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Rational fraction approximations for passive network functions

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Rational fraction approximations for passive network functions
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Johnson, William Joel Dietmar
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Approximation theory
Passive filters
Pade'-chebyshev approximation
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Abstract:
ABSTRACT: In electrical engineering, the designer is often presented with the problem of synthesizing a circuit for which the mathematical specifications are unsuitable for physical realization. Hence, the engineer must approximate as well as possible the prescribed network function by another function which is realizable. This paper describes a new approximation method for solving the problem of realizing passive network transfer functions, where the realization is carried out through the use of passive, reciprocal,lumped, linear, and time-invariant elements.
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Thesis (Ph.D.)--University of South Florida, 2005.
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Includes bibliographical references.
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by William Joel Dietmar Johnson.
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Title from PDF of title page.
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Document formatted into pages; contains 54 pages.
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Includes vita.

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Rational fraction approximations for passive network functions
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ABSTRACT: In electrical engineering, the designer is often presented with the problem of synthesizing a circuit for which the mathematical specifications are unsuitable for physical realization. Hence, the engineer must approximate as well as possible the prescribed network function by another function which is realizable. This paper describes a new approximation method for solving the problem of realizing passive network transfer functions, where the realization is carried out through the use of passive, reciprocal,lumped, linear, and time-invariant elements.
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Adviser: Dr. Arthur David Snider.
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Passive filters.
Pade'-chebyshev approximation.
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Doctoral.
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RationalFractionApproximationsforPassiveNetworkFunction s by WilliamJoelDietmarJohnson Adissertationsubmittedinpartialfulllment oftherequirementsforthedegreeof DoctorofPhilosophy DepartmentofElectricalEngineering CollegeofEngineering UniversityofSouthFlorida MajorProfessor:ArthurDavidSnider,Ph.D. MouradIsmail,Ph.D. EvgueniiA.Rakhmanov,Ph.D. StanleyKranc,Ph.D. KennethA.Buckle,Ph.D. DateofApproval: March21,2005 Keywords:approximationtheory,passivelters,Pade-Chebyshe vapproximation c Copyright2005,WilliamJoelDietmarJohnson

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Dedication Tomywife,Sandy.

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Acknowledgments Iwishtoacknowledgemymainadvisor,Dr.ArthurDavidSnider,f orhisconstant help,guidance,andencouragement.

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TableofContents ListofTables ii ListofFigures iii Abstract v Chapter1Introduction 1 1.1Motivation 1 1.2Background 2 1.3ClassicalFilterMagnitudeDesignMethod 4 1.4Non-traditionalFilters 7 1.5Watanabe'sDoubleBandpassFilterDesign 8 Chapter2Algorithm 10 2.1Strategy 10 2.2Flowchart 11 2.3PadeApproximation 13 2.4Pade-ChebyshevApproximation 16 2.5SpectralFactorization 19 2.6Algorithm 21 Chapter3Examples 24 3.1DoubleBandpassExample 24 3.2RampExample 31 Chapter4ConclusionandFutureDirection35 4.1Conclusion 35 4.2FutureDirection 36 References 37 Appendices 39 AppendixASymbolGlossary 40 AppendixBWatanabe'sDoubleBandpassFilterTheory 41 AppendixCFilterTransformations 45 AbouttheAuthor EndPage i

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ListofTables Table1.PlacementofPolesandZerosforVariousNetworkFunc tions.[8]5 ii

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ListofFigures Figure1.NetworkElements. 3 Figure2.Two-portNetwork. 3 Figure3.FilterTypes. 6 Figure4.ButterworthFilter. 7 Figure5.DoubleBandpassFilterFrequencyResponse.7Figure6.RampLowpassFrequencyResponse.8Figure7.Watanabe'sDesignExample:BandpassNontraditional Frequency Response. 9 Figure8.Errorof e x vs. r 2 ; 2 ( x )FrequencyResponse.16 Figure9.PolesandZerosofanEvenRealRationalFunction.2 0 Figure10.DesignRequirementsforDoubleBandpasslter.24Figure11.EvenFunction. 25 Figure12.MolliedandDesignRequirements.26Figure13. max n 1 j ~ H cont (~ ) j 2 ;A o (Interpret\innity"as A ).27 Figure14.DesignGoalvs.Approximation. 28 iii

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Figure15.Watanabevs.Pade-ChebyshevFrequencyResponse.29Figure16.Watanabevs.Pade-ChebyshevFrequencyResponse.30Figure17.IdealRampFrequencyResponse.31Figure18. max n 1 j ~ H cont (~ ) j 2 ;A o (Interpret\innity"as A ).32 Figure19.Reciprocalvs.Pade-ChebyshevFrequencyResponse. 33 Figure20.IdealRampvs.RationalTransferFunction.34Figure21.FilterTransformations. 45 iv

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RationalFractionApproximationsforPassiveNetworkFunc tions WilliamJoelDietmarJohnson ABSTRACT Inelectricalengineering,thedesignerisoftenpresentedwi ththeproblemofsynthesizingacircuitforwhichthemathematicalspecicationsareu nsuitableforphysical realization.Hence,theengineermustapproximateaswellasp ossibletheprescribed networkfunctionbyanotherfunctionwhichisrealizable.T hispaperdescribesanew approximationmethodforsolvingtheproblemofrealizingpa ssivenetworktransfer functions,wheretherealizationiscarriedoutthroughtheu seofpassive,reciprocal, lumped,linear,andtime-invariantelements. v

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Chapter1 Introduction 1.1MotivationNetworksynthesisisthestudyofobtainingaprescribedinputtoo utputmathematical relationshiputilizingphysicallyrealizableelements.This dissertationwillpresenta newapproximationmethodforobtainingaphysicallyrealiza bletransferfunction fromthestatedrequirements.Themethodisanovelsynthesisofk nowntechniques fromanalyticapproximationtheoryandlterdesign.Wewillc oncernourselveswith thosenetworksthatarepassive,lumped,linear,reciprocal,a ndtime-invariant.The networkshallberealizedasadevicewithasingleinputportan dasingleoutput port,ortwo-port. Transferfunctionsrealizedbypassivenetworksarealwaysstab le,i.e.,forabounded inputthesecircuitsproduceaboundedoutput.Stabilitytra nslatesintoarequirement onthenetworkfunction:itmustbeanalyticintheopenrighth alfofthecomplex plane,andifpolesareontheimaginaryaxistheymustberstord er.Dependingon thedesirednetworkrealization,theremaybeadditionalreq uirementsontheresidues ofthetransferfunction[17].Physicalnetworks,suchaswedescr ibe,requireanadditionalcondition:thetransferfunctionmustbearationalfun ction,numeratordegree denominatordegree,withrealcoecients. Networksynthesisissubdividedintotheapproximationproblem andtherealizationproblem.Therealizationproblem,theprocessofobtain ingthephysicalnetwork 1

