h is derived from all the numeri al and experimental results

ob-tained,isthat theproposed approa hrepresents avery promisingmethodology forUWB

antennadesignbe auseofitsexibilityindealingwithdierentuser-denedrequirements

[1℄ L. Yang and G. B. Giannakis, Ultra-wideband ommuni ations, IEEE Signal

Pro- essing Mag.,pp. 26-94,Nov. 2004.

[2℄ US Federal Communi ation Commission, First Report and Order, Revision of

part 15 of the ommission's rules regarding ultra-wideband transmission systems,

FCC02-48, Apr. 2002.

http://hraunfoss.f .gov/edo s_publi /atta hmat h/FCC-02-48A1.pdf

[3℄ Ele troni Communi ations Committee, Draft ECC/DEC/(06)AA, "On the

har-monised onditions for devi es using UWB te hnology in bands below 10.6 GHz,"

ECC/DEC/(06)AA, 2006.

[4℄ Z. N. Chen, X. H. Wu, H. F. Li, N. Yang, and M. Y. W. Chia, Consideration for

sour e pulses and antennas inUWB radio system, IEEE Trans.Antennas Propag.,

vol. 52,no. 7, pp. 1739-1748, Jul.2004.

[5℄ X.Qing,Z.N.ChenandM.Y.W.Chia,Chara terizationofultrawidebandantennas

using transfer fun tions, Radio S i., vol.41,RS1002, pp. 1-10, 2006.

[6℄ C. C. Lin,Y. C. Kan,L. C. Kuo,and H.R. Chuang, "Aplanartriangularmonopole

antenna for UWB ommuni ation," IEEE Mi rowave Wireless Compon. Lett., vol.

15, no. 10,pp. 624-626, O t. 2005.

[7℄ J. Liang, C. C. Chiau, X. Chen and C. G. Parini, "Study of a printed ir ular dis

monopoleantenna for UWB systems," IEEE Trans. Antennas Propag., vol. 53, no.

11, pp. 3500-3504, Nov. 2005.

[8℄ Y. Ren and K.Chang,"An annular ring antenna for UWB ommuni ations,"IEEE

Antennas Wireless Propag. Lett., vol. 5,pp. 275-276,2006.

[9℄ J.Jung,W.Choi,andJ.Choi,"Asmallwidebandmi rostrip-fedmonopoleantenna,"

rounded diamond antennas for ultrawide-band ommuni ations," IEEE Antennas

Wireless Propag. Lett.,vol.3, pp. 249-252,2004.

[11℄ X.H.WuandZ.N.Chen,DesignandoptimizationofUWBantennasbyapowerfull

CADtool: PULSEKIT, Pro . IEEE AntennasPropag.Int.Symp.,vol.2,pp.

1756-1759, Jun. 2004.

[12℄ K.Kiminami,A.Hirata,andT.Shiozawa,"Double-sidedprintedbow-tieantennafor

UWB ommuni ations,"IEEE Antennas Wireless Propag. Lett.,vol.3,pp. 152-153,

2004.

[13℄ J. M. Johnsonand Y. Rahmat-Samii,"Geneti algorithmsand methodof moments

(GA/MOM) forthe design ofintegratedantennas," IEEE Trans.Antennas Propag.,

vol.47, no. 10,pp. 1606-1614,O t. 1999.

[14℄ F. J.Villegas,T. Cwik, Y.Rahmat-samii,andM.Manteghi,"Aparallel

ele tromag-neti geneti -algorithm optimization (EGO) appli ation for pat h antenna design,"

IEEE Trans.Antennas Propag., vol. 52,no. 9, pp. 2424-2435, Sep. 2004.

[15℄ L. Lizzi, F. Viani, R. Azaro, and A. Massa, Optimizationof a spline-shaped UWB

antenna by PSO, IEEE Antennas Wireless Propag. Lett., vol. 6,pp. 182-185, 2007.

[16℄ C. De Boor,A Pra ti al Guide to Spline. Springer, New York, 2001.

[17℄ J.Robinsonand Y.Rahmat-Samii,"Parti leSwarmOptimizationin

Ele tromagnet-i s," IEEE Trans.Antennas Propag., vol. 52,no. 2, pp. 397-407,Feb. 2004.

