5. Time-Domain Analysis of Discrete-Time Signals a

5.Time-DomainAnalysisofDiscrete-TimeSignalsandSystems5.1.ImpulseSequence(1.4.1)5.2.ConvolutionSum(2.1)5.3.Discrete-TimeImpulseResponse(2.1)5.4.ClassificationofaLinearTime-InvariantDiscrete-TimeSystembyitsImpulseResponse(2.3)5.5.LinearConstant-CoefficientDifferenceEquations(2.4.2)5.1.ImpulseSequence5.1.1.ImpulseSequenceTheimpulsesequenceisdefinedas(5.1).0n,00n,1)n(⎩⎨⎧≠==δTheimpulsesequenceisillustratedinfigure5.1.1Figure5.1.ImpulseSequence.nδ(n)AnAδ(n-n0)n0…………δ(n)hasanimportantsamplingproperty,i.e.,x(n)δ(n-n0)=x(n0)δ(n-n0),(5.2)wheren0isanarbitraryinteger.5.1.2.StepSequenceThestepsequenceisdefinedas.0n,10n,0)n(u⎩⎨⎧≥=Thestepsequenceisillustratedinfigure5.2.Figure5.2.StepSequence.1nu(n)……(5.3)u(n)canbeexpressedastherunningsumofδ(n),i.e.,.)m()n(unm∑-∞=δ=(5.4)δ(n)canbeexpressedasthedifferenceofu(n),i.e.,δ(n)=u(n)-u(n-1).(5.5)5.2.ConvolutionSumTheconvolutionsumofx1(n)andx2(n)isdefinedasNotethatthesummationiscarriedoutwithrespecttoanintroducedvariable,m,andthefinalresultisafunctionofn.Theconvolutionsumsatisfiesthecommutativeproperty.)mn(x)m(x)n(x*)n(xm2121∑∞-∞=-=(5.6)x1(n)*x2(n)=x2(n)*x1(n),(5.7)theassociativeproperty[x1(n)*x2(n)]*x3(n)=x1(n)*[x2(n)*x3(n)],(5.8)andthedistributivepropertyx1(n)*[x2(n)+x3(n)]=x1(n)*x2(n)+x1(n)*x3(n).(5.9)Theconvolutionsumcanbecalculatedinthefollowingsteps.(1)Reflectx2(m)abouttheorigintoobtainx2(-m).(2)Shiftx2(-m)byntoobtainx2(n-m).(3)Calculatetheconvolutionsumatn.Steps(2)and(3)oftenneedtobecarriedoutindifferentwaysfordifferentintervalsofn.Example.Findx1(n)*x2(n),where(2)x1(n)=anu(n),0a1andx2(n)=u(n)..otherwise,02n0,1(n)xandotherwise,01n,20n,5.0(n)x(1)21⎩⎨⎧≤≤=⎪⎩⎪⎨⎧===(4)x1(n)=2nu(-n)andx2(n)=u(n).5.3.Discrete-TimeImpulseResponse5.3.1.DefinitionofDiscrete-TimeImpulseResponseAlineartime-invariantdiscrete-timesystemcanbedescribedbythediscrete-timeimpulseresponse,whichisdefinedastheresponseofthesystemtotheimpulsesequence..otherwise,06n0,a(n)xandotherwise,04n0,1(n)x(3)n21⎩⎨⎧≤≤=⎩⎨⎧≤≤=Alineartime-invariantdiscrete-timesystemcanalsobedescribedbythediscrete-timestepresponse.Itisdefinedastheresponseofthesystemtothestepsequence.5.3.2.I/ORelationbyDiscrete-TimeImpulseResponseTheI/Orelationofalineartime-invariantdiscrete-timesystemcanbeexpressedbyitsimpulseresponse.Assumethatx(n)andh(n)aretheinputandtheimpulseresponseofalineartime-invariantdiscrete-timesystem,respectively.Then,theoutputofthesystemisy(n)=x(n)*h(n).