传媒通信-课件第二章

第二章连续(时间)系统的时域时域分析：对系统的分析与计算均以时间tLTIe(t y(tLTI 连连续系统描述输入、输出关系的数学模型为n阶微分方时域分析法优点：直观、物理概念清时域分析法缺点：对高阶系统或复杂激励计算或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阶微分方系统的微yh(t)y(t) yh(t)yp(t齐次解特解全解yh(t)所含待定解中由初始条件确y(n1)(0)y(0 nniai (t)(ijmb(j连续系统的数学模j(tyh(t)的形式(查P34表inma(inma(ii(t)jb(jj(t齐次方程：ina齐次方程：ina (t)(ii

an1

"a n阶微分方程有n个特征根i(i12n齐次解yh(t)的形式由特征根i的形式决定查P34表2yy(t)CetCh121：求下列方程的齐次解y(t)3y1：求下列方程的齐次解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)的形式由激励e(t)的形式决定查P34表2说明：激励e(t)是在tt00时刻加入系统，因此特解y(t)存在的时间为tt0求当激励为(1)e(t)(t),(2)e(t)e3t(t)两种情况下的特 e(t(t)y(t)pp2(1)t0时激励（（2）e(t)e3t(t∵y(t)Pe3tpyp(t)253(ttina (tina (t)(iijmb(jj(t全解：全解：yt)yht)齐次解Nyp(t特解齐次解中的待定系数Ci在全解中由初始件确定n阶微分方程需要n个初始条件例3：已知系统的微分方程为y(t)4y(t)4y(t)e(t)3e(ty(0) y(0) e(t)et(t

求该系统的全解。yyh(t)(C12C2)(ty(t) (tp全解全解:y(t)yh(typ(t[(3te22) ](t齐次(自由响应特(强迫响应221.2关于系统在t=0-与t=0+状态的讨论（难点ina (t)(ijm(jib (tj求特征根i确定齐yh(t)的形式(查P34表系统的全解y(t)全解y(t) yh(t)yp(t齐次解特解yh(t)中的待定系数全解中由初始条yh(t)含待定系讨论的t<0e

yn1(0)"y(0n y(i)(t)mibe(j)(tjij y(j)(0),j0,1,2,",ne(t)前瞬e(t)加后瞬反映历史信息而与初始条件yj(0)由yj(0)和e(t)共同决定 从 ∼ 即yj

y(jt)可能发生跳变)y(j) 跳变令yj)(0yj)(0yj)(0跳变 初始=初始+跳变

y(j)(0)y(j)(0)+y(j)(0 发生跳变的条件：微分方程右端含(t)及其各阶导 inm (ii(t)jb(jj(ty(t)4y(t)4y(t e(t)3e(t(1)e(t)(t (2)e(t)et(t2）跳变量的确定方法[函数平衡法(匹配法 ((。方方程左端(t)及其各阶导数=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初始条件=初始条件=初始状态+跳变注意：匹配应从微分方程的最高阶项解：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)仅用来表示在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）匹配从方程左端y(i)(t)函数的最高阶项得到匹配1）只匹配(t)及其各阶导数，使方程两边(t)及其各）yi不变）。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(t2.2.2.1零输入ii (t) (t)nzixeiti

y(i)(t)均为单实yy(j)z)y(j)()初始条件=222.2零状态响 方程，解方程，解由yh(t)yp(t)组成） y(t)yy(t)y(t)y(t)Ceity(tnhpspti均为单实根y(j)(y(j)(z))0(j 0,1n1y(j)(e)初始条件=跳变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初始条件=初始状态+物理

零输入响y

小零状态响y

yh(t):yh(t):特征根决定(表2-yp(t):激励决定(表2-y(t)=yzi(t)+ (i

(齐次i数学模齐次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

跳跃量）yi(0yi(0yi中国传媒大中国传媒大 待定系待定系数由初始条件待定系数Csi解中由初始条件待定系数Ci在完解中由初始条例8:某LTI系统数学模型为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冲激响应[用h(t)表示LTe(t)(t yzs(t)h(tLT当e(t)=当e(t)=(t)时系统的零状态 h(t) (t 此时系统方程的形iin h(i)(t) mij(j)(tj (j0,1,2"n初始条件=初始条件：hj0讨论：t>0时系统属于什么响

零输入响应信例9：某一系统数学模型为yt5yt6yt)et2eth(t)h(t)5h(t)6h(t)(t)2(th(0) h(0)h(t)e3t(t10：某一系统数学模型为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：某一系统数学模型 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)的形式与(j)设特征根i为单实

nm h(t)niini

Ceit(tnm h(t)

