1、第一章 信号和系统的概念 信号的概念 基本连续信号 信号的运算与分解 系统的概念ttt)(tf0001tt)(tft)(tft)(tft)(tft)(tft)(tft)(tfTt)(tfTt)(tftdtfETTT2)(lim总能量tdtfTPTTT2)(21lim平均功率tetf21)()()10cos(5)(2tttfJ212)(lim004044221TTttttTdtedtedtedteEW25.622251lim)20cos(1 2251lim)10(cos251lim202022TTdttTdttTPTTTTT2225lim)20cos(1 225lim)10(cos25lim20
2、2022TdttdttETTTTT01P所以 为能量信号,为功率信号。)(1tf)(2tft sAetf)(js AA)sin()cos()()(teAjteAeeAtftttjts=0 指数上升曲线，0 指数衰减曲线，)sin()cos()(tAjtAtf)cos()(RetAtf)sin()(ImtAtf)cos()(ReteAtft)sin()(ImteAtft)(t2)(tt201)(tt01)(lim)(0tt00t01t2)(tpt201)(t)(tt0)1()(lim)(0tpt00t0t00)(tt11)(面积为dttdtt)()(dttdt)()(延迟的阶跃函数定义为：000
3、10)(tttttt用阶跃函数可以表示方波或分段常量波形：)(0tt 1t00tuKt00t1tuKt00t1tK)()()()(1010ttttKttKttKuu1t013)2(5.0)1(5.1)(tttii1t021-12i1t05.1-125.0u1t013)2(5.0)1(5.1)()2()1(5.0)1()(ttttttti)3()1()1()(tttttu)3()1()1()()3()1()1()(ttttttttttu0tt0)(tf)(t)(0tt t0)()(ttf0t)()(00ttttft0)()(ttf0t)()(00ttttf0tt0)(tf)()(0ttt)(tt
4、0)1()()()()();()0()()(000tttftttftfttf)0()()(fdtttft)(0tt 0)1(0t)(0tt t0)1(0t)()()(00tfdttttf)()(tt)(1)(taat)(1)(00attatat)0(1)()(fadtattf)(1)()(00atfadttattf定义：称单位二次冲激函数或冲激偶。dttdt)()(t)(t)1(0)0()()(fdtttf)()()(00tfdttttf)()0()()0()()(tftfttf)()()()()()(00000tttftttftttf)()(tt)sgn(tt011)sgn(t01t01t)
5、(tSat10tttSasin)()0()()()(fdtttfA)()()()(00tfdttttfB)()()()(00tfdttttfC)0()()()(00fdtttttfDC)()()()()()(000tfdtttfdttttfC)()()(ttA0)()(dttCttdD)()()(B)()()(00ttttB)()()(00ttttB绘出下列各时间函数的波形，注意它们的区别：)1()(ttt)1(tt110t110t)1()1()(tttt10t)1()1(tt110tt)(tt绘出下列各时间函数的波形，注意它们的区别：110t)1()()1(ttt)3()2(ttt120t2
6、33)3()2()2(ttt110t231t)1()()(ttt)9()()1(2tSgntf1)(33:0)3)(3()9(2tfttttt时和有时1)(33:0)3)(3()9(2tfttttt时和有时10t331-)(tf)(cos)()2(2tSgntf)(tfTt0113511-tttf10sin)()3()(tft1001.0)10(101010sin10)(tSatttf)42(42tt(1)(2)02)1(4dttt(3)22)5.0()cos(8)22()3(dttttt)2(8)2()5.0()2(4)2()5.0)(4(22tttt0因为(t+1)位于积分范围之外。)1(
7、2)22()3(),1(5.0)22(ttttt825.0sin82)cos(825.0tt原式(1)(2)12(tt)()2()1(sinttt2121)12(tttt120)(1tf101t)(2tf101t)()(21tftf101t2)()(21tftf101t)(tf101t)(tf 01t)1()1()()1(tf101t)(tf101t)1(tf101t)1(tf101t2f(t-t0)将 f(t)延迟时间 t0；即将 f(t)的波形向右移动 t0。f(t+t0)将 f(t)超前时间 t0；即将 f(t)的波形向左移动 t0。)(tf101t)(tf 01t)(tf101t)1(
8、tf101t2 f(-t-1)=f-(t+1)将 f(-t)的波形向左移动1。)(tf 01t)1(tf01t2反折平移平移反折)(tf101t f(-t+1)=f-(t-1)将 f(-t)的波形向右移动1。)(tf 01t反折平移平移反折)1(tf01t)1(tf101t压缩)(tf101t2)2(tf101t25.0扩展)(tf101t2)(21tf104t2压缩)(tf101t2)2(tf101t5.0压缩)2(tf101t2)2(tf 01t5.0反折)22(tf01t5.0)22(tf101t5.0平移平移反折平移反折)(tf101t2反折)2(tf101t2)2(tf 01t5.0
9、压缩)22(tf101t5.0平移平移压缩平移)(tf 101t2)2(tf101t2)3(2)25(ttf0)(dttf)25(tf倍展宽1)6(4)6(22)6(2)3(2)5(2121tttttf)1(46)5(4)()5(5tttftf5左移)1(4)1(4)1(4)(ttttf反折000)1(4)(dttdttf故得：t)25(tf0123)2(t)5(tf0123)4(456t)(tf01)4(t)(tf 012)4()(1)(taat)(1)(00attatat)3(2)25(ttft)25(tf0123)2(t)5(tf0123)4(456t)(tf01)4(t)(tf 012
