Progress In Electromagnetics Research PIER FAST SDM FOR SHAPED REFLECTOR ANTENNA SYNTHESIS - PDF document

ProgressInElectromagneticsResearch,PIER92,361–375,2009FASTSDMFORSHAPEDREFLECTORANTENNASYNTHESISVIAPATCHDECOMPOSITIONSINPOINTEGRALSH.-H.ChouDepartmentofEngineeringCambridgeUniversityCambridgeCB30FA,U.K.H.-T.ChouDepartmentofCommunicationsEngineeringYuanZeUniversityChung-Li320,Taiwan„Thispaperpresentsanapproachofashapedre”ectorantennasynthesisusingasteepestdecentmethod(SDM).It“rstdiscretizesthere”ectorsurfaceintosmallpatchesandthenusesgridnodesasvariablesinthesynthesisprocedure.Eventhoughthenumberofvariablescanbeverylargeforalargere”ectorantenna,theadvantageofprovidingclosed-formsolutionsforthederivativesofacostfunctionpotentiallymakesthisapproachveryecient.ThelargenumberofvariablesalsoassiststhisproceduretoreachamoreglobaloptimumasusuallymetinordinarySDM.Numericalexamplesarepresentedtovalidatethisapproach.1.INTRODUCTIONFastdesignoflargeshapedre”ectorantennasremainschallengingduetotheirincreasingsizesneededintheapplicationsofsatelliteandwirelesscommunications[1…12].Suchchallengesarisefromthecomputationalineciencyinnumericalradiationanalysisthatbecomesdramaticallycumbersomeasthere”ectorsizeoroperationalfrequencyincreaseandneedstoberepeatedlyperformedateveryloopofaniterativesynthesisprocedure.Inthepast,eortshavebeen Correspondingauthor:H.-H.Chou(hsihsirchou@gmail.com). 362ChouandChoufocusedondevelopingfastradiationanalysisorconvergentsynthesistechniquesinarelativelyindependentfashion.Typicalworksforthefastanalysistechniquesarepresentedin[13,14],mostlyusingtheapproximationsofphysicaloptics(PO)orapertureintegration(AI)bynumericallyintegratingradiationintegrals.Thesetechniquesaremostwidelyemployedinthere”ectorantennaanalysisforpracticalapplications.EitherFastFourierTransform(FFT)orexpansionsovertheintegrandsusingproperlyselectedbases,suchasGaussianbeams(GB)[4,5,15],areperformedwithendurablesacri“ceofaccuracy.Previously,successfullyemployedsynthesistechniquesarereferencedin[3,8…12,16…18]includinggeneticalgorithm(GA)[11],steepestdecentmethod(SDM)[3,5,8,9],successiveprojectionmethod(SPM)[10…12,17]andparticleswarmoptimization(PSO)[18].Accelerationeortswereattemptedintwocategories.The“rsttriestousefewervariablesbyrepresentingre”ectorsurfacesintermsofbasisfunctions[7,19,20]andusestheassociatedcoecientsasvariablesforoptimization.Itmayhoweverlossfreedomsforglobaloptimization.Morereductioninthenumberofvariablescausesmorefreedomloss,andtradeosbetweeneciencyandoptimummustbemade.Thesecondattemptstodevelopanalyticformulationsforuseintheoptimizationprocedure[8…12].In[8,9],directsurfacevariationsareconsidered,andanalyticalformulationsforsurfacedeviationsbasedonthegradientsofthecostfunctionde“nedinaframeworkofSDMweredeveloped.Itavoidsnumericalcomputationsonthederivativesofacostfunction,butstillresultsinintegralsthatneedtobenumericallyintegratedusingFFT.In[10…12]there”ectorsurfacewasdiscretizedintopatches.Theradiationsofeachpatchserveasbasisfunctionstosynthesizethedesiredcontouredpatterns,andSPMwasemployedto“ndcoecientsofthebasisfunctionsusinglinearprojectionsintothecommonintersectionofavailablesolutionsets.Thecoecientsaretransformedintothesurfacedeviationsbyconstraininganequalamplitudeofthecoecients.Itprovidesanalyticalandclosedsolutionsforthesurfacevariation.However,SPMdoesnotguaranteetheexistenceofthecommonintersectedsolutions,andthecoecientsmaybequiterandomlydistributed.Inmostcases,notonlysurfacesmoothtechniquesneedtobeperformedinordertoobtaincontinuoussurface,butalsorapidsurfacevariationmayoccur.Thispaperpresentsanusefultechniqueofsynthesisprocedure.Analogoustothatdescribedin[8…12],itprovidesanalyticalformulations.Moreover,itsformulationshaveadvantagesofbeinginclosedformsandcontinuousforthere”ectorsurfacedeviation.Itcompletelyavoidsthenumericalintegrationsandsurfacesmoothing ProgressInElectromagneticsResearch,PIER92,2009363requiredin[3,8…12].It“rstdiscretizesthere”ectorsurfaceintosmallpatcheswhosesizesareselectedsucientlysmalltoaccuratelyapproximatethere”ectorsradiationbyasuperpositionofeachpatchscomponentinthePOradiationintegral.Iftheoptimizedsurfaceisslowlyvarying,thentheradiationanalysiswillretaintheaccuracyasusuallyprovidedbyPO,whichisfoundtobetrueinnumericalexperiences.The-componentsofgridnodescoordinatesareusedasvariablesinconjunctionwithaSDMsynthesistechniquetodeterminetheshapedre”ectorsurface.Itlooks,ata“rstglance,tocomplicatethesynthesisproceduresincethenumberofvariableshasnowbeenincreaseddramaticallytocausecumbersomenumericalcomputationsforthederivativesofacostfunctioninSDM.However,theproposedworkusingpatchdecompositionsfortheradiationintegralexhibitsadvantagesofprovidingapproximatebutinclosed-formsolutionsforthederivatives.Theoverallcomputationaleciencyisfoundtobeimproved.Theorganizationofthispaperisasfollows.Section2.1describestheformulationsofpatchdecompositiontoevaluatethePOradiationintegralofare”ectorantenna,whichreducestheoverallradiationintermsofasuperpositionof“eldsradiatedfromasetofequivalentcurrentmomentslocatedatgridnodes.Section2.2summarizesSDMandpresentstheclosed-formformulationsforthederivativesofacostfunction.Section3analyzesthecomputationalcomplexityofthiswork,andpresentsnumericalexamplestovalidatetheproposedworkinthesynthesisofashapedre”ectorantenna.FinallysomeshortdiscussionsarepresentedinSection4toconcludethiswork.Atimeconvention,jt,isusedthroughoutthispaper.2.THEORETICALDEVELOPMENTS2.1.PatchesDecompositionofPhysicalOpticsRadiationIntegralTheradiationofare”ectorantenna,whenitisfedby()asillustratedinFigure1,canbefoundby[9] jkR where)=2)isde“nedat¯withbeingitsanoutwardunitvectornormaltothesurface.with¯beinga“eldlocation.Equation(1)canbenumericallycomputedby 364ChouandChou Figure1.Geometryfortheshapedre”ectorantennaproblemincontouredbeamapplication.Thefeedradiations().Thesurfaceisdecomposedintosmallplanarpatheswithgridsalsoshowninthis“gure.decomposingintosmallpatchesandgivenby)(2)withjkZ jkR wherewithbeingtheareaofthpatch,and¯with¯beingthephasecenterofpatch.isselectedthat(2)willreachanacceptableaccuracyasprovidedbyPO.Ingeneral4patchwidthissucientinapracticalcase.Asissucientlysmallandbecomesapproximatelyplanar,(3)canbeapproximatedbyaveragingtheirvaluesatpatchcornersbyjkZ jkR RmSmRm×Rm×1 where¯indicates¯thcornerofthpatch,andisthenumberofcornersforthpatch.Thusinthefarzone,(1)canbe