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elementvaluesandtheirconnectedgraph,isnotthefocusof thisdissertationand willnotbefurtherdiscussed.1.2BackgroundTherulesofengagementforcircuittheoryaredetailedinre ferences[1]and[10].In whatfollowswewillpresentonlythosefactspertinenttothis bodyofwork. Insteadofworkingwiththeintegro-dierentialcircuitequa tionsinthe\time domain",engineersfrequentlymakeuseoftheLaplacetransfo rm F ( s )= Z 1 0 ¡ f ( t ) e ¡ st dt andworkinthe\frequencydomain".TheLaplacevariable, s ,isinterpretedas complexfrequency, s = + j! ,where isattenuationand isfrequency( j = p ¡ 1,seeAppendixA).Thistransformallowstheengineertousealgeb raformost computations. Thethreebasiccircuitcomponentsforphysicalrealizationa retheresistor,capacitor,andinductor.Thealgebraicrelationshipbetweenvolt ageandcurrentineachof thesecomponentsisOhm'sLaw V ( s )= Z ( s ) I ( s ) where V ( s )istheLaplacetransformedvoltage, I ( s )istheLaplacetransformedcurrent,and Z ( s )istheimpedance.Figure1showsthesebasicelementsandtheir valuesfor Z ( s ). When R C ,and L elementsareconnectedintoanetworkandonlytwopoints ofaccessareallowed,suchasinFigure2,thecongurationcanb einterpretedasa two-portnetwork.Aportisapointatwhichaccesstoanetwork maybegained. 2

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L Z ( s )= sL R Z ( s )= R Z ( s )= 1 sC C Figure1.NetworkElements. I 2 ( s ) + V L ( s ) Output (Port2) I 1 ( s ) + V g ( s ) Input (Port1) Figure2.Two-portNetwork. Theimportanceofthetwo-portconceptisthatindividualel ementbehaviorcanbe replacedbylarger-scalecollectivebehaviorbydescribingt heoutputasatransformed versionoftheinput.Thereareonlyfourmethodsfordescribin gthesetransformations. Theyare V L ( s ) V g ( s ) ; dimensionlessvoltagetransferfunction, V L ( s ) I 1 ( s ) ; transferimpedancefunction, I 2 ( s ) I 1 ( s ) ; dimensionlesscurrenttransferfunction,(1) 3

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and I 2 ( s ) V g ( s ) ; transferadmittancefunction. Thelawsofcircuittheorydictatethateachofthesetransferf unctionsisarealrational functionof s Itisamatterofchoicetotheengineerwhichnetworktransfer functionistobe utilized.Thesymbolweshalluseinthisbodyofworktodescribea genericnetwork transferfunctionis G ( s ). Anetworkmayberealizedwitheithertwoofthetypesofcircu itcomponent elements( RC RL ,or LC )orallthreeelements( RLC ).Thelocationofthepoles andzerosofthenetworktransferfunctionmayhaveadditiona lconstraints.Table1 summarizestheseconstraints.1.3ClassicalFilterMagnitudeDesignMethodAlterisatwo-portnetworkwhichselectivelypassessomefreque ncieswithlittle attenuation,whilegreatlyattenuatingallotherfrequenc ies.Classicallterdesign techniquesfocusonndingrealizablerationaltransferfunc tions G ratl ( s )whosemagnitudesapproximate,for s = j! ,oneofthe4ideallterspecications, H spec ( ), showninFigure3. Infact,usingwell-knownrational-functiontransformation softhefrequencyvariable,onecantranslatethelatterthree(band-pass,high-pass,b and-stop)specicationstotherst(prototype).Thisresultsinasimplicationkno wnasthelow-pass prototype(LPP).TheLPPdependentvariable,~ s =~ + j ~ ,playsthesameroleasthe originalLaplacevariable s .TherequirementsimposedontheLPParetransformed from H spec ( )tobecome ~ H spec (~ ). Theclassofrationalapproximantstraditionallyusedisrestr ictedtosimplereciprocalsofpolynomials;thesehavetheappropriatenumerat or-denominatordegree 4

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Table1.PlacementofPolesandZerosforVariousNetworkFunc tions.[8] LocationsofPolesandZeros LC a RC or RL a FunctionPolesZerosPolesZeros Transferfunc-tionsofladdernetworks b Simpleon j! axis c Anyorderonj! axis d Simpleonnegativerealaxis c Anyorderonnegativerealaxis d Generaltransferfunctions e Simpleon j! axis c Anyorderon j! axis d ; quadrantalsymmetryinright-andleft-halfplanes Simpleonnegativerealaxis c Anyorderinright-orleft-halfplane d ; f RLC a FunctionPolesZeros Transferfunc-tionsofladdernetworks Anyorderinleft-halfplanes c ; g Anyorderinleft-halfplaneoron j! axis d Generaltransferfunctions Anyorderinleft-halfplane c ; g Anyorderinright-orleft-halfplane d ; f a Excludingmutualinductance. b Laddernetworksareconnectionsofcomponentswhereonlyadjacentneighborsin oneplaneareallowed. c Transferimmittancesmayhavepolesattheoriginandinnitybutdimensionless transferfunctionsmaynot. d Includingtheoriginandinnity. e Generalnetworksareconnectionsofcomponentswherenon-adjacentneighborsin multipleplanesareallowed. f Thepositiverealaxisisexcludedunlessbalancednetworksarepermitted. g Anypolesonthe j! axismustbesimple. 5

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1 highpass 0 0 bandstop 1 0 1 2 1 lowpass(prototype) 0 H spec ( ) 0 1 bandpass 0 1 2 Figure3.FilterTypes. relation,sothepolynomialisconstructedtoapproximatethe reciprocalofthelter, onthelter'ssupport.Butterworthlters,forexample,taketh eform:for~ real, ~ H spec (~ ) 'j ~ G ratl ( j ~ ) j ,let ~ G ratl ( j ~ )= 1 1+ A (~ ) (2) withpolynomial A (~ ), A (0)=0, A (1)=1, A (~ )\maximallyat"at0,Figure4. Thisdissertationaddressesthepossibilityofdevisingrationalt ransferfunctions thatapproximatemoreexoticdesignspecications,suchasindic atedinFigures5 and6.Weacquireadditionalexibilitybyutilizingthe full classofrationalfunctions withtheappropriateasymptoticbehavior(denominatordegr eeexceedingnumeratordegreefordesignspecicationswithboundedsupport).TheP ade-Chebyshev methodologyisusedtoconstructtheapproximantintheinterv alcontainingthesupport(supplementedwithsomeadjustmentsensuringthephysicalr ealizabilityofthe transferfunction). 6

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0 1 1 ~ H spec (~ ) j ~ G ratl ( j ~ ) j ~ Figure4.ButterworthFilter. 1.4Non-traditionalFiltersInequation(2)ofSection1.3,weseethatthetraditional ~ G ratl ( j ~ )isnotthemost generalrationalfunction.Asaresult,therestrictedformhas dicultiesapproximatingmoreexoticlterssuchasinFigure5orinFigure6.Figure5 isanexampleof adoublebandpasslterandFigure6isanexampleofaramplowpa sslter.The classicalapproximationmethodisunsuitedforlterrealizati onssuchasthese.A moresophisticatedmethod,exploitingmoregeneralrational functions,isneeded. gain 1 0.80.60.40.2 omega 0 20 15 10 5 0 ideal double bandpass Figure5.DoubleBandpassFilterFrequencyResponse. 7

