信號(hào)與系統(tǒng)課后答案 第2章 習(xí)題解

第2章習(xí)題d2ddy(t)3y(t)2y(t)0y(0)y(0)2；））））））2dtd2dy(t)4y(t)0y(0)(0)1y；2dtd2ddy(t)2y(t)2y(t)0y(0)y(0)2；2dtd2ddy(t)2y(t)y(t)0y(0)y(0)2；2dtd3d2ddd2y(t)2y(t)y(t)0y(0)y(0)y(0)2。322d2ddy(t)4y(t)0y(0)y(0)2。2dt201,2.212(t)AeBey2t.thd(0)3,y(0)2AB3,A2B2.A8,B5.由y::dt(t)ee:y.t2t40i.21,2(t)Acos(2t)Bsin(2t)y.hd1(0)1,y(0)1:A1,2B1:A1,B.由y2dx1t)cos(2t)sin(2t):y.2220:1i21,2(t)e(Acos(t)Bsin(t))y.thd(0)1,y(0)2A1,BA2,A1,B3.由y::dx(t)e(cos(t)3sin(t)):y.t101.:21,2(t)(AtB)ey.thd(0)1,y(0)2B1,AB2,A3,B1.由y::dx(t)t1)e.:yt010.y:,321,23(t)A(BtC)e.thdd2)y)y)2:AC1,BCC2B2由y.25,B3,C4:A.:y(t)5t4)et.00,4:.212(t)ABey4t.hd31(0)1,y(0)2:AB1,4B2:A,B.由y22dx31t)e22:y.4td2dy(t)5y(t)6y(t)3x(txteut；））t2d2ddut；y(t)3y(t)2y(t)x(t)4x(txte2t2dd(t)3y(t)x(t)xt,xteut）y2；tdtdtd3d2ddy(t)4y(t)8y(t)3x(t)8x(txtut。）32ddyt)6yt)eutt2t)()5yt將x2602,3.212(t)AeBey2t3t.ht)yt)p:y將:ttpdd32()5()6()e():yte.ttttt22t)AeBee.p3:y2t3tt20t0:dd2yt)5yt)6yt)ut):2d2yt)atbtcut()()()'dt2dyt)atbut()()dtyt)aut)()()()5()5()6()3()a'tbtcutatbutaututy(0)y(0)a0。y(0)y(0)b0''3t)AeBee將y2t3tt23AB0A323232A3B02B33t)(eeeut)yt3t2t22ddyt)3yt)2yt)t)2eutt2eut將xt2t2201,2212y(t)AeBet2tht)yt)p:y2:2.ttpt)AeBete:yt2t2t0t0:dd2yt)3yt)2yt)(t)2u(t):2d2yt)t)but)2dyt)aut)()()3()()2()atbutauttut(0)(0)1yya''(0)y(0)b1y(0)221AB3Ay't)AeBeteB3將y2tt2ty(0)4AB81''(t)ee2tey）2t2ttd2t)yt)3yt)t)eut將xtdt303.2(t)Cey3.tht)yt)e2.:y2,:ttpp(t)(Cee)u(t):y32t.t0t0:dyt)3yt)t)ut):dtdt)at)but)ydtyt)aut)()()3()()()1,4atbutauttutaby(0)y(0)11(t)Cee2y32Ctt(0)y(0)44y''(t)(2ee)u(t)y）3t2tddd32t)()4yt()8ytyt)t)ut將x32400,22,22ii32123(t)ABecos(2t)Cesin(2t)y2t2t.h(t)Dty(t)tp:y:.p(t)ABecos(2t)Cesin(2t)t.:y2t2t0t0:ddd32yt)4yt)8y(t)(t)8ut:32d3yt)t)but)dt3d2yt)aut)dt2()()4tut()()8()atbutautdd32y)4,y)3。