IEEE MTT-V053-I05 (2005-05)

283 Pages·2005·21.046 MB·English

by  FANTOMASPING

MAY 2005 VOLUME53 NUMBER 5 IETMAB (ISSN0018-9480)
PAPERS
Chebyshev Collocation and Newton-Type Optimization Methods for the Inverse Problem on Nonuniform Transmission
Lines. ... .... ... ... ... ... ... ... ... ... ... ...... ... .... ... ... ... ... ... ... ... ... ... ..M.Norgren 1561
Four-PortMicrowaveNetworksWithIntrinsicBroad-BandSuppressionofCommon-ModeSignals .. ... ... ... ....
.. ... ... .... ... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... . W.M.FathelbabandM.B.Steer 1569
Ultra-SensitiveDetectionofProteinThermalUnfoldingandRefoldingUsingNear-ZoneMicrowaves.... ... ... ....
.. ... ... .... ... ... ... ... ... ... ... ... ... ... ... .... ... ... ... K.M.TaylorandD.W.vanderWeide 1576
Low-ReflectionSubgridding.. ... ... ... ... ... ... .... ..... .... ... ... ... ... ... ..L.KulasandM.Mrozowski 1587
Table-BasedNonlinearHEMTModelExtractedFromTime-DomainLarge-SignalMeasurements ... ... ... ... ....
.. ... ... .... ... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ...M.C.Currás-Francos 1593
AnAdvancedLow-FrequencyNoiseModelofGaInP–GaAsHBTforAccuratePredictionofPhaseNoiseinOscillators
.. ... ... ..... ..... ... ... ... ..J.-C.Nallatamby,M.Prigent,M.Camiade,A.Sion,C.Gourdon,andJ.J.Obregon 1601
AppropriateFormulationoftheCharacteristicEquationforOpenNonreciprocalLayeredWaveguidesWithDifferentUpper
andLowerHalf-Spaces. ... ... ... ... ... ..... .... ... ... .... ... . R.Rodríguez-Berral,F.Mesa,andF.Medina 1613
CompactMMICCPWandAsymmetricCPSBranch-LineCouplersandWilkinsonDividersUsingShuntandSeriesStub
Loading. . .... ... ... ... ... ... ... ...... ... ... ... ... .... ... ... .K.Hettak,G.A.Morin,andM.G.Stubbs 1624
Synthesis and Design of In-Line -Order Filters With Real Transmission Zeros by Means of Extracted Poles
ImplementedinLow-CostRectangular -PlaneWaveguide. .... ... ... ... ... ... ...... ... ... ... ... ... ....
J.R.Montejo-Garai,J.A.Ruiz-Cruz,J.M.Rebollar,M.J.Padilla-Cruz,A.Oñoro-Navarro,andI.Hidalgo-Carpintero 1636
DigitalSubbandFilteringPredistorterArchitectureforWirelessTransmitters. ... ... ... ... ... ... ... ... ... ....
.. ... ... .... ... ... ... ... ... ... ... ...O.Hammi,S.Boumaiza,M.Jaïdane-Saïdane,andF.M.Ghannouchi 1643
DesignGuidelinesforTerahertzMixersandDetectors . ... ... ....... ... ... .. P.Focardi,W.R.McGrath,andA.Neto 1653
CMOSRFAmplifierandMixerCircuitsUtilizingComplementaryCharacteristicsofParallelCombinedNMOSandPMOS
Devices.. .... ... ... ... ... ... ... ... .... ..... ... ... .... ... ... ... ... ... .. I.Nam,B.Kim,andK.Lee 1662
ALow-Power -BandVoltage-ControlledOscillatorImplementedin200-GHzSiGeHBTTechnology . ... ... ....
.. ... ... .... ..Y.-J.E.Chen,W.-M.L.Kuo,Z.Jin,J.Lee,Y.V.Tretiakov,J.D.Cressler,J.Laskar,andG.Freeman 1672
ModelingDistortioninMultichannelCommunicationSystems. .... .... ..... ... ... .K.M.GharaibehandM.B.Steer 1682
(ContentsContinuedonBackCover)
(ContentsContinuedfromFrontCover)
An AnalyticalTechnique for theSynthesisofCascaded -TupletsCross-CoupledResonators MicrowaveFiltersUsing
MatrixRotations .. ... ... ... ... ... ... ... ...... ... ... .... ... ... ... ... .S.TamiazzoandG.Macchiarella 1693
EffectofMode-StirrerConfigurationsonDielectricHeatingPerformanceinMultimodeMicrowaveApplicators .. ....
