稀疏化表示方法

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DesignofUnequallySpacedArraysforPerformanceImprovement

B.PreethamKumar,SeniorMember,IEEE,andG.R.Branner,Member,IEEE

Abstract—ClassicalantennaarraysynthesistechniquessuchasFourier,Dolph–ChebyshevandTaylorsynthesisefficientlyobtainarraycurrentdistributionsforequallyspacedarraysthatgenerateadesiredfar-fieldradiationpatternfunctionorkeepimportantparameterslikebeamwidthandsidelobelevelwithinprescribedperformancebounds.However,theconceptofoptimizationofthefieldpattern(e.g.,bydecreasingsidelobesorbeamwidth)ofangivenequallyspacedarrayrealizationbyalteringitselementspacingsstillrepresentsachallengingproblemhavingconsiderablepracticaladvantages.Theseincludereductioninsize,weight,andnumberofelementsofthearray.Thispaperdescribesanewapproachtosynthesisofunequallyspacedarraysutilizingasimpleinversionalgorithmtoobtaintheelementspacingsfromprescribedfar-zoneelectricfieldandcurrentdistribution,orcurrentdistributionsfromprescribedfar-zoneelectricfieldandelementspacings.IndexTerms—Antennaarrays.

I.INTRODUCTION

VERthepast60years,thetheoryofuniformlyspacedantennaarrayshasbeenstudiedindepthandiscertainlywelldocumented.Forexample,givenadesiredradiationpat-tern(e.g.,pencil-beam,sectoral,cosecetc.)andthenumberofelements,itispossibletoemploysuchtraditionalsynthesisproceduresasDolph–Chebyshev,Taylor,Fourierinversionornumericaloptimizationtoobtaintherequiredarraycurrentdistributionforauniformlyspacedarray.

TheanalysisofunequallyspacedantennaarraysoriginatedwiththeworkofUnz[1],whodevelopedamatrixformulationtoobtainthecurrentdistributionnecessarytogenerateapre-scribedradiationpatternfromanunequallyspacedlineararray(withprespecifiedgeometry).SubsequenttotheinitialconceptofUnz,recentdesigntechniquesfocusontwocategoriesofnonuniformarrays:arrayswithrandomlyspacedelementsandthinnedarrays,whicharederivedbyselectivelyzeroingsomeelementsofaninitialequallyspacedarray.

Inthefirstcategory,Harrington[2]developedamethodtoreducesidelobelevelsofuniformlyexcited

O

512IEEETRANSACTIONSONANTENNASANDPROPAGATION,VOL.47,NO.3,MARCH1999

(a)

(b)

Fig.2.(a)Sidelobelevelofpencil-beampatternwithabruptskirt.(b)Beamwidthofpencilbeampatternwithabruptskirt.

andCohen[8],whoutilizedstatisticalthinningofarrayswithquantizedelementweightstoimprovesidelobeperformanceinlargecirculararrays.Theirresultsdemonstratedthepossibilityofobtainingconsiderablesidelobereductionbyacombinationofprobabilisticthinninganddiscreteamplitudequantization.Anotherrecentapproachtothinnedlinearandplanararraydesignistheapplicationofgeneticalgorithmstodesignoptimalspacings[9].

Thispaperpresentsanewmethodforunequallyspacedar-raysynthesis,whichyieldsappropriateelementspacingvaluesforaprescribedarrayfactorinasimple,recursivemanner.Thistechniquestartswithaprescribedarraypatternandsynthesizesunequallyspacedarraysundertheconstraintthatadjacentelementspacingsarelimitedbythespacebroadeningfactor

elementlineararrayis

illustratedinFig.1wherethe

and1.0

KUMARANDBRANNER:DESIGNOFUNEQUALLYSPACEDARRAYSFORPERFORMANCEIMPROVEMENT513

(a)

(b)

Fig.3.(a)Sidelobelevelofpencil-beampatternwithlinearskirt.(b)Beamwithofpencil-beampatternwithlinearskirt.