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gain 1 0.80.60.40.2 omega 0 2 1.5 1 0.5 0 ideal ramp Figure6.RampLowpassFrequencyResponse. 1.5Watanabe'sDoubleBandpassFilterDesignIn1961,H.Watanabewrotealtertheorypaperintroducinghis solutiontothe doublebandpasslterproblem[16],similartotheltershowninF igure5.The methodologyisquitecomplex,andissummarizedinAppendixB. Ultimately,the followingtransferfunctionisderived: G ( s )= s 2 ( s 2 + : 717095) ( s 2 + : 162918 s + : 128222)( s 2 + : 072328 s + : 275307) £ ( s 2 + : 332031) ( s 2 + : 693687 s +2 : 103897)( s 2 + : 152983 s + : 984953) ; (3) whichhasafrequencyresponse(shownwithalog-logaxistoscale itsfeatures)given inFigure7. Itisthecontributionofthisdissertationtoputforthanewm ethodforobtaininga doublebandpasslter,whichissubstantiallysimplerthanW atanabe'sinbothconcept andimplementation.Moreover,thisnewmethodallowsformo regeneralshapesthan justdoublebandpasslters(e.g.Figure6). 8

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10-1 100 101 -60 -50 -40 -30 -20 -10 0 10 20 frequency (rad/sec)gain (dB)Watanabe's Design Figure7.Watanabe'sDesignExample:BandpassNontraditional FrequencyResponse. 9

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Chapter2 Algorithm BycombiningpolynomialapproximationtheorywithPadeap proximationmethods, onecanattackdesignspecicationssuchasgivenbyFigures5and 6,ormoregeneral shapes.Unfortunately,Pademethodshaveamajordrawback|th eresultingtransfer functionmaybeunstable.Inthischapter,wewilldescribeano veldesignprocedure, usingtheextraexibilityofthePade-Chebyshevmethodtoget herwiththetechnique ofSpectralFactorization[4],todesignphysicallyrealizab lenetworktransferfunctions forsomeofthespecicationswhichhavebeenconfoundedbyclassi calmethods.The algorithmallowsthedesignertochoose m ,numeratordegree,and n ,denominator degree,suchthatthemagnitudeof G ratl ( s ),for s = j! ,approximatesthedesired specicationin 2 [0 ; 1]andachievesthedesiredattenuationin 2 [1 ; 1 ). 2.1StrategyAnoutlineofourproposedstrategyforobtaining G ratl ( s )fromagivennitelysupportednonnegative H spec ( ),0 !< 1 ,isto 1.Deviseaconvenientanalyticmapping~ = f ( )suchthatthesupportof H spec ( )ismappedinside[ ¡ 1 ; 1]andthetransformedspecication, ~ H even (~ ), isanevenfunctionin( ¡1 ; 1 ).Furtherconditionsonthismappingwillbe elaboratedbelow. 2.Mollify ~ H even (~ )onitssupporttoobtainacontinuousandboundedvariate function, ~ H cont (~ ). 10

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3.ComputeaPade-Chebyshevrationalapproximationto max f 1 j ~ H cont (~ ) j 2 ;A g for somelarge A on 2 [ ¡ 1 ; 1]usingGeddes'algorithm;take m>n inorderto satisfythedesiredattenuation.Denotethisresultas r m;n (~ ). 4.For~ s = j ~ ,dene R m;n (~ s )= 1 r m;n (~ s=j ) andanalyticallycontinue R m;n (~ s )tothe wholeplane. 5.SpectrallyfactorthePade-Chebyshevfunction, R m;n (~ s )toobtain ~ G ratl (~ s ). 6.Returntotheoriginaldomainbymapping ~ G ratl (~ s )to G ratl ( s ).Endwith G ratl ( s ),which a.isrational,physicallyrealizable b. j G ratl ( j! ) j' H spec ( )onsupportof H spec ( ),duetoPade-Chebyshev construction. c. j G ratl ( j! ) j! 0as !1 becausedenominatordegree > numerator degree. 7.Iterateasnecessary.Byeithermodifying H cont ( ), m;n A ,ormapping,tryto bring j G ratl ( j! ) j intocloseragreementwith H spec ( ). Inourexamples,wewillutilizestandardrationalmappingsfr omclassicaldesign methodsforstep2,seeAppendixC.2.2FlowchartThefollowingisaowchartofouralgorithm.Note:thefunctio n ~ H cont (~ )issquared sothatwemayspectralfactorinalaterstep. 11

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H spec ( ) specication H even (~ )even,support 2 [ ¡ 1 ; 1] j ~ H cont (~ ) j 2 continuousandboundedvariation max n 1 j ~ H cont (~ ) j 2 ;A o reciprocate,truncate r m;n (~ ) Pade-Chebyshev:max n 1 j ~ H cont (~ ) j 2 ;A o R m;n (~ s ) analyticextension ~ G ratl (~ s ) spectralfactorization G ratl ( s ) desired: j G ratl ( j! ) j ? = H spec ( ) Yes End No 12

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2.3PadeApproximationPolynomialapproximationisthedominantmethodforobtain ingphysicallyrealizabletransferfunctionsforclassicallters.Polynomialmethod shavesomedistinct advantages: 1.Foranycontinuousfunctiononagivenclosedinterval[ a;b ]andforany > 0, therealwaysexistsanalgebraicpolynomialofsucientdegree thatcanapproximatetheoriginalfunctiontowithinanygiventolerance [2]. 2.Thecoecientsinthepolynomialcanoftenbeobtainedbyal inearsystemof equations. Buttherearesomedistinctdisadvantages: 1.Ahighpolynomialdegreeisgenerallyneededforaccuracy 2.Therestrictedformof ~ G ratl ( j ~ )inEquation(2)cannotapproximatezeroes withinthesupportof ~ H cont (~ ). Amethodthatmayovercomethesedecienciesisrationalappro ximation.Because everypolynomialisarationalfunction,approximationusin grationalfunctionsyields resultsthatarenoworsethanpolynomialapproximation.Anadv antageofrational functionsisthatfunctionswiththenumeratoranddenomina torhavingthesame ornearlythesamedegreewillgenerallyproduceapproximati onresultssuperiorto thosewithpolynomials,forthesameamountofcomputationaleo rt[2].Ingeneral, themethodsforobtainingtheunknowncoecientsarenotline ar.Onecomputationalmethodwherebytheunknowncoecientsoftherational functionareobtained throughalinearsystemofequationsisduetoH.Pade[14].TheP adeapproximation techniqueisanextensionoftheTaylorpolynomialapproxima tionmethod. 13

