Schmidt M Ayyaz Qureshi and Thomas F Eibert Lehrstuhl f57596r Hochfrequenztechnik Technische Universit57572t M57596nchen Arcisstr 21 80333 Munich Germany carstenschmidttumde Abstract Near64257eld transformation algorithms compute the antenna radiat ID: 11144
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3AdaptiveFieldTranslationsInorderfortheplanewaverepresentationtoconverge,theAUTandmeasurementpointboxesmustnotoverlap.Thisrequirementisnotverystrictbutinordertoobtainagoodaccuracyofthetransformedfar- eldpatternalargerseparationbetweenAUTandreceivingboxesisrecommended.Thisisimplementedbyintroducingbuerboxes.Thetransformationisnowextendedsuchthat eldtranslationscanbecarriedouttoboxesondierentlevelsdependingontheirdistancefromtheAUT.Thusitispossibletohavelargerboxesfurtherawayfromtheantennawhilemaintainingtherequirednumberofbuerboxesforthewholesetup.InthiswayonlyboxesclosetotheAUThavetouselowerlevelsfortranslationsandthuslesslevelsintotal.Withadaptive eldtranslationsoneisableto ndagoodcompromisebetweenlargeboxesinthewholesetupgivingsmallcomputationtimeandlessaccuracyversussmallboxesinthewholesetupgivinglargercomputationtimeandhigheraccuracy.AnexampleforaplanarmeasurementsetupisshowninFig.1left.TheblackboxesneartheedgesofthemeasurementplanehavearelativelylargedistancefromtheAUTwhereastheblueboxesclosertothecenterhaveasmallerdistancefromtheAUT.Theboxsizesandnumbersoflevelsarechosenaccordingly.Thesubdivisionintosmallerboxesisjustshownfortheboxontheupperleftcornerforsimplicity.Ifrequired, eldtranslationstomeasurementpointsveryclosetotheAUTcanalsobeconsidereddirectly,asforthepointsinthecenterofthemeasurementplaneshowninFig.1left.4ResultsForvalidationoftheadaptive eldtranslations,aplanarmeasurementsetupisconsideredduetothelargevariationinthemeasurementdistance.TheAUTisahornantennasynthesizedbyadistributionofelectricdipolesat3GHzandtheprobeantennaconsistsoffourelectricdipoles.Thegenerationofthesyntheticdatafollowsthedescriptionin[9].Themeasurementplanehasasizeof4mx4mandissampledequidistantlywithastepsizeof=2.Transformationsarecarriedoutfortwodierentcon gurationsasdescribedinTable1.Columns2-6showthenumberofmeasurementpointswhicharehandledoneachtranslationlevel,e.g.incolumn4thetransformationforthesepointsiscarriedoutusingarecursiveboxstructurewiththreelevels.Column7showstherelativenear- elderrorwhichgivesthedeviationoftheprobesignalscomputedfromthesolutionofthealgorithmwithrespecttothemeasuredprobesignals.Theresiduumoftheiterativesolverwassetto1e-4,theaccuracycanbefurtherenhancedbyloweringtheresiduum.Table1:Con gurationsformultilevelplanewavebasedprocessingofnear- eldmeasurement. Con guration pointsL1pointsL2pointsL3pointsL4pointsL5near- elderror I 000065611.84e-2II 1101640149229164121.23e-3 Thetransformedfar- eldpatternsforbothcon gurationsareshowninFig.2inanE-andH-planecuttogetherwiththeirrespectiveerrorlevelsandthereferencesolution.Theregionofvalidangleisshowninwhite.Anenhancedaccuracyisevidentforthemeasurementutilizingadaptive eldtranslations.5ConclusionTheplanewavebasednear- eldfar- eldtransformationalgorithmwithlownumericalcomplexity,fullprobecorrectioncapabilities,andtheapplicabilityforarbitrarymeasurementgridshasbeenpresented.Inthiscontributionadaptive