10、)4(展宽平移反折t)25(tf0123)2(反折t)5(tf0126)4(展宽平移反折)(tg1t0)(1tfK0t2t)(2tf0123A2t)2()()sin()()(sin)(1tttKtttKtf f 2(t)的第0个周期：)1()(2ttAt f 2(t)的第1周期将第0个周期延迟1：)2()1()1(2tttA f 2(t)的第K个周期：)1()()(2KtKtKtA022)1()()()(kKtKtKtAtft)(tf0kkf1f第个阶跃函数：)(1tf)0()(01fdttdfft第K个阶跃函数：)(ktfk)()(kfdttdffktk当 0,即 为d,而 k 为 。kkt
11、kftf)()()(dtfktkftfk)()()()(lim)(0当 0,即 为d,而 k 为 。t)(tp0221t)(tf0k)(kf)0(f第个脉冲函数：)()0(tpf第K个脉冲函数：)()(ktpkfkktpkftf)()()(dtfktpkftfk)()()()(lim)(0先定义窄脉冲信号：)()(lim0ttp面积为1面积偶分量的定义为：)()(tftfee奇分量的定义为：)()(00tftf任何信号总可写成：)()()()(21)(tftftftftf)()(21)()(21tftftftf)()(0tftfe)()(21)(),()(21)(0tftftftftftfe即
12、：t01)(tf)()(21)(),()(21)(0tftftftftftfet01)(tf t01)(tfet0)(0tft01)(tft021)(tfet0)(0tf21t01)(tft01)(tf t021)(tfet021)(0tft01)(tf)()(21)(tftftfe)()(21)(0tftftf)3(4)2()2(2)(2)(ttttttf)(),1()(),(tfttftft02)(tf34t01)1()(ttf)2(t02)(tf 32)4()22()(1tftf)22()(2tftf(1)(2)t02)(tf222t02)22(tf221t02)22(tf221(3)f
13、e(t)(它的偶部)(4)fo(t)(它的奇部)t02)(tf222t02)(tfe21t02)(tfo2212变性。能满足非时变性质的系统称为非时变系统，变性。能满足非时变性质的系统称为非时变系统，否则为时变系统。否则为时变系统。)()()(trtrtrzszi)(te)(tr)(tet0)(0ttet00t)(trt0)(0ttrt00ttdted)(tdtrd)(tde)(tdr)()()()()()()(0)1(1)(0)1(1)(teBteBteBtrAtrAtrmmmmnnn因果性因果性系统模型为：r(t)=sine(t)(t)()(sin)()(sin)()()()()(sin)
14、()(2211221122112211tteatteatratratteateateateaT故为非线性系统。)()(sin)()()(sin)(00000ttttettrtttetteT故为时变系统。显然输出变化不发生在输入变化之前,故为因果系统。分析如下：系统模型为：r(t)=e(1-t)故为线性系统。故为时变系统。分析如下：)()()1()1()()(221122112211tratrateateateateaT)1()()1()(0000ttettrttetteT设系统的初始状态为x(0)，激励为 f(t)，各系统的全响应y(t)与激励和初始状态的关系如下，试判断下列系统是否为线性的、
15、时不变的？ttdfxety0)(sin)0()(tdfafatfatfaT022112211)()(sin)()(ttdfadfa022011)(sin)(sin)()(2211tyatyazszs)()()sin()(sin)(0000000ttydftdtfttfTzstttt)(2)()(2)()1(tftftyty)()(sin)()2(tftytty解：(1)该方程的所有系数是常数，所有的项都包括了y(t)或 f(t)，故描述的系统是线性时不变系统。(2)该方程的一项系数是 t 的函数，所有的项都包括了y(t)或f(t)，故描述的系统是线性时变系统。(3)该方程的一项系数是y(t)的
16、函数，而y(2t)将使系统 随时间变化，故描述的系统是非线性时变系统。)()2()(4)()3(tftytyty某一线性系统有两个起始条件 和 ，输入为 ，输出为 ，并已知：1x2x)(tf)(ty0)(,2)0(,5)0(21tfxx)57()(tetyt(1)当 时，)15()(tetyt(2)当 时，0)(,4)0(,1)0(21tfxx)1()(tetyt(3)当 时，)()(,1)0(,1)0(21ttfxx)(ty求：当 时的)(3)(,1)0(,2)0(21ttfxx解：零输入响应是初始值的线性函数，故)0()0()(2211xkxktyzi将(1)，(2)条件代入，得：)57(
17、2521tekkt)15(421tekkt解得：ttetek1ttek2所以，零输入响应为tttttzietetexetexty2)(0()(0()(21ttttzizsteetetetytyty2)1()()()()()()(tytytyzszi所以，由(3)零状态响应为：ttttetetetety23)1(2)(故，系统响应为：)(1tf)(1ty)(2tf)(2tyt)(1tf022t)(2tf024t)(2ty02413)(1tf)(1tyt)(1ty024134)(1tf)(1ty)(2tf)(2tyt)(1tf021)(1tf)(1tyt)(1ty0221t)(2tf02141t)(2ty0221234t)(2tf02112t)(2ty02211