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Wereservethevariables ,~ ,and H forourdesignproblemandusethevariable x and f forthepresentexposition.ThePadeapproachisasfollows.If h ( x )isanalytic and h ( k ) (0)=0foreach k =0 ; 1 ; ¢¢¢ ;N; then(4) h hasazeroofmultiplicity( N +1)at x =0[3] : Supposeanarbitraryfunction f satises f ( x ) N X i =0 a i x i + O ( x N +1 ) ; (5) then r m;n isaPadeapproximationoforder( m;n )to f ,if r m;n ( x )= p ( x ) q ( x ) = p 0 + p 1 x + p 2 x 2 + ¢¢¢ + p m x m 1+ q 1 x + q 2 x 2 + ¢¢¢ + q n x n = P p i x i P q i x i with q 0 1(6) and f ( k ) (0) ¡ r ( k ) (0)=0 ; for k =0 ; 1 ; 2 ;:::;N; (7) where N = m + n: Notethatthereare N +1freeparameters( p 0 ;p 1 ; ¢¢¢ ;p m ;q 1 ;q 2 ; ¢¢¢ ;q n ) availableforenforcing(7). Using(6),wehave f ( x ) ¡ r m;n ( x )= f ( x ) ¡ p ( x ) q ( x ) = f ( x ) q ( x ) ¡ p ( x ) q ( x ) = P Ni =0 a i x i P ni =0 q i x i ¡ P mi =0 p i x i P mi =0 q i x i + O ( x N +1 ) : (8) 14

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Fromequation(4),with h = f ¡ r ,werequirethenumeratorofequation(8)tohave notermsofdegree N ,i.e.,thecoecientsof x k =0for k =0 ; 1 ; ¢¢¢ ;N .Or, ( a 0 + a 1 x + ¢¢¢ + a N x N )(1+ q 1 x + ¢¢¢ + q n x n ) ¡ ( p 0 + p 1 x + ¢¢¢ + p m x m )= O ( x N +1 ) : Expandingandcollectingtermsforeach x k yields[5], " k X i =0 a i q k ¡ i ¡ p k # x k =0 ;k =0 ; 1 ; ¢¢¢ ;N (9) (wherewetake p i or q i tobezerowhenthesubscriptisoutofrange). Display(9)isasetof N +1linearequations.Notethatifthecoecients f a i g in theTaylorapproximation(5)to f arereal,thenthecoecients f p i ;q i g inthePade approximationwillalsobereal.Example:For n = m =2,ndthePadeapproximationfor f ( x )= e x .TheTaylor seriesexpansionfor e x is e x = 1 X i =0 x i i : Therst5termsare e x 1+ x + x 2 2 + x 3 6 + x 4 24 : Enforcingequation(9)with n = m =2; x 4 : 1 2 q 2 + 1 6 q 1 + 1 24 =0 x 3 : q 2 + 1 2 q 1 + 1 6 =0 x 2 : q 2 + q 1 + 1 2 = p 2 x 1 : q 1 +1= p 1 x 0 :1= p 0 : 15

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Solvingtheabovesystemyields q 0 1 ;q 1 = ¡ 1 2 ;q 2 = 1 12 p 0 =1 ;p 1 = 1 2 ;p 2 = 1 12 r 2 ; 2 ( x )= 1 12 x 2 + 1 2 x +1 1 12 x 2 ¡ 1 2 x +1 = x 2 +6 x +12 x 2 ¡ 6 x +12 : Aplotoftheerror, r 2 ; 2 ( x ) ¡ e x ,isshowningure8. 0.004 0.6 0.0030.002 0.4 0.001 0 0.2 0 x 1 0.8 error Figure8.Errorof e x vs. r 2 ; 2 ( x )FrequencyResponse. 2.4Pade-ChebyshevApproximationAproblemwiththeregularPademethodisthatitcanyieldpo orapproximations overan interval ,becausetheregularPademethodrequiresmatchingderivat ivesat onlyonepoint.Inanattempttodecreasetheapproximationer roroveraninterval, 16

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onecanexpandthegiven f ( x )usingChebyshevpolynomialsoftherstkind, f ( x )= 1 X k =0 0 c k T k ( x ) ; (10) where T k ( x )=cos( k cos ¡ 1 x ) and 1 X k =0 0 u k = 1 2 u 0 + u 1 + u 2 + ::: and c k = 2 Z 1 ¡ 1 f ( x ) T k ( x ) p 1 ¡ x 2 dx; (11) andtruncatethisexpansion.If f ( x ) 2 C [ ¡ 1 ; 1]andhasboundedvariationonthe interval[ ¡ 1 ; 1],then P 10k =0 c k T k ( x )convergesuniformlyto f ( x )[12]. ThePade-Chebyshevapproximantto f ( x )isdenedasthatrationalfunction, r m;n ( x )= P mk =0 p k x k P nk =0 q k x k ; (12) suchthatforsomeinteger N f ( x ) ¡ r m;n ( x )= 1 X k = N +1 d k T k ( x ) : ThealgorithmduetoGeddes[6],is 1.Withassumptionson f ( x ),statedabove,chooseaninteger N andform ~ f N ( x ) bytruncatingtheChebyshevseriesfor f ( x )after N +1terms ~ f N ( x ) N X k =0 0 c k T k ( x ) ; (13) with c k asin(11). 17

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2.Carrythevaluesof ~ f N ( x )fromtheinterval[ ¡ 1 ; 1]inthe x ¡ domaintotheunit circle C = f z : j z j =1 g inthe z ¡ domainusingthetwo-to-onemapping z = x § j p 1 ¡ x 2 toobtain ^ f N ( z ) N X k =0 0 c k z k (14) wherethepolynomialcoecientsinequation(14)areprecisel ythesameasthe coecientsinequation(13). 3.Thepolynomialinequation(14)isthenapproximatedbyth eregularPade methodtoyield r m;n ( z )aspreviouslydiscussedinSection2.2.Forreasonsto bediscussedbelow,inourapplicationwetake m n 4.TransformtheregularPadefunctionusingthemapping R m;n ( x )= r m;n ( x + j p 1 ¡ x 2 )+ r m;n ( x ¡ j p 1 ¡ x 2 ) 2 toyieldthedesiredPade-Chebyshevapproximation, R m;n ( x ).Theexplanation whythiscalculationyieldsareal,rationalfunctionwitht hestateddegreesis giveninGeddes[6].Notefurtherthatif f ( x )iseven,then R m;n ( x )willalsobe even. TheabovealgorithmstatesamethodforcomputingthePade-C hebyshevapproximationfor f ( x )ifthefollowingthreeconditionshold[6]. 1. m>n .ThisensuresthatthenumeratordegreeinthenalPade-Cheb yshev approximationis,infact, m 18

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2.ThePadeapproximant r m;n ( z )satisfying ~ f N ( z ) ¡ r m;n ( z )= O ( z m + n +1 ) canbefound(i.e.,itis\normal"). 3.ThePadeapproximant r m;n ( z )hasnopoleslyingintheclosedunitdisk.This ensuresthat r m;n ( z )isunique. BecausethetruncatedChebyshevseriesisanear-minimaxpolyn omialapproximation [13],itisreasonabletoexpectthatthePade-Chebyshevrati onalapproximationwill alsobenear-minimax[9].2.5SpectralFactorizationIfarationalfunction, F ( s ),withrealcoecientshastheproperty F ( s )= F ( ¡ s )and F (0) > 0,thenthereexistsafactorization F m;n ( s )= ^ G ( s ) ^ G ( ¡ s ) suchthat ^ G ( s )hasallitspolesandzerosintheclosedleft-halfofthecomp lexplane and ^ G ( ¡ s )hasitspolesandzerosintheclosedright-halfcomplexplan e[5].This factorizationisknownasspectralfactorization. Thisfollowsbecausethepolesandzerosofatypicallyreal,e ven F m;n ( s )occurin quadrantalsymmetrywithrespecttoboththerealaxisandthei maginaryaxis;e.g., seeFigure9.Thatis,thecomplexrootsnotontheimaginaryaxi soccuringroups offour,rootsontheimaginaryaxisoccurinpairs,androotso ntherealaxisoccur twoatatime. 19