ProgressInElectromagneticsResearch,PIER92,2009365re-expressedasasuperpositionofthecontributionsfromcornersbyjkZ jkr r×r×Ncqr·¯rqNqmSm jkZ jkr eq,q)(5)whereisthenumberofcornersformedbythepatchesandthenumberofpatchesassociatedwiththcornerat¯.Theequivalentcurrentmoment,eq,q),in(5)isde“nedbyeq,q eq,qeq,qwitheq,qeq,q whichisanequivalentsurfacevectorassociatedwiththcorner.In(6)and(7))=2)withbeingthpatch.Equation(5)isdependentonthecornersparameters,andallowsonetoemploythecornerlocationsastheoptimizationvariables.Apracticalimplementationselectsthelocationsofcorners“rst,whichgloballydistributeoverthere”ectingsurface,anddeterminethepatches.2.2.EcientSynthesisProcedureviaSteepestDecentMethodSDMgivesgradualparameterschangesinre”ectorsurfacevariations,andresultsinasmoothre”ectorsurface.It“rstde“nesacostfunctionintermsofsampledradiationspatternsas[3]whereisanumberofsamplesandisthenormalizedgainat“eldpointwithbeingitsdesiredvalue.Itwillbeminimizedtooptimizethere”ectorsurface.Bothsidelobeandcross-polarizationlevelsarecontrolledbyspecifyinginthesampledpoints,andareconsideredasdierentsampledgains.isintroducedin(8)to 366ChouandChouemphasizesomedesiredgains,whichisusefulforside-lobeandcross-polarizationsuppressionsincetheirvaluesareverysmall.InSDMthevariables,)withbeingthenumberofvariablestode“nethere”ectorsurface,isoptimallychangedinaniterativefashionusing(9),thatis,at(+1)thiterationisfoundby[6]+1)+1)+1) 1=1(i) 2=2(i)... whereisascalefactortoproperminimize(8).TheimplementationprocedureisdemonstratedinFigure2.Numericalcomputationsofthederivativesin(9)arecumbersomebecauseaschanges,indicatinganewsurfaceshapeofre”ector,(1)needstobere-calculated,whichincreasesdramaticallyasincreases.Inthepast,thevalueofneedstobe“rstminimizedbyexpandingthesurfaceusingeitherglobalorlocalbasisfunctionssuchasJacobi-Fourierseries[7]andspline[14],respectively.Twodisadvantagesarefound.First,repeatedlycomputationsof(1)areextremelycumbersomesincechangesfrequentlyduringSDM.Secondly,asmalllimitsthefreedomofsurfacevariation.Thiswork,employingconers Figure2.SDMsynthesisprocedure. ProgressInElectromagneticsResearch,PIER92,2009367ofthepatchedasoptimizationvariables,allowsamaximumfreedominvaryingthesurfaceforaglobaloptimization.Thesurfaceshapeisdeterminedbycornerlocations.Inparticular,ofthethcornerscoordinate,¯),areusedasvariables(i.e.,)while“xing()toretaintheprojectedapertureofthere”ector.Ata“rstglance,thisworkincreaseuptoseveralordersandmayworsentheoptimization.Ithoweverexhibitsadvantagesofprovidingclosed-formsolutionsforthederivativesin(9)becauseeachcorner,(),isassociatedwithveryfewpatchesasillustratedinFigure1.Thederivativein(9)canbefoundbydierentiating(8) sf where =8 Z0Pr¯E)·v¯E) andusing(5)gives =k2Z0(b3Ša3) jkr eq,qIn(11),istheradiationpower,)and )with¯beingthefeedslocation.istheunitvectorindicatingthepolarizationofinterest,andmaybeusedtospecifyco-orcross-polarizations.Notethat(10)isaclosed-formsolution,andiscontinuousoverthesurface.Notonlythederivativesof(8)canbeecientlycomputed,butalsothesynthesizedsurfaceretainssmoothalongtheiterativeprocedure.Oncethecornerlocationsaredeterminedand“xed,thesurfacesbetweenthesecornerscanbeinterpolatedandsmoothusinglocalbasisfunctionssuchasthesedescribedin[19,20]withoutlosingtheaccuracyoftheradiationpatterns.Asaresult,ifthedistributionsofthecornersareshowntobecontinuousandsmooth,thenthesmoothnessoftheoverallsurfacecanbeassured.2.3.AUsefulCriteriontoChooseSDMuses(9)toupdateiteratively.Thechangesof,ineachiterationarecontrolledbyaproperselectionof,andresultinphasechangesfortheequivalentmomentsin(6)inawaythatthesuperpositionoftheirradiationswillapproachthedesiredcontoured 368ChouandChoupatterns.Itcanbeachievedbyallowingvaryingwithinahalfwavelength.Thevalueofisselectedsuchthatmaximumeachiterationislessthanahalfwavelength.Agoodvalueislessthan0.25wavelengththatrepresentsamaximumphasechangeof180degrees(causesamaximumphasechangeof180degrees)sothattheoverallmaximumafterthecompletionofsynthesismayretainlessthanahalfwavelength.Inpractice,shouldbecontinuouslydecreasesasthesynthesisproceedssinceSDMconvergesveryfastinthe“rstfewroundsofiterationandwillbecomegraduallysmaller.Onemaysimplydecreasetoaquarteror0.1ofitsvalueinthepreviousiterationwhenthevalueofcostfunctionin(8)isfoundtoincrease,andretainthisvalueforthenextiteration.2.4.AdvantagesandLimitationsoftheProposedWorkIncomparisonwithpreviousSDMworks[3,5,8,9],thecurrentapproachexhibitsadvantages.In[3],thegridsoftheequivalentaperturebasedonAIareusedasoptimizationvariables.Itrequiresperformingray-tracingtodeterminethere”ectorssurfacewhichneedstobesmoothedtoretainacontinuessurface.Theproposedworkcompletingavoidsthese.In[5],are”ectorsurfaceisrepresentedbyasetofglobalbasisfunctionstoreducethenumberofoptimizationvariablesandcomputationaltime.Theproposedworkusessurfacegridsasoptimizationvariables.Ithasamuchlargernumberofoptimizationvariables,butresultsinlesscomputationaltimeastobeshowninSection3.Incomparisonwith[8,9],whichalsoprovidesclosedformsolutions,theproposedworkdoesntrequireperformingFFTincomputingthesolution.Thelimitationsoftheproposedworkcanbeobserved.First,thenumberofoptimizationvariablesincreasesinanorderofre”ectorssurfacesize.Second,theincreasewillfurtherincreasethesizeofcomputersmemory.Theselimitationdonotcauseanyinconvenienceinpracticalapplicationswithtodayscomputertechnologiesinhand.3.NUMERICALVALIDATIONANDDISCUSSION3.1.AnalysisofComputationalComplexityThecomputationalcomplexity,justi“edbycountingthenumberoftermstheoperationofsummationsineachiterationofnumericalevaluation,isexamined.One“rstexaminesthecomputationcomplexityinatraditionalSDM[5].Assumingthatbasisfunctionsareusedtorepresentthere”ectorsurfaceinaconventionalSDM,thenumberoftermsiscountedinthefollowing.Using(5)to“ndthe ProgressInElectromagneticsResearch,PIER92,2009369electromagnetic“eldsatpointsfor(8)requiresterms.Ineachiterationoneneedsto“ndin(9))derivativesnumericallyfor(9),anditrequiresterms.Similarlythenumberoftermsintheproposedworkiscountedinthefollowing.Using(5)to“ndthetheelectromagnetic“eldatpointsfor(8)needsterms.Using(9)to“ndderivatives(for(9)needstocomputeterms.Thereforeifthenumberofiterationsisnotconsidered,theproposedworkapparentlyhasabettereciencybycuttingthecomputationalcomplexityinanorderof3.2.NumericalExamplesTheexampleshownin[3]isre