IEEE MTT-V055-I03 (2007-03)

156 Pages·2007·19.858 MB·English

by  FANTOMASPING

MARCH2007 VOLUME55 NUMBER3 IETMAB (ISSN0018-9480)
PAPERS
LinearandNonlinearDeviceModeling
VolterraBehavioralModelforWidebandRFAmplifiers ........................................................................
.............................................................. C.Crespo-Cadenas,J.Reina-Tosina,andM.J.Madero-Ayora 449
ActiveCircuits,SemiconductorDevices,andICs
DesignandAnalysisofTransmit/ReceiveSwitchinTriple-WellCMOSforMIMOWirelessSystems ....................
........................................................................................................... A.PohandY.P.Zhang 458
AHigh-PerformanceCMOSVoltage-ControlledOscillatorforUltra-Low-VoltageOperations .............................
........................................................................................................ H.-H.HsiehandL.-H.Lu 467
A15/30-GHzDual-BandMultiphaseVoltage-ControlledOscillatorin0.18- mCMOS.....................................
........................................................................................... H.-H.Hsieh,Y.-C.Hsu,andL.-H.Lu 474
DesignofClassEAmplifierWithNonlinearandLinearShuntCapacitancesforAnyDutyCycle .........................
................................................................................. A.Mediano,P.Molina-Gaudó,andC.Bernal 484
AnalysisandDesignofaDynamicPredistorterforWCDMAHandsetPowerAmplifiers ...................................
.............................................. S.Yamanouchi,Y.Aoki,K.Kunihiro,T.Hirayama,T.Miyazaki,andH.Hida 493
Millimeter-WaveandTerahertzTechnologies
TheDirectDetectionEffectintheHot-ElectronBolometerMixerSensitivityCalibration...................................
.................................................. S.Cherednichenko,V.Drakinskiy,T.Berg,E.L.Kollberg,andI.Angelov 504
FieldAnalysisandGuidedWaves
ModelingEffectsofRandomRoughInterfaceonPowerAbsorptionBetweenDielectricandConductiveMediumin3-D
Problem ...................................................................................X.Gu,L. Tsang,andH.Braunisch 511
Modelingof3-DSurfaceRoughnessEffectsWithApplicationto -CoaxialLines ...... M.V.Lukic´ andD.S.Filipovic 518
(ContentsContinuedonBackCover)
(ContentsContinuedfromFrontCover)
CADAlgorithmsandNumericalTechniques
EquivalentSPICECircuitsWithGuaranteedPassivityFromNonpassiveModels .........A.LameckiandM.Mrozowski 526
SingularTetrahedralFiniteElementsforVectorElectromagnetics .................................................J.P.Webb 533
Space-MappingOptimizationWithAdaptiveSurrogateModel................................. S.KozielandJ.W.Bandler 541
FiltersandMultiplexers
CeramicLayer-By-LayerStereolithographyfortheManufacturingof3-DMillimeter-WaveFilters........................
......................................................... N.Delhote,D.Baillargeat,S.Verdeyme,C.Delage,andC.Chaput 548
Packaging,Interconnects,MCMs,Hybrids,andPassiveCircuitElements
HeatConductioninMicrowaveDevicesWithOrthotropicandTemperature-DependentThermalConductivity...........
............................................................................................................................. J.Ditri 555
Low-LossPatternedGroundShieldInterconnectTransmissionLinesinAdvancedICProcesses ...........................
................................................................. L.F.Tiemeijer,R.M.T.Pijper,R.J.Havens,andO.Hubert 561
CompactLeft-HandedTransmissionLineasaLinearPhase–VoltageModulatorandEfficientHarmonicGenerator .....
................................................................ H.Kim,A.B.Kozyrev,A.Karbassi,andD.W.vanderWeide 571
Microwave Photonics
AnLTCC-BasedWirelessTransceiverforRadio-Over-FiberApplications....................................................
............................................................................L.Pergola,R.Gindera,D.Jäger,andR.Vahldieck 579
Biological,Imaging,andMedicalApplications
Developmentofa2.45-GHzLocalExposureSystemforInVivoStudyonOcularEffects...................................
