版《数字信号处理(英)》课件Chap-4-Digital-Processing-of-CT-Signals

Chap4DigitalProcessingofCTSignalsDiscrete-TimeSignalProcessingofCTS;SamplingofCTSignals;AnalogLowpassFilterDesign;1Chap4DigitalProcessingofMostsignalsintherealworldarecontinuousintime;4.1IntroductionDTsignalprocessingalgorithmsarebeingusedincreasingly;DigitalprocessingofaCTsignalinvolves3basicsteps:SampleaCTsignalintoaDTsignal;

(analog-to-digital(A/D)

converter)ProcesstheDTsignal(binaryword);ConverttheprocessedDTsignalbackintoCTsignal.

(digital-to-analog(D/A)

converter)2MostsignalsintherealworSimplifiedBlockdiagramofaCTsignalprocessedbyDTsystemSincetheA/Dconversionusuallytakesafiniteamountoftime,asample-and-holdcircuitisusedtoensurethattheanalogsignalattheinputoftheA/Dconverterremainsconstantinamplitudeuntiltheconversioniscompletetominimizetheerrorinitsrepresentation;

C/DConverterDiscrete-TimeProcessor

D/CConverterFig.4.2BlockdiagramofCTsignalprocessedbyDTsystem3SimplifiedBlockdiagramof

OtheradditionalcircuitsTopreventaliasing,ananaloganti-aliasingfilterisemployedbeforetheS/Hcircuit;TosmooththeoutputsignaloftheD/Aconverter,whichisastaircase-likewaveform,ananalogreconstructionfilterisused.SampleandholdDigitalSystemD/A

Anti-aliasingfilterA/DcompensatedreconstructionfilterFig.4.1CompletedblockdiagramrepresentationofaCTsignalprocessedbyDTsystem4OtheradditionalcircuitsToNormalizeddigitalangularfrequencyExample

:Normalizeddigitalangularfrequency0:ThesampledDTsignal:CTsignal:5NormalizeddigitalangularfEffectofsamplingintheFrequency-DomainSupposeacontinuous-timesignal:

ga(t)Samplingsequence:

g

[n]

samplingperiod:

T

;

samplingfrequency:FT=1/T

CTFTGa(j)ofsignal

ga(t)is:6EffectofsamplingintheFreq

EffectofsamplingintheFrequency-DomainTheDTFTG(ej)ofasequenceg[n]isgivenbyRelationbetweenG(ej)andGa(j)

Conversionfromimpulsetraintodiscrete-timesequencega(t)gp(t)p(t)g[n]=ga(nT)PeriodsamplinginMathematics7EffectofsamplingintheFre

PeriodsamplinginMathematics8PeriodsamplinginMathematiCTFTGp(j)ofgp(t)

AccordingtothedefinitionofCTFTAccordingtothemodulationtheoremofCTFT9CTFTGp(j)ofgp(t)Accord

Effectofsamplinginthefrequency-domain

Gp(j)isaperiodicfunctionoffrequencyconsistingofasumofshiftedandscaledreplicasofGa(j),

shiftedbyintegerof

Tandscaledby1/T.Basebandsignal:

thetermontheright-handsideofEq.(4.16)for

k=0iscalledbasebandportionofGp(j).Baseband/Nyquistband:frequencyrange

T/2<T/210EffectofsamplinginthefreIllustrationofthefrequency-domainEffectsoftime-domainsamplingNooverlap11Illustrationofthefrequency

EffectofsamplingintheFrequency-DomainItisevidentfromthefigurethatifT2m,thereisnooverlapbetweentheshiftedreplicasof

Ga(j)generating

Gp(j).

IfT

2m

,ga(t)

canberecoveredexactly

fromgp(t)by

passingifthroughan

ideallowpassfilter

Hr(j)

with

gain

T

anda

cutofffrequency

c

greaterthanm

andlessthanTm

.