32(t)ABecos(2t)Cesin(2t)t將y2t2t138A=,B8t)yC13t)e(cos(2t)sin(2t))tCy2t88dydt(t)2y(t)xt）xt3txteuteut）eutxteut）32tt2t）xteutT2tTdt)2yt)eutyt2dt202ytCe()2tht)yt)py(t)(Cete)u(t)2:y2tt2t2tp0-到0+y(0)0:y(t)teu(t)2tydt)2yt)eutt3dt202ytCe()2tht)yt)epy(t)(Cee)u(t)3:y3tt2t3tp0-到0+y(0)0:y(t)(ee)u(t)2t3t）dt)2yt)euteut3ty2tdtdy(t)Ae0t0yt)at)but)，2tdtyt)aut)a1,b2，y(t)e2:th(t)eu(t)2ty(t)x(t)*h(t)(e()ute())*(ute())ut2t3t2t=teu(t)(ee)u(t)2t2t3tydt)2yt)eutTtT2dt202y(t)Ce2tht)ktT)eyt)tT)ep設(shè)特解為y2(),帶入得tT2()，零狀態(tài)響應(yīng)tTp(t)Ceu(t)(tT)e)utT):y2t2(tT到T+yT)0:y(t)(tT)e2(tT)u(tT)d）y(t)y(t)x(t),y0xtut；dtd）ydtd(t)3y(t)x(t)2xt,y0xtut；dt（3）d2dddty(t)5y(t)6y(t)2x(t)3x(ty0y0xteut；2d2ddd2ty(t)6y(t)9y(t)x(t)x(ty0y0xteut；））2d2dddy(t)2y(t)5y(t)3x(t)5x(ty0y0xtut。2）(t)Aey(0)y(0)Ae1.y:0th(t)e:y.tzi(t)B1y(t)Ae1設(shè)y:B,tpzsd0t0yt)ut)1y(t)e1zs在設(shè):AA:.tdt(t)yy(t)(e1e)u(t)u(t):y）ttzizsdt)3yt)t)utydt(t)Aey(0)y(0)Ae2y(t)2eziy3t:0:3t.h22(t)BByt)Ceut)設(shè)y:故3t33p2112y(0)1C1，Cyt)(eut)3t333312t)yt)yt)(eut)2ey3。t3t33）(t)AeBeyAB2:y2t3thdy(0)2A3B0:y(t)6e4e.2t3tdtzi323t)Byt)AeBee0t0零狀態(tài)響應(yīng):設(shè)y,代入得:t.,在2t3tt2pdd2時(shí),yt)5yt)6yt)6(t)u(t)設(shè)2dd2yt)t)but),ydtt)aut),29e23yt)0:a6,b:yt)ete.23tt293t)yyt)eeeut)6e4e:y）。2t3tt2t3t22d(t)(AtB)ey(0)1,y(0)1,解系統(tǒng)的齊次解：y3t,零輸入響應(yīng):由hdt4,B1y(t)(4te3.:A.tzit)1y(t)((AtB)ee)u(t)0t0:設(shè)y2C.t3t2tpzsdd2yt)6yt)9yt)t)u(t),,2dd2yt)t)but),yt)aut)yt)0a1,b7,解得:則,.得2dt2,B1y(t)((2tee)u(t).:A.3t2tzs(t)yy(t)y(t)((AtB)ee)u(t)(4te:y）3t2t3tzizszs(t)e(Acos(2t)Bsin(2t))y:thd(0)1,y(0)1,A1,B1y(t)e(cos(2t)sin(2t))由y:.tdtzi(t)C1y(t)e(Acos(2t)Bsin(2t))1:y:C.tpzsdd20t0yt)2yt)5yt)t)5u(t),在,2dd2t)at)but),ytaut:ab.1()()設(shè)y2dt1,B1y(t)e(cos(2t)sin(2t))1:A..tzs:y(t)e(cos(2t)sin(2t))[e(cos(2t)sin(2t))1]u(t)ttht)ddd2dd）y(t)4y(t)x(t)；dtdt）y(t)3y(t)2y(t)x(t)3x(t)；2d2dy(t)4y(t)4y(t)x(t)；）2dd2d）y(t)2y(t)x(t)3x(t)3x(t)；2）(t)Aey4t，hd0t0yt)t)t)cut)yt)at)but)，，在dt1,b4,c:A4.a(t)(t)4eu(t)y）4t(t)AeBe0t0y2tthdd2yt)t)t)cut)()()()()(),yttbut,ytaut設(shè)2dt'()()btcut()(()atbut