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ModernCharacterizationof ElectromagneticSystemsandIts AssociatedMetrology
ModernCharacterizationof ElectromagneticSystemsandIts AssociatedMetrology
TapanK.Sarkar†
SyracuseUniversity
11WexfordRoad,Syracuse,NewYork13214
MagdalenaSalazar-Palma
CarlosIIIUniversityofMadrid
Avda.delaUniversidad30,28911Leganés,Madrid,Spain
MingDaZhu
XidianUniversity
No.2SouthTaibaiRoad,Xi’an,Shaanxi,China
HengChen
SyracuseUniversity
211LafayetteRd.Room425,Syracuse,NY,USA
Thiseditionfirstpublished2021
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LibraryofCongressCataloging-in-PublicationData
Names:Sarkar,Tapan(TapanK.),editor.|Salazar-Palma,Magdalena, editor.|Zhu,MingDa,editor.|Chen,Heng,editor.
Title:Moderncharacterizationofelectromagneticsystemsandits associatedmetrology/editedbyTapanK.Sarkar,Magdalena Salazar-Palma,MingDaZhu,HengChen.
Description:Hoboken,NJ:Wiley,2020.|Includesbibliographical referencesandindex.
Identifiers:LCCN2020008264(print)|LCCN2020008265(ebook)|ISBN 9781119076469(hardback)|ISBN9781119076544(adobepdf)|ISBN 9781119076537(epub)
Subjects:LCSH:Electromagnetism–Mathematics.|Electromagnetic waves–Measurement.
Classification:LCCQC760.M532020(print)|LCCQC760(ebook)|DDC 537/.12–dc23
LCrecordavailableathttps://lccn.loc.gov/2020008264
LCebookrecordavailableathttps://lccn.loc.gov/2020008265
CoverDesign:Wiley
CoverImage:©zfL/GettyImages
Setin10/12ptWarnockbyStraive,Pondicherry,India 10987654321
Contents
Preface xiii
Acknowledgments xxi
TributetoTapanK.Sarkar – MagdalenaSalazarPalma,MingDaZhu, andHengChen xxiii
1MathematicalPrinciplesRelatedtoModernSystemAnalysis 1
Summary 1
1.1Introduction 1
1.2Reduced-RankModelling:BiasVersusVarianceTradeoff 3
1.3AnIntroductiontoSingularValueDecomposition(SVD)andthe TheoryofTotalLeastSquares(TLS) 6
1.3.1SingularValueDecomposition 6
1.3.2TheTheoryofTotalLeastSquares 15
1.4Conclusion 19 References 20
2MatrixPencilMethod(MPM) 21
Summary 21
2.1Introduction 21
2.2DevelopmentoftheMatrixPencilMethodforNoiseContaminated Data 24
2.2.1ProcedureforInterpolatingorExtrapolatingtheSystemResponse UsingtheMatrixPencilMethod 26
2.2.2IllustrationsUsingNumericalData 26
2.2.2.1Example1 26
2.2.2.2Example2 29
2.3ApplicationsoftheMPMforEvaluationoftheCharacteristic ImpedanceofaTransmissionLine 32
2.4ApplicationofMPMfortheComputationoftheS-Parameters WithoutanyAPrioriKnowledgeoftheCharacteristicImpedance 37
2.5ImprovingtheResolutionofNetworkAnalyzerMeasurements UsingMPM 44
2.6MinimizationofMultipathEffectsUsingMPMinAntenna MeasurementsPerformedinNon-AnechoicEnvironments 57
2.6.1ApplicationofaFFT-BasedMethodtoProcesstheData 61
2.6.2ApplicationofMPMtoProcesstheData 64
2.6.3PerformanceofFFTandMPMAppliedtoMeasuredData 67
2.7ApplicationoftheMPMforaSingleEstimateoftheSEM-Poles WhenUtilizingWaveformsfromMultipleLookDirections 74
2.8DirectionofArrival(DOA)EstimationAlongwithTheir FrequencyofOperationUsingMPM 81
2.9EfficientComputationoftheOscillatoryFunctionalVariationinthe TailsoftheSommerfeldIntegralsUsingMPM 85
2.10IdentificationofMultipleObjectsOperatinginFreeSpaceThrough TheirSEMPoleLocationsUsingMPM 91
2.11OtherMiscellaneousApplicationsofMPM 95
2.12Conclusion 95 Appendix2AComputerCodesforImplementingMPM 96 References 99
3TheCauchyMethod 107
Summary 107
3.1Introduction 107
3.2ProcedureforInterpolatingorExtrapolatingtheSystemResponse UsingtheCauchyMethod 112
3.3ExamplestoEstimatetheSystemResponseUsingtheCauchy Method 112
3.3.1Example1 112
3.3.2Example2 116
3.3.3Example3 118
3.4IllustrationofExtrapolationbytheCauchyMethod 120
3.4.1ExtendingtheEfficiencyoftheMomentMethodThrough ExtrapolationbytheCauchyMethod 120
3.4.2InterpolatingResultsforOpticalComputations 123
3.4.3ApplicationtoFilterAnalysis 125
3.4.4BroadbandDeviceCharacterizationUsingFewParameters 127
3.5EffectofNoiseContaminatingtheDataandItsImpactonthe PerformanceoftheCauchyMethod 130
3.5.1PerturbationofInvariantSubspaces 130
3.5.2PerturbationoftheSolutionoftheCauchyMethodDue toAdditiveNoise 131
3.5.3NumericalExample 136
3.6GeneratingHighResolutionWidebandResponsefromSparseand IncompleteAmplitude-OnlyData 138
3.6.1DevelopmentoftheInterpolatoryCauchyMethodfor Amplitude-OnlyData 139
3.6.2InterpolatingHighResolutionAmplitudeResponse 142
3.7GenerationoftheNon-minimumPhaseResponsefrom Amplitude-OnlyDataUsingtheCauchyMethod 148
3.7.1GenerationoftheNon-minimumPhase 149
3.7.2IllustrationThroughNumericalExamples 151
3.8DevelopmentofanAdaptiveCauchyMethod 158