.. ... ... .... ... ... ... . P.Plaza-González,J.Monzó-Cabrera,J.M.Catalá-Civera,andD.Sánchez-Hernández 1699
Compact Planar and Vialess Composite Low-Pass Filters Using Folded Stepped-Impedance Resonator on
Liquid-Crystal-PolymerSubstrate.. ... ...... ... . S.Pinel,R.Bairavasubramanian,J.Laskar,andJ.Papapolymerou 1707
PrecisionOpen-EndedCoaxialProbesforInVivoandExVivoDielectricSpectroscopyofBiologicalTissuesatMicrowave
Frequencies...... .D.Popovic,L.McCartney,C.Beasley,M.Lazebnik,M.Okoniewski,S.C.Hagness,andJ.H.Booske 1713
Experimental Class-F Power Amplifier Design Using Computationally Efficient and Accurate Large-Signal pHEMT
Model... .... ... ... ... ... ... ... ... .... ..... ... ... .... ... ... ... ... ... ... . M.WrenandT.J.Brazil 1723
RadiometricMillimeter-WaveDetectionviaOpticalUpconversionandCarrierSuppression .. ... ... ... ... ... ....
.. ... ... .... ... ... ... ... ... ... ... ... ...C.A.Schuetz,J.Murakowski,G.J.Schneider,andD.W.Prather 1732
ExperimentalValidationofAnalysisSoftwareforTunableMicrostripFiltersonMagnetizedFerrites.. ... ... ... ....
.. ... ... .... ... ... ... ... ... ... ... ... ... ... ... .... .. G.León,M.J.Freire,R.R.Boix,andF.Medina 1739
SystematicLinearityAnalysisofRFICsUsingaTwo-PortLumped-Nonlinear-SourceModel. ... ... ... ... ... ....
.. ... ... .... ... ... ... ... ... ... ... ... ... ... ... ....Q.Liang,J.M.Andrews,J.D.Cressler,andG.Niu 1745
MeasurementoftheDielectricConstantsofMetallicNanoparticlesEmbeddedinaParaffinRodatMicrowaveFrequencies
.. ... ... .... ... ... ... ... ... .... ..... ... ... ... ... .... ... ... ... ..Y.-S.Yeh,J.-T.Lue,andZ.-R.Zheng 1756
ComputationalApproachBasedonaParticleSwarmOptimizerforMicrowaveImagingofTwo-DimensionalDielectric
Scatterers. .... ... ... ... ... ... ... ... ... ...... ... ... .... ... ... ... ... ... ... .M.DonelliandA.Massa 1761
DesignofDielectric-FilledCavityFiltersWithUltrawideStopbandCharacteristics .. ... .... ..... ... ... . C.Rauscher 1777
SimplifiedAnalysisTechniquefortheInitialDesignofLTCCFiltersWithAll-CapacitiveCoupling .. ... ... ... ....
.. ... ... .... ... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... .. K.RambabuandJ.Bornemann 1787
-BandOrthomodeTransducerWithWaveguidePortsandBalancedCoaxialProbes.. ....G.EngargiolaandA.Navarrini 1792
Optimum Operation of Asymmetrical-Cells-Based Linear Doherty Power Amplifiers—Uneven Power Drive and Power
Matching. .... ... ... ... ... ... ... ... ...... ... ... ... .... ... ... ... ... J.Kim,J.Cha,I.Kim,andB.Kim 1802
MiniaturizedParallelCoupled-LineBandpassFilterWithSpurious-ResponseSuppression... ... ... ... ... ... ....
.. ... ... .... ... ... ... ... ... ... ... ... ... ... ... .... ... ... ... . P.Cheong,S.-W.Fok,andK.-W.Tam 1810
Periodically Nonuniform Coupled Microstrip-Line Filters With Harmonic Suppression Using Transmission Zero
Reallocation .. ... ... ... ... ... ... ... ... ... ..... .... .... ... ... ... ... ... ... ... ... S.SunandL.Zhu 1817
ParallelCoupledMicrostripFiltersWithFloatingGround-PlaneConductorforSpurious-BandSuppression... ... ....
.. ... ... .... ... ... ... ... ... ... ... ... ... ..M.delCastilloVelázquez-Ahumada,J.Martel,andF.Medina 1823
LETTERS
Commentson“OnDeembeddingofPortDiscontinuitiesinFull-WaveCADModelsofMultiportCircuits”andRelated
Comments.... ... ... ... ... ... ... ... ... ... ... ...... .... ... ... ... ... ... ... ... ... ... ...M.Farina 1829
Authors’Reply .. ... ... ... ... ... ...... ... ... ... ... ... . V.I.Okhmatovski,J.D.Morsey,andA.C.Cangellaris 1829
Comments on “Toward Functional Noninvasive Imaging of Excitable Tissues Inside the Human Body Using Focused
MicrowaveRadiometry”... ... ... ... ... ... ... ... ... ....... ... ... ... ... ... ... ... ... ... . A.N.Reznik 1829
Authors’Reply .. ... ... ... ... ..... .... ... ... ... ... . I.S.Karanasiou,N.K.Uzunoglu,andC.C.Papageorgiou 1831
Correctionson“ModeDiscriminatorBasedonMode-SelectiveCoupling”... ... ... ... ..... .... ... ... ... .W.Wang 1833
InformationforAuthors.. ... ... ... ... ... ... ... ... ... .... ...... ... ... ... ... ... ... ... ... ... ... .... 1834
CALLSFORPAPERS
14thTopicalMeetingonElectricalPerformanceofElectronicPackaging.... ... ... ... ... ...... ... ... ... ... .... 1835
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IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES,VOL.53,NO.5,MAY2005 1561
Chebyshev Collocation and Newton-Type
Optimization Methods for the Inverse Problem
on Nonuniform Transmission Lines
Martin Norgren, Member, IEEE
Abstract—Afrequency-domaininverseproblemforthenonuni- the CG against the QN method has been confirmed for some
form transmissionlineisconsidered.Theparametersof classesofinverseproblems,in[8]forreconstructingpiecewise
thenonuniform lineareinterpolatedbyChebyshevpolynomials,
constantprofilesandin[9]foropticaltomography.