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j! £ £ £ £ £ £ oo oo o o £ -poles o-zeros Figure9.PolesandZerosofanEvenRealRationalFunction. Thezerosandpolescanbecomputedusingarootndingalgorith monthenumeratoranddenominatorof F m;n ( s ).Thedesiredrationalfactor ^ G ( s )isobtained byselectingalloftheleft-half-planezerosandpoles,andby selectingone-halfofthe imaginaryaxiszerosandpoles[7].Let F m;n ( s )= l Q ( s ¡ z i )( s + z i ) Q ( s ¡ p i )( s + p i ) with Re ( z i ) ;Re ( p i ) 0 ; thenthespectralfactorof F m;n ( s )is ^ G ( s )= k Q ( s ¡ z i ) Q ( s ¡ p i ) 20

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2.6AlgorithmAsoutlinedinsection2.1,ouralgorithmhassevenparts.Thestep sareelaborated below. 1. Deviseamappingofthefrequencyaxistorecastthespecicat ion H cont ( ) 2 [0 ; 1 ) ,asanevenfunction ~ H even (~ ) ~ 2 ( ¡1 ; 1 ) ,withsupporton [ ¡ 1 ; 1] Themappingmustbeanalyticallyextendabletothewholecomp lex plane;itmustbeexplicitlyinvertible,andwhencomposedwit hrationalfunctions,theresultmustyieldarationalfunction.Furthermore, ifthemapping isnonlinear,compositionwithitmayalterthenumeratorand denominator degreesoftherationalapproximatingfunctionandthezero sandpolesofthe latterwillbemoved.Therefore,thedesignermustaccorddued iligencewhen choosingthemapping. 2. Mollify ~ H even (~ ) toacontinuousandboundedvariatefunction. Atpointsof discontinuitywithin[ ¡ 1 ; 1],thedesignrequirementsaremodiedinorderto convert ~ H even (~ )to ~ H cont (~ ),suchthat ~ H cont (~ )iscontinousandofbounded variation.Thismollicationisanengineeringdesigndecisio nanddependson theparticularproblemtobesolved.Generally,themodicati onisobtainedby converting ~ H even (~ )toacontinuousfunctionatitspointsofdiscontinuity. 3. ComputePade-Chebyshevrationalapproximation. Becausewewillspectrally factorourPade-Chebyshevapproximationtoachievethedesi rednetworkfunction G ratl ( s ),thefunctiontobeapproximatedisnot j ~ H cont (~ ) j but j ~ H cont (~ ) j 2 Inorderfor G ratl ( s )toattenuatefor s = j! 2 [1 ; 1 ),thedegreeofitsnumeratormustbelessthanthedegreeofitsdenominator.However ,asstated inSection2.4,thePadeapproximation r m;n ( )constructedbyGeddesrequires 21

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m>n [6].Therefore,wereciprocateand,ifnecessary,truncate j ~ H cont (~ ) j 2 to formmax n 1 j ~ H cont (~ ) j 2 ;A o asthe\target"fortherationalapproximation. Inourexperience,thethreeconditionsofSection2.4forsuc cessfulexecution ofthePade-Chebyshevconstructionhavealwaysbeensatisedin networkdesignapplications.Wesuspectthatincreasingthenumeratorandd enominator degreesoftheapproximantwouldovercomeproblemsinthisa reaiftheyarose. Choose m and n ,with m n ,andcalculate c k = 2 Z 1 ¡ 1 max ( 1 j ~ H cont (~ ) j 2 ;A ) T k (~ ) p 1 ¡ ~ 2 d ~ !: Let r m;n ( z )bethePadeapproximationto P m + n k =0 0 c k z k ,andcomputethePadeChebyshevapproximation, ^ R m;n (~ )= r m;n (~ + j p 1 ¡ ~ 2 )+ r m;n (~ ¡ j p 1 ¡ ~ 2 2 : 4. Dene R m;n (~ s = j ~ )= 1 ^ R m;n (~ s=j ) andanalyticallycontinue R m;n (~ s ) tothewholeplane 5. SpectrallyfactorthePade-Chebyshevfunction. Forreasonswehavediscussed, R m;n (~ s )willbeevenwithrealcoecients[11].Thusitcanbespectral lyfactored;retainthefactorcontainingitsleft-halfplanepol esandzerostoform ~ G ratl (~ s ). 6. Returntotheoriginaldomainbyinvertingthemappinginstep 1toyield G ratl ( s ),which a.isrational,physicallyrealizable 22

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b. j G ratl ( j! ) j' H spec ( )onsupportof H spec ( ),duetoPade-Chebyshev construction. c. j G ratl ( j! ) j! 0as !1 becausedenominatordegree > numerator degree. 7. Iterateasnecessary. Iftheresultantmagnitudefrequencyresponsedoesnot satisfythedesignspecications,theengineermayiteratethisal gorithmby1) mollifying H cont ( ),2)adjusting m;n ,or( m;n );whilekeeping m>n instep3, 3)adjustingthevalueof A ,or4)choosingadierentmappingofthefrequency axis. 23

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Chapter3 Examples 3.1DoubleBandpassExampleProblem:Determinearealizabletransferfunctionimplemen tingadoublebandpass lterwithcenterfrequencyof 0 =0 : 75rad/sec,rstpassbandstartingat =0 : 4, endingat =0 : 5;secondpassbandstartingat =1 : 0,endingat =1 : 3;and notchattenuation ¡ 30dB(gain 0 : 0316),Figure10. 1 0.80.60.40.2 0 omega 2 1.5 1 0.5 0 Hspec(w) Figure10.DesignRequirementsforDoubleBandpasslter. 24

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Step1usesastandardbandpassmapping~ =1 : 1111 ¡ : 5444 :(0 ; 1 ) ( ¡1 ; 1 )[15]todeneanevenfunction ~ H even (~ )withsupporton[ ¡ 1 ; 1],Figure11. FromStep2ofouralgorithm,wemollify ~ H even ( ),Figure12.Inordertocompute thePade-Chebyshevapproximation,weformthereciprocalo f j ~ H cont (~ ) j 2 or 1 j ~ H cont (~ ) j 2 truncatedat A =(1 =: 0316) 2 =1001,Figure13. 1 0.80.60.40.2 0 omega 2 1 0 -1 -2 Heven(w) Figure11.EvenFunction. Equation15showstheresultofapplyingStep3toFigure13.Inc omputing r 12 ; 8 ( ),weusetheChebpadecommandinMaple tm .Acomparisonofthe 1 j ~ H cont (~ ) j 2 versusthePade-Chebyshevrationalapproximation, r m;n ( ),isshowninFigure14. r 12 ; 8 (~ )= 32 : 858~ 12 ¡ 152 : 94~ 10 +288 : 19~ 8 ¡ 274 : 50~ 6 +140 : 74~ 4 ¡ 36 : 430~ 2 +3 : 7000 2 : 1918~ 8 ¡ 0 : 8989~ 6 +0 : 3653~ 4 ¡ 0 : 0472~ 2 +0 : 0038 (15) Forstep4,wesubstitute~ =~ s=j andreciprocateasdepictedinequation(16). 25