3.8.1Introduction 158
3.8.2AdaptiveInterpolationAlgorithm 159
3.8.3IllustrationUsingNumericalExamples 160
3.8.4Summary 171
3.9EfficientCharacterizationofaFilter 172
3.10ExtractionofResonantFrequenciesofanObjectfromFrequency DomainData 176
3.11Conclusion 180 Appendix3AMATLABCodesfortheCauchyMethod 181 References 187
4ApplicationsoftheHilbertTransform – ANonparametricMethodfor Interpolation/ExtrapolationofData 191
Summary 191
4.1Introduction 192
4.2ConsequenceofCausalityandItsRelationshiptotheHilbert Transform 194
4.3PropertiesoftheHilbertTransform 195
4.4RelationshipBetweentheHilbertandtheFourierTransforms fortheAnalogandtheDiscreteCases 199
4.5MethodologytoExtrapolate/InterpolateDataintheFrequency DomainUsingaNonparametricMethodology 200
4.6InterpolatingMissingData 203
4.7ApplicationoftheHilbertTransformforEfficientComputation oftheSpectrumforNonuniformlySpacedData 213
4.7.1FormulationoftheLeastSquareMethod 217
4.7.2HilbertTransformRelationship 221
4.7.3MagnitudeEstimation 223
4.8Conclusion 229 References 229
5TheSourceReconstructionMethod 235
Summary 235
5.1Introduction 236
5.2AnOverviewoftheSourceReconstructionMethod(SRM) 238
5.3MathematicalFormulationfortheIntegralEquations 239
5.4Near-FieldtoFar-FieldTransformationUsinganEquivalent MagneticCurrentApproach 240
5.4.1DescriptionoftheProposedMethodology 241
5.4.2SolutionoftheIntegralEquationfortheMagneticCurrent 245
5.4.3NumericalResultsUtilizingtheMagneticCurrent 249
5.4.4Summary 268
5.5Near-FieldtoNear/Far-FieldTransformationforArbitrary Near-FieldGeometryUtilizinganEquivalentElectricCurrent 276
5.5.1DescriptionoftheProposedMethodology 278
5.5.2NumericalResultsUsinganEquivalentElectricCurrent 281
5.5.3Summary 286
5.6EvaluatingNear-FieldRadiationPatternsofCommercial Antennas 297
5.6.1Background 297
5.6.2FormulationoftheProblem 301
5.6.3ResultsfortheNear-fieldToFar-fieldTransformation 304
5.6.3.1ABaseStationAntenna 304
5.6.3.2NFtoFFTransformationofaPyramidalHornAntenna 307
5.6.3.3ReferenceVolumeofaBaseStationAntennaforHuman ExposuretoEMFields 310
5.6.4Summary 311
5.7Conclusions 313 References 314
6PlanarNear-FieldtoFar-FieldTransformationUsingaSingleMoving ProbeandaFixedProbeArrays 319 Summary 319
6.1Introduction 320
6.2Theory 322
6.3IntegralEquationFormulation 323
6.4FormulationoftheMatrixEquation 325
6.5UseofanMagneticDipoleArrayasEquivalentSources 328
6.6SampleNumericalResults 329
6.7Summary 337
6.8DifferencesbetweenConventionalModalExpansionandthe EquivalentSourceMethodforPlanarNear-FieldtoFar-Field Transformation 337
6.8.1Introduction 337
6.8.2ModalExpansionMethod 339
6.8.3IntegralEquationApproach 341
6.8.4NumericalExamples 344
6.8.5Summary 351
6.9ADirectOptimizationApproachforSourceReconstruction andNF-FFTransformationUsingAmplitude-OnlyData 352
6.9.1Background 352
6.9.2EquivalentCurrentRepresentation 354
6.9.3OptimizationofaCostFunction 356
6.9.4NumericalSimulation 357
6.9.5ResultsObtainedUtilizingExperimentalData 358
6.9.6Summary 359
6.10UseofComputationalElectromagneticstoEnhancetheAccuracy andEfficiencyofAntennaPatternMeasurementsUsingan ArrayofDipoleProbes 361
6.10.1Introduction 362
6.10.2DevelopmentoftheProposedMethodology 363
6.10.3PhilosophyoftheComputationalMethodology 363
6.10.4FormulationoftheIntegralEquations 365
6.10.5SolutionoftheIntegro-DifferentialEquations 367
6.10.6SampleNumericalResults 369
6.10.6.1Example1 369
6.10.6.2Example2 373
6.10.6.3Example3 377
6.10.6.4Example4 379
6.10.7Summary 384
6.11AFastandEfficientMethodforDeterminingtheFarField PatternsAlongthePrincipalPlanesUsingaRectangular ProbeArray 384
6.11.1Introduction 385
6.11.2DescriptionoftheProposedMethodology 385
6.11.3SampleNumericalResults 387
6.11.3.1Example1 387
6.11.3.2Example2 393
6.11.3.3Example3 397
6.11.3.4Example4 401
6.11.4Summary 406
6.12TheInfluenceoftheSizeofSquareDipoleProbeArray MeasurementontheAccuracyofNF-FFPattern 406
6.12.1IllustrationoftheProposedMethodologyUtilizingSample NumericalResults 407
6.12.1.1Example1 407
6.12.1.2Example2 411
6.12.1.3Example3 416
6.12.1.4Example4 419
6.12.2Summary 428
6.13UseofaFixedProbeArrayMeasuringAmplitude-Only Near-FieldDataforCalculatingtheFar-Field 428
8.2TheConjugateGradientMethodwithFastFourierTransformfor ComputationalEfficiency 495 Contents x
6.13.1ProposedMethodology 429
6.13.2SampleNumericalResults 430
6.13.2.1Example1 430
6.13.2.2Example2 434
6.13.2.3Example3 437
6.13.2.4Example4 437
6.13.3Summary 441
6.14ProbeCorrectionforUsewithElectricallyLargeProbes 442