andtheTelegraphersequationsaresolvedbyacollocationmethod
Inthispaper,weconsiderthepossibilityofQNmethodsasan
usingthesamepolynomials.Theinterpolationcoefficientsforthe
unknownparametersarereconstructedbymeansofNewton-type alternativewhenreconstructingthelineparametersfromwide-
optimizationmethodsforwhichtheJacobianmatrixhasbeencal- banded measurement data in the frequency domain. The anal-
culated explicitly. For the reconstruction of one or two parame- ysisisbasedontheChebyshevpolynomialsasaninterpolation
ters,thealgorithmistestedonsyntheticdata,andthenecessityto
basisforboththeunknownparametersandthevoltageandcur-
useregularizationisdiscussed.Finally,thealgorithmistestedwith
rent,inthetransmission-lineequations.
measured reflection data to reconstruct shunt capacitances with
piecewiseconstantprofiles. Chebyshevpolynomialstogetherwiththecollocation(pseu-
dospectral)method[10],[11]havebeenappliedsuccessfullyin
IndexTerms—Collocation,inverseproblem,optimization,trans-
[12] for analyzing transient wave propagation on nonuniform
missionline.
multiconductortransmissionlines.Here,thistechniqueisused
tosolvethe propagationproblem inthe frequencydomain. Of
I. INTRODUCTION similarreasonsasin[12],wechosethecollocationmethod:itis
THE INVERSE problem of parameter reconstruction on suitablefornonconstantcoefficientdifferentialequationsandis
nonuniformtransmissionlinesisofimportanceinvarious expectedtohaverapidconvergenceforsmoothparameterpro-
sensor applications. For example, a medium, e.g., like soil or files [10].
snow, can be diagnosed using the reflection/transmission data Forinverseproblemsthathavebeensolvedbymeansofthe
from a submerged transmission line,which has beendesigned CGmethod[1],[4]–[6],thegradienthasbeencalculatedexplic-
sothattheparametersaresensitivetothepropertiesofthesur- itlybyformingtheFréchetdifferentialoftheobjectivefunction
roundingmedia(see,e.g.,[1]). wherethegradientisobtainedfromasacontinuousfunctionthat
For nonuniform transmission lines, analytical inversion isdiscretizedwhenimplementednumerically.However,thatap-
methods have been developed [2], [3], but the existence or proach does not provide any information about higher order
tractability of an analytical method depends largely on the derivatives.Usingthecollocationmethod,thenecessarytrunca-
particular parameter model used. Hence, many inverse prob- tion results already from the beginning ina finite-dimensional
lemsfortransmissionlineshaveinsteadbeensolvedbymeans parameter space with a quite low dimension for smooth pa-
of optimization methods both in time-domain [4], [5] and rameters.Theresultingalgebraicframeworkfacilitatesexplicit
frequency-domain [6], [7] settings. Optimization methods calculations of derivatives, in principal, of any order. Hence,
exhibit a large versatility for different parameter models and thedirectsolverbasedonChebyshevcollocationcaneasilybe
are also easy to implement due to the availability of well-de- extended to higher order optimization methods like, e.g., full
veloped numerical software packets. The most frequently Newtonmethods.
used optimization method for reconstructing transmission-line This paper is organized as follows. The collocation method
parametershasbeentheconjugategradient(CG)method(see, isdescribedinSectionIIandthecalculationoftheexplicitJa-
e.g., [1], [4]–[6]). However, CG utilizes information from the cobianmatrixisdescribedinSectionIII.InSectionIV,recon-
first-orderderivativesonlyandisthuslikelytobemoreslowly struction results are presented, from synthetic as well as mea-
convergent than, e.g., quasi-Newton (QN) methods, which sureddata.SectionVcontainstheconclusions.
utilizeapproximatesecond-orderderivatives.Theinferiorityof
II. DIRECTPROBLEM
We consider a nonuniform transmission line with the series
ManuscriptreceivedMay28,2004;revisedOctober24,2004.Thisworkwas
supportedbyTheFifthFrameworkProgrammeoftheEuropeanCommission inductance , shunt capacitance , series resistance
underagrant. , and shunt conductance , where is the position
The author is with the Division of Electromagnetic Theory, Kungliga
alongtheline.Forconvenienceoftheanalysis,weassumethat
TekniskaHögskolan,SE-10044Stockholm,Sweden.