.... K.Wake,H.Hongo,S.Watanabe,M.Taki,Y.Kamimura,Y.Yamanaka,T.Uno,M.Kojima,I.Hata,andK.Sasaki 588
LETTERS
Commentson“TOAEstimationforIR-UWBSystemsWithDifferentTransceiverTypes”.................. S. Nadarajah 597
Authors’Reply ........................................................................ I. Guvenc,Z.Sahinoglu,andP.V.Orlik 598
InformationforAuthors 599
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Editors-In-Chief
DYLAN WILLIAMS AssociateEditors
NIST
Boulder,CO80305USA DANIELDEZUTTER YOSHIO NIKAWA ZOYA POPOVIC RICHARD SNYDER
Phone:[phone] Universiteit Gent Kokushikan Univ. Univ.ofColorado,Boulder RSMicrowaveCompany
Fax:[phone] Belgium Japan USA USA
email:[email protected] email:[email protected] email:[email protected] email:[email protected] email:[email protected]
AMIRMORTAZAWI KENJI ITOH JOSÉ PEDRO SANJAY RAMAN RUEY-BEEI WU
Univ. of Michigan MitsubishiElectronics Univ. of Aveiro VirginiaPolytech.Inst.andStateUniv. NationalTaiwanUniv.
AnnArbor,MI48109-2122USA Japan Portugal USA Taiwan, R.O.C.
Phone:[phone] email:[email protected] email:jcp.mtted.av.it.pt email:[email protected] email:[email protected]
Fax:[phone] JENSHAN LIN ALEXANDERYAKOVLEV
email:[email protected] Univ. of Florida Univ. of Mississippi
USA USA
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IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES,VOL.55,NO.3,MARCH2007 449
Volterra Behavioral Model for
Wideband RF Amplifiers
CarlosCrespo-Cadenas, Associate Member, IEEE, JavierReina-Tosina, Associate Member, IEEE, and
María J.Madero-Ayora
Abstract—This paper proposes a behavioral modeling ap- is its high computationalcomplexity [5]. Thegreat number of
proachforthedescriptionofnonlinearitiesinwidebandwireless coefficientsrequiredforthedescriptionofsystemswithastrong
communication circuits with memory. The model is formally
nonlinearityandlongmemoryhassteeredtheworkofmanyre-
derived exploiting the dependence on frequency of the amplifier
searchers in order to reduce the number of model parameters.
nonlinear transfer functions and reduce the number of param-
eters in a general Volterra-based behavioral model. To validate Probably the most manageable solution to reduce the number
the proposed approach, a commercial amplifier at 915 MHz, ofcoefficientsisthememorypolynomial(MP)modelproposed
exhibiting nonlinear memory effects, has been widely charac- in [1], the structure of which presents a notable truncation in
terized using different stimuli, including two tones, quadrature
thenumberofparameters.Althoughthereductionobtainedwith
phase-shift keying wideband code division multiple access, and
thissimplifiedVolterramodelisimportant,thenumberofcoef-
16-quadrature amplitude modulation signals with rectangular
androot-raisedcosineconformingpulses.Thetheoreticalresults ficientsremainshigh,particularlyinthecaseofamplifierswith
have been compared with experimental data demonstrating that long memory. To achieve a further reduction, an extension of
the model performance is comparable to the well-established theMPmodelwithasparsedelaytapstructurewasproposedin
memory polynomial model. Calculated and measured baseband
[2].
waveforms, signal constellation, spectral regrowth and adjacent
Itisnotclearhowtheparameterreductionoftheseparticular
channelpowerratioaretightlycoincidentinallcases,emphasizing
therelevanceoftheproposedmodel. structuresaffectstheattainableaccuracyofthemodelowingto
thepossibleimportanceofotherunderestimatedterms.Specif-
Index Terms—Behavioral models, microwave amplifiers, non-
ically, the need for considering those neglected terms was the
linearmemoryeffects,Volterraseries.
purpose of the novel structure reported in [3]. That new ap-
proachisbasedonthepruningofredundantkernelsinthefull
I. INTRODUCTION
Volterra-seriesmodelsothatthecoefficientswithlesseffecton
AS A fundamental block in wireless communications sys- the output signal are discarded following an a posteriori pro-
tems,thepoweramplifier(PA)hasundergoneexhaustive cedure.Despitethesignificanceofthecitedmodels,itisdesir-
studyofitscharacteristics,inparticularthoserelatedwithnon- abletodevelopanapproachwithanoptimizednumberofcoef-
linear memory effects. Many efforts have been devoted to ob- ficients,sustainedontheoreticalprinciplesandwithnoneedof
tainbehavioralmodelsformicrowavePAs,forwhichtheoutput apreviousempiricalselection.Thatwastheaimoftheauthors’
of this black-box method is predicted without knowledge of initialstudyofanamplifierwithoneFETbasedonitssimpli-
the nonlinear device internal structure. The goals of these ap- fiedequivalentcircuit.Athird-ordermodelwasvalidatedwith
proaches are, on the one hand, reduction of complexity main- experimentaldataandpartialresultswerepresentedin[6].