12EffectofsamplingintheFre

Hr(j)ga(t)gp(t)p(t)ga(t)

EffectofsamplingintheFrequency-Domain13Hr(j)ga(t)gp(t)p(t)ga(t)Illustrationofthefrequency-domaineffectsoftime-domainsamplingOverlap14Illustrationofthefrequency

EffectofsamplingintheFrequency-DomainOntheotherhand,ifT<2m,thereisanoverlap

ofthespectraoftheshiftedreplicasofGa(j)

generating

Gp(j).

IfT

<2m

,duetotheoverlap

oftheshiftedreplicas

of

Ga(j),thespectrumGp(j)

cannotbeseparatedbyfilteringtorecover

Ga(j)

becauseofthedistortioncausedbyapartofreplicasimmediatelyoutsidethebasebandbeing

foldedBackor

aliasedintothebaseband.15EffectofsamplingintheFreSamplingTheoremSupposethat

ga(t)beaband-limitedsignalwiththenga(t)isuniquelydeterminedbyitssamplesg[n]=ga(nT),n=0,±1,±2,···,if

Nyquistconditions:

Foldingfrequency:16SamplingTheoremSupposethatSamplingTheorem

NyquistFrequency:

m

Nyquistrate:2mAndthenpassingifthroughanideallowpassfilter

Hr(j)

withagain

Tandacutofffrequency

c

satisfyingGiven{g[n]=ga(nT)

},wecanrecoverexactlyga(t)bygeneratinganimpulsetrain,17SamplingTheoremNyquistFrSeveralSamplingOversampling:Thesamplingfrequencyishigher

thantheNyquistrateUndersampling:Thesamplingfrequencyislower

thantheNyquistrateCriticalsampling:ThesamplingfrequencyisequaltotheNyquistrateNote:Apuresinusoidmaynotberecoverablefromitscriticallysampledversion.18SeveralSamplingOversamplinExamplesofsamplingIndigitaltelephony,a3.4kHzsignalbandwidthisadequatefortelephoneconversation;Hence,asamplingrateof8kHz,whichisgreaterthantwicethesignalbandwidth,isused.Inhigh-qualityanalogmusicsignalprocessing,abandwidthof20kHzisusedforfidelity;Hence,inCDmusicsystems,asamplingrateof44.1kHz,whichisslightlyhigherthantwicethesignalbandwidth,isused.19ExamplesofsamplingIndigi

ExamplesofsamplingExample4.3Consider3CTsinusoidalsignals:ThecorrespondingDTFTsare:TheyaresampledatarateofT=0.1sec,orsamplingfrequencyT=20rad/sec.20ExamplesofsamplingExample4TheCTFTofthethreesignals:TheCTFTofthethreesignalTheCTFTofthesampledimpulsetrains:TheCTFTofthesampledimp

Commentsonexample4.3Inthecaseofg1(t),thesamplingratesatisfiestheNyquistconditionandthereisnoaliasing;ThereconstructedoutputispreciselytheoriginalCTsignalg1(t);Intheothertwocases,thesamplingratedoesnotsatisfytheNyquistcondition,resultinginaliasing,andoutputssreallequaltothealiasedsignalg1(t)=cos(6t);23Commentsonexample4.3InInthefigureofG2p(j),theimpulseappearingat

=6inthepositivefrequencypassbandofthelowpassfilterresultsfromthealiasingoftheimpulseinG2(j)at=14;InthefigureofG3p(j),theimpulseappearingat

=6inthepositivefrequencypassbandofthelowpassfilterresultsfromthealiasingoftheimpulseinG3(j)at=26;

Commentsonexample4.324InthefigureofG2p(j),thRelationbetweenG(ej)andGa(j)Since:therefore:or:25RelationbetweenG(ej)andSoorG(ej)isobtainedfromGp(j)simplybyscalingaccordingtotherelationRelationbetweenG(ej)andGa(j)26SoorG(ej)isobtainedfromGRecoveryoftheAnalogSignalSampleandholdDigitalSystemD/A