eldtranslationshavebeenutilizedtohandlevariabledistancesbetweenAUT PlaneWaveBasedNear-FieldFar-FieldTransformationwithAdaptiveField Translations CarstenH.Schmidt , M.AyyazQureshi ,and ThomasF.Eibert LehrstuhlfürHochfrequenztechnik,TechnischeUniversitätMünchen, Arcisstr.21,80333Munich,Germany, carstenschmidt@tum.de Abstract Near- eldtransformationalgorithmscomputetheantennaradiationpatternfromameasurementinthera- diatingnear eld.Theplanewavebasednear- eldtransformationalgorithmhasalownumericalcomplexity whileachievingfullprobecorrectionanditisfurthermoreapplicabletoarbitrarymeasurementgridsgivinga huge exibilityfornear- eldprobing.Thetransmissionequationforasetofmeasurementpointsisevaluated inamultilevelfashionandthe eldtranslationsarenowcarriedoutondierentlevelsdependingonthe distanceofthemeasurementpointfromtheantenna.Theadaptive eldtranslationsallowtoenhancethe accuracyofthetransformedfar- eldpattern. 1Introduction Theradiationpatternofanantennaundertest(AUT)iscommonlymeasuredbyfar- eld,compact range,ornear- eldtechniques[1].Far- eldmeasurementssuerfromthehugespacethatisrequiredto measureelectricallylargeantennasandalsofromenvironmentaleectsifmeasuredoutdoors.Compact rangemeasurementscanbecarriedoutindoorsincontrolledmeasurementfacilitiesbutespeciallythelarge re ectorsformeasurementsatlowerfrequenciesareratherexpensive.Near- eldmeasurementsareconducted inanechoicchambersandthe elddistributionismeasuredintheradiatingnear- eld.Thereforeapost- processingnear- eldfar- eldtransformationisrequiredtocomputethefar- eldpattern[2].Near- eld transformationalgorithmsmodeltheradiationbehavioroftheAUTbyasetofequivalentsources.These equivalentsourcesaresetintorelationtothemeasuredprobesignalsbyatransmissionformulaandare obtainedassolutionoftheinverse eldproblem.Formeasurementsinthenear eldthein uenceofthe eld probehastobeconsideredsinceitisratherintegratingthe eldstrengtharoundthemeasurementpoint thanmeasuringatadiscretepointduetoits niteextension.Thishastobeconsideredwhensettingupthe transmissionformulaandiscalledprobecorrection. Classicalalgorithmsfornear- eldfar- eldtransformationuseeigenmodeexpansionsoftheunderlying coordinatesystemofthemeasurementsystemsuchassphericalmultipolesforsphericalnear- eldmeasure- ments[3]andplanewavesinahalfspaceforplanarnear- eldmeasurements[4].Thesetechniquesevaluate orthogonalitypropertiesoftheeigenmodesandthusrequiremeasurementsonregularlysampledgrids. Inthispaperthemultilevelplanewavebasednear- eldfar- eldtransformationwithlownumericalcom- plexity,fullprobecorrectioncapabilities,andtheabilitytohandlearbitrarymeasurementgridsisrevisited. Thefocusistoenhancethe exibilityofthe eldtranslationssothattheycanbeutilizedonvariouslevels. Thisallowstooptimizetheaccuracyofthetransformedfar- eldpatternsandtheruntime. 