[18℄ R. F. Harrington, Field Computation by Moment Methods. Malabar, FL: Robert E.

Krieger, 1987.

[19℄ C. A. Balanis, Advan ed Engineering Ele tromagneti s. New York, NY: John Wiley

& Sons, 1989.

[20℄ S. Rao, D. Wilton, and A. Glisson, "Ele tromagneti s attering by surfa es of

Sons, 1982.

[22℄ C. G.Montgomery,R.H.Di ke,andE.M.Pur ell,Prin iplesof Mi rowaveCir uits.

New York: M Graw-Hill,1948.

[23℄ R.E.Collin,Foundationsfor Mi rowaveEngineering.New York: Wiley-IEEEPress,

2000.

[24℄ S. Silver, Mi rowave AntennaTheory and Design. New York: M Graw-Hill,1949.

[25℄ D. M. Kerns, Plane-Wave S attering-Matrix Theory of Antennas and

Antenna-AntennaIntera tions. Nat.Bur. of Stand.Monog.162.Boulder: NationalBureau of

Standards, 1981.

[26℄ J.Kennedy,R.C. Eberhart,and Y.Shi, Swarm Intelligen e.San Fran is o: Morgan

Kaufmann Publishiers, 2001.

[27℄ F. van den Bergh, An analysis of parti le swarm optimizers, Ph. D. dissertation,

Dept. Comput. S i., Univ. Pretoria,Pretoria, SouthAfri a, 2002.

[28℄ S. H. Lee, J. K. Park, and J. N. Lee, A novel CPW-fed ultra-wideband antenna

•

Figure 1. Des riptive parameters of the spline-based antennarepresentation.

•

Figure 2. S hemati representation of (a) the system onstituted by a ouple of identi al antennas, (b) the antenna and ree ting surfa e onguration, ( ) the

equivalent image onguration, and (d) the single antenna radiatinginfree-spa e.

•

Figure 3. Test Case 1 (

f

(min)

_{= 6 GHz}

,

f

(max)

_{= 9 GHz}

, ECC standard,

N

= 4

) - Evolution of the antenna shape: (a)

k

= 0

, (b)

k

= 10

, ( )

k

= 19

, and (d)

k

= k

conv

= 28

.

•

Figure 4. Test Case1 (

f

(min)

_{= 6 GHz}

,

f

(max)

_{= 9 GHz}

,ECC standard,

N

= 4

) -Plotof (a)



s

11



AGG ς

(k)





, (b)



s

21



AGG ς

(k)





, ( )

∠s

21



AGG ς

(k)

,and (d)

τ



AGG ς

(k)

versus the frequen y atdierent iterationsof the optimization

pro ess.

•

Figure 5. Test Case 1 - Parametri approa h - Des riptive parameters for the ir ular monopolereferen e geometry.

•

Figure 6. Test Case 1 - Parametri Approa h (

f

(min)

_{= 6 GHz}

,

f

(max)

_{= 9 GHz}

,

ECC standard) - Evolution of the antenna shape: (a)

k

= 0

, (b)

k

= 100

, ( )

k

= 300

, and (d)

k

= K = 400

.

•

Figure 7. Test Case 1 - Parametri Approa h (

f

(min)

_{= 6 GHz}

,

f

(max)

_{= 9 GHz}

,

ECCstandard)-Plotof(a)



s

11



AGG ς

(k)





,(b)



s

21



AGG ς

(k)





,( )

∠s

21



AGG ς

(k)

, and (d)

τ



AGG ς

(k)

versusthe frequen y atdierentiterationsof the optimiza-tion pro ess.

•

Figure 8. Test Case1 (

f

(min)

_{= 6 GHz}

,

f

(max)

_{= 9 GHz}

,ECC standard,

N

= 4

) -Comparison between parametri approa h and spline-based approa h - Plot of the

optimal value of the ost fun tion

Ω



AGG ς

(k)



and orresponding terms versus

the iteration number

k

.

•

Figure 9. Test Case1 (

f

(min)

_{= 6 GHz}

,

f

(max)

_{= 9 GHz}

,ECC standard)-

points

N

.

•

Figure 10. Test Case 2 (

f

(min)

_{= 4 GHz}

,

f

(max)

_{= 9 GHz}

, FCC standard,

N

= 7

) - Antenna prototype: (a)front view and (b)ba k view.