2()'()()auttt:at+=b0,c2A2,B1y(t)(2ee)u(t):a）:t2t(t)(AtB)ey2，thdd20t0yt)at)but),yt)aut)()0,yt:ab4在2dt1B0.:A,y(t)te）2t(t)Aey2t,hd0t0yt)''(t)t)(t)du(t)()'()()(),ytatbtcut在時(shí)dt''()'()()()2('()()btctdutatbtcut())''()'()()ttt則at:a=,1,b1,c1,d2:A1(t)eu(t)(t)t):y2tgt)d）ydtdd2dd(t)y(t)x(t)xt；()7yty(t)y(t)x(t)；）dt2d2ddy(t)2y(t)y(t)3x(t)2x(t)；））2d2dd3d2y(t)8y(t)y(t)x(t)2x(t)5x(t)；232）(t)Aey(t)Bp1:Bythd0t0yt)at)bu(t)yt)aut),,在dt():atbut)aut)t)ut),1,b0A0:a:yt)ut)）(t)AeBey3t4t，hdd20t0yt)t)but),yt)aut)()0,yt在:2dt():atbut)7aut()()a1,b7A1,B1=t:(t)(ee)u(t):yt34t（3）1(t)e(Acos3tBsint)C:yt5hdd20t0yt)t)but),yt)aut)()0在:,yt.2dt11415():at()2()()2()a3,b4,tauttutAB=::,51141t)e(cos3tsin3tut)ut):yt5155）5(t)(AtB)eC:y4:t16hdd2yt)''(t)t)t)t),yt)t)t)cu(t)設(shè)2dtyt)t)but):a1,b6,cd,2710116271015,Byt)(t:)et():A4t441616dy(t)5y(t)x)f(t)x(t)f(t)eu(t)3(t)tdtt)h。dt)t)yt)5yt)eut)t)y(t)xtdtzidt)yt)yt)5yt)t)hdt17t)eu(t)h(t)ht)*(eu(t)2(t))(eeut)h.=5t5ttt44(t)u(t)y(t)6eu(t)x(t)(t)2LTIx3t11y2(t)(t)(t)y；(t)eu(t)y(t)x。t33(t)yx(t)2xy1zs1zs2dy(t)yx(t)x(t)()yx,.133zs312dtzs2A(t)yAeu(t)yAeut)t)x:=3t.3t31zs1zs2(t)y(t)y6eu(t)x(t)y(t)y(t),對(duì)x:對(duì):3t1zizs12zizs2y6eu(t)(t):y,3tzs1zs2A:Ae3tut)(Aeut)t))6eu(t)(t):3t3t3A3yeu(t),3tzs16eyeu(t)h(t)(t)eu(t)）y）g）3t3t3tzizs1t)h)eut)t3t0(t)eu(t)xt331=h=3ty(t)*x(t)(eeut)t22zs333191t)yt)y(eeut)eut)(eeut)y=3tt3t3tt22223zs3(t)eu(t)y(t)1xt1x(t)teu(t)2y(t)e2u(t)ht)。tt21dt)xt)2xt)x,2dt11dt)yt)2yt).y2dt11d11t)4yt)eut)yt)(eeut):y:,2t2t4tdt11122因x(t)x(t)(t)h(t)3y(t)eu(t)2t,2111113t)(3(ee)eut)(eeut):h2t4t2t2t4t2222(t)eu(t)y(t)1xt1x(t)e2u(t)y(t)e2u(t)ht)。tt21x(tx(t)（提示：由于知道有不是簡(jiǎn)單的各階導(dǎo)數(shù)及其線性組合關(guān)系，所以不能用2-9題的方法。但根據(jù)條件12y(t)eu(t)*htt(p2-10-1)1y(t)eu(t)eu(t)*ht2t2t(p2-10-2)1即eu(t)*hteu(t)eu(t)*htt2t2t(p2-10-3)(p2-10-4)e2u(t)eu(t)e2u(t)*htttt對(duì)式子(p2-10-4)求兩次導(dǎo)數(shù)，并利用卷積的微分性質(zhì)有2()teu(t)2e2utht()*te2utttt(p2-10-5)(p2-10-6)這些'4()'teu(t)4e2utht()*tte2uttttteu(t),e2u(t)tt通過(guò)將式(p2-10-4)(p2-10-5)(p2-10-6)乘上適當(dāng)?shù)某?shù)再相加，可以消去方程右端ht普通函數(shù)和的卷積項(xiàng)。