2PlaneWaveBasedNear-FieldFar-FieldTransformation Theplanewavebasednear- eldtransformation[5,6]utilizesplanewaves ~ J ( ^ k ) ,whichareassumedfor allspatialdirections,asequivalentsources.Theprobesignaliscomputedbytranslatingtheplanewavesto themeasurementpoint r M utilizingthediagonaltranslationoperator T L ( ^ k; ^ r M ) ,knownfromfastmultipole methods(FMM)[7].Theplaneswavesarethenweightedwiththeprobefar- eldpattern � P ( ^ k; r M ) and (a)E-planecut. (b)H-planecut.Figure2:Comparisonoftransformedfar- eldpatternsforcon gurationsIandIIaccordingtoTable1.andthemeasurementpointswithhierarchicalboxstructuresofdierentnumbersoflevels.Inthiswaytheaccuracyofthetransformedfar- eldpatterncanbeoptimizedasacompromisebetweenlargetranslationboxesresultinginlessbuerboxes,smallruntime,andlessaccuracyandsmalltranslationboxesresultinginmorebuerboxes,largerruntime,andhigheraccuracy.6References1.C.A.Balanis,ModernAntennaHandbook,JohnWiley&Sons,Inc.,2008.2.A.D.Yaghjian,AnOverviewofNear-FieldAntennaMeasurements,"IEEETrans.AntennasPropag.,vol.34,no.1,January1986,pp.3045.3.J.E.Hansen,SphericalNear-FieldAntennaMeasurements,PeterPeregrinus,London,1988.4.D.Kerns,Plane-WaveScattering-MatrixTheoryofAntennasandAntenna-AntennaInteractions,"NationalBureauofStandards,BoulderCO,1981.5.C.H.Schmidt,M.M.Leibfritz,andT.F.Eibert,FullyProbe-CorrectedNear-FieldFar-FieldTrans-formationEmployingPlaneWaveExpansionandDiagonalTranslationOperators,"IEEETrans.AntennasPropag.,vol.56,no.3,March2008,pp.737746.6.C.H.SchmidtandT.F.Eibert,MultilevelPlaneWaveBasedNear-FieldFar-FieldTransformationforElectricallyLargeAntennasinFree-SpaceorAboveMaterialHalfspace,"'IEEETrans.AntennasPropag.,vol.57,no.5,May2009,pp.13821390.7.W.Chew,J.Jin,E.Michielssen,andJ.Song,FastandE cientAlgorithmsinComputationalElectromagnetics,ArtechHouse,Inc.2001.8.Y.Saad,IterativeMethodsforSparseLinearSystems,2nded.,SocietyforIndustrialandAppliedMathematics,2003.9.C.H.Schmidt,D.T.Schobert,andT.F.Eibert,ElectricDipoleBasedDataGenerationforProbe-CorrectedNear-FieldAntennaMeasurements,"5thEuropeanConferenceonAntennasandPropagation,Rome,Italy,April2011. superimposedcoherently.InordertoachievealownumericalcomplexityofO(NlogN)themeasurementpointsaregroupedinahierarchicalmultilevelboxstructure,asseeninFig.1. Figure1:Setupformultilevelplanewavebasedprocessingofnear- eldmeasurement.Left:hierarchicalmeasurementpointgrouping.Thesubdivisionintosmallerboxesisjustshownfortheboxontheupperleftcornerforsimplicity.Right:disaggregation.The eldtranslations~JiNN(^k)=TL(^k;^rbox)�I�^k^k~J(^k)(1)arethencarriedouttotheboxcentersonthehighestleveliNonlyinsteadtoallmeasurementpointsexplicitly.Theplanewavespectra~Jinn(^k)=�Dinn(k;rinn)�I�^k^k~Jin+1n+1(^k)(2)arefurtherprocessedthroughthelevelsbydisaggregationandanterpolationwhere�Dinn(k;rinn)=�Vn(^k)e�jrinnk(3)isthedisaggregationandanterpolationoperator.Disaggregationisasimplephaseshiftbetweentheboxcentersonlevelin+1andlevelinor nallytheboxcenteronthelowestlevelandthemeasurementpoint(seeFig.1right).Anterpolationcanbeseenasadjointoperationtointerpolationandreducesthesamplingrateoftheplanewavespectrawithdecreasingboxsize.Thisispossiblesincethespectralcontentoftheplanewavespectraisproportionaltotheboxsize.TheprobesignalU(rM)=�j!0 4Xk'Xk#Wk#;L#WL'e�j~rMk�P(^k;rM)�I�^k^k~Ji00(k';k#)(4)isobtainedbyapplyingthe nalphaseshifttothemeasurementpointsandweightingtheplanewaveswiththeprobe'sfar- eldpattern.TheintegralisevaluatedbynumericalquadraturewiththequadratureweightsWk#;L#andWL'.Thealgorithmisimplementedwithinaniterativegeneralizedminimumresidual(GMRES)solver[8]inanon-the- ymanner.