•

Figure 11. Test Case 2 (

f

(min)

_{= 4 GHz}

,

f

(max)

_{= 9 GHz}

, FCC standard,

N

= 7

) - Numeri al and measured values of (a)

|bs

11

|

, (b)

|bs

21

|

, ( )

∠bs

21

, and (d)

τ

b

versus the frequen y

f

.

•

Figure 12. Test Case 3 (

f

(min)

_{= 4 GHz}

,

f

(max)

_{= 10 GHz}

,

ϕ

(max)

1

= 35 mm

,

ϕ

(max)

₂

= 15 mm

, FCC standard,

N

= 7

) - Numeri al and measured values of (a)

|bs

11

|

, (b)

|bs

21

|

, ( )

∠bs

21

, and (d)

b

τ

versus the frequen y

f

.

•

Figure 13. Test Case 3 (

f

(min)

_{= 4 GHz}

,

f

(max)

_{= 10 GHz}

,

ϕ

(max)

1

= 35 mm

,

ϕ

(max)

₂

= 15 mm

, FCC standard,

N

= 7

) - Antenna prototype: (a) front view and

(b) ba k view.

Table Captions

•

Table I. Test Case 2 (

f

(min)

_{= 4 GHz}

,

f

(max)

_{= 9 GHz}

, FCC standard,

N

= 7

) -Geometri des riptors and ele tri parameters of the synthesized antenna.

•

Table II. Test Case 3 (

f

(min)

_{= 4 GHz}

,

f

(max)

_{= 10 GHz}

,

ϕ

(max)

1

= 35 mm

,

ϕ

(max)

₂

= 15 mm

, FCC standard,

N

= 7

) - Geometri des riptors and ele tri

ϕ

₁

ϕ

2

z

ϕ

₃

y

x

ϕ

4

P

3

P

N −1

P

1

P

2

..

..

..

Port

Input

P

N

a

2

a

1

[S]

b

1

b

2

d

Metallic Plate

a

′

1

b

′

1

d/2

(a) (b)

a

′

1

b

′

1

a

′

2

b

′

2

J

J

J

J

[S

M P

_]

a

0

b

0

( ) (d)

ϕ

1

=

7

.9

7

cm

ϕ

₂

= 1.59 cm

ϕ

₂

= 2.33 cm

ϕ

1

=

3

.6

7

cm

(a) (b)

ϕ

₂

= 2.46 cm

ϕ

1

=

3

.5

7

cm

ϕ

₂

= 2.47 cm

ϕ

1

=

3

.5

7

cm

( ) (d)

-25

-20

-15

-10

-5

9

8.5

8

7.5

7

6.5

6

|s

11

| [dB]

Frequency [GHz]

k = 0

k = 10

k = 19

k = k

_conv

-42

-40

-38

-36

-34

-32

-30

-28

9

8.5

8

7.5

7

6.5

6

|s

21

| [dB]

Frequency [GHz]

k = 0

k = 10

k = 19

k = k

_conv

(a) (b)

-180

-90

0

90

180

6

6.5

7

7.5

8

8.5

9

Arg(S

21

) [deg]

Frequency [GHz]

k = 0

k = 10

k = 19

k = k

_conv

0.8

0.7

0.6

0.5

0.4

9

8.5

8

7.5

7

6.5

6

τ

[ns]

Frequency [GHz]

k = 0

k = 10

k = 19

k = k

_conv

( ) (d) 4 -L. Lizzi et al. , "A Spline-based Shaping Approa h for Ultra-Wideband ..." 23

ϕ

₁

ϕ

₂

z

ϕ

₃

y

x

ϕ

4

P

1

Port

Input

r

ϕ

2

= 1.8

cm

ϕ

1

=

7.

5

cm

ϕ

1

=

7.

2

cm

ϕ

2

= 1.9

cm

(a) (b)

ϕ

1

=

7.

2

cm

ϕ

2

= 1.9

cm

ϕ

1

=

7.