Anaffirmativeanswertotheaboveassertionisbasedonthefactthatsincedifferentsetsofnonuniformelementspacingsgeneratedifferentradiationpatterns,onesuchpatterncouldprovideimprovementontheequallyspacedarraypattern(e.g.,byprovidinglowerpeaksidelobelevel(PSLL),narrowerbeamwidthorclosermean-squaredfittotheprescribedpatternresponse).Althoughthispossibilityexists,generalnonuniformspacedarraydesignismorechallengingthanuniformlyspaceddesignbasedonseveralconsiderations.

1)Sincetheelementspacingsoccurasexponentialortrigonometricfunctions,elementspacingsynthesisisanonlinearproblemwhereasthearraycurrentsynthesisisalinearproblem.

2)Constraintshavetobeplacedonthesolutionsfortheelementspacings;viz.,theycannotbecomplexnumbersandtheadjacentelementpositionsshouldbe

toreducethearrayelementcount.

Inordertodelineatethesynthesisalgorithm,thedesired

overtheintervalarraypatternisspecifiedas

,thesynthesisproblemis

addressedonlyovertheinterval

uniformly

514IEEETRANSACTIONSONANTENNASANDPROPAGATION,VOL.47,NO.3,MARCH1999

(a)

(b)

Fig.4.(a)Comparisonofpencil-beampatterns(desiredpatternwithabruptskirt)N=9.(b)Comparisonofpencil-beampatterns(desiredpatternwithabruptskirt)N=19.

spacedpointsovertheobservationrange.Thesecondstepcontainsthekeydevelopmentinthesynthesistechnique,viz.,theLegendretransformationofthearrayfactor.InStep3,thelimitingpropertyoftheLegendrepolynomialsiseffectedandthisleadstothegenerationofatriangularsystemofequations.TheactualdesignequationsforelementspacingvaluesarepresentedinStep4.Thefollowingparagraphspresentthesestepsindetail.

StepI.DefinitionoftheSynthesisProblem

elementnonpe-Theactualarraypattern

riodicsymmetricarrayofpointsources(Fig.1)isgivenby[15]

(1)

where

StepII.LegendreTransformationoftheDesiredArrayPattern

pointsintheinterval

.

Thefollowingstepsdefinetheprocedurefortransformingthearrayfunction

(3)

KUMARANDBRANNER:DESIGNOFUNEQUALLYSPACEDARRAYSFORPERFORMANCEIMPROVEMENT515

(c)

(d)

Fig.4.(Continued.)(c)Comparisonofpencil-beampatterns(desiredpatternwithlinearskirt)N=9.(d)Comparisonofpencil-beampatterns(desiredpatternwithlinearskirt)N=19.

whereistheLegendrefunctionofhalfintegerorder.Therangeandvalues

willbeselectedinaccordancewithof

theelementpositionsaswillbedescribedshortly.Substituting(1)and(2)for

Applyingthepropertyin(5)to(4)permitsthetransformedarrayfunctiontobeexpressedas

(6)

where

ismotivatedbyconsiderationofthefollowinglimitingrelationfortheLegendrepolynomialoffractionalorder[16]:

islarge.Equation(8)yieldsthelimitingcondition

on(5)

516IEEETRANSACTIONSONANTENNASANDPROPAGATION,VOL.47,NO.3,MARCH1999

(a)(i)

(a)(ii)

Fig.5.(a)(i)Peaksidelobelevelofflat-topbeamwithabruptskirt.(a)(ii)Firstsidelobelevelofflat-topbeamwithabruptskirt.

Asanillustration,iftheadjacentelementspacingsofthearrayarelimitedto

(12)

Thissystemisinvertibleasfollows:

gridistheimportantconstituentinthe

reconstructionofthearraycurrentsandpositionsinrecursiveform.Thisgridisdefinedbythefollowingrelation:

(13)

and

StepIV.ApplicationofInversionAlgorithminStepIIItoSynthesizeCurrentsandPositions

Thealgorithmdescribedin(1)–(13)isutilizedtoyieldthesynthesizedarrayspacingsinthefollowingmanner.