-examinedtoproduceaCONUSbeamwithpowerconstraintswithin3dB.Theinitialsurfaceisparabolicwithafocallength25.Theradiusofaprojectedcircularapertureontheplaneis12,withanosetdistance3toavoidafeedblockage.Theoperationalfrequencyis11.811GHz.Thefeedhasacosradiationpatternwitharighthandcircularpolarization.Thisinitialre”ectorsurfacewillradiatefar“eldswithadirectivityof38.0dBwithroughly5squaredegreesbeamarea.ToachievethebeamconstraintsshowninFigure3,themaximumdirectivityisroughly30dB.Thusthedesiredgoalofdirectivityinthespeci“edareaofFigure3willbelargerthan28dB,wherein(8)issettooneforsimpli“cation.Themaximumdeviation,,isinitiallysettobe0.2wavelength,whichisgraduallydecreasedbyafactorof0.25whenthecostfunctionisfoundtoincreaseintheiterativeprocedure.Inapracticalimplementation,thesurfacechangeat“rstiterationrepresentsthesteepestsurfacevariationstominimizethecostfunctionbydefocusingthefocused“eldsradiatedfromaparabolicre”ector.Thisrapidsurfacechangeforenergydefocusingmayeventuallyresults Figure3.DesiredCONUScontouredpattern. 370ChouandChouinarapidsurfacevariationthatisnotconsideredtobesucientsmoothfromapointofviewinarealisticmanufacture.Figure4(a)showsthederivativesofthecostfunctionusing(11),whichisnormalizedinawaythatthevectorformedbythesederivativeshasaunitnormandrepresentsthechangingrateof.AlsoFigure5(a)showsthecontouredradiationpatternsafterthissurfacechange.Notethatthemaximumchangelengthofisrestrictedto0inthiscaseasmentionedinthepreviousparagraph.Inparticular,Figure4(a)showsthatthesurfacedistortswithavariationsimilartoasinefunction.IttendstospreadtheenergyoutinupwardanddownwarddirectionsasshowninFigure5(a),wheretwobeamswereformedinthe Figure4.ComparisonofinitialderivativesofthecostfunctionusedinSDM. (a) Using (11)(b) Using (13)Figure5.Contouredradiationpatternsat“rstiteration. ProgressInElectromagneticsResearch,PIER92,2009371coveragearea.Toavoidthisphenomenon,analternativeformulationisusedat“rstiteration.Theideaistoconsiderthebehaviorofthepowerderivativesusedin(11),wherearealpartisusedandexhibitsasinefunctionbehavior.Thusitcanbeconjecturedthatitsimaginarypartwillhaveacosinefunctionbehavior.Figure4(b)showsthesurfacechangingratecorrespondingtoFigure4(a)exceptnow(11)isreplaced =Š8 Z0Pr¯E)·v¯E) where-Žsignisemployedtoassurepositivesurfacevariationsforconvenience.Thecorrespondingradiationpatternat“rstiterationisshowninFigure5(b).Itisobservedthattheenergyspreadsoutconcentrically.Thisgradualenergydistributionallowsthesynthesistoreachabettersmoothnessforitsuseinmanufacture.Afterward,(11)isemployedtosynthesizethesurface.Tofurtherexaminethesmoothnessofthere”ector,changingratesat2nd,7thand21thiterationsareshowninFigures6(a)(c),respectively.First,Figure6(a)showsthatthebehaviorofchangingrateissimilartothatinFigure4(b).itindicatesthatSDMcontinuestodefocustheradiatingenergyafteritsinitialiteration.Oncetheenergydefocusinghassucientlycoverthedesiredarea,SDMstartstomodifythesurfacesuchthattheradiationwillformtheshapedpatternasshowninFigures6(b)and(c).Notethatthemaximumofchanginglengthdecreasesalongthesynthesisprocedure.Figure7(a)showsthechangesofmaximumduringthesynthesisprocedure,whichmakesthecostfunctioncontinuouslydecreases.Thedecreasesonisnecessaryinordertoavoidwastingtimeincomputingthevaluesofcostfunctionwhichtendstoincreasealongthesynthesisprocedureifthemaximumisretainedsameineachiteration.AlsoFigure7(b)showstheconvergenceofthecostfunctionalongthesynthesis.Thecostfunctionconvergesveryfastinthe“rstfewroundsofiterationandtherateslowsdownafter50iterations.Thusafter50iterations,theslowconvergenceratewillresultinsmallerchangesinthecostfunctionandsurfacevariationsofthere”ectoraswell,whichjusti“estheneedtouseasmaller.Thisisadvantageousoverothertechniqueswhereglobalorlocalbasisfunctionsareusedtorepresentthere”ectorsurfacesincenopriorknowledgeontheselectionofappears.Inthecurrentexample,only60iterationsareperformedtoachievethecontouredpatternsshowninFigure8(a),where73sampled“eldpointsareconsidered.Themaximumdirectivityis30.5dBinthiscase.Thecomputationof(5)usestriangularpatcheswitha0.25wavelengthsamplinglength,whichissucienttocomputethe“eldslocatedintheangularareaofinterest(),andresultsin8154corners(also8154variables)in 372ChouandChouthesynthesisprocedure.Thisnumberofvariablesisfarlargerthanthenumberof“eldpointstobesynthesized.Thusitreducesthepossibilityofcostfunctionsconvergencesstuckinalocalminimumbeforeareasonableresultisachieved. (a) 2nd iteration (b) 7th iteration (c) 21th iteration Figure6.Derivativesofthecostfunctionwithrespecttotheoptimizationvariablesusing(11).Thecomputationaleciencycanbefurtherimproved.Notethat(5)issimplyusedtocomputetheradiation“elds,whichcanbefurtheracceleratedifotherecienttechniquesareavailablesuchastheanalysistechniquesbasedonaGaussianbeamexpansion[4,5,10].Thedeviationoftheshapedsurfacefromtheoriginal(initial)parabolicre”ectorisalsoshowninFigure8(b)forcomparison.Asdescribedearlierthatthederivativesofthecostfunctionwithrespecttoarecontinuous,whichassuresthecontinuityandsmoothnessofthesynthesizedsurfaceasdemonstratedinFigure8(b)wheresmoothsurfacedeviationisobserved.FinallytheCPUtime,runningonanAcerNotebookwithIntel2.2GHzCore2DuoProcessorT7500,is0.28secondsforthecomputationinaniterationwithtotaltimelessthan20secondto ProgressInElectromagneticsResearch,PIER92,2009373 (a) Variation of maximum
z q (b) Convergence curve Figure7.Changesofmaximumandtheconvergencecurvealongthesynthesisprocedure. Figure8.Theachievedcontouredpatternandthesurfacedeviation(unit:wavelength)afterthecompleteofsynthesis.complete60iterationsofsynthesisandachievetheresultsshowninFigure8(a).4.CONCLUSIONThisproposedworkisveryeectiveinthefastsynthesisofshapedre”ectorantennastoradiatecontouredbeams.Thismethodexhibitslargestfreedomsbyusingalargenumberofvariablesinthesynthesis,wherethesurfacegridnodesareused.Closedformandcontinuoussolutionsofcostfunctionsderivativesaredeveloped,whichallowsthesurfacevaryingsmoothlywhile,inthemeantime,retainingthecomputationaleciency.Numericalexamplesshowthatthe 