Anti-aliasingfilterA/DcompensatedreconstructionfilterFig.4.1DetailedblockdiagramrepresentationofaCTsignalprocessedbyDTsystemThelowpassreconstructionfilterHr(j):TheinputtoHr(j)isimpulsetraingp(t);

27RecoveryoftheAnalogSignalSTheimpulseresponsehr(t)ofthelowpassreconstructionfilterisobtainedbytakingtheinverseCTFTofHr(j):RecoveryoftheAnalogSignal28Theimpulseresponsehr(t)oTheoutputofHr(j)isgivenbyga(t)ga(t)Withassuming:c=T/2=/T.RecoveryoftheAnalogSignal29TheoutputofHr(j)4.3SamplingofBandpassSignalsBandpassCTSignalBandwidth:=HL;Assuming:H=M•(),Misaninteger;Samplingrate:T=2()=2H/M;CTFTofthesampledimpulsetrain:304.3SamplingofBandpassSigIllustrationofBandpassSamplingT=2()NoaliasingBandpassfilterRecoverthebandpasssignal31IllustrationofBandpassSamp

FrequencytranslationFrequencytranslation:Anyofthereplicasinthelowerfrequencybandscanberetainedbypassinggp(t)throughbandpassfilterswithpassbands:providingatranslationoftheoriginalbandpasssignaltolowerfrequencyranges.32FrequencytranslationFreque4.4AnalogLowpassFilterDesignFilterSpecification

:Passband:Stopband:Passbandedgefrequency:p;Stopbandedgefrequency:s.334.4AnalogLowpassFilterDeFrequency-DomainCharacterizationoftheLTIDTSystemPeakpassbandripple:

p=20log10(1p)

dB(4.35)Minimumstopbandattentuation:

s=20log10(s)

dB(4.36)RipplesareusuallyspecifiedindBas:Peakripplevalueinthepassband:p;Peakripplevalueinthestopband

:s.34Frequency-DomainCharacterizatNormalizedspecificationsforanaloglowpassfilterThemaximumvalueofthemagnitudeinthepassbandisassumedtobeunity(1);Passbandripple:ThemaximumvalueofthemagnitudeinthepassbandMaximumstopbandripple:35NormalizedspecificationsfoTwoadditionalparametersTransitionratio/selectivityparameter:

k<1forlowpassfilterDiscriminationparameter:usually,36TwoadditionalparametersTButterworthApproximationN-thorderbutterworthfilter:alsocalledamaximallyflatmagnitudefilterGainindB:Atdc,i.e.,

=0:At

=c:c3dBcutofffrequency.37ButterworthApproximationN-tTypicalmagnituderesponsewithc=1Twoparameters:the3-dBcutofffrequency

c

andtheorderN

completelycharacterizeaButterworthfilter.candNaredeterminedfrom:p,,s,1/A.38Typicalmagnituderesponsew

ObtaincandN

Solvetheequationsof(4.40)39ObtaincandNSolvetheeqTransferfunctionofButterworthlowpassfilterwhereThedenominatorDN(s)isknownastheButterworthpolynomialoforderN.40TransferfunctionofButterwoExampleanaloglowpassButterworthFilterExample4.8DeterminethelowestorderofatransferfunctionHa(s)havingamaximallyflatcharacteristicwitha1-dBcutofffrequencyat1kHzandaminimumattenuationof40dBat5kHz.Solution:Obtain:41ExampleanaloglowpassButterObtainA:OrderN:LetNbetheminimuminteger,soN=4.ObtainA:OrderN:LetNbetChebyshevApproximationType1ChebyshevApproximation

:ChebyshevpolynomialoforderN:orrecurrencerelationofChebyshevpolynomial:43ChebyshevApproximationType1TypicalType1Chebyshevlowpassfilterpassbandripple:stopbandattenuation:

1/Aats44TypicalType1Chebyshevlow

ObtainNandtransferfunctionPolepl

oftransferfunctionHa(s)

45ObtainNandtransferfuncti

ObtaintransferfunctionChebyshevIfiltertransferfunction:46ObtaintransferfunctionCheType2ChebyshevApproximationType2ChebyshevApproximation

:(4.52)47Type2ChebyshevApproximationExampleofChebyshevIIlowpassFilterExample4.9DeterminetheminimumorderNrequiredtodesignalowpassfilterwithatype1Chebyshevortype2Chebyshev(specifications:a1-dBcutofffrequencyat1kHzandaminimumattenuationof40dBat5kHz).Solution:Obtain:48ExampleofChebyshevIIlowpaObtainA:OrderN:LetNbetheminimuminteger,soN=3.Note:

ChebyshevorderislowerthanButterworthorder.ObtainA:OrderN:LetNbet4.5DesignofothertypeanalogfiltersSpectraltransformationmethodisusedtodesignothertypesoffilters;Stepsfordesignothertypesoffilters;Step1:DevelopthespecificationsofaprototypeanaloglowpassfilterHLP(s)

fromthespecificationsoftheDesiredanalogfilterHD(s)

usingfrequencytransformation;Step2:Designtheprototypeanaloglowpassfilter;Step3:DeterminethetransferfunctionHD(s)

ofthedesiredanalogfilterbyapplyingtheinverseoffrequencytransformationtoHLP(s).504.5Designofothertypeana

MarksoftheprototypeanddesiredfiltersToeliminatetheconfusion,Sign:theprototypeanaloglowpassfilter:HLP(s)

Laplacetransformvariables

thedesiredanalogfilter:HD(s)

Laplacetransformvariable

sTransformbetweenHLP(s)andHD(s)

or51MarksoftheprototypeanddethepassbandedgefrequencyofdesiredanaloghighpassfilterHHP(s).

TransformationtotheHighpassFilterwherethepassbandedgefrequencyofprototypeanaloglowpassfilterHLP(s);

Ontheimaginaryaxis,52thepassbandedgefrequenMappingofimaginaryaxisins-domainto-domainsLowpassfilterpassbandHIghpassfilterpassband53MappingofimaginaryaxisinsExampleofHighpassfilterdesignExample4.18DesignananalogButterworthhighpassfilter,withspecifications:Solution:Passbandedgefrequency:4kHz,passbandripple:0.1dB;Stopbandedgefrequency:1kHz,stopbandattenuation:40dB;Forprototypelowpassfilter,let54ExampleofHighpassfilterdeExample4.18Stopbandedgefrequency:Specificationsforprototypelowpassfilter:p=1,p

=0.1dB;s=4,s

=40dB;MATLABcodefragments:55Example4.18StopbandedgefMATLABcodes[N,wn]=buttord(wp,ws,afap,afas,‘s’);[B,A]=butter(N.wn,’s’);[num,den]=lp2hp(B,A,2*pi*4000);56MATLABcodes[N,wn]=buttord4.6Anti-AliasingFilterdesignAnti-aliasingfilter=analoglowpassfilter;Requirementforanti-aliasingfilter:Idealanti-aliasingfilter574.6Anti-AliasingFilterdesAnti-AliasingFilterdesignSpectrumofaliasedcomponentofinput58Anti-AliasingFilterdesignS4.7ReconstructionFilterdesignReconstructionfilter=analoglowpassfilter;Idealreconstructionfilterisnoncausalandunrealizable:Idealreconstructionfilter594.7ReconstructionFilterde4.7ReconstructionFilterdesignzero-orderholdfrequencyresponse;Reconstructionfilterwhere604.7ReconstructionFilterdeSummary(I)DTSPofCTS;SamplingofCTSignals;61Summary(I)DTSPofCTS;SSummary(II)AnalogLowpassFilterDesignPassbandedgefrequencypandripplep;Stopbandedgefrequencys