6.14.1DevelopmentoftheProposedMethodology 443
6.14.2FormulationoftheSolutionMethodology 446
6.14.3SampleNumericalResults 447
6.15Conclusions 449 References 449
7SphericalNear-FieldtoFar-FieldTransformation 453 Summary 453
7.1AnAnalyticalSphericalNear-FieldtoFar-Field Transformation 453
7.1.1Introduction 453
7.1.2AnAnalyticalSphericalNear-FieldtoFar-Field Transformation 454
7.1.3NumericalSimulations 464
7.1.3.1SyntheticData 464
7.1.3.2ExperimentalData 465
7.1.4Summary 468
7.2RadialFieldRetrievalinSphericalScanningforCurrent ReconstructionandNF–FFTransformation 468
7.2.1Background 468
7.2.2AnEquivalentCurrentReconstructionfromSpherical MeasurementPlane 470
7.2.3TheRadialElectricFieldRetrievalAlgorithm 472
7.2.4ResultsObtainedUsingThisFormulation 473
7.2.4.1SimulatedData 473
7.2.4.2UsingMeasuredData 475
7.3Conclusion 482 Appendix7AAFortranBasedComputerProgramforTransforming SphericalNear-FieldtoFar-Field 483 References 489
8DeconvolvingMeasuredElectromagneticResponses 491 Summary 491
8.1Introduction 491
8.2.1Theory 495
8.2.2NumericalResults 498
8.3TotalLeastSquaresApproachUtilizingSingularValue Decomposition 501
8.3.1Theory 501
8.3.2TotalLeastSquares(TLS) 502
8.3.3NumericalResults 506
8.4Conclusion 516 References 516
9PerformanceofDifferentFunctionalsforInterpolation/ ExtrapolationofNear/Far-FieldData 519 Summary 519
9.1Background 520
9.2ApproximatingaFrequencyDomainResponsebyChebyshev Polynomials 521
9.3TheCauchyMethodBasedonGegenbauerPolynomials 531
9.3.1NumericalResultsandDiscussion 537
9.3.1.1ExampleofaHornAntenna 537
9.3.1.2Exampleofa2-elementMicrostripPatchArray 539
9.3.1.3ExampleofaParabolicAntenna 541
9.4Near-FieldtoFar-FieldTransformationofaZenith-Directed ParabolicReflectorUsingtheOrdinaryCauchyMethod 543
9.5Near-FieldtoFar-FieldTransformationofaRotatedParabolic ReflectorUsingtheOrdinaryCauchyMethod 552
9.6Near-FieldtoFar-FieldTransformationofaZenith-Directed ParabolicReflectorUsingtheMatrixPencilMethod 558
9.7Near-FieldtoFar-FieldTransformationofaRotatedParabolic ReflectorUsingtheMatrixPencilMethod 564
9.8Conclusion 569 References 569
10RetrievalofFreeSpaceRadiationPatternsfromMeasuredData inaNon-AnechoicEnvironment 573 Summary 573
10.1ProblemBackground 573
10.2ReviewofPatternReconstructionMethodologies 575
10.3DeconvolutionMethodforRadiationPatternReconstruction 578
10.3.1EquationsandDerivation 578
10.3.2StepsRequiredtoImplementtheProposedMethodology 584
10.3.3ProcessingoftheData 585
10.3.4SimulationExamples 587
10.3.4.1ExampleI:OnePECPlateServesasaReflector 587
10.3.4.2ExampleII:TwoPECPlatesNowServeasReflectors 594
10.3.4.3ExampleIII:FourConnectedPECPlatesServeasReflectors 598
10.3.4.4ExampleIV:UseofaParabolicReflectorAntennaastheAUT 604
10.3.5DiscussionsontheDeconvolutionMethodforRadiationPattern Reconstruction 608
10.4EffectofDifferentTypesofProbeAntennas 608
10.4.1NumericalExamples 608
10.4.1.1ExampleI:UseofaYagiAntennaastheProbe 608
10.4.1.2ExampleII:UseofaParabolicReflectorAntennaastheProbe 612
10.4.1.3ExampleIII:UseofaDipoleAntennaastheProbe 613
10.5EffectofDifferentAntennaSize 619
10.6EffectofUsingDifferentSizesofPECPlates 626
10.7ExtensionoftheDeconvolutionMethodtoThree-Dimensional PatternReconstruction 632
10.7.1MathematicalCharacterizationoftheMethodology 632
10.7.2StepsSummarizingfortheMethodology 635
10.7.3ProcessingtheData 636
10.7.4ResultsforSimulationExamples 638
10.7.4.1ExampleI:FourWidePECPlatesServeasReflectors 640
10.7.4.2ExampleII:FourPECPlatesandtheGroundServeas Reflectors 643
10.7.4.3ExampleIII:SixPlatesForminganUnclosedContourServeas Reflectors 651
10.7.4.4ExampleIV:AntennaMeasurementinaClosedPECBox 659
10.7.4.5ExampleV:SixDielectricPlatesFormingaClosedContour SimulatingaRoom 662
10.8Conclusion 673
AppendixA:DataMappingUsingtheConversionbetweenthe SphericalCoordinateSystemandtheCartesianCoordinate System 675
AppendixB:Descriptionofthe2D-FFTduringtheData Processing 677
References 680
Index 683
Preface