KUMARANDBRANNER:DESIGNOFUNEQUALLYSPACEDARRAYSFORPERFORMANCEIMPROVEMENT517

(b)

Fig.5.(Continued.)(b)Beamwidthofflat-topbeamwithabruptskirt.

TABLEI

SYNTHESISALGORITHM

FLOWCHART

OF

thespacebroadening

factorismaintainedlessthan0.5

inordertoreducemutual

couplingeffectsandintheupperlimit,theadjacentelementspacingislessthan

areselectedasfollows:

(14)

Equation(14)isthefirstdesignequationofthearray.Inordertosynthesizethesecond-elementcurrent/position,isselectedas

(15)

Equation(15)istheseconddesignequationofthearraysincetheleft-handsideoftheequationcontainsboththearraycurrent

thdesignequationofthearrayisobtainedas

1)Thefirstarrayelementispositionedat

518IEEETRANSACTIONSONANTENNASANDPROPAGATION,VOL.47,NO.3,MARCH1999

(a)(i)

(a)(ii)

Fig.6.(a)(i)Peaksidelobelevelofflat-topbeamwithlinearskirt.(a)(ii)Firstsidelobelevelofflat-topbeamwithlinearskirt.

TheproposedsynthesistechniquecanberepresentedbytheflowchartformasshowninTableI.Theorganizationoftheflowchartfollowstheanalysisgivenaboveasafour-stepprocessoriginatingwiththedesiredarrayfieldpatternandculminatingwiththegenerationofarrayelementpositions.

Theabovedevelopment[(14)–(16)]providesameanstosolvethefollowingdistinctsynthesisproblems.

1)SynthesisofArrayCurrentsofaNonuniformlySpacedArraywithPrescribedElementPositions:Thisproblemisformulatedasfollows.Givenaprescribedsymmetricalpattern

overtheinterval

elementsym-metricarray,obtaintheappropriatesetofcurrents

expressedintheinterval

elementarray,obtain

theappropriatenonuniformsetofelementpositions

KUMARANDBRANNER:DESIGNOFUNEQUALLYSPACEDARRAYSFORPERFORMANCEIMPROVEMENT519

(b)

Fig.6.(Continued.)(b)Beamwidthofflat-topbeamwithlinearskirt.

(16)as

Thedesiredarraypatternfunctioncangenerallybeex-pressedas

(20)

delineatesthemainlobeandsidelobewheretheboundary

regions.

1)SynthesisofPencil-BeamPatternUtilizingUniformCur-rentDistribution:Asalludedtoabove,inthisfirstapplicationthearraycurrentdistributionisuniformandtheinitialarraygeometryconsistsof(19)

.and

Toexcludeinfeasiblesolutions,thefollowingconditionsareplacedonthesevaluesof

(21)

,then

,then

are

,

b)Prescribedpatternwithlinearskirt:

)oftheinitial

equallyspacedarray.

Thesynthesisalgorithm(SectionII)hasbeenappliedinbothcasesa)andb)toobtaintheappropriateadjacentelementspacings.Forcasea)Fig.2(a)describestheresultsobtainedforthevariationoffirstsidelobelevel(FSLL)andpeakside-lobelevel(PSLL)asafunctionofthespacebroadeningfactor

pointsfor.

520IEEETRANSACTIONSONANTENNASANDPROPAGATION,VOL.47,NO.3,MARCH1999

(a)

(b)

Fig.7.(a)Comparisonofflat-topbeampatterns(desiredpatternwithabruptskirt)N=9.(b)Comparisonofflat-topbeampatterns(desiredpatternwithabruptskirt)N=19.

broadeningfactor:3.185,1.592for

andrespectively.SimilarlyFig.3(a)illustrates

thesidelobecharacteristicsofthesynthesizednonuniformarraysforcaseb)utilizingtheprescribedfunctionin(22).The3-dBbeamwidth[Fig.3(b)]ofthenonuniformarrayremainsunchangedasincasea),withvariationofthespacebroadening

,1.592forfactor:3.185

ofthespacebroadeningfactor

forbothcasesa)andb).