andripples.Peakpassbandripple:

p=20log10(1p)

dB(4.35)Minimumstopbandattentuation:

s=20log10(s)

dB(4.36)62Summary(II)AnalogLowpassN-thorderButterworthfilter:N-thorderButterworthfilterExercisesPage166:4.3,4.7Page167:4.12(*);4.16;4.2364ExercisesPage166:Page167:6Chap4DigitalProcessingofCTSignalsDiscrete-TimeSignalProcessingofCTS;SamplingofCTSignals;AnalogLowpassFilterDesign;65Chap4DigitalProcessingofMostsignalsintherealworldarecontinuousintime;4.1IntroductionDTsignalprocessingalgorithmsarebeingusedincreasingly;DigitalprocessingofaCTsignalinvolves3basicsteps:SampleaCTsignalintoaDTsignal;

(analog-to-digital(A/D)

converter)ProcesstheDTsignal(binaryword);ConverttheprocessedDTsignalbackintoCTsignal.

(digital-to-analog(D/A)

converter)66MostsignalsintherealworSimplifiedBlockdiagramofaCTsignalprocessedbyDTsystemSincetheA/Dconversionusuallytakesafiniteamountoftime,asample-and-holdcircuitisusedtoensurethattheanalogsignalattheinputoftheA/Dconverterremainsconstantinamplitudeuntiltheconversioniscompletetominimizetheerrorinitsrepresentation;

C/DConverterDiscrete-TimeProcessor

D/CConverterFig.4.2BlockdiagramofCTsignalprocessedbyDTsystem67SimplifiedBlockdiagramof

OtheradditionalcircuitsTopreventaliasing,ananaloganti-aliasingfilterisemployedbeforetheS/Hcircuit;TosmooththeoutputsignaloftheD/Aconverter,whichisastaircase-likewaveform,ananalogreconstructionfilterisused.SampleandholdDigitalSystemD/A

Anti-aliasingfilterA/DcompensatedreconstructionfilterFig.4.1CompletedblockdiagramrepresentationofaCTsignalprocessedbyDTsystem68OtheradditionalcircuitsToNormalizeddigitalangularfrequencyExample

:Normalizeddigitalangularfrequency0:ThesampledDTsignal:CTsignal:69NormalizeddigitalangularfEffectofsamplingintheFrequency-DomainSupposeacontinuous-timesignal:

ga(t)Samplingsequence:

g

[n]

samplingperiod:

T

;

samplingfrequency:FT=1/T

CTFTGa(j)ofsignal

ga(t)is:70EffectofsamplingintheFreq

EffectofsamplingintheFrequency-DomainTheDTFTG(ej)ofasequenceg[n]isgivenbyRelationbetweenG(ej)andGa(j)

Conversionfromimpulsetraintodiscrete-timesequencega(t)gp(t)p(t)g[n]=ga(nT)PeriodsamplinginMathematics71EffectofsamplingintheFre

PeriodsamplinginMathematics72PeriodsamplinginMathematiCTFTGp(j)ofgp(t)

AccordingtothedefinitionofCTFTAccordingtothemodulationtheoremofCTFT73CTFTGp(j)ofgp(t)Accord

Effectofsamplinginthefrequency-domain

Gp(j)isaperiodicfunctionoffrequencyconsistingofasumofshiftedandscaledreplicasofGa(j),

shiftedbyintegerof

Tandscaledby1/T.Basebandsignal:

thetermontheright-handsideofEq.(4.16)for

k=0iscalledbasebandportionofGp(j).Baseband/Nyquistband:frequencyrange

T/2<T/274EffectofsamplinginthefreIllustrationofthefrequency-domainEffectsoftime-domainsamplingNooverlap75Illustrationofthefrequency

EffectofsamplingintheFrequency-DomainItisevidentfromthefigurethatifT2m,thereisnooverlapbetweentheshiftedreplicasof

Ga(j)generating

Gp(j).

