傳媒通信-課件第二章

第二章連續(xù)(時間)系統(tǒng)的時域時域分析：對系統(tǒng)的分析與計算均以時間tLTIe(t y(tLTI 連連續(xù)系統(tǒng)描述輸入、輸出關系的數(shù)學模型為n階微分方時域分析法優(yōu)點：直觀、物理概念清時域分析法缺點：對高階系統(tǒng)或復雜激勵計算或inm 或inm (ii(t)jb(jj(te(tb0y(taay''(t) y'(t) y(t)be(t)be(t y''(t)ay'(t)ay(t)be''(t)be'(t)be(t aany(n)(t)n1y(n1)(t)ay(t) y(t10 e(m)(t)mme(m1)(t)be(t)be(t10n階微分方系統(tǒng)的微yh(t)y(t) yh(t)yp(t齊次解特解全解yh(t)所含待定解中由初始條件確y(n1)(0)y(0 nniai (t)(ijmb(j連續(xù)系統(tǒng)的數(shù)學模j(tyh(t)的形式(查P34表inma(inma(ii(t)jb(jj(t齊次方程：ina齊次方程：ina (t)(ii

an1

"a n階微分方程有n個特征根i(i12n齊次解yh(t)的形式由特征根i的形式?jīng)Q定查P34表2yy(t)CetCh121：求下列方程的齊次解y(t)3y1：求下列方程的齊次解y(t)3y(t)2y(t)2e(ty(t)2y(t)y(t)e(ty(t)2y(t)5y(t)2e(t yh(tet[C1cos2tC2sin2tAetcos(2t)inma(inma(ii(t)jb(jj(t特解yp(t)的形式由激勵e(t)的形式?jīng)Q定查P34表2說明：激勵e(t)是在tt00時刻加入系統(tǒng)，因此特解y(t)存在的時間為tt0求當激勵為(1)e(t)(t),(2)e(t)e3t(t)兩種情況下的特 e(t(t)y(t)pp2(1)t0時激勵（（2）e(t)e3t(t∵y(t)Pe3tpyp(t)253(ttina (tina (t)(iijmb(jj(t全解：全解：yt)yht)齊次解Nyp(t特解齊次解中的待定系數(shù)Ci在全解中由初始件確定n階微分方程需要n個初始條件例3：已知系統(tǒng)的微分方程為y(t)4y(t)4y(t)e(t)3e(ty(0) y(0) e(t)et(t

求該系統(tǒng)的全解。yyh(t)(C12C2)(ty(t) (tp全解全解:y(t)yh(typ(t[(3te22) ](t齊次(自由響應特(強迫響應221.2關于系統(tǒng)在t=0-與t=0+狀態(tài)的討論（難點ina (t)(ijm(jib (tj求特征根i確定齊yh(t)的形式(查P34表系統(tǒng)的全解y(t)全解y(t) yh(t)yp(t齊次解特解yh(t)中的待定系數(shù)全解中由初始條yh(t)含待定系討論的t<0e

yn1(0)"y(0n y(i)(t)mibe(j)(tjij y(j)(0),j0,1,2,",ne(t)前瞬e(t)加后瞬反映歷史信息而與初始條件yj(0)由yj(0)和e(t)共同決定 從 ～ 即yj

y(jt)可能發(fā)生跳變)y(j) 跳變令yj)(0yj)(0yj)(0跳變 初始=初始+跳變

y(j)(0)y(j)(0)+y(j)(0 發(fā)生跳變的條件：微分方程右端含(t)及其各階導 inm (ii(t)jb(jj(ty(t)4y(t)4y(t e(t)3e(t(1)e(t)(t (2)e(t)et(t2）跳變量的確定方法[函數(shù)平衡法(匹配法 ((。方方程左端(t)及其各階導數(shù)=方程右端(t)及其各例5例5y'(t)3y(t)3e(t已知y0)求(1)e1(t)(t (2)e2(t)(t) y(0初始條件=初始條件=初始狀態(tài)+跳變注意：匹配應從微分方程的最高階項解：1）y(0)y(0ye(0)002）y(0)y(0)ye(0)03例6:y(t4y(t3y(te(t2e(te(t已知e(t)(t)，y(0) y(0) 求y(0),y(0),e在此(t)僅用來表示在t0解：y(0)y(0ye(0)12y(0)y(0)ye(0)01y(0)y(0)ye(0)01例6:y(t4y(t3y(te(t2e(te(t已知e(t)(t)，y(0) y(0) 求y(0),y(02）匹2）匹配從方程左端y(i)(t)函數(shù)的最高階項得到匹配1）只匹配(t)及其各階導數(shù)，使方程兩邊(t)及其各）yi不變）。例7yt3y(例7yt3y(t3(t),求ye注意注意例y'(t3y(t3e(t)，y(0e(t(t求yye(0)y(t)3e3t(ty(t)3e3t30y(t)3e3t(y(t)3e3t303(t)9e3t(t2.2.2.1零輸入ii (t) (t)nzixeiti