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1 0.80.60.40.2 omega 0 2 1 0 -1 -2 Heven(w) Hcont(w) Figure12.MolliedandDesignRequirements. R (~ s )= 2 : 1918~ s 8 +0 : 8989~ s 6 +0 : 3653~ s 4 +0 : 0472~ s 2 +0 : 0038 32 : 858~ s 12 +152 : 94~ s 10 +288 : 19~ s 8 +274 : 50~ s 6 +140 : 74~ s 4 +36 : 430~ s 2 +3 : 7000 (16) Thespectralfactorof R (~ s )withtheappropriatezerosandpolesfromStep5is revealedinequation(17). ~ G ratl (~ s )=0 : 2569 (~ s 2 +0 : 6432~ s +0 : 3327)(~ s 2 +0 : 3082~ s +0 : 1267) (~ s 2 +0 : 4249~ s +1 : 4897)(~ s 2 +0 : 0251~ s +0 : 3355) (~ s 2 +0 : 3969~ s +0 : 6713) (17) Finally, ~ G ratl (~ s )istranslatedbacktotheoriginaldomainviatheinverseband passmapping[15],asrevealedinequation(18)andsimpliedin equation(19). 26

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omega 2 1 0 -1 -2 infinity 0 reciprocal |Hcont(w)|^2 Figure13. max n 1 j ~ H cont (~ ) j 2 ;A o (Interpret\innity"as A ). G ratl ( s )= 0 : 2569 1 : 1111 s + 0 : 5444 s 2 +0 : 7147 s + 0 : 3502 s +0 : 3327 ¢ 1 : 1111 s + 0 : 5444 s 2 +0 : 3425 s + 0 : 1678 s +0 : 1267 1 : 1111 s + 0 : 5444 s 2 +0 : 4721 s + 0 : 2313 s +1 : 4897 ¢ 1 : 1111 s + 0 : 5444 s 2 +0 : 4410 s + 0 : 2161 s +0 : 6713 ¢ 1 : 1111 s + 0 : 5444 s 2 +0 : 0278 s + 0 : 0136 s +0 : 3355 (18) 27

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200 0 0 -2 -4 omega 1000 4 800600 2 400 |Hcont(w)|^2 reciprocalpade'-chebyshev Figure14.DesignGoalvs.Approximation. G ratl ( s )= 0 : 3916 s 10 +0 : 3353 s 9 +0 : 9762 s 8 +0 : 5444 s 7 +0 : 7794 s 6 +0 : 2673 s 5 +0 : 2343 s 4 +0 : 0394 s 3 +0 : 0225 s 2 1 : 8816 s 12 +1 : 4343 s 11 +9 : 6258 s 10 +5 : 1749 s 9 +17 : 0275 s 8 +6 : 2399 s 7 +12 : 8429 s 6 +3 : 0575 s 5 +4 : 0883 s 4 +0 : 6088 s 3 +0 : 5549 s 2 +0 : 0405 s +0 : 0260 (19) Figures15and16compareWatanabe'sresulttoourresult(wher ealog-logaxis areusedtoscalefeatures).Themaindierenceliesinthefactth atallofWatanabe's polesandzerosliesolelyonthe j! axis.Thisresultsinsignicantovershootatthe edgesoftransitionofthepassbands.Incontrast,ourresultusesth eLeft-Handside ofthecomplexplaneforitspolesandzeroswhichresultsinsmo othedgetransitions. 28

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0 -10-20-30-40 omega .1e2 5. 1. .5 .1 2010 Watanabe vs. Pade'-Chebyshev Design watanabe Pade'-Chebyshev Figure15.Watanabevs.Pade-ChebyshevFrequencyResponse. 29

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100 -60 -40 -20 0 20 frequency (rad/sec) Watanabe's Designgain (dB) 100 -60 -40 -20 0 20 frequency (rad/sec) Pade'-Chebyshev Designgain (dB) -1 -0.5 0 -3 -2 -1 0 1 2 3 Watanabe Pole-Zero Map Real AxisImaginary Axis -1 -0.5 0 -3 -2 -1 0 1 2 3 Pade'-Chebyshev Pole-Zero Map Real AxisImaginary Axis Figure16.Watanabevs.Pade-ChebyshevFrequencyResponse. 30

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3.2RampExampleProblem:Determinearealizabletransferfunctionimplemen tingaramplter.In Figure17isshowntheidealrampfunction,oftenusedinthepro cessofimpedance matchingoftransistorstages.Forthisexample,thersttwostepsa renotcompletely needed,sincethespecicationisalreadycontinuousonitssupp ortandstatedasa \lowpass".Instead,weneedonlyextend H spec ( )to ~ H even (~ ),~ 2 ( ¡1 ; 1 )with supportin[-1,1]. gain 1 0.80.60.40.2 omega 0 2 1.5 1 0.5 0 ideal ramp Figure17.IdealRampFrequencyResponse. BeforewecomputethePade-Chebyshevrationalapproximati on r m;n (~ ),thereciprocalisformed(A=10,000),Figure18;withtheresultshown inequation(20). Figure19comparesthe r 10 ; 4 (~ )withthereciprocalof j ~ H cont (~ ) j 2 31

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omega 2 1 0 -1 -2 gain infinity 0 reciprocated ramp Figure18. max n 1 j ~ H cont (~ ) j 2 ;A o (Interpret\innity"as A ). r 10 ; 4 (~ )= 1 : 8826 ¡ 2 : 0314~ 2 +1 : 6512~ 4 +0 : 4611~ 6 ¡ 0 : 0107~ 8 +0 : 0002~ 10 1 : 0000 ¡ 0 : 2835~ 2 +0 : 9891~ 4 (20) Reciprocatingandextending r 10 ; 4 (~ )totheentirecomplexplane;i.e.,replacing ~ with~ s=j ,resultsinequation(21). R 4 ; 10 (~ s )= 1 : 0000+0 : 2835~ s 2 +0 : 9891~ s 4 1 : 8826+2 : 0314~ s 2 +1 : 6512~ s 4 ¡ 0 : 4611~ s 6 ¡ 0 : 0107~ s 8 ¡ 0 : 0002~ s 10 : (21) Finally,thecomputedtransferfunctionisrevealedinequat ion(22)andshownin Figure20alongwiththeidealramp. 32

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1 0.5 0 omega 4 2 0 -2 -4 gain 3 2.5 2 1.5 inverted ramp pade'-chebyshev ramp Figure19.Reciprocalvs.Pade-ChebyshevFrequencyResponse. Theramptransferfunctionwasachievedwitharelativelylow orderapproximation.Whilethisisgoodforimplementation(especiallyfort ransistorcircuits),if thegivenproblemrequiresgreateraccuracyonecouldaccom plishthisbyincreasing eitherm,n,or(m,n). G ratl ( s )=79 s 2 +1 : 3131 s +1 : 0055 ( s +2 : 0844)( s 2 +6 : 4428 s +55 : 7993)( s 2 +0 : 8637 s +0 : 9328) : (22) 33

