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1、Chapte2 Poblm nswers2.1 Twosequeces and are, resctvey, given byand .Find thesquence of the saple-ded sumofto seuncs.Soltion: Thesampl-adde smis give .2 Finthesmpl-aumulated seuene of a genquence.Soltion: For ,th ampl-cumuat suence i given b or , w have.Tus, the samleaccumated sqencis.2.3uth a equenc
2、e s e ae s that in problem2.2. Dermine the first-order forwrdan barddiference equencesof , repectivelySolution: The firs-order forwarddfference unce i given by,n th fs-ordard difrence squenc is en .4 Deermi whethr r noteh of folowing sigalss periodic. Ifa sgl,in h ae, is erioi, speifys undamtal peri
3、d.()()(3) (4) Sltio: (1) Conideig,and taking ,we see thtthe seque isperdic with prio .Sluti: (2)Thissquece i iodic Sc,wecnak andoba he p of th sequenc a .olution: (3) Conidering,ntaking,we eethatth perod of the equce is Solin: (4) Snce, isnot integ fo ay iteger alue of. hu, thegieneuen s nopeidic.2.
4、10For eh of he flowing sytems, dtrmine whether he sstem is (a) stable, (b) causal, () linear, (d) timeinvarian, and (e) emorlss.下列系统,确定系统是否(a)稳定,(b)因果,()线性,()时不变,和(E)无记忆。 (1)() (3) () (5) (6) (7)Sluton: () Th systm isfinedby .(a) Cnider aninput icbounded by , the systemspnse o ti inpu is detrmined a
5、s(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).ecauseth syem s usble. 系统不稳定() Sin he systemouput alu at presen time, does not depend on n fturvalue f the inpt , e ssmis cusal.因为系统输出值y(n)目前时间,并不依赖于任何未来值的输入(n),系统是因果。 (c) onidring thathe se liner.该系统是线性的 (d) Asue ata w put s . Thesstem resose othis ne
6、wsial s gven 假设一个新的输入。系统对这个新的信号产生的响应为 .Tis imls ha . Tu, tsystem istime-ivriant.这意味着。因此,该系统是时不变的(e) Becausesystem oututat pesen tm depndso the pesent and pas ie vlues ofthe nut, the sysem is dynam.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。Solutn: (2) e sysem is efined by .(a) or anyboue iutsisfyg,we aeTheref
7、orthe ys is stable(b) Bcase the otpt vaueat prsent ime doesnodend on an futurevlueofthe inu,thesystem i usal.() Cnidrng tatth smis olinea.(d) Let he sytem respose tothis nw signal s v bywhich satisis . Thus,th system s timnvan.(e) Bause te een outpuae ees only o thepesent alue of the nt, the system
8、is mmorylesSolu: () The sseis fined y .(a)If anip saisies ,thn.Tereoe thsstemis tabl.(b) ecase the presnuput value is unate to anfure ipt ale, th systems causal.(c) onsirig thaWe see htthe syte is line.() . The ystem respone tohis ne gl is gvey.hismeas ht Tus, the systemis time-varing.(e) inc te res
9、entoutput alue epends nly onthepresnt vaue f e inpt, te system is moyless.Sotion: () osierg thathe systeis dned y .(a)Letteiput of te system be wic is undd y . Fr and , thsstmrespns to thsit i dtermined asor both case, havThushe yem sunstab(b) A h defiitn f the sytem,weaways assum tat. Therfor the p
10、rsenouuvaue does ot eend onany fuure valueof te nput This means tha theysem s causl.() Becauseth sytem sline.(d) Let .Thessem reone otis ew sgnals gve by.euse, the sym is timevayig(e) Genall, hprsnt output vale is late to th inut vaue t other me. Terore, theyse is dynamc.Souion:(5) Cosderintht the s
11、ystem i ined () For bouninpt whichsatisies , we hae.Therefo,i , he . This ws hat theytem is tbe.() Bease he peset output valueisunlated toyfuurevalueof heiput, th sysem is usal.(c) CondringhtWe tht hestems inar() Le . he systmrespose t ts e ignal s given y.Bcase, the syem is ime-vaing.(e)Since he pe
12、setotp aue deped only onthepreetinptvle, thstem is mryes.Solio: ()onsierngthat te ystem is efiney (a) If, then.Thus, thsystem istale.(b) f, hen heprset output alue sunretd o the utureinput valu. Ths the syst i ausal. therwise, the system inoncausal.(c) Becasete systm is lear() Ltting, e v.T,the syemsis meinvarit.() If , he the ouuval depends nlyo th esent nput value,i.e., t