taininganaccuracycomparabletotheresultsobtainedwithcir- In this paper, the authors introduce the demonstration of a
cuit-levelsimulationsand,ontheotherhand,asimplemethod fifth-order Volterra model for a general amplifier with band-
toextractthemodelparameters.Anindicatoroftheimportance widthlargerthantheRFsignalband.Theapproachallowsthe
oftheseapproachesisthevaluableworkpresentedoverthelast analysis of nonlinear memory effects from a model based on
years, e.g., [1]–[3], and the recent publication of an extensive thetransferfunctionswithnorestrictionsinthenumberorkind
revisionrelatedtothistopic[4]. ofnonlineardevicescomposingtheamplifier.InSectionII,the
Exploitingthebandlimitedcharacterofwirelesssignals,PA studyofawidebandamplifierandthecompletionofitsbehav-
descriptioncanbetranslatedintoanenveloperepresentationand ioral model based on a Volterra-series approach is described.
frequentlyhasbeendeducedasVolterraseries,aprocedurethat The frequency independence of the amplifier response inside
treats this problem in a strictly and an orderly way. However, the RF bandwidth is exploited to reduce the order model and
onedifficultyofamplifiermodelingusingthisVolterraapproach torevealthedependenceofcoefficientsextractiononthesam-
plingrate.SectionIIIdescribestheprocedureofparameterex-
ManuscriptreceivedAugust8,2006;revisedOctober26,2006.Thiswork traction,whichhasbeensimplifiedbecauseoforderreduction
wassupportedbytheSpanishNationalBoardofScientificandTechnological inthemodelstructure.Applicationofthemethodtoacommer-
ResearchunderProjectTEC2004-06451-C05-03.
cialamplifierandcomparisonwiththeMPmodel,andalsowith
TheauthorsarewiththeDepartamentodeTeoríadelaSeñalyComunica-
ciones,EscuelaSuperiordeIngenieros,UniversidaddeSevilla,41092Seville, experimentaldatausingdifferenttypesofinputsignals,ispre-
Spain(e-mail:[email protected];[email protected];[email protected]). sentedtovalidatethedemonstratedtheoreticalresults.Finally,
Colorversionsofoneormoreofthefiguresinthispaperareavailableonline
ageneralizationofthemodeltoanyorderisproposedandsome
athttp://ieeexplore.ieee.org.
DigitalObjectIdentifier10.1109/TMTT.2006.890514 relevantstatementsarecommentedupon.
0018-9480/$25.00©2007IEEE
450 IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES,VOL.55,NO.3,MARCH2007
for . In (2), is a vector with its first
componentsequal to and the remaining componentsare
equalto isacolumnvectorhaving
basebandfrequencycomponents, isthetransposeof ,and
isacolumnvectorwithits componentsequalto .
The bandlimited condition of the input signal allows to ne-
glecttheintegraloutsidetwo -dimensionalboxeswithlength
andcenteredat .Consequently,itispossibletosubsti-
tute withanequivalenttransferfunctionbandlimitedinto
these -dimensional hypercubes , from which the ban-
Fig.1. Generalschematicofanonlinearcircuit. dlimited equivalent Volterra kernels are obtained making use
of a multidimensional inverse Fourier transform. This equiva-
lenttransferfunctionandthediscrete-timecorrespondingkernel
II. VOLTERRAMODELFORAWIDEBANDAMPLIFIER satisfytherelation
A. Volterra-BasedBehavioralModelBackground (3)
Letageneralamplifierberepresentedbythecircuitshownin
Fig.1.Nonlinearitiesareconstitutedbytheirlinearcomponents, Substitution in (2) allows to separate the integrals and to ob-
includedintheassociatedlinearcircuit,andnonlinearsources, tain the output component . Therefore, after sampling at
whichareassumedtobedependentontwocontrolvoltages instants ,(1)isimmediatelyderived.
and . Let the input be an RF current excitation and The Volterra model (1) is a very general result, but it has a
betheoutputvoltagecorrespondingtothefundamentalfre- highdegreeofdifficultyduetothelargenumberofparameters
quencyzone,centeredat .Makinguseofthefactthatawire- and numerical operations involved [5]. The complexity of the
lesssignalcommonlyhasabandwidth negligiblewithrespect problemisrevealedinthefactthatthekernels forman
to the carrier frequency , the discrete time-domain complex -dimensionalgriddefined bythe discretedelaysineach axis
envelope Volterra model for this general nonlinear system can ofthemultidimensionalspace ;hence,itisdesirable
beexpressedas toreducethenumberofthesedelays.