y(i)(t)均為單實yy(j)z)y(j)()初始條件=222.2零狀態(tài)響 方程，解方程，解由yh(t)yp(t)組成） y(t)yy(t)y(t)y(t)Ceity(tnhpspti均為單實根y(j)(y(j)(z))0(j 0,1n1y(j)(e)初始條件=跳變2.2.32.2.3 y(i)(t)be(j)(t i jy(t)yh(t)yp(t) yzi(t) yzs(ty(t)niC iy(t)niC ii (t)npxeinseiiyzs(t (tpiyh(typ(tyzi(t初始條件=初始狀態(tài)+物理

零輸入響y

小零狀態(tài)響y

yh(t):yh(t):特征根決定(表2-yp(t):激勵決定(表2-y(t)=yzi(t)+ (i

(齊次i數(shù)學模齊次i

ay(i)(t)

ai (t)

ay(i)(t)be(j)y(t)y(t)yh(t)yp(t

初始

yy(t)yh(t完全完全y(i)(0)完全完全

y(i)(0)

y(t)yh(t)yp(ty(i)(0)初始

y(i)(0)y(i)(0

y(i)(0)(e

跳躍量）yi(0yi(0yi中國傳媒大中國傳媒大 待定系待定系數(shù)由初始條件待定系數(shù)Csi解中由初始條件待定系數(shù)Ci在完解中由初始條例8:某LTI系統(tǒng)數(shù)學模型為y''(t3y'(t2y(t2(t已知y(0)1,y'(0) 求y(t),y zs解yy(t) et e2t)(t)(5et4e2t)(t xye(0)ye(0)y(t)(Ce

e

3)(t)(8e

7e

3)(t s12y(t)(Ce12

Ce

3)(t y(t)yzi(t)yzs(t)(33e

3e2

)(tina (t)ina (t)(im(jib (t)j2.2.3.1沖激響應[用h(t)表示LTe(t)(t yzs(t)h(tLT當e(t)=當e(t)=(t)時系統(tǒng)的零狀態(tài) h(t) (t 此時系統(tǒng)方程的形iin h(i)(t) mij(j)(tj (j0,1,2"n初始條件=初始條件：hj0討論：t>0時系統(tǒng)屬于什么響

零輸入響應信例9：某一系統(tǒng)數(shù)學模型為yt5yt6yt)et2eth(t)h(t)5h(t)6h(t)(t)2(th(0) h(0)h(t)e3t(t10：某一系統(tǒng)數(shù)學模型為yt)5yt)6yt)et)2eth(t)5hh(t)5h(t)6h(t)(t)2(t。h(0) h(0)h(t)(t)3e3t例11：某一系統(tǒng)數(shù)學模型 y(t)5y(t)6y(t)e(t)2e(tina ina (t)(ijmb (t(jijh(t)5h(t)6h(t)(t)2(th(0) h(t)(t)5(t)(12e2th(t)的形式與(j)設特征根i為單實

nm h(t)niini

Ceit(tnm h(t)

Ceit(t)C(tn nninn :h(t)Ceit(t)

mC

(k)(tii

k2.2.3.2階躍響應[用g(t)表示inaina (t)(iijmb(jj(tLTe(t)(t yzs(t)g(tLT當e(t當e(t)=(t)時系統(tǒng)的零狀態(tài) g(t) 0,(t)此時系統(tǒng)方程的一般iin (i(t)jmb(jij(t初始狀態(tài)：g(j)(0) (j0,1,2"n初始條件：gj0h(t與g(t)