具體來(lái)說(shuō)，就是假設(shè)(p2-10-4)，(p2-10-5)，(p2-10-6)三個(gè)式子乘的系數(shù)分別是a,,c，則要求abc0(p2-10-7)2b4c0a可以得到一個(gè)解abc1。將式(p2-10-4)，(p2-10-5)，(p2-10-6)分別乘以，再相加得到''ttht*tt(p2-10-8)即d2htht'ttdt(p2-10-9)ddt2htht'tt3ht)Aet0t02ddt'()()()()()atbtcuthtatbut設(shè)ht,,111412143,b,A()ht()tt()eut2:a:24利用上節(jié)的方法求解本題。有一系統(tǒng)對(duì)激勵(lì)為(t)u(t)時(shí)的完全響應(yīng)為y(t)2u(t)x(t)euty(t)2eut2t2t122y(t)；(t)eu(t)y(t)xt332u(t)y(t)u(t)*h(t)2eu(t)y(t)eu(t)*h(t),,2t2tziziu(t)2eu(t)[u(t)eu(t)]*h(t),22tt4eu(t)[2(t)2eu(t)]*h(t),t22tut)[t)u(th(t),t)t)th(t)dhdtt)ht)2t)h(t)2eu(t),t(t)y(t)h(t)*x(t)2(1e)u(t)xt1zs1(t)y(t)y(t)2eu(t)y=tzi1zs(t)2eu(t)ht(t)2eu(t)2eu(t)*eu(t)(2e2te)u(t)y=+=ttttt3t(t)()2)exdxyttQ()(()ytxthtxhtd)()tyt)ex2)de2)()utdtxtetp2x(putp2)dpyt)2pyt)eut2)x)dppt2故ht2(t)eu(t2)。eutvttSvt。oits2vtvtovtso4題圖2-13題圖2-14I(t)I(t)v(t)v(t)有=-sss0dvvv，即1[vt)vt)]5[vt)vt)]vt)dtLs00sssdvdt)vt)vt)4v(t)dt00dtss(t)(t)eu(t),h5t31v(t)ht)*vt)[eeut)t5t440s零起始狀態(tài)電路如題圖所示，求該電路的單位沖激響應(yīng)。若激勵(lì)為ittutt1ut1vt。Sovt)vt)1ddt)5vt)vt)0it)由節(jié)點(diǎn)電流定理得i00dt02dt105ss1t)eut)h,t511vt=h(t)*i(t)=tut)(1etut)]t1)ut1)eut1)]t55ost0t0S與S21(t)vS122S112V2vt題圖2-15設(shè)經(jīng)過(guò)電感的電流為i(t),則:1vt)dddit)1,vt)lit),vt)cit)vt),it)cvt)2dtdtdtl1c1ccdd2根據(jù)KCL:i(t)i(t)i(t),代入得:vt)2vt)vt)it)s1c2s得齊次解:y(t)(AtB)eu(t),thd零輸入響應(yīng):開(kāi)關(guān)位于”1”穩(wěn)定后,v,因?yàn)閕(t)0,所以cvt)0,易求dtcc得零輸入響應(yīng):y(t)(10t10)e,tzi開(kāi)關(guān)位于”2”時(shí)，i(t)u(t),零狀態(tài)響應(yīng):y(t)(AtB)eu(t)Bu(t)tszs易求得零狀態(tài)響應(yīng):y(t)(6t6)eu(t)6u(t),tzs完全響應(yīng):y(t)(6t6)eu(t)u(t)(10t10)ettt0”且已達(dá)到穩(wěn)態(tài)，t0i(0)i(0)i(0)i(0),,和0tit)）寫(xiě)出tt00）i(0)0,i'(0)0,v(0)10,i(0)0,cdlit)dti'(0)vldd2由電路易得系統(tǒng)微分方程:it)it)it)(t)2eu(t)零輸入響應(yīng)為:0,2t23零狀態(tài)響應(yīng)為:[ett)2eu(t)。