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ASSBANDANDINTHESTOPBANDAREGIVENWITHSOMEACCEPTABLETOLERANCESINADDITION,ATRANSITIONBANDISSPECIFIEDBETWEENTHEPASSBANDANDTHESTOPBANDTOPERMITTHEMAGNITUDETODROPOFFSMOOTHLYFOREXAMPLE,THEMAGNITUDEOFALOWPASSFILTERMAYBEGIVENASSHOWNINFIGUREJEG71ASINDICATEDINTHEFIGURE,INTHEPASSBANDDEFINEDBY0,WEREQUIRETHATTHEPMAGNITUDEAPPROXIMATESUNITYWITHANERROROF,IE,PPPJPFOREG,11INTHESTOPBAND,DEFINEDBY,WEREQUIRETHATTHEMAGNITUDEAPPROXIMATESZEROSWITHANERROROFE,IS,FOR,SJESTHEFREQUENCIESANDARE,RESPECTIVELY,CALLEDTHEPASSBANDEDGEFREQUENCYANDTHEPSSTOPBANDEDGEFREQUENCYTHELIMITSOFTHETOLERANCESINTHEPASSBANDANDSTOPBAND,PAND,AREUSUALLYCALLEDTHEPEAKRIPPLEVALUESNOTETHATTHEFREQUENCYRESPONSESOFADIGITALFILTERISAPERIODICFUNCTIONOF,ANDTHEMAGNITUDERESPONSEOFAREALJEGCOEFFICIENTDIGITALFILTERISANEVENFUNCTIONOFASARESULT,THEDIGITALFILTERSPECIFICATIONSAREGIVENONLYFORTHERANGE0DIGITALFILTERSPECIFICATIONSAREOFTENGIVENINTERMSOFTHELOSSFUNCTION,INDBHERETHEPEAKPASSBANDRIPPLEANDTHEMINIMUMLOG201JEGPSTOPBANDATTENUATIONAREGIVENINDB,IE,THELOSSSPECIFICATIONSOFADIGITALFILTERARESGIVENBY,DBPP1LOG20SS091PRELIMINARYCONSIDERATIONSASINTHECASEOFANANALOGLOWPASSFILTER,THESPECIFICATIONSFORADIGITALLOWPASSFILTERMAYALTERNATIVELYBEGIVENINTERMSOFITSMAGNITUDERESPONSE,ASINFIGURE72HERETHEMAXIMUMVALUEOFTHEMAGNITUDEINTHEPASSBANDISASSUMEDTOBEUNITY,ANDTHEMAXIMUMPASSBANDDEVIATION,DENOTEDAS1/,ISGIVENBYTHEMINIMUMVALUEOF21THEMAGNITUDEINTHEPASSBANDTHEMAXIMUMSTOPBANDMAGNITUDEISDENOTEDBY1/AFORTHENORMALIZEDSPECIFICATION,THEMAXIMUMVALUEOFTHEGAINFUNCTIONORTHEMINIMUMVALUEOFTHELOSSFUNCTIONISTHEREFORE0DBTHEQUANTITYGIVENBYMAXDB1LOG202MAXISCALLEDTHEMAXIMUMPASSBANDATTENUATIONFOR1,ASISTYPICALLYTHECASE,ITPCANBESHOWNTHATP21LOG20MAXTHEPASSBANDANDSTOPBANDEDGEFREQUENCIES,INMOSTAPPLICATIONS,ARESPECIFIEDINHZ,ALONGWITHTHESAMPLINGRATEOFTHEDIGITALFILTERSINCEALLFILTERDESIGNTECHNIQUESAREDEVELOPEDINTERMSOFNORMALIZEDANGULARFREQUENCIESAND,THESEPCIFIEDCRITICALPSFREQUENCIESNEEDTOBENORMALIZEDBEFOREASPECIFICFILTERDESIGNALGORITHMCANBEAPPLIEDLETDENOTETHESAMPLINGFREQUENCYINHZ,ANDFPANDFSDENOTE,RESPECTIVELY,THETFPASSBANDANDSTOPBANDEDGEFREQUENCIESINHZTHENTHENORMALIZEDANGULAREDGEFREQUENCIESINRADIANSAREGIVENBYTFPTP2FSSS912SELECTIONOFTHEFILTERTYPETHESECONDISSUEOFINTERESTISTHESELECTIONOFTHEDIGITALFILTERTYPE,IE,WHETHERANIIRORANFIRDIGITALFILTERISTOBEEMPLOYEDTHEOBJECTIVEOFDIGITALFILTERDESIGNISTODEVELOPACAUSALTRANSFERFUNCTIONHZMEETINGTHEFREQUENCYRESPONSESPECIFICATIONSFORIIRDIGITALFILTERDESIGN,THEIIRTRANSFERFUNCTIONISAREALRATIONALFUNCTIONOF1ZHZNMDZZDPP210MOREOVER,HZMUSTBEASTABLETRANSFERFUNCTION