One of the most extended methods proposed to achieve a
more manageable number of parameters is the MP model de-
scribed in [1]. For this model, the reduction in the number of
coefficientsisobtainedbyselectingonlythedelayspositioned
in the diagonal, i.e., the delays along the direction defined by
.Moreover,ifasparsedelaytapstructure
isadoptedandonlythemostsignificantdelaysareretainedfol-
lowing an a posteriori procedure, a further important cutback
(1)
inthenumberofcoefficientscanbeprocured[2].However,the
model precision can be diminished due to the possible impor-
where and arecomplexenvelopesamplesoftheinput
tanceofnondiagonalterms.Followingamorerelaxedpruning
andoutputRFsignals,respectively, representsthedis-
approach,whichalsoretainsthetermsnearthediagonal,amore
crete Volterra kernels of order , and is an -dimensional
recentmodelwasproposedwithaconsequentimprovementin
vectorcomposedoftheinteger-valueddelays
precision at the expense of a moderate increase in the number
[3].AlthoughasamplingrateequaltotheinputsignalRFband-
ofcoefficients[3].
widthissufficientformemorylessnonlinearsystemidentifica-
Although the MP model has proven to be effective and the
tion [7],itshould beincreased according tothe broadeningof
reduction of coefficients is considerable, the lack of a theoret-
theoutputbandwidth ifaliasinghastobeavoided.Asacon-
ical justification originates the need of these empirical-based
sequence,foranadequaterepresentationoftheoutput ,the
methods. Additionally, important issues as the adequate sam-
samplingtimehastobereducedcorrespondinglyto .
plingrateorthedependenceofthediscretekernelsonthissam-
Equation (1) is a discrete-time Volterra series derived from
plingratein(1)shouldbeaddressedbyabehavioralmodel.
therepresentationusingthemultidimensionalnonlineartransfer
functions (NLTFs) [8]. In continuous-time form, the
B. NLTFsforaWidebandAmplifier
th-ordertermoftheoutputsignalcanbewrittenas
A formal reduction in the number of coefficients in (1) can
beobtainedundertheonlyassumptionofawidebandamplifier,
i.e.,anamplifierwithapassbandlargerthantheRFsignalband-
width.Thissuppositiondoesnotintroduceanyimportantlossof
(2) generalitysincemanywirelessamplifierspresentingnonlinear
memory effects have frequency responses essentially constant
CRESPO-CADENASetal.:VOLTERRABEHAVIORALMODELFORWIDEBANDRFAMPLIFIERS 451
Fig.2. IllustrativeexampletoshowfrequencydependenceoftheNLTFs.(a)Elementarynonlinearnetwork.(b)Associatedlinearnetworkexcitedbyappropriate
nonlinearcurrentsandspectrumofthegeneratedsecond-ordertransferfunctions.(c)Thesameassociatedlinearnetworkproducingthethird-ordertransferfunc-
tionsandtheirrelatedspectrum.
intheirrespectiveRFsignalbands,see,e.g.,[2].Underthisas- The wideband condition of the amplifier allows to approxi-
sumption, completion of the particular frequency dependence matethelinearfunction asacoefficientindependentof
ofthe transferfunctions foratypical circuitcanbeac- basebandfrequencies.Thesameistrueforthetransferfunction
complished by a combination of the probing method and the relatingthelinearpartoftheoutputwiththeinput .
nonlinearcurrentsmethod,awidespreadprocedure[10],[11]. The spectral components function can depend on the
For the general circuitof Fig. 1, the vector formed with the sumofbasebandfrequencies ,asisthecaseofanon-
NLTFsoforder relatingthevoltagesoftheindependentports linear capacitance, or is independent of baseband frequencies
of the associated linear network can be obtained by using the forothernonlinearities.Inanycase,sincethecomponentsofthe
followingequation: admittancematrixhavethissamedependence,thesecond-order
NLTF dependson (with
(4) and )inthedczone,andcanbeconsideredasa
constantcoefficientinthesecond-harmoniczone(seeTableI).