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

mC

(k)(tii

k2.2.3.2阶跃响应[用g(t)表示inaina (t)(iijmb(jj(tLTe(t)(t yzs(t)g(tLT当e(t当e(t)=(t)时系统的零状态 g(t) 0,(t)此时系统方程的一般iin (i(t)jmb(jij(t初始状态：g(j)(0) (j0,1,2"n初始条件：gj0h(t与g(t)

初始条件=∵(t) t初始条件=

tg(t) h(xt∵(t) d(t

h(t) g(t 例10：某一系统数学模型为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所示系统的阶跃响应gg(t)5g(t)6g(t)(t)2(tg(0g(0)1,g(0)g(t)e3t(t 卷积(积分)是特殊的积分运算,在该课程中求解系统零状态一、卷积(积分)的(数学) f1(t)与f2(t)是定义在（）区间的连续时间函ff(t)f1()f2(t)d(267f(t) f(t)f(tf1(t)与f2(t)的卷积积分，积分结是以时间t为变量的函数f(t12f1(t)与f2(t)卷积简记符二、二、ff(t)f(t)12(t)f1()f2(t)d(267 f1(t)与f2(t)是定义在（）区间的连续时间函当当f1(t)和f2(t)的时间没有限制时，卷积积分的积分限从–+，当f1(t)和f2(t)几种特殊

若t0 f1(t) f2(t)不受限 f(t)0f() (t f(t)0f() (t)d12（26则则f(t)tf1()f2(t)d(269若t0 f1(t)f2(t)则则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(−)在轴上平移t得f2(tf1()和f2(t–相乘后积例 f1(t)和f2(t)的波形如图所示,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、右边缘值(即积分上下限)中两函数没 即表示两函数相乘值为零2

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卷积中积分限取决于两个波 部卷积结果卷积积h(t0eth(t0线性时不LTLTLTLT(t)(tt0

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

t

中国传媒大学许信

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任意信号f(t)可以分解为无穷多个连续的冲激信 之ff(t)+0nf(n)(tn)f()(t)d(2732、利用卷积积分2、利用卷积积分求解系统的零状态响应(卷积积分的物理意义LT对于LTI系LT对于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卷积的物理意义卷积的物理意义系统对任意激励e(t)的零状态响应yzs(t)为激励信号e(t)与系yzs(t)e(t) (2yyz(t)e)h(t)(275解解得 (t)e(t)h(t)1(1e2t)(tz225卷积积分的11、卷积的代数运f(t)f(t)f1(t)f2(t)f2(t)f1(t (2在卷积积分中两函数的位置可以互换说明反折函数可f1(t202ff1(t202f2012f2(2 02f1f220 选反折函数时要考虑1反折表达式简单的函2反折边界与纵轴重合的函数t、坐标原点一。(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则yzs(te(th(t)e1th(te2th(t"en(th(ty1zs(ty2zs(tynzs(tn个信号相加作状态响应,等于n个信号分别作用系统产生的零状态响应之若h(t[h1(th2(thn(t)]e(t则yzs(th(te(te(th1(t若h(t[h1(th2(thn(t)]e(t则yzs(th(te(te(th1(te(th2(t"e(thn(tyzse(tyzs(th(t)h1(t)h2(t)"hn(tn个子系统并联组成的系统响应等于各子系统冲激响h(t…b)b)设e(tf1(th(tf2tf3th1(th2(t则yzs(t)e(th(te(t[h1th2t3）结合

f1(t)f2(t)f3(t)f1(t)f2(t)f3(t) (2设e(t设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个子系统级联组成的系响应等于各子系统冲激响h(th2(th(t)h1(t)h2(t)"hn(t例16：求图所示复合系统的冲激响应为yzsh(th4(th(t)[h1(t)h2(t)h3(t)]h4(t2、函数与冲激函数的卷 f(t)(t)f(t9）函数f(t)与(t)卷积的 f(t)(tt0)f(tt0（29）的结果是把原函数f(t)延时t0推

f(t)(t)， f(t)(t)(t)(t)(t(t)(tt1)(tt1例17：f1(t)和f2(t)的波形如图所示,求f1(t)∗f2(t)1101f202f1(t)20123tT(ttnT梳妆函数f0f010T……0T函数与冲激函数卷积特性的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数的另fT(t)f0(t)T1

fT(t)f0(t)T1… T

大学许信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)对两函数卷积的结果中任一函数先求导后再做卷ff(t)(t)f(t(2函数f(t)与冲激偶卷积相当于对f(t)求导(2)卷积的 f(t)f1(t)f2(t)f2(t)f1(t则（-(t) f(则（-(t) f(t)f12f(t)1f(2(t)(1(t)f2(2

ff(t)[(t)](1)f(t)(t)tf((2函数f(t)与(t)(3)卷积的微、积分性质(在卷积运算中的应用 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)与f2(t卷积等于先对其中任一函数求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：已知某一系统e