Theareaofelectromagneticsisanevolutionaryone.Intheearlierdaysthe analysisinthisareawaslimitedto11separablecoordinatesystemsforthesolutionofHelmholtzequations.Theelevencoordinatesystemsarerectangular, circularcylinder,ellipticcylinder,paraboliccylinder,spherical,conical, parabolic,prolatespheroidal,oblatespheroidal,ellipsoidalandparaboloidal coordinates.However,Laplace ’sequationisseparablein13coordinatesystems, theadditionaltwobeingthebisphericalandthetoroidalcoordinatesystems. Outsidethesecoordinatesystemsitwasnotpossibletodevelopasolution forelectromagneticproblemsintheearlierdays.However,withtheadvent ofnumericalmethodsthissituationchangedanditwaspossibletosolvereal practicalproblemsinanysystem.Thisdevelopmenttookplaceintwodistinct stagesandwasprimarilyaddressedbyProf.RogerF.Harrington.Inthefirst phaseheproposedtodevelopthesolutionofanelectromagneticfieldproblem intermsofunknowncurrents,bothelectricandmagneticandnotfieldsby placingsomeequivalentcurrentstorepresenttheactualsourcessothatthese currentsproduceexactlythesamedesiredfieldsineachregion.Fromthese currentshecomputedtheelectricandthemagneticvectorpotentialsinany coordinatesystem.Intheintegralrepresentationofthepotentialsintermsof theunknowncurrents,thefreespaceGreen’sfunctionwasusedwhichsimplifiedtheformulationconsiderablyasnocomplicatedformoftheGreen’sfunctionforanycomplicatedenvironmentwasnecessary.Fromthepotentials,the fields,bothelectricandmagnetic,weredevelopedbyinvokingtheMaxwellHertz-Heavisideequations.Thismadethemathematicalanalysisquiteanalytic andsimplifiedmanyofthecomplexitiesrelatedtothecomplicatedGreen’stheorem.Thiswasthemainthemeinhisbook “TimeHarmonicElectromagnetic Fields”,McGrawHill,1961.Attheendofthisbookhetriedtodevelopavariationalformforalltheseconceptssothatanumericaltechniquecanbeapplied andonecansolveanyelectromagneticboundaryvalueproblemofinterest.This themewasfurtherdevelopedinthesecondstagethroughhissecondclassic book “FieldComputationsbyMomentMethods”,MacmillanCompany,1968. Inthesecondbookheillustratedhowtosolveageneralelectromagneticfield
problem.Thisgradualdevelopmenttookalmosthalfacenturytomature.Inthe experimentalrealm,unfortunately,nosuchprogresshasbeenmade.Thismay bepartiallyduetodecisionstakenbythepastleadershipoftheIEEEAntennas andPropagationSociety(AP-S)whofirstessentiallydisassociatedmeasurementsfromtheirprimaryfocusleadingantennameasurementpractitioners toformtheAntennaMeasurementsTechniquesAssociation(AMTA)asan organizationdifferentfromIEEEAP-S.AndlateroneventhenumericaltechniquespartwasnotconsideredinthemainthemeoftheIEEEAntennasand PropagationSocietyleadingtotheformationoftheAppliedComputational ElectromagneticSociety(ACES).However,inrecenttimestheseshortcomings ofthepastdecisionsoftheAP-Sleadershiphavebeenaddressed.
Theobjectiveofthisbookistoadvancethestateoftheartofantennameasurementsandnotbeinglimitedtothesituationthatmeasurementscanbe madeinoneoftheseparablecoordinatesystemsjustlikethestateofelectromagneticsoverhalfacenturyago.Weproposetocarryoutthistransformation intherealmofmeasurementfirstbytryingtofindasetofequivalentcurrents justlikewedointheoryandthensolvefortheseunknowncurrentsusingthe Maxwell-Hertz-HeavisideequationsviatheMethodofMomentspopularized byProf.Harrington.Sincetheexpressionsbetweenthemeasuredfieldsand theunknowncurrentsareanalyticandrelatedbyMaxwell-Hertz-Heaviside equations,themeasurementscanbecarriedoutinanyarbitrarygeometry andnotjustlimitedtotheplanar,cylindricalorsphericalgeometries.The advantageofthisnewmethodologyaspresentedinthisbookthroughthetopic “SourceReconstructionMethod” isthatthemeasurementofthefieldsneed notbedoneusingaNyquistsamplingcriteriawhichopensupnewavenues particularlyintheveryhighfrequencyregimeoftheelectromagneticspectrum whereitmightbedifficulttotakemeasurementsampleshalfawavelength apart.Secondlyaswillbeillustratedthesemeasurementsamplesneednot evenbeperformedinanyspecifiedplane.Alsobecauseoftheanalyticalrelationshipbetweenthesourcesthatgeneratethefieldsandthefieldsthemselves itispossibletogobeyondtheRaleighresolutionlimitandachievesuperresolutioninthediagnosisofradiatingstructures.IntheRaleighlimittheresolutionislimitedbytheuncertaintyprincipleandthatisdeterminedbythe lengthoftheaperturewhoseFouriertransformwearelookingatwhereas inthesuperresolutionsystemthereisnosuchrestriction.Anotherobjective ofthisbookistooutlineaverysimpleproceduretorecoverthenon-minimum phaseofanyelectromagneticsystemusingamplitude-onlydata.Thissimple procedureisbasedontheprincipleofcausalitywhichresultsintheHilbert transformrelationshipbetweentherealandtheimaginarypartsofatransfer functionofanylineartimeinvariantsystem.Thephilosophyofmodelorder reductioncanalsobeimplementedusingtheconceptsoftotalleastsquares alongwiththesingularvaluedecomposition.Thismakestheill-poseddeconvolutionproblemquitestablenumerically.Finally,itisshownhowto
interpolateandextrapolatemeasureddataincludingfillingupthegapofmissingmeasurednear/far-fielddata.