初始條件=∵(t) t初始條件=

tg(t) h(xt∵(t) d(t

h(t) g(t 例10：某一系統(tǒng)數(shù)學模型為y(t5y(t6y(t)e(t2e(th(0)h(0) h(0)h(t)(t)h(t)5h(t)6h(t)(t)h(t)h(t)dg(t

LTyzs(LT 例12：求例10所示系統(tǒng)的階躍響應gg(t)5g(t)6g(t)(t)2(tg(0g(0)1,g(0)g(t)e3t(t 卷積(積分)是特殊的積分運算,在該課程中求解系統(tǒng)零狀態(tài)一、卷積(積分)的(數(shù)學) f1(t)與f2(t)是定義在（）區(qū)間的連續(xù)時間函ff(t)f1()f2(t)d(267f(t) f(t)f(tf1(t)與f2(t)的卷積積分，積分結是以時間t為變量的函數(shù)f(t12f1(t)與f2(t)卷積簡記符二、二、ff(t)f(t)12(t)f1()f2(t)d(267 f1(t)與f2(t)是定義在（）區(qū)間的連續(xù)時間函當當f1(t)和f2(t)的時間沒有限制時，卷積積分的積分限從–+，當f1(t)和f2(t)幾種特殊

若t0 f1(t) f2(t)不受限 f(t)0f() (t f(t)0f() (t)d12（26則則f(t)tf1()f2(t)d(269若t0 f1(t)f2(t)則則f(t)t0f()f(t)d12(270 f1(t)=t(t

f2(t)(t

求f(tf1tf2t解解 f(t)f(t) (t)1t2(t122例14: f1(t)=

2t(t

f2(t)(t

f(t解解 f(t) f(t)f(t)1(1e2t)(t122ff(t)f(t)12(t)f1()f2(t(2242f(t) 1(t)2(t)f1()f2(t)d反折f2()f2f2(?)在軸上平移t得f2(tf1()和f2(t–相乘后積例 f1(t)和f2(t)的波形如圖所示,求f1(t)f2(t204f1(t204

f2(t

f1(t)2[(t)(tf(t)3t[(t)(t

2)反折f2(f2(-2)反折f2(f2(-2

2tt

tt33f2(-)在t得f2(t2

2

f2(t

2t–2

t4

0t

2t

4t4f1()和f24f1()和f2(t–相乘后

(t)f

(t)

f1()f

(tff1()f2(t) (t34tt

的左、右邊緣值(即積分上下限)中兩函數(shù)沒 即表示兩函數(shù)相乘值為零2

2

f2(t

2t–2

t4

0t

2t

4tf(t)f(t)

f(t)f(t)34(t4)2f1(t2

f2

f1(t)f2(t3

f(t)f(t)f(t)f(t)3124卷積中積分限取決于兩個波 部卷積結果卷積積h(t0eth(t0線性時不LTLTLTLT(t)(tt0

h(th(tt00h(tt00LTh(t)h(ttLT

t

中國傳媒大學許信

ff

f

1+

0

(tt t0 0 2 3

nf(t)lim+0nf(n)(tnnf(t)lim+0nf(n)(tn)f()(t)d(273nf(t) f(tf(t) f(tg+(tn)(t任意信號f(t)可以分解為無窮多個連續(xù)的沖激信 之ff(t)+0nf(n)(tn)f()(t)d(2732、利用卷積積分2、利用卷積積分求解系統(tǒng)的零狀態(tài)響應(卷積積分的物理意義LT對于LTI系LT對于LTI系e(t)lim+ne(n)e(t)lim+ne(n)(tn) e)(t)d e(t)(tn+) yzs(t)h(tn+)e(t)e(n+)(tn+) yzs(t)e(n+)h(tn+) e(t)lime(n)(tn) yzs(t)lime(n)h(tn)+0nyzs(t)e()h(t (275)yzs(t)e(t)h(t (2

