数字信号处理作业(第二章答案)蔡坤宝英文版版

Chapter2ProblemAnswers2.1Twosequences)(1nxand)(2nxare,respectively,givenby1012)(1)(11n,n,/nxnand010(2))(2n,nn,nxn.Findthesequence)(nyofthesample-addedsumoftwosequences.Solution:Thesample-addedsumisgivenby0)1((1/2)12312)()()(121n,nn,/n,nxnxnynn.2.2Findthesample-accumulatedsequence)(nyofagivensequence1012)(1)(1n,n,/nxn.Solution:For1n,thesample-accumulatedsequenceisgivenby0)()(nkkxny.For1n,wehave1211)21(2)21(1)21(1)21()()(nnnkknk////kxny.Thus,thesample-accumulatedsequenceis101212)(1n,n,nyn.2.3Assumethatasequence)(nxisthesameasthatinproblem2.2.Determinethefirst-orderforwardandbackwarddifferencesequencesof)(nx,respectively.Solution:Thefirst-orderforwarddifferencesequenceisgivenby2)21(2120)()1()(2n,/n,n,nxnxnxn,andthefirst-orderbackwarddifferencesequenceisgivenby1)21(1110,)1()()(1n,/n,-nnxnxnxn.2.4Determinewhetherornoteachoffollowingsignalsisperiodic.Ifasignal,inthecase,isperiodic,specifyitsfundamentalperiod.(1))010cos()(n.nx(2))3sin()(πnnx(3))873cos()(πnπAnx(4))8()(πnjenxSolution:(1)Consideringminminminkk./k/N)200()0102()2(0,andtaking1mink,weseethatthesequenceisperiodicwithperiod200N.Solution:(2)Thissequenceisperiodic.Sinceminminmink/k/k/N)32()32()2(0,wecantake3minkandobtaintheperiodofthesequenceas2N.Solution:(3)Consideringminminmink/k//k/N)314()]73(2[)2(0,andtaking3mink,weseethattheperiodofthesequenceis14N.Solution:(4)Sinceminminminkk//k/N)16()]81(2[)2(0,Nisnotanintegerforanyintegervalueofmink.Thus,thegivensequenceisnonperiodic.2.10Foreachofthefollowingsystems,determinewhetherthesystemis(a)stable,(b)causal,(c)linear,(d)timeinvariant,and(e)memoryless.下列系统，确定系统是否(a)稳定，（b）因果，（C）线性，（D）时不变，和（E）无记忆。(1)nmmxny)()((2)2)]([)(nxny(3))792)sin(()(πnπnxny(4)nnmmxnxT0)()]([(5)given.)(with)()()]([ngnxngnxT(6))()]([0nnxnxT(7))()]([nxenxTSolution:(1)Thesystemisdefinedbynmmxny)()(.(a)Consideraninput)()(nunxwhichisboundedby1|)(xBnx|,thesystemresponsetothisinputisdeterminedas(考虑一个输入x(n)=u(n)这是有界的x(n)||=B=1,系统响应这个输入是确定))1()()()(0nmumunynmnm.Because|1|lim|)(limnny|nnthesystemisunstable.系统不稳定(b)Sincethesystemoutputvalue)(nyatpresenttimen,doesnotdependonanyfuturevalueoftheinput)(nx,thesystemiscausal.因为系统输出值y(n)目前时间n,并不依赖于任何未来值的输入x(n),系统是因果。(c)Consideringthat)]([)]([)()()]()([)]()([21212121nxTnxTmxmxmxmxnxnxTnmnmnmthesystemislinear.该系统是线性的(d)Assumethatanewinputis)()(01nnxnx.Thesystemresponsetothisnewsignalisgivenby假设一个新的输入)()(01nnxnx。系统对这个新的信号产生的响应为0)()()()]([)(0111nnknmnmkxnmxmxnxTny.Thisimpliesthat)()(01nnyny.Thus,thesystemistime-invariant.这意味着)()(01nnyny。因此,该系统是时不变的(e)Becausethesystemoutputatpresenttimedependsonthepresentandpasttimevaluesoftheinput,thesystemisdynamic.因为系统输出当前时间取决于当前和过去的时间值的输入,系统是动态的。Solution:(2)Thesystemisdefinedby2)]([)(nxny.(a)ForanyboundedinputsatisfyingnBnx|xallfor|)(,wehaven,Bnx|ny|xallfor|)(|)(22Thereforethesystemisstable.(b)Becausetheoutputvalueatpresenttimedoesnotdependonanyfuturevalueoftheinput,thesystemiscausal.(c)Consideringthat)]([)]([)]([)()(2)]([)]()([)]()([212222121222121nxTnxTnxnxnxnxnxnxnxnxTthesystemisnonlinear.(d)Let)()(01nnxnx.Thesystemresponse)(1nytothisnewsignalisgivenby202111)]([)]([)]([)(nnxnxnxTnywhichsatisfies)()(01nnyny.Thus,thesystemistime-invariant.(e)Becausethepresentoutputvalue)(nydependsonlyonthepresentvalueoftheinput)(nx,thesystemismemoryless.Solution:(3)Thesystemisdefinedby)792)sin(()(πnπnxny.(a)Ifaninputsatisfiesn,Bnx|xallfor|)(,thenn,Bnnx|ny|xallfor|)792)sin((|)(.Thereforethesystemisstable.(b)Becausethepresentoutputvalueisunrelatedtoanyfutureinputvalue,thesystemiscausal.(c)Consideringthat)]([)]([)]792)sin(([)]792)sin(([)792)]sin(()([)]()([21212121nxTnxTnnxnnxnnxnxnxnxTWeseethatthesystemislinear.(d)Let)()(01nnxnx.Thesystemresponse)(1nytothisnewsignalisgivenby)792)sin(()792)sin(()(011πnπnnxπnπnxny.Thismeansthat]7)(92)sin[()()(0001nnnnxnnyny.Thus,thesystemistime-varying.(e)Sincethepresentoutputvalue)(nydependsonlyonthepresentvalueoftheinput)(nx,thesystemismemoryless.Solution:(4)ConsideringthatthesystemisdefinedbynnmmxnxTny0)()]([)(.(a)Lettheinputofthesystembe)()(nunxwhichisboundedby1|)(xBnx|.For0nand0nn,thesystemresponsetothisinputisdeterminedas0),1(0),1()()(0000nnnnnmunynnm.Forbothcases,wehave0|1|lim0|1|lim|)(lim000n,nnn,nny|nnnThusthesystemisunstable.(b)Asthedefinitionofthesystem,wealwaysassumethat0nn.Thereforethepresentoutputvalue)(nydoesnotdependonanyfuturevalueoftheinput)(nx.Thismeansthatthesystemiscausal.(c)Because)]([)]([)()()]()([)]()([21212121000nxTnxTmxmxmxmxnxnxTnnmnnmnnmthesystemislinear.(d)Let)()(11nnxnx.Thesystemresponse)(1nytothisnewsignalisgivenby11011000)()()()()]([)(1111nnnnmnnnnknnmnnmmxkxnmxmxnxTny.Because10)()()(11nnnmmxnnyny,thesystemistim