Thebookcontainstenchapters.InChapter1,themathematicalpreliminaries aredescribed.Inthemathematicalfieldofnumericalanalysis,modelorder reductionisthekeytoprocessingmeasureddata.Thisalsoenablesustointerpolateandextrapolatemeasureddata.Thephilosophyofmodelorderreduction isoutlinedinthischapteralongwiththeconceptsoftotalleastsquaresandsingularvaluedecomposition.
InChapter2,wepresentthematrixpencilmethod(MPM)whichisamethodologytoapproximateagivendatasetbyasumofcomplexexponentials.The objectiveistointerpolateandextrapolatedataandalsotoextractcertainparameterssoastocompressthedataset.Firstthemethodologyispresented followedbysomeapplicationinelectromagneticsystemcharacterization. Theapplicationsinvolveusingthismethodologytodeembeddevicecharacteristicsandobtainaccurateandhighresolutioncharacterization,enhance networkanalyzermeasurementswhennotenoughphysicalbandwidthis availableformeasurements,minimizeunwantedreflectionsinantennameasurementsand,whenperformingsystemcharacterizationinanon-anechoic environment,toextractasinglesetofexponentsrepresentingtheresonant frequencyofanobjectwhendatafrommultiplelookanglesaregivenand computedirectionsofarrivalestimationofsignalsalongwiththeirfrequencies ofoperation.Thismethodcanalsobeusedtospeedupthecalculationofthe tailsencounteredintheevaluationoftheSommerfeldintegralsandinmultiple targetcharacterizationinfreespacefromthescattereddatausingtheir characteristicexternalresonancewhicharepopularlyknownasthesingularity expansionmethod(SEM)poles.Referencestootherapplications,including multipathcharacterizationofapropagatingwave,characterizationofthe qualityofpowersystems,inwaveformanalysisandimagingandspeedingup computationsinatimedomainelectromagneticsimulation.Acomputer programimplementingthematrixpencilmethodisgivenintheappendixso thatitcaneasilybeimplementedinpractice.
Innumericalanalysis,interpolationisamethodofestimatingunknowndata withintherangeofknowndatafromtheavailableinformation.Extrapolationis alsotheprocessofapproximatingunknowndataoutsidetherangeoftheknown availabledata.InChapter3,wearegoingtolookattheconceptoftheCauchy methodfortheinterpolationandextrapolationofbothmeasuredandnumericallysimulateddata.TheCauchymethodcandealwithextendingtheefficiency ofthemomentmethodthroughfrequencyextrapolation.Interpolatingresults foropticalcomputations,generationofpassbandusingstopbanddataandvice versa,efficientbroadbanddevicecharacterization,effectofnoiseontheperformanceoftheCauchymethodandforapplicationstoextrapolatingamplitudeonlydataforthefar-fieldorRCSinterpolation/extrapolation.Usingthis methodtogeneratethenon-minimumphaseresponsefromamplitude-only
data,andadaptiveinterpolationforsparselysampleddataisalsoillustrated.In addition,ithasbeenappliedtocharacterizationoffiltersandextractingresonantfrequenciesofobjectsusingfrequencydomaindata.Otherapplications includenon-destructiveevaluationoffruitstatusofmaturityandqualityoffruit juices,RCSapplicationsandtomultidimensionalextrapolation.Acomputer programimplementingtheCauchymethodhasbeenprovidedintheAppendix againforeaseofunderstanding.
Theprevioustwochaptersdiscussedtheparametricmethodsinthecontext oftheprincipleofanalyticcontinuationandprovideditsrelationshiptoreduced rankmodellingusingthetotalleastsquaresbasedsingularvaluedecomposition methodology.Theproblemwithaparametricmethodisthatthequalityofthe solutionisdeterminedbythechoiceofthebasisfunctionsanduseofunsuitable basisfunctionsgeneratebadsolutions.Aprioriitisquitedifficulttorecognize whataregoodbasisfunctionsandwhatarebadbasisfunctionseventhough methodologiesexistintheoryonhowtochoosegoodones.Theadvantageof thenonparametricmethodspresentedinChapter4isthatnosuchchoices ofthebasisfunctionsneedtobemadeasthesolutionprocedurebyitselfdevelopsthenatureofthesolutionandnoaprioriinformationisnecessary.Thisis accomplishedthroughtheuseoftheHilberttransformwhichexploitsoneofthe fundamentalpropertiesofnatureandthatiscausality.TheHilberttransform illustratesthattherealandimaginarypartsofanynonminimumphasetransfer functionforacausalsystemsatisfythisrelationship.Inaddition,someparametrizationcanalsobemadeofthisprocedurewhichcanenableonetogeneratea nonminimumphasefunctionfromitsamplituderesponseandfromthatgeneratethephaseresponse.Thisenablesonetocomputethetimedomain responseofthesystemusingamplitudeonlydatabarringatimedelayinthe response.Thisdelayuncertaintyisremovedinholographyasinsuchaprocedureanamplitudeandphaseinformationismeasuredforaspecificlookangle thuseliminatingthephaseambiguity.Anoverviewofthetechniquealongwith examplesarepresentedtoillustratethismethodology.TheHilberttransform canalsobeusedtospeedupthespectralanalysisofnonuniformlyspaceddata samples.Therefore,inthissectionanovelleastsquaresmethodologyisapplied toafinitedatasetusingtheprincipleofspectralestimation.Thiscanbeapplied fortheanalysisofthefar-fieldpatterncollectedfromunevenlyspacedantennas. Theadvantageofusinganon-uniformlysampleddataisthatitisnotnecessary tosatisfytheNyquistsamplingcriterionaslongastheaveragevalueofthesamplingrateislessthantheNyquistrate.Accurateandefficientcomputationof thespectrumusingaleastsquaresmethodappliedtoafiniteunevenlyspaced dataisalsostudied.
InChapter5,thesourcereconstructionmethod(SRM)ispresented.Itisa recenttechniquedevelopedforantennadiagnosticsandforcarryingout near-field(NF)tofar-field(FF)transformation.TheSRMisbasedontheapplicationoftheelectromagneticEquivalencePrinciple,inwhichoneestablishesan
equivalentcurrentdistributionthatradiatesthesamefieldsastheactualcurrentsinducedintheantennaundertest(AUT).Theknowledgeoftheequivalentcurrentsallowsthedeterminationoftheantennaradiatingelements,aswell asthepredictionoftheAUT-radiatedfieldsoutsidetheequivalentcurrents domain.Theuniquefeatureofthenovelmethodologypresentedisthatit canresolveequivalentcurrentsthataresmallerthanhalfawavelengthinsize, thusprovidingsuper-resolution.Furthermore,themeasurementfieldsamples canbetakenatspacinggreaterthanhalfawavelength,thusgoingbeyondthe classicalsamplingcriteria.Thesetwodistinctivefeaturesarepossibleduetothe choiceofamodel-basedparameterestimationmethodologywherethe unknownsourcesareapproximatedbyabasisinthecomputationalMethod ofMoment(MoM)contextand,secondly,throughtheuseoftheanalyticfree spaceGreen’sfunction.Thelatterconditionalsoguaranteestheinvertibilityof theelectricfieldoperatorandprovidesastablesolutionforthecurrentseven whenevanescentwavesarepresentinthemeasurements.Inaddition,the useofthesingularvaluedecompositioninthesolutionofthematrixequations providestheuserwithaquantitativetooltoassessthequalityandthequantity ofthemeasureddata.Alternatively,theuseoftheiterativeconjugategradient (CG)methodinsolvingtheill-conditionedmatrixequationsfortheequivalent currentscanalsobeimplemented.Twodifferentmethodsarepresentedinthis section.Onethatdealswiththeequivalentmagneticcurrentandthesecond thatdealswiththeequivalentelectriccurrent.Iftheformulationissound,then eitherofthemethodologieswillprovidethesamefar-fieldwhenusingthesame near-fielddata.Examplesarepresentedtoillustratetheapplicabilityandaccuracyoftheproposedmethodologyusingeitheroftheequivalentcurrentsand appliedtoexperimentaldata.Thismethodologyisthenusedfornear-fieldto near/far-fieldtransformationsforarbitrarynear-fieldgeometrytoevaluate thesafedistanceforcommercialantennas.
InChapter6,afastandaccuratemethodispresentedforcomputingfar-field antennapatternsfromplanarnear-fieldmeasurements.Themethodutilizes near-fielddatatodetermineequivalentmagneticcurrentsourcesoverafictitiousplanarsurfacethatencompassestheantenna,andthesecurrentsareused toascertainthefarfields.Undercertainapproximations,thecurrentsshould producethecorrectfarfieldsinallregionsinfrontoftheantennaregardless ofthegeometryoverwhichthenear-fieldmeasurementsaremade.Anelectric fieldintegralequation(EFIE)isdevelopedtorelatethenearfieldstotheequivalentmagneticcurrents.Methodofmoments(MOM)procedureisusedto transformtheintegralequationintoamatrixone.Thematrixequationissolved usingtheiterativeconjugategradientmethod(CGM),andinthecaseofarectangularmatrix,aleast-squaressolutioncanstillbefoundusingthisapproach forthemagneticcurrentswithoutexplicitlycomputingthenormalformofthe equations.Near-fieldtofar-fieldtransformationforplanarscanningmaybe efficientlyperformedundercertainconditionsbyexploitingtheblockToeplitz
structureofthematrixandusingtheconjugategradientmethod(CGM)andthe fastFouriertransform(FFT),therebydrasticallyreducingcomputationtime andstoragerequirements.Numericalresultsarepresentedforseveralantenna configurationsbyextrapolatingthefarfieldsusingsyntheticandexperimental near-fielddata.Itisalsoillustratedthatasinglemovingprobecanbereplaced byanarrayofprobestocomputetheequivalentmagneticcurrentsonthe surfaceenclosingtheAUTinasinglesnapshotratherthantediouslymoving asingleprobeovertheantennaundertesttomeasureitsnear-fields.Itis demonstratedthatinthismethodologyaprobecorrectionevenwhenusing anarrayofdipoleprobesisnotnecessary.Theaccuracyofthismethodology isstudiedasafunctionofthesizeoftheequivalentsurfaceplacedinfrontof theantennaundertestandtheerrorintheestimationofthefar-fieldalongwith thepossibilityofusingarectangularprobearraywhichcanefficientlyandaccuratelyprovidethepatternsintheprincipalplanes.Thiscanalsobeusedwhen amplitude-onlydataarecollectedusinganarrayofprobes.Finally,itisshown thattheprobecorrectioncanbeusefulwhenthesizeoftheprobesisthatofa resonantantennaanditisshownthenhowtocarryitout.
