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(b)Forthebilineartransformation,thereisnoamplitudescalingofthefrequencyresponse;however,thereisthefollowingfrequencytransformation=2arctan(-2) wTABSTRACT.ToillustratesomeoftheideasdevelopedinLecture,weintroduceinthislectureasimpleandparticularlyusefulclassoffiltersreferredtoas Butter-worthfiltersItwasfirstdescribedinbytheBritishengineerandphysicistStephenButterworthinhispaperentitled"OntheTheoryofFilterWritethe expressionforthemagnitudesquaredofthefrequencyTheButterworthfilterisatypeofsignalprocessingfilterdesignedtohaveafrequencyresponsethatisasflat aspossibleinthepassbandThenewmaximallyatlowpassIIRltershaveanunequalnumberofzerosandpolesandpossessaspeciedhalf-magnitudefrequency AButterworthfilterhasamonotonicresponsewithoutripple,butarelativelyslowtransitionfromthepassbandtothestopbandThefiltertobedesignedisa Stepped-impedanceLowpassFilterwithmaximallyflatresponseorButterworthresponse,cutofffrequency=ButterworthandChebyshevfiltersarespecialcases ofellipticalfilters,whicharealsocalledCauerfilters.Thefrequencyresponseofthesefiltersismonotonicbutterworth[p]:=Module[{},n=2*p;denom=1; For[k=0,kT=butterworth[3]s2+p2s+1s2+(p)sp+1s2+(1+3)sp+1LafonctiondetransfertetlediagrammedeBode:H[x]:=T/.s->I*x; GdB[x]:=20*Log[10,Abs[H[x]]];phase[xButterworthFiltersSolutionsSSincetherelationbetweenandwislinear,theshapeofthefrequencyresponseis preservedFiltersinthisclassarespecifiedbytwoparameters,thecutofffrequencyandthefilterorderItisalsoreferredtoasamaximally1FILTRESDE FREQUENCESACTIFSPASSE-BASDEBUTTERWORTH1èrePARTIE:ETUDETHEORIQUEOnappellefiltredefréquencespasse-basidéal(figure1), unButterworthFiltersThenewmaximallyat1)Stepped-impedanceLowpassFilterAChebyshevfilterhasarapidtransitionbuthasrippleineitherthestopband orpassbandVascoQuartinBastosdeAlmeidadeCarvalhoItisalsoreferredtoasamaximallyflatmagnitudefilterButterworthFiltersSolutionsSSincethe relationbetweenandwislinear,theshapeofthefrequencyresponseispreserved.Ingeneral,anellipticalfilterhasrippleinboththestopbandandthePDothe followingforafifth-orderButterworthfilterwithcutofffrequencyofkHzandtransferfunctionB(s)(b)Forthebilineartransformation,thereisnoThispaper presentsaformula-basedmethodforthedesignofIIRfiltershavingmorezerosthan(nontrivial)polesThispaperpresentsaformula-basedmethodforthedesign ofIIRfiltershavingmorezerosthan(nontrivial)polesThefiltersaredesignedsothattheirsquaremagnitudefrequencyThenewformulasintroducedinthispaper unifytheclassicaldigitalButterworthlterandthewellknownmaximallyatFIRlterdescribedbyHerrmann[3]Thefiltersaredesignedsothattheirsquare magnitudefrequencyresponsesaremaximally-flatatandatandaretherebygeneralizationsofclassicaldigitalButterworthfiltersThenewformulasintroducedin thispaperunifytheclassicaldigitalButterworthlterandthewellknownmaximallyatFIRlterdescribedbyHerrmann[3]IST-UniversidadedeLisboa,Av RoviscoPais,LisboaE-mail:vascoqbac@Abstract MicrowavefiltersplayaveryimportantroleinRadioFrequencyandMicrowaveapplications,suchason communicationsorradarsystemsTheyarewidelyusedtofulfillthefundamentalrequirementsTheButterworthfilterisatypeofsignalprocessingfilterdesignedto haveafrequencyresponsethatisasflataspossibleinthepassband