where is the admittance matrix of the associated linear Noticethatifothernonlinearitiesarepresent,theircontribution
network and is a vector with the spectral components issuperposedatthesamefrequenciesand,therefore,thetypeof
of the nonlinear currents exciting the independent ports. The dependenceremainsunchanged.
bandlimited condition of the wireless signal allows to extend Bothinthefundamentalfrequencyandinthethird-harmonic
thewidebandamplifierassumptiontoalltheharmoniczonesso zones, the admittance matrix is a function only of the carrier
that the admittance matrix can be approximated by frequency because the baseband frequency dependence of the
for at all relevant values of . third-ordertransferfunction isduetothespectralcompo-
Foran th-orderapproximation,thebandofinterestinthefirst nentsof .Takingintoaccountthatthisdependencecomes
harmoniczoneshouldbesupposedheretobe . from , in the fundamental frequency zone, it is possible
For clarity, let consider only , the transfer functions to determine two types of terms: a baseband frequency-inde-
relatingtheinputwiththevoltageattheportofonenonlinearity, pendent term and terms of the form . In
typically a voltage-controlled current source, conductance, or thethird-harmoniczone,thistransferfunctiondoesnotpresent
capacitance. To illustrate the procedure, an elementary circuit dependenceonbasebandfrequencies.Thetypeofdependence
withanonlinearcurrentsourceisshowninFig.2andasketched describedaboveis in agreement with previouslypublished re-
summaryofthemethodisattached.Intheexample,anonlinear sults[12]–[16].
currentsourcedependentonvoltage isconsideredasthemain Althoughtheextensionoftheanalysisfocusedtothededuc-
nonlinearity. For each order, the associated linear network is tionofclosed-formexpressionsforhigherordertransferfunc-
excitedbyappropriatenonlinearcurrentsinordertoobtainthe tionsisalmostbeyondtheboundsofpossibility,keepingtrack
transferfunctionsrelatingvoltage withtheinput.Spectrashow oftheirfrequencydependenceisamorefeasibleexercise.Next,
theprevalentcomponentsfororders2and3. thisstudyiswidenedtohigherorderNLTFs.
452 IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES,VOL.55,NO.3,MARCH2007
TABLEI TABLEII
TYPESOFBASEBANDFREQUENCYDEPENDENCEFORK~ ((cid:24)(cid:24)(cid:24)) TYPESOFBASEBANDFREQUENCYDEPENDENCEFORH~ ((cid:24)(cid:24)(cid:24))
These results can be exploited to reduce the number of pa-
rametersinthebehavioralmodel(1)withouttheadditionofany
otherrestriction.
C. KernelsoftheReduced-OrderBehavioralModel
1) Third-Order Kernel: Based on the previous deductions,
the third-order term of the output voltage is obtained by
substituting in (2) the corresponding components and
Thecauseoffrequencydependencein istwofold.On
.Thefirsttypegeneratesthememorylessterm,
the one hand, the admittance matrix can be approximated by
andthesecondtypegivesrisetothegenericexpression
in the dc zone, and by a frequency-
independentfunctioninthesecondandfourthharmoniczones.
Ontheotherhand,thecomponent presentsseveralterms
withproductsofthetransferfunctions ,and . (5)
Summarizing, in the dc zone, the fourth-order transfer
function is composed by one term that can be expressed Althoughthisisadoubleintegral,thetransferfunctionisaone-
by , a second type of terms dimensionalfunction,afactthatcanbeexplicitlydisplayedwith
,andathirdtype achangetothenewvariables and
forwhich sothat
oftermswiththeform
.Ithasbeenconsideredthat
and ,for and with
and (see Table I). Note that is represented as the (6)
productoftwoseparatedfunctions and ,andinthesame
Relyingonthebandlimitedassumptionof ,theintegralin
form, is denoted by the product of the functions and
isnegligibleoutsideanybandwidth ,allowingthe
. In these expressions, is a generic function with the
definitionofanequivalenttransferfunction confinedto
specific dependence on the sum of four baseband frequencies
thisband.MakinguseoftheFouriertransformandforfrequen-
and are generic functions dependent on the sum of
ciesinside canbeexpressedintermsofitsdiscrete
two baseband frequencies. The significance of this particular
impulseresponse
frequencydependenceisdiscussedbelow.