InChapter7,twomethodsforsphericalnear-fieldtofar-fieldtransformation arepresented.Thefirstmethodologyisanexactexplicitanalyticalformulation fortransformingnear-fielddatageneratedoverasphericalsurfacetothefarfieldradiationpattern.Theresultsarevalidatedwithexperimentaldata. Acomputerprograminvolvingthismethodisprovidedattheendofthechapter.Thesecondmethodpresentstheequivalentsourceformulationthroughthe SRMdescribedearliersothatitcanbedeployedtothesphericalscanningcase whereonecomponentofthefieldismissingfromthemeasurements.Againthe methodologyisvalidatedusingothertechniquesandalsowithexperimentaldata.
TwodeconvolutiontechniquesarepresentedinChapter8toillustratehow theill-poseddeconvolutionproblemhasbeenregularized.Dependingonthe natureoftheregularizationutilizedwhichisbasedonthegivendataonecan obtainareasonablygoodapproximatesolution.Thetwotechniquespresented herehavebuiltinself-regularizingschemes.Thisimpliesthattheregularization process,whichdependshighlyonthedata,canbeautomatedasthesolution procedurecontinues.Thefirstmethodisbasedonsolvingtheill-poseddeconvolutionproblembytheiterativeconjugategradientmethod.Thesecond methodusesthemethodoftotalleastsquaresimplementedthroughthesingularvaluedecomposition(SVD)technique.Themethodshavebeenappliedto measureddatatoillustratethenatureoftheirperformance.
Chapter9discussestheuseoftheChebyshevpolynomialsforapproximating functionalvariationsarisinginelectromagneticsasithassomeband-limited propertiesnotavailableinotherpolynomials.Next,theCauchymethodbased onGegenbauerpolynomialsforantennanear-fieldextrapolationandthefarfieldestimationisillustrated.Duetovariousphysicallimitations,thereareoften
missinggapsintheantennanear-fieldmeasurements.However,themissing dataisindispensableifwewanttoaccuratelyevaluatethecompletefar-fieldpatternbyusingthenear-fieldtofar-fieldtransformations.Toaddressthisproblem,anextrapolationmethodbasedontheCauchymethodisproposedto reconstructthemissingpartoftheantennanear-fieldmeasurements.Asthe near-fielddatainthissectionareobtainedonasphericalmeasurementsurface, thefarfieldoftheantennaiscalculatedbythesphericalnear-fieldtofar-field transformationwiththeextrapolateddata.Somenumericalresultsaregivento demonstratetheapplicabilityoftheproposedschemeinantennanear-field extrapolationandfar-fieldestimation.Inaddition,theperformanceofthe GegenbauerpolynomialsarecomparedwiththatofthenormalCauchymethod usingPolynomialexpansionandtheMatrixPencilMethodforusingsimulated missingnear-fielddatafromaparabolicreflectorantenna.
Typically,antennapatternmeasurementsarecarriedoutinananechoic chamber.However,agoodanechoicchamberisveryexpensivetoconstruct. Previousresearcheshaveattemptedtocompensatefortheeffectsofextraneous fieldsmeasuredinanon-anechoicenvironmenttoobtainafreespaceradiation patternthatwouldbemeasuredinananechoicchamber.Chapter10illustrates adeconvolutionmethodologywhichallowstheantennameasurementundera non-anechoictestenvironmentandretrievesthefreespaceradiationpatternof anantennathroughthismeasureddata;thusallowingforeasierandmore affordableantennameasurements.Thisisobtainedbymodellingtheextraneousfieldsasthesystemimpulseresponseofthetestenvironmentandutilizinga referenceantennatoextracttheimpulseresponseoftheenvironmentwhichis usedtoremovetheextraneousfieldsforadesiredantennameasuredunderthe sameenvironmentandretrievetheidealpattern.Theadvantageofthisprocess isthatitdoesnotrequirecalculatingthetimedelaytogateoutthereflections; therefore,itisindependentofthebandwidthoftheantennaandthemeasurementsystem,andthereisnorequirementforpriorknowledgeofthetest environment.
Thisbookisintendedforengineers,researchersandeducatorswhoareplanningtoworkinthefieldofelectromagneticsystemcharacterizationandalso dealwiththeirmeasurementtechniquesandphilosophy.Theprerequisiteto followthematerialsofthebookisabasicundergraduatecourseinthearea ofdynamicelectromagnetictheoryincludingantennatheoryandlinearalgebra. Everyattempthasbeenmadetoguaranteetheaccuracyofthecontentofthe book.Wewouldhoweverappreciatereadersbringingtoourattentionany errorsthatmayhaveappearedinthefinalversion.Errorsand/oranycomments maybeemailedtooneoftheauthors,atsalazar@tsc.uc3m.es,mingda.zhu@live. com,hchen43@syr.edu.
Acknowledgments
GratefulacknowledgementismadetoProf.PramodVarshney,Mr.PeterZaehringerandMs.MarilynPoloskyoftheCASECenterofSyracuseUniversityfor providingfacilitiestomakethisbookpossible.ThanksarealsoduetoProf.Jae Oh,Ms.LauraLawsonandMs.RebeccaNobleoftheDepartmentofElectrical EngineeringandComputerScienceofSyracuseUniversityforprovidingadditionalsupport.ThankstoMichaelJamesRice,Systemsadministratorforthe CollegeofEngineeringandComputerScienceforprovidinginformationtechnologysupportinpreparingthemanuscript.AlsothanksareduetoMr.Brett Kurzmanforpatientlywaitingforustofinishthebook.