SynthesizedfieldpatternsaredepictedinFig.4(a)–(d)forthetwocasesoftargetfunctionsdefinedin(21)and(22).Ineachfigure,acomparisonismadeamongthespecifiedtarget

,thepatternofthepattern

(inwavelengths)rangesfrom0.1to0.5

6.5dBoverthatofauniformarrayforboth

and,

respectively,correspondingtocaseb).Inthesefigures,itisobservedthatincomparisonwiththeequallyspacedarray,

increases,

thePSLLreachesitslowestvalueforasmallervalue

KUMARANDBRANNER:DESIGNOFUNEQUALLYSPACEDARRAYSFORPERFORMANCEIMPROVEMENT521

(c)

(d)

Fig.7.(Continued.)(c)Comparisonofflat-topbeampatterns(desiredpatternwithlinearskirt)N=9.(d)Comparisonofflat-topbeampatterns(desiredpatternswithlinearskirt)N=19.

thereductionofthePSLLis

pointssources

having

(25)

,Inbothcasesa)andb)

,andthesecondboundary

522IEEETRANSACTIONSONANTENNASANDPROPAGATION,VOL.47,NO.3,MARCH1999

TABLEII

(a)SYNTHESIZEDELEMENTSPACINGS(󰀕)PENCIL-BEAMPATTERNWITHABRUPTSKIRT.(b)SYNTHESIZEDELEMENTSPACINGS(󰀕)PENCIL-BEAMPATTERNWITHLINEARSKIRT

(a)

(b)

positionofthefirstnullinthepatternofthecorrespondingequallyspacedarray.

Fig.5(a)(i)and(a)(ii)describesthevariationofPSLLandFSLL,respectively,ofthesynthesizednonuniformarrayforcasea)asafunctionofthespace-broadeningfactor.Similarly,Fig.6(a)and(b)

illustratesthevariationofthesidelobelevelsandbeamwidth,respectively,forthecaseb).Analysisrevealsthatthepatternsynthesisoftheflat-topbeamishighlysensitivetothespace-broadeningfactorandasaconsequence,hence,therangeof

increasesfrom0–0.14

.

2)The3-dBbeamwidthessentiallyremainsunchangedasafunctionof

,

whereitisobservedthatthePSLLisimprovedoverthatoftheuniformarrayby

and

,respectively.ThePSSLisagainreducedby7dBforand

6.5dBincomparison

withuniformlyexcitedpencil-beamarraysandupto

KUMARANDBRANNER:DESIGNOFUNEQUALLYSPACEDARRAYSFORPERFORMANCEIMPROVEMENT523

forflat-topbeamarrayswhileessentiallymaintainingthesamebeamwidth.Thedesignofunequallyspacedarraysbygenetic

20-dBpeaksidelobelevelforthinnedalgorithms[9]yields

linearandplanararraysoptimizedoverbothscanangleandbandwidthofoperation.Anothercomparisoncanbemadebetweenthecurrentmethodandrecentoptimizationmethods

18.5dB[17],wherethepeaksidelobelevelachievedis

20dBforcurrentcasewitha14-elementarray).A(

primaryadvantageofthemethodpresentedinthispaperisthatitisnoniterativeinnatureand,hence,lesspronetoerrors.Optimizationandgeneticalgorithms,ontheotherhand,areinherentlyiterativetechniques,withtheassociatedpotentialforspeed,convergence,andaccuracyproblems.

Theextensionofthemethodtolowsidelobesymmetricplanararraysisstraightforwardandcanbeconsideredasthetwo-dimensionalcounterpartofthealgorithmdescribedinSectionII.Intheplanarcase,atwo-dimensionalLegendretransformationcanbeeffectedonadesiredarraypattern