In the second-harmonic zone, there are two type of terms,
(7)
i.e., and ,andinthefourth-harmoniczone,
there is onlyone (frequencyindependent) term, i.e., .The
explicit dependence has been omitted in the components that sothatthesamplingtime shouldbe,atleast,halfthe
are only functions of the carrier frequency. These arguments symbolperiod.Substitutingin(6)andchangingnowtotheorig-
are sufficient to deduce the frequency behavior of the relevant inalvariables,thetwointegralsbecomeseparableasfollows:
outputtransferfunctions and .
Thetransferfunction hasafrequencydependencesim-
ilarto ,discussedabove.Inthecaseof ,considering
thattheadmittancematrixdoesnotintroduceanyfrequencyde-
pendence,itsbehaviorisdeterminedby or,equivalently,
bythefunctions to .Recallingthatthezoneofin-
terest is the fundamental frequency zone, the relevant transfer
(8)
functionscanberepresentedbythesixtypesoftermsshownin
TableII.Observethatnowtherearefourdifferenttypesofterms
withacompletedependenceonallthefrequenciesfrom to Finally, after adding the memoryless part and sampling at in-
. stants ,thethird-ordertermoftheoutputcomplexenve-
CRESPO-CADENASetal.:VOLTERRABEHAVIORALMODELFORWIDEBANDRFAMPLIFIERS 453
lopecanbeexpressedinadiscrete-timeform Substitutingin(12)andchangingtotheoriginalvariables,the
fourintegralsarenowseparable and,samplingat instants ,
(9) theoutputindiscrete-timeformcanbewrittenas
When the memoryless term is included in (9), we obtain the
followingexpressionforthethird-ordercoefficients:
(14)
with
for
(10)
(15)
for
Itisimmediatetonotethat ,relatedwiththethirdtype ,
2) Fifth-OrderKernel: Accordingtothepreviousdiscussion,
isaparticularcaseofthisresultinwhichthetransferfunction
the fifth-order transfer function is composed of six different
isdirectlyseparable.
types of terms, which produce a particular set of kernels after
substitution in (2). As in the third-order transfer function, one More revealing is the fourth type , which involves the
ofthefiveintegralsinvolvedgivesriseagainto ,remaining sumofallfrequenciessothatitpresentsthehighestdegreeof
inthiscaseaquadrupleintegral,forwhichthefirsttypeofterms symmetryandproducestermsgivenby
canbecomputeddirectlytocontributeonlytomemorylessnon-
(16)
lineareffects.Thesecondtypeoftermsdependson so
thatonlytwointegralscanbecomputeddirectlyandtheother
twocanbehandledasinthethird-ordercaseproducingone-di- with
mensionalkernelsofthetype
(17)
(11)
Notice that, for this particular coefficients, the sampling rate
shouldbeatleastfourtimesthesymbolrate.
Fortheotherfourtypes,itispossibletowriteagenericfifth-
In conclusion, let us observe that the demonstrated Volterra
order transfer function with a frequency dependence given by
behavioral model incorporates a substantial reduction in the
,where and aredefinedaround
number of parameters, when compared to (1), with the only
and ,respectively.Letconsiderthedifferentproperties
assumption of a wideband amplifier. Surprisingly, the de-
ofsymmetrythatthisfunctioncanpresent,beginningwiththe
scribedrepresentationexhibitsanexclusively“outofdiagonal”
moregeneralconditioncorrespondingtothefifthandsixthtypes
structure, different to other well-known published behavioral
ofterms.Atafirstglance,themultipleintegralin(2)isnegli-
models[1],[2].
gibleoutsideafour-dimensionalcubeoflength .However,
To corroborate these new results, an amplifier has been
the particular symmetry of involves a bidimensional fre-
tested and the model parameters have been extracted from
quencydependencethatisexhibitedclearlyafterthechangeof
experimentaldata,aswillbediscussedinSectionIII.