,ANDFORREDUCEDCOMPUTATIONALCOMPLEXITY,ITMUSTBEOFLOWESTORDERNONTHEOTHERHAND,FORFIRFILTERDESIGN,THEFIRTRANSFERFUNCTIONISAPOLYNOMIALIN1ZNNNZHZH0FORREDUCEDCOMPUTATIONALCOMPLEXITY,THEDEGREENOFHZMUSTBEASSMALLASPOSSIBLEINADDITION,IFALINEARPHASEISDESIRED,THENTHEFIRFILTERCOEFFICIENTSMUSTSATISFYTHECONSTRAINTNHTHEREARESEVERALADVANTAGESINUSINGANFIRFILTER,SINCEITCANBEDESIGNEDWITHEXACTLINEARPHASEANDTHEFILTERSTRUCTUREISALWAYSSTABLEWITHQUANTIZEDFILTERCOEFFICIENTSHOWEVER,INMOSTCASES,THEORDERNFIROFANFIRFILTERISCONSIDERABLYHIGHERTHANTHEORDERNIIROFANEQUIVALENTIIRFILTERMEETINGTHESAMEMAGNITUDESPECIFICATIONSINGENERAL,THEIMPLEMENTATIONOFTHEFIRFILTERREQUIRESAPPROXIMATELYNFIRMULTIPLICATIONSPEROUTPUTSAMPLE,WHEREASTHEIIRFILTERREQUIRES2NIIR1MULTIPLICATIONSPEROUTPUTSAMPLEINTHEFORMERCASE,IFTHEFIRFILTERISDESIGNEDWITHALINEARPHASE,THENTHENUMBEROFMULTIPLICATIONSPEROUTPUTSAMPLEREDUCESTOAPPROXIMATELYNFIR1/2LIKEWISE,MOSTIIRFILTERDESIGNSRESULTINTRANSFERFUNCTIONSWITHZEROSONTHEUNITCIRCLE,ANDTHECASCADEREALIZATIONOFANIIRFILTEROFORDERWITHALLOFTHEZEROSONTHEUNITIRNCIRCLEREQUIRES33/2MULTIPLICATIONSPEROUTPUTSAMPLEITHASBEENSHOWNTHATIRNFORMOSTPRACTICALFILTERSPECIFICATIONS,THERATIONFIR/NIIRISTYPICALLYOFTHEORDEROFTENSORMOREAND,ASARESULT,THEIIRFILTERUSUALLYISCOMPUTATIONALLYMOREEFFICIENTRAB75HOWEVER,IFTHEGROUPDELAYOFTHEIIRFILTERISEQUALIZEDBYCASCADINGITWITHANALLPASSEQUALIZER,THENTHESAVINGSINCOMPUTATIONMAYNOLONGERBETHATSIGNIFICANTRAB75INMANYAPPLICATIONS,THELINEARITYOFTHEPHASERESPONSEOFTHEDIGITALFILTERISNOTANISSUE,MAKINGTHEIIRFILTERPREFERABLEBECAUSEOFTHELOWERCOMPUTATIONALREQUIREMENTS913BASICAPPROACHESTODIGITALFILTERDESIGNINTHECASEOFIIRFILTERDESIGN,THEMOSTCOMMONPRACTICEISTOCONVERTTHEDIGITALFILTERSPECIFICATIONSINTOANALOGLOWPASSPROTOTYPEFILTERSPECIFICATIONS,ANDTHENTOTRANSFORMITINTOTHEDESIREDDIGITALFILTERTRANSFERFUNCTIONGZTHISAPPROACHHASBEENWIDELYUSEDFORMANYREASONSAANALOGAPPROXIMATIONTECHNIQUESAREHIGHLYADVANCEDBTHEYUSUALLYYIELDCLOSEDFORMSOLUTIONSCEXTENSIVETABLESAREAVAILABLEFORANALOGFILTERDESIGNDMANYAPPLICATIONSREQUIRETHEDIGITALSIMULATIONOFANALOGFILTERSINTHESEQUEL,WEDENOTEANANALOGTRANSFERFUNCTIONAS,SDPHAAWHERETHESUBSCRIPT“A“SPECIFICALLYINDICATESTHEANALOGDOMAINTHEDIGITALTRANSFERFUNCTIONDERIVEDFORMHASISDENOTEDBYZPGTHEBASICIDEABEHINDTHECONVERSIONOFANANALOGPROTOTYPETRANSFERFUNCTIONHASINTOADIGITALIIRTRANSFERFUNCTIONGZISTOAPPLYAMAPPINGFROMTHESDOMAINTOTHEZDOMAINSOTHATTHEESSENTIALPROPERTIESOFTHEANALOGFREQUENCYRESPONSEAREPRESERVEDTHEIMPLIESTHATTHEMAPPINGFUNCTIONSHOULDBESUCHTHATATHEIMAGINARYJAXISINTHESPLANEBEMAPPEDONTOTHECIRCL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