1

cm

ϕ

2

= 2.0

cm

( ) (d)

-20

-15

-10

-5

0

6

6.5

7

7.5

8

8.5

9

|s

11

| [dB]

Frequency [GHz]

k = 0

k = 100

k = 300

k = K

-50

-45

-40

-35

-30

-25

6

6.5

7

7.5

8

8.5

9

|s

21

| [dB]

Frequency [GHz]

k = 0

k = 100

k = 300

k = K

(a) (b)

-180

-90

0

90

180

6

6.5

7

7.5

8

8.5

9

Arg(S

21

) [deg]

Frequency [GHz]

k = 0

k = 100

k = 300

k = K

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

6

6.5

7

7.5

8

8.5

τ

[ns]

Frequency [GHz]

k = 0

k = 100

k = 300

k = K

( ) (d) 7 -L. Lizzi et al. , "A Spline-based Shaping Approa h for Ultra-Wideband ..." 26

10

-3

10

-2

10

-1

10

0

10

1

10

2

0

50

100

150

200

250

300

350

400

Ω

[arbitrary unit]

Iteration No., k

Spline-based Approach

Parametric Approach

0

50

100

150

200

250

300

350

400

4

5

6

7

8

9

10

0.4

0.8

1.2

1.6

2

2.4

2.8

k

conv

F

11

, F

21

, F

GD

[x 10

-1

]

Number of Control Points, N

k

_conv

F

₁₁

F

₂₁

F

_GD

(b)

-35

-30

-25

-20

-15

-10

-5

0

9

8.5

8

7.5

7

6.5

6

5.5

5

4.5

4

|s

11

| [dB]

Frequency [GHz]

Simulated Data

Measured Data

-50

-45

-40

-35

-30

-25

-20

-15

-10

9

8.5

8

7.5

7

6.5

6

5.5

5

4.5

4

|s

21

| [dB]

Frequency [GHz]

Simulated Data

Measured Data

(a) (b)

-180

-90

0

90

180

4

4.5

5

5.5

6

6.5

7

7.5

8

8.5

9

Arg(s

21

) [deg]

Frequency [GHz]

Simulated Data

Measured Data

2

1.5

1

0.5

0

-0.5

-1

9

8.5

8

7.5

7

6.5

6

5.5

5

4.5

4

τ

[ns]

Frequency [GHz]

Simulated Data

Measured Data

( ) (d) 11 -L. Lizzi et al. , "A Spline-based Shaping Approa h for Ultra-Wideband ..." 30

-30

-25

-20

-15

-10

-5

10

9

8

7

6

5

4

|s

11

| [dB]

Frequency [GHz]

Simulated Data

Measured Data

-50

-45

-40

-35

-30

-25

-20

10

9

8

7

6

5

4

|s

21

| [dB]

Frequency [GHz]

Simulated Data

Measured Data

(a) (b)

-180

-90

0

90

180

4

5

6

7

8

9

10

Arg(s

21

) [deg]

Frequency [GHz]

Simulated Data

Measured Data

2

1.5

1

0.5

0

-0.5

-1

10

9

8

7

6

5

4

τ

[ns]

Frequency [GHz]

Simulated Data

Measured Data

( ) (d) 12 -L. Lizzi et al. , "A Spline-based Shaping Approa h for Ultra-Wideband ..." 31

(b)

Control Points Coordinates [mm℄

x

1

y

1

x

2

y

2

x

3

y

3

x

4

y

4

5.4 51.6 6.9 57.6 9.0 60.6 7.0 65.1

x

5

y

5

x

6

y

6

x

7

y

7

2.7 67.3 2.0 64.5 0.0 56.0 Geometri Variables [mm℄

ϕ

1

ϕ

2

ϕ

3

ϕ

4

69.2 10.0 5.4 10.9 Performan e

△f [GHz]

△ |s

21

| [dB]

△τ [ns]

5 (4-9) 5 0.5

Control Points Coordinates [mm℄

x

1

y

1

x

2

y

2

x

3

y

3

x

4

y

4

5.0 14.7 5.2 17.1 3.1 22.5 5.1 25.7

x

5

y

5

x

6

y

6

x

7

y

7

3.3 24.7 3.8 23.0 0.0 27.2 Geometri Variables [mm℄

ϕ

1

ϕ

2

ϕ

3

ϕ

4

33.3 12.2 5.0 12.2 Performan e

△f [GHz]

△ |S

21

| [dB]

△τ [ns]

6 (4-10) 12 0.15