DigitalObjectIdentifier10.1109/TMTT.2005.847045 thenonuniformlineoccupiestheregion .
0018-9480/$20.00©2005IEEE
1562 IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES,VOL.53,NO.5,MAY2005
TheTelegrapher’sequationsforthevoltage andthecur- The collocation points shall be located at the extrema of
rent are viz. , (cf. [10]).
Note that the point is omitted since it is where the
(1) boundary condition (5) is enforced. For evaluation of , ,
andtheirderivativesatthecollocationpoints,weintroducean
(2) matrix
where an time dependence has been assumed and sup-
pressed.Thenonuniformlineisexcitedat fromauni- ... ... ... (11)
formlinewithacharacteristicimpedance ,andisterminated
with a load impedance at . The voltages and cur-
rentsatthefeedingandloadendsaredenoted Atthecollocationpoints,wethusobtain
(3)
(4) ... ... (12)
The direct problem is posed as a one-ended boundary-value
problem,wheretheboundaryvaluesat arechosenas
(openend) and analogous expressions for the current . The method re-
(short-circuitedend) quires the values of , , , and at the collocation points
otherwise only.However,forthepurposeofsolvingtheinverseproblem,
(5) it is convenient to interpolate the values using the same set of
where istheloadimpedancethatterminatesthetransmis- Chebyshevpolynomialsasfortheexpansionofthevoltageand
sion line. From the solution of (1), (2), and (5), the reflection current.Hence,wealsointroducethecorrespondingcoefficient
coefficientatthefeedingendbecomes vectors , etc. for the parameters. With the
abbreviations and ,theenforce-
(6) ments of the Telegrapher’s equations at the collocation points
become
A. CollocationMethod
(13)
Thedirectproblemissolvedbyaspectralcollocationmethod,
similar to the one used in [12]. In the region , the where denotes the element-wise product between matrices:
voltageandcurrentareexpandedas , and is an row vector
with unit elements. At the load end, (5) and (7) together with
(7) theproperty imply
(14)
where is the th-order Chebyshev
polynomial, and and are the expansion co- Theexpansioncoefficientvectors and arefinallydetermined
efficients.The derivativesof and areexpandedsimilarly fromtheequation
as follows:
(15)
(8)
where
Thecoefficientsin(8)and(7)arerelatedas[11]
(16)
(9)
where and , .
Inthenumericalimplementation,wetruncatetheexpansion (17)
afterthe th-orderpolynomial.Theexpansioncoefficientsare
collectedinto columnvectors ,
etc., whereby the linear relations in (9) can be written as the isan rowvectorwithunitelementsand isan
followingmatrixmultiplications: columnvector withnullelements.
(10) III. OPTIMIZATIONAPPROACHTOTHEINVERSEPROBLEM
where the matrix represents the derivative (higher order Using Chebyshev interpolation for the parameters, the
derivativesfollowanalogously: ). inverse problem is to determine a subset of the coefficients
NORGREN:CHEBYSHEVCOLLOCATIONANDNEWTON-TYPEOPTIMIZATIONMETHODS 1563
from reflection data measured at dif- is solved. Introducing the row vector , it
ferent frequencies. For brevity, we introduce the parameter nowfollowsfrom(21)–(23)that
vector ,containingallinterpolationcoefficients
(24)
Note that the computational efficiency when solving (23) can
be increased one order if the matrix has been LU factor-
izedwhen(15)wassolved.Letthe vectorsobtainedfromthe
Let denote the calculated reflection coefficient and let derivativesof withrespecttothecoefficients
denotethemeasuredreflectioncoefficient,atthean- and ,respectively,buildupthecolumnsinthe
gular frequency . If the reflection coefficient has been mea- matrices and ,respectively.Itthenfollowsfrom(16)and
sured over a set of frequencies, we obtain the fol- (23)that and areobtainedfromtheequation
lowingvectorwithnonlinearequations:
.
. (18)
.
(25)
Thecorrespondingderivativematricesforthereactiveparame-
tersareobtainedas
The inverse problem is formulated as the minimization of the
(26)
scalarcostfunction
Now it follows that the row–vector gradient of the reflection
(19)
coefficientbecomes[cf.(24)]
Manyavailableoptimizationroutinesincorporatethefirst-order
(27)
derivatives and approximate second-order derivatives through
the Jacobian matrix . The accuracy and, especially, the com-
putationalefficiency,canthenbeimprovedconsiderablyifthe wherebywefinallyobtaintheJacobianmatrix
Jacobian is calculated explicitly rather than by numerical per-
turbations in the parameters. With the Jacobian available, the
gradientofthecostfunctionbecomes
.