374ChouandChoucomputationalcomplexityinthisworkcanbereducedbyanorder(thenumberofvariablesinthetraditionalSDMusingnumericalcomputationsto“ndthederivativesofacostfunction).ACKNOWLEDGMENTFinancialsupportforthisworkfromtheNationalScienceCouncil,Taiwan,isacknowledged.REFERENCES1.Rudge,A.W.,K.Milne,A.D.Olver,andP.Knight,HandbookofAntennaDesign,Vol.1,PeterPeregrinus,London,2.Rusch,W.V.T.,Thecurrentstateofthere”ectorantennaart,ŽIEEETrans.AntennasPropagat.,Vol.32,313…329,1984.3.Cherrette,A.R.,S.W.Lee,andR.J.Acosta,Amethodforproducingashapedcontourradiationpatternusingasinglere”ectorandasinglefeed,ŽIEEETrans.AntennasPropagat.Vol.37,No.6,698…702,Jun.1989.4.Chou,H.T.,P.H.Pathak,andR.J.Burkholder,NovelGaussianbeammethodfortherapidanalysisoflargere”ectorantennas,ŽIEEETrans.AntennasPropagat.,Vol.49,880…893,Jun.2001.5.Chou,H.-T.andP.H.Pathak,Fastgaussianbeambasedsynthe-sisofshapedre”ectorantennasforcontouredbeamapplications,ŽIETProceeding-Microwave,AntennaandPropagations,Vol.151,No.1,13…20,Feb.2004.6.Theunissen,W.,Recon“gurablecontouredbeamsynthesisusingamechanicalfemsurfacedescriptionofdualosetre”ectorantennasurfaces,ŽPh.D.dissertation,UniversityofPretoria,SouthAfrica,1999.7.Duan,D.andY.R.-Samii,Ageneralizeddiractionsynthesistechniqueforhighperformancere”ectorantennas,ŽIEEETrans.AntennasPropagat.,Vol.43,27…40,Jan.1995.8.Westcott,B.S.andA.A.Zaporozhets,Singlere”ectorsynthesisusingananalyticalgradientapproach,ŽElectronicsLettersVol.30,No.18,1462…1463,Sep.1,1994.9.Westcott,B.S.andA.A.Zaporozhets,Dual-re”ectorsynthesisbasedonanalyticalgradient-iterationprocedures,ŽIETProceeding-Microwave,AntennaandPropagations,Vol.142,No.2,129…135,Apr.1995. ProgressInElectromagneticsResearch,PIER92,200937510.Poulton,G.T.andS.G.Hay,Ecientdesignofshapedre”ectorsusingsuccessiveprojections,ŽElectronicsLetters,Vol.27,2156…2158,1991.11.Hay,S.G.,Dual-shaped-re”ectordirectivitypatternsynthesisusingthesuccessiveprojectionsmethod,ŽIETProceeding-Microwave,AntennaandPropagations,Vol.146,No.2,119…124,Apr.1999.12.Stirland,S.J.,Fastsynthesisofshapedre”ectorantennasforcontouredbeams,ŽProceedingsoftheICAP,18…21,Edinburgh,UK,1993.13.Bucci,O.M.,A.Capozzoli,andG.DElia,Aneectivepowersynthesistechniqueforshaped,double-re”ectormultifeedantennas,ŽProgressInElectromagneticsResearch,PIER39,93…123,2003.14.Pathak,P.H.,Techniquesforhigh-frequencyproblems,ŽAntennaHandbook—Theory,ApplicationandDesign,(byY.T.LoandS.W.Lee),Chapter4,VanNostrandReinholdCompany,NewYork,1988.15.Chou,H.T.andP.H.Pathak,Uniformasymptoticsolutionfortheemre”ectionanddiractionofanarbitrarygaussianbeambyasmoothsurfacewithanedge,ŽRadioScience,1319…1336,Jul.…Aug.,199716.Agastra,E.,G.Bellaveglia,L.Lucci,R.Nesti,G.Pelosi,G.Ruggerini,andS.Selleri,Geneticalgorithmoptimizationofhigh-eciencywide-bandmultimodalsquarehornsfordiscretelenses,ŽProgressInElectromagneticsResearch,PIER83,335…352,2008.17.Poulton,G.T.,Powerpatternsynthesisusingthemethodofsuccessiveprojections,ŽIEEEAntennaandPorpagationSocietyInternationalSymposium,Vol.2,667…670,1986.18.Mikki,S.M.andA.A.Kishk,Physicaltheoryforparticleswarmoptimization,ŽProgressInElectromagneticsResearch,PIER75,171…207,2007.19.Pereira,L.andF.Hasselmann,Approximationofshapedre”ectorsurfacesbypseudo-splines,ŽIEEEAntennasandPropagationSocietyInternationalSymposium,Vol.22,327…329,Jun.1984.20.Teixeira,F.L.,J.R.Bergmann,andC.Rego,Acomparisonbetweentechniquesforglobalsurfaceinterpolationinshapedre”ectoranalysis,ŽIEEETrans.onAntennasPropagat.,Vol.42,No.1,47…53,Jan.1994.