IfT

2m

,ga(t)

canberecoveredexactly

fromgp(t)by

passingifthroughan

ideallowpassfilter

Hr(j)

with

gain

T

anda

cutofffrequency

c

greaterthanm

andlessthanTm

.

76EffectofsamplingintheFre

Hr(j)ga(t)gp(t)p(t)ga(t)

EffectofsamplingintheFrequency-Domain77Hr(j)ga(t)gp(t)p(t)ga(t)Illustrationofthefrequency-domaineffectsoftime-domainsamplingOverlap78Illustrationofthefrequency

EffectofsamplingintheFrequency-DomainOntheotherhand,ifT<2m,thereisanoverlap

ofthespectraoftheshiftedreplicasofGa(j)

generating

Gp(j).

IfT

<2m

,duetotheoverlap

oftheshiftedreplicas

of

Ga(j),thespectrumGp(j)

cannotbeseparatedbyfilteringtorecover

Ga(j)

becauseofthedistortioncausedbyapartofreplicasimmediatelyoutsidethebasebandbeing

foldedBackor

aliasedintothebaseband.79EffectofsamplingintheFreSamplingTheoremSupposethat

ga(t)beaband-limitedsignalwiththenga(t)isuniquelydeterminedbyitssamplesg[n]=ga(nT),n=0,±1,±2,···,if

Nyquistconditions:

Foldingfrequency:80SamplingTheoremSupposethatSamplingTheorem

NyquistFrequency:

m

Nyquistrate:2mAndthenpassingifthroughanideallowpassfilter

Hr(j)

withagain

Tandacutofffrequency

c

satisfyingGiven{g[n]=ga(nT)

},wecanrecoverexactlyga(t)bygeneratinganimpulsetrain,81SamplingTheoremNyquistFrSeveralSamplingOversampling:Thesamplingfrequencyishigher

thantheNyquistrateUndersampling:Thesamplingfrequencyislower

thantheNyquistrateCriticalsampling:ThesamplingfrequencyisequaltotheNyquistrateNote:Apuresinusoidmaynotberecoverablefromitscriticallysampledversion.82SeveralSamplingOversamplinExamplesofsamplingIndigitaltelephony,a3.4kHzsignalbandwidthisadequatefortelephoneconversation;Hence,asamplingrateof8kHz,whichisgreaterthantwicethesignalbandwidth,isused.Inhigh-qualityanalogmusicsignalprocessing,abandwidthof20kHzisusedforfidelity;Hence,inCDmusicsystems,asamplingrateof44.1kHz,whichisslightlyhigherthantwicethesignalbandwidth,isused.83ExamplesofsamplingIndigi

ExamplesofsamplingExample4.3Consider3CTsinusoidalsignals:ThecorrespondingDTFTsare:TheyaresampledatarateofT=0.1sec,orsamplingfrequencyT=20rad/sec.84ExamplesofsamplingExample4TheCTFTofthethreesignals:TheCTFTofthethreesignalTheCTFTofthesampledimpulsetrains:TheCTFTofthesampledimp

Commentsonexample4.3Inthecaseofg1(t),thesamplingratesatisfiestheNyquistconditionandthereisnoaliasing;ThereconstructedoutputispreciselytheoriginalCTsignalg1(t);Intheothertwocases,thesamplingratedoesnotsatisfytheNyquistcondition,resultinginaliasing,andoutputssreallequaltothealiasedsignalg1(t)=cos(6t);87Commentsonexample4.3InInthefigureofG2p(j),theimpulseappearingat

=6inthepositivefrequencypassbandofthelowpassfilterresultsfromthealiasingoftheimpulseinG2(j)at=14;InthefigureofG3p(j),theimpulseappearingat

=6inthepositivefrequencypassbandofthelowpassfilterresultsfromthealiasingoftheimpulseinG3(j)at=26;