. (28)
.
(20)
TobuildtheJacobianmatrix,weneedthegradientofthere-
flectioncoefficientwithrespecttothevector .Let denotea
singlecomponentof ,andletusconcentrateonthereflection
A. Regularization
coefficientfromoneparticularfrequency(hence,the -depen-
Tosuppresserroneousrapidoscillationsinthereconstructed
denceisleftoutforbrevity).From(6),itfollowsthat
parameterprofiles,duetomodelerrorsandnoise-contaminated
measurement data, we must add regularization terms to the
(21)
equation system (18). The conventional Tikhonov regulariza-
tion,i.e.,suppressingthesquarednormofthespatialderivative
From the property , we introduce the vector oftheparameter,isenforcedbyextendingthevector in(18)
,when(3)and(7)yield withthefollowingvectorofequations[cf.(10)]:
(22) (29)
From(15),weobtain wheretheextensionoftheJacobianis
(23)
(30)
where
and is the regularization parameter. Note that any desired
th-orderregularizationcanbeappliedbyinsteadusing .
1564 IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES,VOL.53,NO.5,MAY2005
IV. RECONSTRUCTIONRESULTS
Fortheparticularmodelwithfournondispersivereal-valued
parameters , , , and , it has been shown in [13] that at
mosttwoparameterscanbereconstructedsimultaneously.Thus,
wewillonlyconsiderreconstructionofoneortwoparameters.
Mathematically,therearetwodifferentcasesofone-parameter
reconstructions, or , or , and four different cases of
Fig.1. Reconstructionofshuntcapacitance.Thetrueprofileisdepictedwitha
two-parameter reconstructions ( , or , or , solidline.(a)Withoutnoise:(cid:11)=0dottedline,(cid:11)=0:01dashedline.(b)With
). noise:(cid:11)=0dottedline,(cid:11)=0:1dashedline.
A. ReconstructionsFromSyntheticData
In all reconstructions from synthetic data, we consider a
scaled problem where the line occupies the region
. The characteristic impedance of the feeding line
andloadimpedanceare .Thefrequencyrange
goes from 0.025 to 1.000 in steps of 0.025, i.e., 40 frequency
pointsareused.Inthespectralapproximationandinterpolation Fig.2. Reconstructionofshuntconductance.(a)Withoutnoise:(cid:11)=0dotted
oftheparameters,weuse ,i.e.,thehighestChebyshev line,(cid:11) = 0:005dashedline.(b)Withnoise:(cid:11) = 0dottedline,(cid:11) = 0:1
dashedline.
polynomialusedis .
Toavoidtheinversecrime,thesyntheticdatahasbeengen-
This reconstruction error is a consequence of the numerical
eratedbynumericalintegrationofthe Telegrapher’sequations
differencebecauseadifferentdirectsolverhasbeenusedwhen
(1)and(2)(cf.[6])insteadofusingthespectralmethod,which
generating the data (if the inverse crime is committed, the re-
isusedasadirectsolverintheoptimizationprocedure.
constructionwouldbevirtuallyperfect).However,byimposing
Asinitialguessesoftheparameters,wehaveusedtheirmean
a small amount of regularization, the effect of this systematic
values,whichmaybeestimatedfromlow-frequencyimpedance
error can be effectively suppressed. The smallest deviation in
measurementswhenthefarendofthecableiseithershort-cir-
the reconstructed profile was obtained with , which
cuited ( and ) or open-ended ( and ). If a good initial
yieldsavirtuallyperfectreconstruction;seethedashedlinein
guess is unavailable, one can run the algorithm several times
Fig.1(a).Withnoisydata,weseeinFig.1(b)that,withoutreg-
starting from different initial guesses and take the best fit as
ularization,thereconstructionfailsevenmore.Tostabilizethe
the reconstruction result. A more systematic approach, which,
reconstruction,wethusneedmoreweightontheregularization
in practice, has been proven to reduce the problem with local
termthaninthenoise-freecase.Thebestreconstructionresult
minima even if one starts from a poor initial guess, is to start
fromnoisydatawasobtainedwith [seethedashedline
with a limited setof data from the lowerend of the frequency
inFig.1(b)].
band to obtain a rough reconstruction of the large-scale varia-
Theresultswhenreconstructingtheshuntconductance are
tions.Onethengraduallyincorporatesdatafromhigherfrequen-
similartotheoneswhenreconstructing .Again,regularization
ciestoobtainthesmall-scalevariations(see,e.g.,[8]).
is needed to handle model errors and noise contaminated data
When evaluating the reconstruction algorithm against noise
(seeFig.2).