t2t3（3）系統(tǒng)方程為y(t)y(t)y(t)9(t)2eu(t)。'''2t理應(yīng)沖擊函數(shù)匹配法有：()()()9aytatbut''b7即y(t)從0-到0+有-9的跳變，yt)沒(méi)有跳變。'yt)aut)'d2dy(t)5y(t)6y(t)eu(t)t2(t)Ceu(t)r(0)r(0)Cy和t11t)(eeeut)yt2t3t22(t)(AeBe)u(t)y2t3tzi112(t)Ceu(t)A1,B,Cyt212ABr)122A3Br)'htxt11yttutut2xt,hthtxtxt,hthtxt1111xt2,htht1xtxt3,hthtxt1111dxtxt2,htht1xtxt,ht2htdtxt11111dddxt,hthtxtxt,hthtdtdtxt1dt111xt)tut)utxt)tu(t)u(t（3）xt)(t1)[u(t1)u(t（4）（5）xt)t1)[ut1)u(t3)](t1)[u(t1)u(t（6）()()(()(2)]（7）()()(()(xtututttxtututtt（8）xt)t)tt)t2esin2tutxt3t2xte2sin2tut3txt12xtxty(t)zsy(t)(2esin2t)u(t)y(t)2y(t)(e2sin2t)u(t)y(t)zi,3t3tzszizs(esint)u(t)y(t)eu(t)y(t)zs3t3tzi1y(t)(ezs1sin2(t1))u(t1),xt3(t1)eu(t)y(t)zi,3ty(t)(e3(t1)sin2(tu(t1)eu(t)3t12y(t)2y(t)(4e2sin2t)u(t)y(t)23tzizsft)與ft)ft)*ft)1122u(t),f(t)eu(t)f(t)13t2tu(tf(t)u(t)f(t)12tu(tf(t)u(t)ut2f(t)12f(t)tu(tf(t)u(t2)12(t2),f(t)cos(t45)f(t)12ft)tut)utft)utut2)12tf(t)(t(tf(t)12eu(t),f(t)sintu(t)f(t)12t211ft)*ft)(1eut)ft)ft)*ft)tut)ft)ft)t2321212111tut)t2)ut2)ft)t1)u(t1)222222cos((t2)45)u(t2)f(t)0,t1121t,1t222(7)ft)1)t)1)tut)ft)321tt,2t3220,3t211t)(sintcosteut)f2t333t)ut)utst)ft)ft)）f）ft)utut2)st)ft)ft)0,t00,t2t,0t1t2,2t3st)st)2t,1t24t,3t40,t20,t4(t)s(t)s。1t)ututft)ttft)t)f，，1232t)ft)ft)）s112(t)f(t)f(t)f(t)）s2122(t){[f(t)f(t)][u(t5)u(t5)]}f(t)）s3122(t)f(t)f(t)）s413t)ft)ft)ututututs=112（2）s(t)u(t11)u(t9)2[u(t1)u(t1)]u(t9)u(t11)2u(t10)u(t9)u(t1)u(t1)u(t9)u(t10)s(t)=s(t)2331ft)ft)u(t)u(t)st)42213令xtut3ut5和h(t)eut3tdgt和ytxthtgtxtht）求yt）dt11（1）yt)eut3)[1eut5)3(t3(t33d（2）gtxthtut3)ut5))'*eut)3tdt=e3(t3)u(t3)e3(t5)ut5)（3）g(t)y(t)'0t1xtxt/1和ht其他xthtytdytdt0,t0t,0t1yt),1t1t,1t10,1t1y(t)eu(t)(tk)03r(t)Ae設(shè)tAtttk0000t3):teut()(3)(3)ee3ktk=e3)utk=t(tkkkk11=et=AetA1e31e3t)ft)ft)f(t)與f,1212113a2023210123tt2b101012t23tftft212cet11101t0t21sind0t101ttut12e0t101ttut21f1001tt1234561g1012143t012t2-26y