+0n例15 e(t)e2t(t

h(t)(t

求yzst卷積的物理意義卷積的物理意義系統(tǒng)對任意激勵e(t)的零狀態(tài)響應yzs(t)為激勵信號e(t)與系yzs(t)e(t) (2yyz(t)e)h(t)(275解解得 (t)e(t)h(t)1(1e2t)(tz225卷積積分的11、卷積的代數(shù)運f(t)f(t)f1(t)f2(t)f2(t)f1(t (2在卷積積分中兩函數(shù)的位置可以互換說明反折函數(shù)可f1(t202ff1(t202f2012f2(2 02f1f220 選反折函數(shù)時要考慮1反折表達式簡單的函2反折邊界與縱軸重合的函數(shù)t、坐標原點一。(2)分配h(tf1(t)f2(t)f3(t)f1(t)f2(t)f1(t)f3(t)（2h(t物理

yzs e(tf2tf3te1(te2th(tf1(t yzs(t)h(t)e(t)e1(t)h(t)e2(t)h(ty1zs(t y2zs(t若若e(t[e1te2t"en(t則yzs(te(th(t)e1th(te2th(t"en(th(ty1zs(ty2zs(tynzs(tn個信號相加作狀態(tài)響應,等于n個信號分別作用系統(tǒng)產(chǎn)生的零狀態(tài)響應之若h(t[h1(th2(thn(t)]e(t則yzs(th(te(te(th1(t若h(t[h1(th2(thn(t)]e(t則yzs(th(te(te(th1(te(th2(t"e(thn(tyzse(tyzs(th(t)h1(t)h2(t)"hn(tn個子系統(tǒng)并聯(lián)組成的系統(tǒng)響應等于各子系統(tǒng)沖激響h(t…b)b)設e(tf1(th(tf2tf3th1(th2(t則yzs(t)e(th(te(t[h1th2t3）結合

f1(t)f2(t)f3(t)f1(t)f2(t)f3(t) (2設e(t設e(tf1(th1(tf2th2tf3t yzs(t)e(t)h1(t)h2(t)e(t)h1(t)h2(t)e(t)h(te(t)h1yzsyzsh(t)h1(t)h2(tn個子系統(tǒng)級聯(lián)組成的系響應等于各子系統(tǒng)沖激響h(th2(th(t)h1(t)h2(t)"hn(t例16：求圖所示復合系統(tǒng)的沖激響應為yzsh(th4(th(t)[h1(t)h2(t)h3(t)]h4(t2、函數(shù)與沖激函數(shù)的卷 f(t)(t)f(t9）函數(shù)f(t)與(t)卷積的 f(t)(tt0)f(tt0（29）的結果是把原函數(shù)f(t)延時t0推

f(t)(t)， f(t)(t)(t)(t)(t(t)(tt1)(tt1例17：f1(t)和f2(t)的波形如圖所示,求f1(t)?f2(t)1101f202f1(t)20123tT(ttnT梳妝函數(shù)f0f010T……0T函數(shù)與沖激函數(shù)卷積特性的fT(t) f0(t)fT(t) f0(t)T(tf0(t)(tnTf0(tnT(2971 本例提供了周期本例提供了周期函數(shù)的另fT(t)f0(t)T1

fT(t)f0(t)T1… T

大學許信3 f3f(1)(tdef(1)(tdedf(tdtf(x)d f(t)f1(t)f2(t)f2(t)f1(tf(1)(tf(t)f(t)](1)f(t)f(1)(tf(1)(t)f121212f(2)(t)對兩函數(shù)卷積的結果中任一函數(shù)先求導后再做卷ff(t)(t)f(t(2函數(shù)f(t)與沖激偶卷積相當于對f(t)求導(2)卷積的 f(t)f1(t)f2(t)f2(t)f1(t則（-(t) f(則（-(t) f(t)f12f(t)1f(2(t)(1(t)f2(2

ff(t)[(t)](1)f(t)(t)tf((2函數(shù)f(t)與(t)(3)卷積的微、積分性質(在卷積運算中的應用 f(t)f1(t)f2(t)f2(t)f1(tff(t)f1(t)f2(tf(1)(t)f(1)(t)f(1)(t)f(1)(t1212(2f1(t)與f2(t卷積等于先對其中任一函數(shù)求f(t)f(i)(t)f(i)(t)f(i)(t)f(i)(t

是否成立f1(t2102tf210232102 5t例 f1(tf1(t2102tf210232102 5tt例19e2tt(tt[e2t(t)(t)](t)(t)2e2t(t例20：已知某一系統(tǒng)e