contaminated data, random noise has been added to the cal-
2) Comparison Between Different Optimization
culated reflection coefficient. The magnitude of the complex-
Methods: All reconstruction results presented have been
valuednoisedatahasbeengeneratedfromaGaussiandistribu-
obtained using a trust-region (TR) optimization method,
tion with a mean value zero and a standard deviation of 0.05.
provided in the MATLAB numerical software [14]. In the
Theargumentofthenoisedatahasbeengeneratedfromauni-
reconstruction of , we compared with other methods:
formdistributionintherangefrom0to .
the Gauss–Newton (GN), the Levenberg–Marquardt (LM)
1) Reconstructionof or : Whenstudyingone-parameter
method, the Broyden–Fletcher–Goldfarb–Shanno (BFGS)
reconstruction,wechosethefollowingprofilesfortheparame-
method, another TR method (denoted TRH) using the exact
ters:
Hessianmatrix,wheretheHessianwascalculatedexplicitly,as
described in the Appendix. All these methods were available
inthenumericalsoftware,andfordetails,wereferto[14]and
[15].Furthermore,wecomparedthemwiththecommonlyused
CGmethod,whichwasimplementedbytheauthor.Theresults
( denotes the th-order Bessel function). Allof the one-pa- ofthecomparisonarepresentedbelowinTableI.
rameterreconstructionsarefromreflectiondatawhenexciting The fastest method is TR, while the CG is the slowest. For
thetransmissionlineattheleft-handside . noise-freedata,GNisfast,butslowerfornoisydata,whichisin
Theshuntcapacitance wasfirstreconstructedfromnoise- accordancewiththeGNbeingmoreefficientforproblemswith
free data without using any regularization. As can been seen smallresiduals[15].TRHisratherslowduetothecalculation
fromthedottedlineinFig.1(a),thereconstructionfailsdueto of the Hessian. The large number of cost-function evaluations
superimposedrapidoscillationsinthereconstructedprofile. indicate that the CG has a slow rate of convergence, although
NORGREN:CHEBYSHEVCOLLOCATIONANDNEWTON-TYPEOPTIMIZATIONMETHODS 1565
TABLE I
COMPARISONBETWEENDIFFERENTOPTIMIZATIONMETHODS,WHEN
RECONSTRUCTINGTHESHUNTCAPACITANCEC(x)
Fig. 5. Reconstruction of series inductance and shunt capacitance from
double-sideddata(losslesscase).Noise-freedatawith(cid:11)=0:006:dottedline.
Noisydatawith(cid:11)=0:09:dashedline.
Fig. 6. Reconstruction of series inductance and shunt capacitance from
double-sideddata(lossycase).Noise-freedatawith(cid:11) = 0:004:dottedline.
Noisydatawith(cid:11)=0:05:dashedline.
Fig. 3. Reconstruction of shunt capacitance and shunt conductance
from noise-free data. Left-hand-sided data with (cid:11) = 0:02: dotted line.
Right-hand-sideddatawith(cid:11) = 0:004:dashedline.Double-sideddatawith
(cid:11) =0:005:dashed–dottedline.
Fig.7. Reconstructionofseriesresistanceandshuntconductance.Noise-free
datawith(cid:11)=0:003:dottedline.Noisydatawith(cid:11)=0:04:dashedline.
Fig.4. Reconstructionofshuntcapacitanceandshuntconductancefromnoisy
data.Left-hand-sideddatawith(cid:11) = 0:05:dottedline.Right-hand-sideddata We first consider the lossless case, when . The
with(cid:11)=0:03:dashedline.Double-sideddatawith(cid:11)=0:09:dashed–dotted reconstructionsfor and fromdouble-sideddataaredepicted
line.
inFig.5.Thereconstructionsexhibitcleardeviationsforboth
noise-freeandnoisydata.
thenumbercanbeslightlyreducedwithanimprovedline-search Next, we introduce losses by setting (still ).
procedure. Comparedwiththepreviouscase,weseeinFig.6thatthede-
3) Simultaneous Reconstruction of and : Using the viationsaresmaller,especiallywhenusingnoise-freedata.
same values of the parameters as when reconstructing one The above results conform with the uniqueness result re-
parameter,wenextconsiderthesimultaneousreconstructionof portedin[13],inwhich,inthelosslesscase,onecannotobtain
and . auniquesimultaneousreconstructionof and .
Thepreviousstudyin[6]indicatedthatatwo-parameterre- 5) SimultaneousReconstructionof and : Whenrecon-
construction is unstable using only one-sided data. To investi- structingbothdissipativeparameters,wechose
gateinmoredetailaboutthatindication,wetestthealgorithm
withone-sidedreflectiondata,bothfromtheleft-andright-hand
side,aswellasdouble-sideddata.
The results using noise-free data are depicted in Fig. 3.
