FiniteImpuseResponseFiltersFiltersAfilterisasystemthatprocessesasignalinsomedesiredfashion.Acontinuous-timesignalorcontinuoussignalofx(t)isafunctionofthecontinuousvariablet.Acontinuous-timesignalisoftencalledananalogsignal.Adiscrete-timesignalordiscretesignalx(kT)isdefinedonlyatdiscreteinstancest=kT,wherekisanintegerandTistheuniformspacingorperiodbetweensamplesTypesofFiltersTherearetwobroadcategoriesoffilters:Ananalogfilterprocessescontinuous-timesignalsAdigitalfilterprocessesdiscrete-timesignals.Theanalogordigitalfilterscanbesubdividedintofourcategories:LowpassFiltersHighpassFiltersBandstopFiltersBandpassFiltersIdealFiltersPassbandStopbandStopbandPassbandPassbandPassbandStopbandLowpassFilterHighpassFilterBandstopFilterPassbandStopbandStopbandBandpassFilterM(w)M(w)wwwwwcwcwc1wc1wc2wc2Discrete-TimeSignalsDiscrete-TimeSystemT{}x[n]y[n]=T{x[n]}inputoutputThediagramsuggeststhattheoutputsequenceisrelatedtotheinputsequencebyaprocessthatcanbedescribedmathematicallybyanoperatorT.MovingAverageFilterAsimple,butusefulfilteristhemovingaveragefilter.Assumewehavethefollowinginputs,x[n]:246A3-pointaverageforafinite-lengthsignalofthevalues{x[0],x[1],x[2]}={2,4,6}givestheanswer?

(2+4+6)=4.Thisvaluedefinesoneoftheoutputvalues.Thenextoutputvalueisobtainedbyaveraging{x[1],x[2],x[3]}={4,6,4}thatyieldsavalueof14/3.y[0]=?(x[0]+x[1]+x[2])y[1]=?

(x[1]+x[2]+x[3])whichgeneralizestothefollowinginput-outputequationy[n]=?(x[n]+x[n+1]+x[n+2])Thisequationiscalledadifferenceequation.Forthetriangularinput,theresultisthesignaly[n]astabulatedbelow:Notethatthevaluesboldtypeinthex[n]rowarethenumbersinvolvedinthecomputationofy[2].Theoutputsequenceisplottedbelow:246Notethattheoutputsequenceislongerthantheinputsequenceandsomewhatrounded.Ingeneral,valuesfromeitherthepresentorfutureorbothcanbeusedFIRfiltercalculations.Afilterthatusesonlythepresentandpastvaluesoftheinputiscalledacausalfilter.Afilterthatusesfuturevaluesoftheinputiscalledanon-causalfilter.Analternativeoutputindexingschemecanproduceafilterthatiscausal.Forthepreviousproblem,theoutputbecomes:Notethatthisformissimplyatime-shiftedformoftheoriginalform.TheGeneralFIRFilterThegeneralformfortheFIRfilteris:ImplementationofFIRFiltersRecallthatthegeneralformfortheFIRfilteris:TheUnitImpulseTheunitimpulseisperhapsthesimplestsequencebecauseithasonlyonenon-zerovalue,whichoccursatt=0.Themathematicalnotationis:1TheShiftedImpulse1kTheshiftedimpulse,

d[n-k],isnon-zerowhenitsargumentiszero,i.e.,n-k=0,orwhenn=0.Theshiftedimpulseisaveryusefulconceptforrepresentingsignalsandsystems.Forexample,246Inordertoimplementthisform,weneedthefollowing: (1)ameansofmultiplyingdelayed-input signalsbythefiltercoefficients; (2)ameansofaddingthescaledsequence values; (3)ameansofobtainingdelayedversions oftheinputsequence.Itisusefultorepresenttheseoperationsinblockdiagramform.×x[n]y[n]by[n]=bx[n]+Multiplierx1[n]x2[n]y[n]Addery[n]=x1[n]+x2[n]x[n]y[n]Delayy[n]=x[n-1]UnitDelayExample:TheFIRfilteriscompletelydefinedoncethesetoffiltercoefficients{bk}isknown.Forexample,ifthe{bk}arethenwehavealength4filterwithM=3.Thisexpandsintoa4-pointdifferenceequation:Toillustratetheutilityoftheresultsthatweobtained,considerthecascadeoftwosystemsdefinedby:TheoverallcascadesystemhastheimpulseresponseconvolutionIntheanalogworld,convolutionisdescribedbytheequation:Constantwithrespecttot.Rotatedaboutthey-axis.Movesalongthex-axis.Inordertofindtheoverallimpulseresponsewemustconvolveh1[n]withh2[n].Thus,theequivalentimpulseresponseis

n 012345h_1[n] 1111h_2[n] 111h_1[0]h_2[n] 1111h_1[1]h_2[n] 1111h_1[2]h_2[n] 1111h[n] 123321Thus,theequivalentimpulseresponseiswhere{bk}isthesequence{1,2,3,3,2,1}.Thismeansthatasystemwithimpulseresponseh[n]canbeimplementedbythesingledifferenceequationUseconvolutiontocomputetheoutputy[n]forthelength4filterthathavethecoefficientsbk={1,-2,2,-1}.Usetheinputsignalshownbelow.FindthesystemfunctionH(z)ofaFIRfilterwhoseimpulseresponseis:FindthesystemfunctionH(z)ofaFIRfilterwhoseimpulseresponseis:CompuationalstructureforafirstorderFIRfilte