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0.2 6 0 4 2 0 gain omega 1 10 0.80.6 8 0.4 ideal ramp pade'-chebyshev Figure20.IdealRampvs.RationalTransferFunction. 34

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Chapter4 ConclusionandFutureDirection 4.1ConclusionClassicalltertheoryusesrationalapproximationsthatarere ciprocalsofpolynomials, i.e.,theyhaveonlypolesandnozeros.(Note:Ellipticltersp ossesszerosandpoles, buttheyrestricttheplacementofthezeros,i.e.,thezerosar einverselyrelatedtothe poles.)Asaresult,theseapproximantsaretooinexibletoappro ximatefunctions whichhaveinteriorzeros,e.g.,thedoublebandpasslter. Herein,wehavedevelopedanewmethodthatenableslterdesign erstoapproximatemoregeneralltershapesbyallowingarbitraryzerosan dextendingtheuse ofthePadealgorithm.Previousattemptstomatchnetworkt ransferfunctionsto generalltermagnitude-frequencydesignrequirementsnece ssitatedextremelycomplicatedtheoryandformulation[16].Watanabe'smethod(se esummaryinAppendix B)producesapproximationsthataresharper(duetoimaginar yaxispolesandzeros); however,theauthor'smethodproducesapproximationsthat aresmoother. Theauthor'salgorithmissimplerinbothconceptandimpleme ntation.However, computingthePadeapproximationinStep3isnotwithoutit snumericdiculties. HighorderPadeapproximationsometimesrequirestheinversi onofnearlysingularmatrices.Inthesesituations,carefulattentioninchoosing thematrixinversion algorithmisrequired. Thechoiceofthedegreesof( m;n )andthemollicationof H spec ( )aremadeso astomakethephysicalimplementationof G ratl ( s )asaccurateaspossible.However, 35

PAGE 44

thelowertheorderof( m;n ),thelessthenumberofcomponentsrequired.Therefore, thisconictisresolvedbyanappropriatecompromise(problem specic)fortheselectionofthedegreesandmollication.Theattenuationreq uiredof j G ratl ( s = j! ) j isachievedbyincreasingthedenominatordegreewhilekeepi ngthenumeratordegree thesame.Thetradespaceamongthedesigndecisionsthenis( m;n ), H cont ( ),and attenuation.Withexperience,theengineerwillreapsuccessf uldesignsforabroader classofproblemspreviouslyunattainablewithclassicalmetho ds. 4.2FutureDirectionThealgorithmpresentedhereinisexecuted,initspresentcon guration,manually; i.e.,the(m,n)and ~ H cont (~ )arenotsoftwaredriven,ratheruserdriven.Anobvious extensionistoautomateourmethodsothatthedesignerisnotin cumberedbythese manipulations. Anotherdirectionistousethistheorytocomputephase-freque ncyspeciedrationalfunctions.Thesetypesoffunctions(lters)arequiteoften usedincommunication systemstocorrectfortimingerrors(e.g.,symbolerrors). Eventhoughthispaperhasdescribedamethodoflterapproxim ation,without modicationitcouldbeemployedinanothercloselyrelatedel d|FrequencyDomain SystemIdentication.Heretheobjectistondarealizabletran sferfunctionclosely approximatingagivensetofmeasuredata,[ F ( k ) ;! k ];where k isavectorofdiscrete frequencepointsatwhichthesystem, F ( ),ismeasured. 36

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References [1]NormanBalabanianandTheodoreBickart. LinearNetowrkTheory:Analysis, Properties,DesignandSynthesis .MatrixSeriesinCircuitsandSystems.Matrix Publishers,Chesterland,Ohio,secondedition,1981. [2]RichardL.BurdenandJ.DouglasFaires. NumericalAnalysis ,section8.4,pages 517{529.WadsworthGroup,PacicGrove,California,seventhe dition,2001. [3]RichardL.BurdenandJ.DouglasFaires. NumericalAnalysis ,section2.4, page86.WadsworthGroup,PacicGrove,California,seventhed ition,2001. [4]Wai-KaiChen. PassiveandActiveFiltersTheoryandImplementations ,section 1.2,pages10{119.JohnWiley&Sons,Inc.,NewYork,NewYork,sec ondedition, 1986. [5]J.C.Doyle,B.A.Francis,andA.Tannebaum. FeedbackControlTheory ,section 12.2,page207.MacMillanPub.Co.,NewYork,NewYork,1992. [6]K.O.Geddes.Blockstructureinthechebyshev-padetable. SIAMJournalon NumericalAnalysis ,18:844{861,1981. [7]T.N.T.Goodman,C.A.Micchelli,G.Rodriguez,andS.Seatz u.Spectral factorizationoflaurentpolynomials. preprint ,1996. [8]LawrenceP.Huelsman. ActiveandPassiveAnalogFilterDesign:AnIntroduction ,chapter4,page185.McGraw-HillSeriesinElectricalandCo mputer Engineering.McGraw-Hill,Inc.,NewYork,NewYork,1993. [9]GeorgeA.BakerJr.andPeterGraves-Morris. PadeApproximants ,volume59of EncyclopediaofMathematicsandItsApplications ,section7.4,page386.CambridgeUniversityPress,NewYork,NewYork,secondedition,1996. [10]ShlomoKarni. AppliedCircuitAnalysis .JohnWileyandSons,NewYork,New York,1988. [11]R-C.Li.Alwayschebyshevinterpolationinelementaryfu nctioncomputations. preprint ,2002. [12]J.C.MasonandD.C.Handscom. ChebyshevPolynomials ,section5.5,page119. Chapman&Hall,BocaRaton,Florida,2003. 37

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[13]J.C.MasonandD.C.Handscom. ChebyshevPolynomials ,section5.5,page125. Chapman&Hall,BocaRaton,Florida,2003. [14]H.Pade.Surlarepresentationappocheed'unefonctio npardesfractionsrationelles. Ann.Scientiquedel'EcoleNormaleSuperieure ,9:1{93,1892. [15]KendallL.Su. AnalogFilters ,section4.2,page78.Chapman&Hall,London, UK,1996. [16]HitoshiWatanabe.Approximationtheoryforlter-network s. IRETransactions onCircuitTheory ,pages341{356,Sep1961. [17]L.Weinberg. NetworkAnalysisandSynthesis ,section7.6,pages331{333. McGraw-Hill,Inc.,NewYork,NewYork,1962. 38

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Appendices 39

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AppendixA : SymbolGlossary SymbolMeaning! Frequencyinrad/sec ~ Lowpassprototypefreqnuecyinrad/sec s Laplacedomainvariable ~ s Lowpassprototypedomainvariable A Reciprocalofmaximumattenuationconstant G ratl ( s )Desirednetworktransferfunction H spec ( )Magnitude-squareddesignrequirements ~ H cont (~ )Lowpassprototypedesignrequirements A (~ )PolynomialforLPPlters f ( x )Genericfunctionof x ~ f ( x )TaylororChebyshevseriesapproximationof f ( x ) T k ( x )Chebyshevfunctionof k thorder ~ f ( z )Transformed ~ f ( x )tounitcircle r ( x )Paderationalapproximationto ~ f ( x ) R (~ s )Pade-ChebyshevrationalapproximationinLPPdomain ~ G ratl (~ s )Spectralfactorof R (~ s ) 40