Even though regularization has been used to suppress the
influence of small numerical errors, the reconstructions from
one-sided data clearly exhibit deviations. The reconstructions
The reconstructions for and from double-sided data are
fromdouble-sideddataare,ontheotherhand,virtuallyperfect.
depictedinFig.7.Thedeviations,usingnoise-freeornoisydata,
TheresultswhenusingnoisydataaredepictedinFig.4,where
arelargerthanintheotherconsideredcases.
the reconstructions from one-sided data exhibit much larger
deviationsthantheonesfromdouble-sideddata.
B. ReconstructionsFromMeasurementData
4) SimultaneousReconstructionof and : Whenrecon-
structingbothreactiveparameters,wechose Finally, we test the reconstruction algorithm on measured
data.Thetransmissionlinestobeinvestigatedarebuiltfromthe
samekindofflatbandcablethathasbeenusedforsoilmoisture
1566 IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES,VOL.53,NO.5,MAY2005
TABLE II
NOMINAL VALUES FOR THE DIELECTRIC MEDIA
AROUNDTHEFLATBANDCABLE
Fig.8. Resistorsattheloadend.
Fig.10. Connectoratthefeedingend.
Near the load end, we can expect an increased pile up of
chargesontheconductors,whichyieldsanextraamountofca-
pacitanceattheload.Theloadimpedancethusbecomes
(32)
Fig.9. FlatbandcablesandwichedbetweentwoblocksofPlexiglas.
where pFisthestraycapacitanceattheloadend.
andsnowwatercontentdeterminationsattheInstituteforMete-
The measurements were performed with an
orologyandClimateresearch(IMK),Karlsruhe,Germany(see
HP8510C/HP8517A network analyzer system. The
[1] and [5]). The band cable consists of three strip conductors
feeding coaxial cable from the network analyzer has a
ofcopperembeddedinaplasticbandofpolyethylene(see,e.g.,
characteristic impedance . At the feeding end, the
Fig. 8).
outerconductorsofthebandcableareattachedtothescreenof
Theseriesinductancefortheevenmodeisestimatedto
the coaxial connector via two wires, which are approximately
nH/m[1].Foracablesubmergedintoadielectricmedium,
2cminlength(seeFig.10).
experimentsatIMKindicatethattheshuntcapacitanceforthe
Atthefeedingpoint,thesewiresyieldsanincreasedamount
evenmodecanbeestimatedwiththeformula
of shunt capacitance, which is modeled as a capacitor
pF.Tocompensateforthisstraycapacitance,wemustuse
(31)
thefollowingreflectioncoefficientastheinputtothereconstruc-
tionalgorithm:
where is the relative permittivity in the surrounding
media,ofwhichpropertiesvaryalongthecable. , and
(33)
arepartcapacitanceswiththevalues[1]
where isthereflectioncoefficientrecordedbythenetwork
pF/m pF/m pF/m.
analyzer.
Bothofthestraycapacitances and wereestimated
To obtain a varying , i.e., a nonuniform shunt capacitance,
frompriormeasurementsonshortersamplesofthecable,which
certainsectionsofthecablearesandwichedbetweenblocksof
weresurroundedbyairandeithershortcircuitedorresistively
plasticmaterial(seeFig.9).
loadedattherearend.
Thetwokindsofplasticmaterialsusedintheexperimentsare
2) ReconstructionsofRelativePermittivityFromOne-Sided
Plexiglas and polyethylene. We neither had information from
ReflectionData: In the first experiment,a 775-mm section of
anymanufactureraboutthedielectricconstantsforthesemedia,
thecablewassandwichedbetweentwoblocksofPlexiglas(cf.
nordidwemeasuredthedielectricconstantbyanyothermeans.
Fig. 9). The order and lengths of the sections along the cable
Thus,theonlyinformationwecomparewithinthisstudyisthe
were
followingnominalvaluesfoundin[16]and[17](seeTableII).
Sincetheplasticmaterialshavenegligiblelosses,wetakethe air mm Plexiglas mm air mm
shunt conductance . The seriesresistance inthe copper
conductorsisalsosmalland,hence,neglected,i.e., . Measurement data from 382 evenly spaced frequencies in the
1) MeasurementSetupandCompensationforStrayCapac- rangeof45–500MHzwereusedforthereconstruction.Inthe
itances: The length of the band cable was 2.00 m, and it was spectral approximation, we used . The inversion al-
terminated with two 390- resistors between the central con- gorithm was tested with constant initial guesses for the shunt
ductortoeachoftheouterconductors,yieldingaloadresistance capacitance in the range of 20–30 pF/m in steps of 2 pF/m.
fortheevenmode(seeFig.8). The smallest value of the cost function and at the same time
NORGREN:CHEBYSHEVCOLLOCATIONANDNEWTON-TYPEOPTIMIZATIONMETHODS 1567
Fig. 11. Reconstruction of shunt capacitance along a cable sandwiched Fig. 12. Reconstruction of shunt capacitance along a cable sandwiched
betweenPlexiglasin0:500m<x<1:275m. between Plexiglas in 0:250 m < x < 1:025 m and polyethylene in
1:300m < x < 1:658m.
thesmallestnumberofiterationswereobtainedwiththeinitial
guessof26pF/m.Theresultofthereconstructionisdepictedin , i.e., quite close to the value in the previous case.