(5.10)Proof.Since,)mn()m(x)n(*)n(x)n(xm∑∞-∞=-δ=δ=(5.11)theoutputofthesystemcanbeexpressedasSincethesystemislinear,then.)mn()m(xT)]n(x[T)n(ym⎥⎦⎤⎢⎣⎡-δ==∑∞-∞=(5.12)(5.13)Bydefinition,h(n)=T[δ(n)].Sincethesystemistime-invariant,thenh(n-m)=T[δ(n-m)].(5.14)Substituting(5.14)into(5.13),oneobtains(5.10).5.4.ClassificationofaLinearTime-InvariantDiscrete-TimeSystembyitsImpulseResponse5.4.1.MemorylessSystemsversusSystemswithMemoryAssumethath(n)istheimpulseresponseofalineartime-invariantdiscrete-timesystem.Thesystemismemorylessifandonlyif[].)mn(T)m(x)n(ym∑∞-∞=-δ=h(n)=0,n≠0.(5.15)5.4.2.CausalSystemsversusNoncausalSystemsAssumethath(n)istheimpulseresponseofalineartime-invariantdiscrete-timesystem.Thesystemiscausalifandonlyifh(n)=0,n0.(5.16)5.4.3.StableSystemsversusUnstableSystemsAssumethath(n)istheimpulseresponseofalineartime-invariantdiscrete-timesystem.Thesystemisstableifandonlyifh(n)isabsolutelysummable.Thatistosay,thereisafiniteconstantBhsuchthat.B|)n(h|hn≤∑∞-∞=(5.17)Proof.Considerthesufficiencyfirst.Letx(n)bebounded,i.e.,|x(n)|≤Bx,(5.18)whereBxisafiniteconstant.Then,.BB|)m(h|B|)mn(x||)m(h|)mn(x)m(h|)n(y|hxmxmm≤≤-≤-=∑∑∑∞-∞=∞-∞=∞-∞=(5.19)Thatis,y(n)isalsobounded.Thus,thesystemisstable.Considerthenecessitynext.Fortheinput,0)n(h,00)n(h|,)n(h|/)n(h)n(x*⎩⎨⎧=-≠---=(5.20)theoutputofthesystematn=0is∑∑≠-∞-∞=-=-=0)m(hm)m(h)m(x)m(h)m(x)0(y.|)n(h||)m(h||)m(h|nm0)m(h∑∑∑∞-∞=∞-∞=≠-=-=-=(5.21)Thesystemisassumedtobestable.Then,sincex(n)isbounded,y(n)isalsobounded.Thus,thereexistsafiniteconstantBhsuchthat.B|)n(h|hn≤∑∞-∞=(5.22)Example.Determinewhetherthefollowingsystemsarestable:(1)h(n)=δ(n-n0).(2)h(n)=u(n)(3)h(n)=0.5nu(n).(4)h(n)=2nu(n).(5)h(n)=0.5nu(-n-1).5.4.4.InvertibleSystemsversusNoninvertibleSystemsWeassumethattwolineartime-invariantdiscrete-timesystemsAandBhavetheimpulseresponsesg(n)andh(n),respectively.AandBaremutuallyinverseifandonlyifg(n)*h(n)=δ(n).(5.23)(5.23)canbeusedtoconstructtheinverseofagivensystem.5.5.LinearConstant-CoefficientDifferenceEquationsAdiscrete-timesystemmaybecharacterizedbyalinearconstant-coefficientdifferenceequation.However,itneedbementionedthatonlyalinearconstant-coefficientdifferenceequationcannotspecifyadiscrete-timesystemuniquely.Otherconditions,suchassomeoutputsamplesunderagiveninputorthestatementsaboutlinearity,time-invariance,causalityandstability,arealsorequired.Adiscrete-timesystemisoftencharacterizedbyalinearconstant-coefficientdifferenceequation,aright-sidedinputandsomeinitialconditions.Wewillfocusonthesecases.5.5.1.HomogeneousSolutionandParticularSolutionAlinearconstant-coefficientdifferenceequationcanbesolvedinthefollowingsteps.(1)Findthehomogeneoussolution.α,acharact