variables and
asfollows:
III. MODELPARAMETERSEXTRACTIONANDVALIDATION
The commercial amplifier MAX2430 manufactured by
MAXIM Integrated Products Inc., Sunnyvale, CA, has been
modeled with the current structure. It is a wideband amplifier
at 915 MHz; however, in the experimental characterization
with two tones separated by 2 MHz, the amplifier exhibited
an asymmetry in the intermodulation distortion (IMD) prod-
ucts, a clear indication of the existence of nonlinear memory
(12) effects. The measurement setup used in this study is basically
the same as that presented in [15]. However, the excitations
Therefore, isnegligibleoutsideanysquareoflength taken into account are diverse in order to test the proposed
and can be substituted by its equivalent function. model with a wide variety of signals. In particular, standard
Itis possibletoexpressthis bandlimitedfunctionasarelation two-tone,aswellasdigitallymodulatedsignalslikequadrature
betweenthecorrespondingdiscrete-timekernels phase-shiftkeying(QPSK)widebandcodedivisionmultipleac-
cess (WCDMA), 16-quadratureamplitude modulation(QAM)
withrectangularpulsesandroot-raisedcosinepulseshavebeen
(13) used as input stimuli. These signals have been loaded in the
internalmemoryofanSMIQ02Bsignalgeneratorwithbuilt-in
454 IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES,VOL.55,NO.3,MARCH2007
(a)
Fig.3. Normalizederrorasafunctionoftheinputlevel.ProposedVBWmodel
(squaresandsolidline)andMPmodel(circlesanddotted–dashedline).
(b)
arbitrarywaveformfacilityandtheE4407Bspectrumanalyzer
Fig. 4. Normalized complex envelope in-phase component for a 16-QAM
withamodulationanalysisoptionhasbeenusedtoacquirethe
signal.(a)Rectangular pulses.(b)Root-raised cosinepulses.Acquireddata:
basebandsignalattheamplifier’soutput. dots.VBWmodel:solidline.MPmodel:dotted–dashedline.
A. 16-QAMSignalWithRectangularPulses
As was demonstrated in Section II, it is necessary to use a correspondstotheresultsforathird-orderVBWmodelwitha
samplingrateofapproximatelyfourtimesthesymbolrateifpa- memoryof samples.Inbothcases,theerrorgrowsup
rametersuptothefifthorderhavetobeextracted.Inthecaseof astheamplifierentersinamorenonlinearcondition,indicating
root-raisedcosinemodulatingpulses,eachsampleisdependent in the first case the presence of nonlinear memory and, in the
onpreviousandfuturesamples,includedthosemanysymbols secondcase,thatnotallthenonlinearmemorycoefficientshave
away.Eveninthecaseofamemorylessamplifier,theoutputwill been extracted. On the contrary, the fifth-order VBW model
displaymemoryanditis,therefore,reasonabletousefornon- depicted in solid line shows a very low error, constant in all
linearcharacterizationmodulatingpulseswithoutintersymbol the range of input levels, which is an evident confirmation of
interference, i.e., with length no longer than a symbol period. a correct parameters identification of the memory nonlinear
Consequently, the use of an RF signal modulated with rectan- model. To corroborate the validity of the current results, the
gular pulses as an input stimulus guarantees that memory ef- well-establishedMPmodeldescribedin[2]wasextractedwith
fects,ifpresentintheoutput,havebeencausedbythenonlinear agenerousnumberofdelays,andtakenasareliablereference.
memory of the device. The modulation format is also relevant The error for this model with seven delays is also represented
becauseitiswellknownthatforphase-shiftkeying(PSK)sig- in Fig.3 (dashed–dotted line), demonstrating a similar perfor-
nals,athird-ordermodelcancapturesomeofthehigherorder mance with respect to the VBW model. Furthermore, in the
nonlinear characteristics and produce degeneration in the pa- range of higher levels, the most relevant to this context, the
rameter-extractionprocess[9]. VBWmodeloutperformstheMPmodel.
According to the above considerations, a 915-MHz car- The corresponding time-domain in-phase component for
rier modulated with a random train of rectangular pulses at an input level of 11 dBm is shown normalized in Fig. 4(a).
2 Msymb/s and a 16-QAM format was selected as the first The acquisitions (dots), the prediction of the MP model
soundingsignal.Thearbitrarywaveformgeneratorcanhandle (dashed–dotted line), and the prediction of the present VBW
up to 40 Msamp/s so that the shape of the pulses was rather model (solid line) have been represented. The average NMSE
rectangular.Sinceintherecoverypartthesetuphasasampling oftheVBWmodelis 30.5dB,representing1dBofimprove-
rateof15Msamp/s,the acquiredoutputsignalsweresampled mentcomparedtotheMPmodel.Thevectorrepresentationof
at 7.5 samples per symbol, amply suitable for signal repre- themodeledcomplexenvelopeisplottedwithdotsinFig.5and
sentation and fifth-order parameters extraction. A fifth-order compared with the input envelope (squares) and the acquired
Volterrabehavioralmodelforwidebandamplifiers(VBW)was output(crosses).