TapanK.Sarkar
MagdalenaSalazarPalma(salazar@tsc.uc3m.es) Ming-daZhu(mingda.zhu@live.com) HengChen(hchen43@syr.edu) Syracuse,NewYork
TributetoTapanK.Sarkar
byMagdalenaSalazarPalma,MingDaZhu,andHengChen
ProfessorTapanK.Sarkar,PhD,passed awayon12March2021.Thereviewofthe proofsofthisbookisprobablythelasttask hewasabletoaccomplish.Thus,forus,his coauthors,thisbookwillbealwayscherished andvaluedashislastgifttothescientific community.
Dr.SarkarwasborninKolkata,India,in August1948.HeobtainedhisBachelorof Technology(BT)degreefromtheIndian InstituteofTechnology(IIT),Kharagpur, India,in1969,theMasterofScienceinEngineering(MSCE)degreefromtheUniversity ofNewBrunswick,Fredericton,NB,Canada, in1971,andtheMasterofScience(MS)and Doctoral(PhD)degreesfromSyracuseUniversity,Syracuse,NY,USA,in1975.Hejoined thefacultyoftheElectricalEngineeringandComputerScienceDepartmentat SyracuseUniversityin1979andbecameFullProfessorin1985.Priortothat,he waswiththeTechnicalApplianceCorporation(TACO)DivisionoftheGeneral InstrumentsCorporation(1975–1976).HewasalsoaResearchFellowatthe GordonMcKayLaboratoryforAppliedSciences,HarvardUniversity,Cambridge,MA,USA(1977–1978),andwasfacultymemberattheRochesterInstituteofTechnology,Rochester,NY,USA(1976–1985).ProfessorSarkar receivedtheDoctorHonorisCausadegreefromUniversitéBlaisePascal,ClermontFerrand,France(1998),fromPolytechnicUniversityofMadrid,Madrid, Spain(2004),andfromAaltoUniversity,Helsinki,Finland(2012).Hewasnow emeritusprofessoratSyracuseUniversity.ProfessorSarkarwasaprofessional engineerregisteredinNewYork,USA,andthepresidentofOHRNEnterprises, Inc.,asmallbusinessfoundedin1986andincorporatedintheStateofNew York,USA,performingresearchforgovernment,private,andforeignorganizationsinsystemanalysis.
TributetoTapanK.Sarkar – MagdalenaSalazarPalma,MingDaZhu,andHengChen
Dr.Sarkarwasagiantinthefieldofelectromagnetics,aphenomenal researcherandteacherwhoalsoprovidedaninvaluableservicetothescientific communityinsomanyaspects.
Dr.Sarkarresearchinterestsfocusedonnumericalsolutionstooperator equationsarisinginelectromagneticsandsignalprocessingwithapplication toelectromagneticsystemsanalysisanddesignandwithparticularattention tobuildingsolutionsthatwouldbeappropriateandscalableforpracticaladoptionbyindustry.Amonghismanycontributionstogetherwithhisstudentsand coworkers,itmaybementionedthedevelopmentofthegeneralizedpencil-offunction(GPOF)method,alsoknownasmatrixpencilmethod,forsignalestimationwithcomplexexponentials.BasedonDr.Sarkargroup’sworkonthe originalpencil-of-functionmethod,thetechniqueisusedinelectromagnetic analysesoflayeredstructures,antennaanalysis,andradarsignalprocessing. HeisalsocoauthorofthegeneralpurposeelectromagneticsolverHOBBIES (HigherOrderBasisBasedIntegralEquationSolver).ThelistofProfessorSarkar’soriginalandsubstantivecontributionstothefieldofcomputationalelectromagneticsandantennatheoryisquitelong.Justtonameafew,theseinclude methodsofevaluatingtheSommerfeldintegrals,thealreadymentionedmatrix pencilmethodforapproximatingafunctionbyasumofcomplexexponentials, theconjugategradientmethodandfastFouriertransformmethodfortheefficientnumericalsolutionofintegralequationshavingconvolutionalkernels,the introductionofhigherorderbasisfunctionsinthenumericalsolutionofintegralequationsusingthemethodofmoments,thesolutionoftimedomainproblemsusingtheassociatedLaguerrefunctionsasbasisfunctions,theapplication oftheCauchymethodtothegenerationofaccuratebroadbandinformation fromnarrowbanddata,broadbandantennadesignandanalysis,andnear-field tofar-fieldtransformation,andmanymore.Dr.Sarkar’sworkhasmodernized manysystemsthatincludewirelesssignalpropagation,hasmadepossiblethe designofantennasconsideringtheeffectsoftheplatformswheretheyare deployedforthecurrentandnextgenerationsofairbornesurveillancesystem, andhasdevelopedadaptivemethodologiesthatmadeperformanceofadaptive systemspossibleinrealtime.Hisadvancedcomputationaltechniqueshave beenimplementedforparallelprocessingonsupercomputersforfastandefficientsolutionofextremelylargeelectromagneticfieldproblems.Hehasalso developedantennasystemsandprocessingforultrawidebandapplications. Heappliedphotoconductiveswitchingtechniquesforgenerationofkilovolts amplitudeelectricalpulsesofsubnanoseconddurationwithapplicationsin manyfieldsincludinglowprobabilityinterceptradarsystems.Itisremarkable thatDr.Sarkarhasbeenabletokeepinnovatingforsuchasustainedperiodof timethroughouthiscareer.ProfessorSarkarhasauthoredorcoauthoredmore than380journalarticles,innumerablecontributionsforconferencesandsymposia,16booksand32bookchapters,with24549citationsandh-indexof74 (GoogleScholar).Inthepast,hewaslistedamongtheISIHighlyCited