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AppendixB : Watanabe'sDoubleBandpassFilterTheory Theattenuationandphaseofthelterfunctionaredescribedas w ( )= u ( )+ jv ( ),where w ( )istheidealtransmissionfunction, u ( )isthe attenuationinnepers,and v ( )isthephaseinradians.Forthetype oflters(attenuationonlydesigns)presentedinhiswork,only u ( )had specialconditionsthat w ( )hadtoprovide.Theseare 1. u ( )satises u ( )= A k for 2 B k where B k 'saregivenregionsontheimaginaryaxisofthe -plane, i.e., B k =[ § b 2 k ; § b 2 k +1 ], k =0 ; 1 ; 2 ;:::;n ¡ 1. 2.Thefunction [ u ¡ log j ¡ a i j ] isregularatgivenpoints a i i =1 ; 2 ;:::;m 3.Otherwise, u ( )isaharmonicfunction. A w ( )iscallednon-poleifitsatisescondition1)and3)only. Theorem1 Theidealtransmissionfunction, w ( ) isanAbelianintegral ofthethirdkind,andanon-pole w ( ) isanAbelianintegraloftherst kind.Theycanbeexpressedas w ( )= Z ¡ ¢ d see[16]forproof. Thismeansthatthedierential, ¢ d ,hasthefollowingproperties: 1. ¢ d haspolesofrstorderatthenitenumberofgivenpoints a i 's. 41

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AppendixB(Continued) 2. ¢ d hasbranchpointsoforder2atthenitenumberofgiven points jb 2 k 'sand jb 2 k +1 's. Letting S ( p )denotethetransmissionfunctionand ( p )denotethecharacteristicfunction G ( p 2 )= S ( p ) S ( ¡ p )=1+ ( p 2 )=1+ ( p ) ( ¡ p ) : Assumethatthestandardformofidealtransmissionfunction, w ( ), ofadoublebandpasslterhaspassbands[ ¡ jk 1 ; + jk 1 ]and[ § j; 1 ]and attenuationpoles, Q 's,intheniteregionsonthefundamentalplane. FromthegeneraltheoryinSectionIIIin[16],theidealtran smissionfunctionisgivenas w ( )= Z 0 ( d 1 + m X =1 d 2 + Q 2 ) d q ( 2 + k 2 1 )( 2 +1) (23) where i = d i + jQ i : Thisintegralisanellipticintegralofthethirdkind. Itisknownthatthecharacteristicfunction, ( p ),ofadoublebandpass ltercanbefoundwithoutsolvinganytranscendentalequation sifand onlyiftheintegral(23)isexpressedintheformof w ( )= N X =1 w ( ) w ( )= sinh ¡ 1 X ( ) where X ( )isanalgebraicfunctionof and isaconstant. w ( )is denedasthecanonicalformof w ( )foradoublebandpasslter. 42

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AppendixB(Continued) Theorem2 Thenecessaryandsucientconditionsforanidealtransmissionfunction, w ( ) ,foradoublebandpassltertobeincanonicalform are: 1.Allresiduesofdierential, dw ( ) are § 1 2.Thereexistpolynomials A ( ) B ( ) ,and R ( ) thatsatisfy ( 2 + k 2 1 )( 2 +1) R 2 ( )= A ( ) B ( ) ; (24) m Y =1 ( 2 + Q 2 )= A ( ) ¡ B ( ) : (25) See[16]forproof.Corollary3 Thecanonicalformof w ( ) foradoublebandpassltercan bewritten w ( )=2sinh ¡ 1 vuut ¡ A ( ) Q m =1 ( 2 + Q 2 ) : FromTheorem6,itisknownthatonepolynomialof A ( )or B ( )is oforder2 m andtheotherisoforder,atmost,(2 m ¡ 2).Hence,from (24), R ( )mustbeanoddpolynomialoforder,atmost,(2 m ¡ 3),andis expressedas R ( )= R ¢ m ¡ 2 Y i =1 ( 2 + 2 i )(26) where R and i are( m ¡ 1)unknownparameters.By(24),(25),and(26); weget m relationsfortheparameters R i Q ,and k 1 F k ( R; i ;Q ;k 1 )=0 ;k =1 ; 2 ;:::;m: (27) 43

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AppendixB(Continued) For(27)tobevalid,itisnecessaryforoneconditiontobesatise d F ( Q ;k 1 )=0 ; (28) whichisobtainedbyeliminating( m ¡ 1) R and i unknownparameters. Condition(28)isdenedasthecharacteristicconditionfort hedouble bandpasslter.Thus anydoublebandpasslterwithattenuationpoles, Q ,andbandratio, k 1 ,becomespossibleofrealizationwithoutsolvingany transcendentalequations,providedthatthecharacteristi ccondition(28) isestablished Bytakingalinearcombinationofvariouscanonicalidealtr ansmission functionshavingthesame bandfactor, k 1 ,andvariousattenuationpoles, Q ,whichsatisfy(28),thegeneralidealtransmissionfunctioncan be written w ( )= N X =1 ¢ 8<: 2sinh ¡ 1 vuut ¡ A ( ) Q m =1 ( 2 + Q 2 ) 9=; (29) Equation(29)canbetransformed,seeTheorem2in[16],by ( )= H cosh w ( ) into ( )= H RA [ Q N =1 f q A ( )+ q B ( ) g 2 ] Q N =1 ¦ m =1 ( 2 + Q 2 ) ; (30) where RA meansrationalpartofthefunction.AccordingtoTheorem2, (30)willexhibitTschebysheperformanceinthepassband,but notin thestopband. 44

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AppendixC : FilterTransformations Band-passnetwork L LPP =B B=L LPP 2 0 L LPP B=C LPP 2 0 C LPP =B C LPP B 1 2 0 = p 1 2 Band-stopnetwork L LPP L LPP B=! 2 0 1 =L LPP B C LPP 1 =C LPP B C LPP B=! 2 0 B 1 2 0 = p 1 2 LPPtoband-pass transformation LPPtoband-stop transformation LPPtoLow-passnetwork L LPP C LPP L LPP =! 0 C LPP =! 0 0 LPPto high-pass transformation High-passnetwork L LPP 1 =L LPP 0 C LPP 1 =C LPP 0 0 Figure21.FilterTransformations. 45

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AbouttheAuthor WilliamJoelDietmarJohnsonisawellbalancedindividualwi thplentyofexperience ineverything.HeisanexpertmarksmanaswellasanexpertRFde signer.Hecan packdishwashersoptimallyaswellascartrunksandmovingbox es.Hetendstoaccumulateexpensivethingslikebooks,motorcycles,andrearms.Hea lsospeaksforeign languagesandhaslivedandtraveledextensivelythroughout theworld,including Asia.