extracted from the acquired output complex envelope samples
B. Two-ToneSignal
through the minimization of the average normalized mean
squareerror(NMSE)betweenmeasuredandmodeledoutputs. Another set of experiments was performed using an input
AfirstresultisshowninFig.3,wherethe normalizederroris signalformedbytwotonesofequalmagnitudeandphase,with
plottedasafunctionoftheinputlevel.Theerrorisrepresented afrequencyseparationof2MHz,andmeasuringtheupperand
as a dotted line for a memoryless model and the dashed line lowerthird-orderIMproducts.Whentheamplifierisworkingin
CRESPO-CADENASetal.:VOLTERRABEHAVIORALMODELFORWIDEBANDRFAMPLIFIERS 455
Fig.7. Outputspectrumofa16-QAMsignalwith2Msymb/s.Root-raised
Fig.5. Vectorrepresentationofthe16-QAMsignalwithrectangularpulses. conformingpulses.P =(cid:0)11dBm.Spectrumanalyzertrace(dots)andVBW
Inputsignal:squares.Acquireddata:crosses.VBWmodel:dots. modelprediction(solidline).
measured and calculated lower IM3 at 14 dBm is caused by
setuplimitationatlowinputlevels.Onthecontrary,thesignifi-
cantcoincidenceinsidethefaithfulrangeisalsorevealedwhen
measuredandpredictedasymmetriesarecompared,asisshown
inFig.6(b).
C. 16-QAMSignalWithRoot-RaisedCosinePulses
Anothertypeofsounding signalemployed inthe extraction
(a)
processhasbeenacarrierat915MHzmodulatedina16-QAM
formatwitha2-Msymb/strainofsymbolsusingroot-raisedco-
sinepulses.Theacquirednormalizedin-phasecomponentand
thecorrespondingwaveformobtainedwiththeextractedmodel
are represented in Fig. 4(b) (dots and solid line, respectively).
Forcomparativepurposes,thewaveformobtainedwiththeMP
model is also depicted in Fig. 4(b) (dashed–dotted line). As a
further test, the extracted model was used to predict the spec-
trum of the signal and the adjacent-channel power (ACP) in
ordertobecomparedwithotheralternativemeasurementsusing
(b) theconventionalspectrumanalyzerwithouttheacquisitionfa-
cility. The results are plotted in Figs. 7 and 8 using marks for
Fig.6. MeasuredandsimulatedresultsforIM3whentonespacingis2MHz.
theexperimentaldataandsolidlinesforthemodeledoutput.It
Acquireddata:triangles.VBWmodel:solidline.
is worth noting that the model was first extracted from an ex-
perimentalacquisitionofbasebandsamplesandservedtopre-
thelinearmode,theIMproductsarenegligiblecomparedtothe dict the output signal spectrum. Although the marks represent
nonsystematic errors of the measurement setup so that model ameasurementprocessindependentoftheacquisition,thepre-
parametersextractedfromacquisitionsatlowsignallevelsare dictionisfavorablycomparedinFig.7.Thesecondfigurecor-
irregular.Thisisnotspeciallyinconvenientbecausetheinterest roboratesthisoutcomeshowingagoodmatchbetweentheACP
is in the range of high signal levels, where the nonlinear ef- measuredandcalculated.Thepredictionisalsoabletoestimate
fects are more relevant and model parameters can be reliably adequatelytheACPasymmetrybetweenupperandlowerchan-
extracted. For that reason, after rejection of meaningless data, nels. As a reference, the dashed–dotted line represents the re-
the experimental points represented in Fig. 6 belong to levels sultsoftheMPmodel.
near the 1-dB compression point. In Fig. 6(a), the measured
D. QPSK-WCDMASignal
outputofthefundamentaltonesandthethird-orderintermodu-
lation(IM3)productsarerepresentedwithmarks.AlsoinFig.6, Finally, a carrier at 915 MHz modulated with a WCDMA
theresultsoftheextractedmodelaredepictedviaasolidline, signal compliant with the Universal Mobile Telecommunica-
reflectingaremarkablecorrespondencewiththeacquireddata. tionsSystem(UMTS)standardwasemployedasinputsignal.In
According to the previous discussion, the difference between thiscase,theconditionoffourtimesthesymbolratetocorrectly