Fig. 11. For the polyethylene, we get a reconstructed value around
From the prior knowledge that the permittivity of the sur- 33 pF/m, which yields , i.e., below the nominal
rounding media is piecewise constant, we see that the recon- valueinTableII.However,althoughtherearesystematicerrors
structionissuccessful.TheboundariesoftheblocksofPlexiglas due to the air gaps, the proportion between the reconstructed
are located correctly and, elsewhere, the profile tends to fluc- values for Plexiglas and polyethylene are in accordance with
tuatearoundpiecewiseconstantvalues.Wherethecableissur- theproportionbetweenthenominalvaluesinTableII.
roundedbyair,theshuntcapacitanceisapproximately18pF/m,
which,accordingtotheexperimentallydeterminedmixingfor-
mula (31), yields in the surrounding medium (air). V. DISCUSSIONANDCONCLUSIONS
InthePlexiglasregion,theshuntcapacitanceisapproximately
For the direct problem, the convergence of the Chebyshev
37 pF/m, which yields for the Plexiglas; a value
collocation method is very fast for smooth parameter profiles.
slightlybelowthenominalvalueinTableII.Therearemainly
Thus,theoptimization-basedalgorithmisveryefficientforre-
two reasons for this difference: the mixing formula (31) has
constructingsmoothprofiles,requiringamoderatenumber of
beendeterminedusingmediahavingalargeextentoutsidethe
expansionfunctions.Itisstraightforwardtochangethe
cable. Here, the blocks of Plexiglas has a thickness of 24 mm
modeltootherparameterdependenciesbymodifyingthe ma-
(see Fig. 9) whereby the air region outside the blocks reduces
trix in (15). Such dependencies are, e.g., a surface-resistance
theshuntcapacitance.Therearealsosmallairgapsbetweenthe
modelfor [6]oradispersionmodelforthedielectricmedium,
cable and the blocks of Plexiglas. As can be seen in Fig. 10,
whichdeterminesboth and [7].
thecablehasindentationsintheinsulationneartheconductors,
The simulations on synthetic data show that stable recon-
wherethefieldisstrong.Hence,betweenthePlexiglasandthe
structions of one parameter can be achieved using one-sided
insulation,therewillbeairgaps,whichreducetheshuntcapac-
reflection data. For two parameters, one must, in practice, use
itance.
double-sided reflection data and the only cases with reason-
Inthesecondexperiment,onesectionofthecablewassand-
able stability are the simultaneous reconstructions of a reac-
wiched between the same two blocks of Plexiglas that were
tiveparametertogetherwithadissipativeparameter.Inallcir-
used in the first experiment. Another section of the cable was
cumstances,onehastouseregularizationtoachievestablere-
sandwichedbetweentwoblocksofpolyethylene.Theorderand
constructions. The comparison between different optimization
lengthsofthesectionsalongthecablewere
methodsindicatethatNewton-typemethodsaremoreefficient
thantheCGmethod.
air mmPlexiglas mmair mm
The reconstructions from measured reflection data show
polyethylene mmair mm
thatthemethodmayalsobeusedforreconstructingpiecewise
Measurement data from 633 evenly spaced frequencies in the constantparameters,buttheconvergencethenrequiresalarger
rangeof45–800MHzwereused.Inthespectralapproximation, number of expansion functions, which slows down the com-
weused .Thealgorithmwastestedwithconstantinitial putation. The reconstructions of the permittivities of different
guesses in the range of 26–32 pF/m in steps of 2 pF/m. The materials around the band cable yields consistent results, al-
smallestvalueofthecostfunctionandthesmallestnumberof though the values are slightly too high in the air regions and
iterationswereobtainedwiththeinitialguessof30pF/m.The slightlytoolowintheregionssurroundedbyplasticmaterials.
resultisdepictedinFig.12. Theseerrorsmaypartlybeexplainedbytheairgaps.Also,for
Also in this more difficult case, the reconstruction is suc- thebandcable,thereareuncertaintiesinthevaluesoftheseries
cessful. The boundaries between the different regions are inductance and the part capacitances in the mixing formula
located correctly and, elsewhere, the profile tend to fluctuate (31) (see, e.g., [18] where slightly different values have been
around piecewise constant values. For the regions where proposed and used). Another source of error for this kind of
the cable is surrounded by air, we obtain slightly different open cable is radiation, which introduces dissipation, which
values of the shunt capacitance in the range of approximately cannot be easily included in the model. The introduction of
18-19 pF/m, which yields –1.11. For the Plexiglas, stray capacitances at the endpoints is necessary to account for
we get a reconstructed value around 36 pF/m, which yields thegeometricalmismatchesattheconnectorandload.