Commentsonexample4.388InthefigureofG2p(j),thRelationbetweenG(ej)andGa(j)Since:therefore:or:89RelationbetweenG(ej)andSoorG(ej)isobtainedfromGp(j)simplybyscalingaccordingtotherelationRelationbetweenG(ej)andGa(j)90SoorG(ej)isobtainedfromGRecoveryoftheAnalogSignalSampleandholdDigitalSystemD/A

Anti-aliasingfilterA/DcompensatedreconstructionfilterFig.4.1DetailedblockdiagramrepresentationofaCTsignalprocessedbyDTsystemThelowpassreconstructionfilterHr(j):TheinputtoHr(j)isimpulsetraingp(t);

91RecoveryoftheAnalogSignalSTheimpulseresponsehr(t)ofthelowpassreconstructionfilterisobtainedbytakingtheinverseCTFTofHr(j):RecoveryoftheAnalogSignal92Theimpulseresponsehr(t)oTheoutputofHr(j)isgivenbyga(t)ga(t)Withassuming:c=T/2=/T.RecoveryoftheAnalogSignal93TheoutputofHr(j)4.3SamplingofBandpassSignalsBandpassCTSignalBandwidth:=HL;Assuming:H=M•(),Misaninteger;Samplingrate:T=2()=2H/M;CTFTofthesampledimpulsetrain:944.3SamplingofBandpassSigIllustrationofBandpassSamplingT=2()NoaliasingBandpassfilterRecoverthebandpasssignal95IllustrationofBandpassSamp

FrequencytranslationFrequencytranslation:Anyofthereplicasinthelowerfrequencybandscanberetainedbypassinggp(t)throughbandpassfilterswithpassbands:providingatranslationoftheoriginalbandpasssignaltolowerfrequencyranges.96FrequencytranslationFreque4.4AnalogLowpassFilterDesignFilterSpecification

:Passband:Stopband:Passbandedgefrequency:p;Stopbandedgefrequency:s.974.4AnalogLowpassFilterDeFrequency-DomainCharacterizationoftheLTIDTSystemPeakpassbandripple:

p=20log10(1p)

dB(4.35)Minimumstopbandattentuation:

s=20log10(s)

dB(4.36)RipplesareusuallyspecifiedindBas:Peakripplevalueinthepassband:p;Peakripplevalueinthestopband

:s.98Frequency-DomainCharacterizatNormalizedspecificationsforanaloglowpassfilterThemaximumvalueofthemagnitudeinthepassbandisassumedtobeunity(1);Passbandripple:ThemaximumvalueofthemagnitudeinthepassbandMaximumstopbandripple:99NormalizedspecificationsfoTwoadditionalparametersTransitionratio/selectivityparameter:

k<1forlowpassfilterDiscriminationparameter:usually,100TwoadditionalparametersTButterworthApproximationN-thorderbutterworthfilter:alsocalledamaximallyflatmagnitudefilterGainindB:Atdc,i.e.,

=0:At

=c:c3dBcutofffrequency.101ButterworthApproximationN-tTypicalmagnituderesponsewithc=1Twoparameters:the3-dBcutofffrequency

c

andtheorderN

completelycharacterizeaButterworthfilter.candNaredeterminedfrom:p,,s,1/A.102Typicalmagnituderesponsew

ObtaincandN

Solvetheequationsof(4.40)103ObtaincandNSolvetheeqTransferfunctionofButterworthlowpassfilterwhereThedenominatorDN(s)isknownastheButterworthpolynomialoforderN.104TransferfunctionofButterwoExampleanaloglowpassButterworthFilterExample4.8DeterminethelowestorderofatransferfunctionHa(s)havingamaximallyflatcharacteristicwitha1-dBcutofffrequencyat1kHzandaminimumattenuationof40dBat5kHz.Solution:Obtain:105ExampleanaloglowpassButterObtainA:OrderN:LetNbetheminimuminteger,soN=4.ObtainA:OrderN:LetNbetChebyshevApproximationTy