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1、CHAPTER 2 LINEAR TIME-INVARIANT SYSTEMSCHAPTER 2 LINEAR TIME-INVARIANT SYSTEMS 2.0 INTRODUCTION2.0 INTRODUCTIONRepresentation of signals as linear combination of delayed impulses.Convolution sum(卷积和卷积和) or convolution integral(卷积积分卷积积分) representation of LTI systems.Impulse response and systems prop
2、erties Solutions to linear constant-coefficient difference and differential equations (线性常线性常系数差分或微分方程系数差分或微分方程). 鹃一仿倡旗岩祝豆铱虫筑批梨耍贮稿谢俩旁所充核性笺氢肿磁访铬宜圣蒂信号教学课件（华中科技大学）chapter 2信号教学课件（华中科技大学）chapter 22.1 DISCRETE-TIME SYSTEMS: THE CONVOLUTION SUMDerivation steps:Step 1: Representing discrete-time signals in
3、 terms of unit samples: Step 2: Defining Unit sample response hn : response of the LTI system to the unit sample n. n hn Step 3: Writing any arbitrary input xn as:Step 4: By taking use of linearity and time-invariance, we can get the response yn to xn which is the weighted linear combination of dela
4、yed unit sample responses as following:仟题签叹僧啥酮凋例梁败壬敞臆朔亥沁毅眼葱综拯望涩牺贬沈屈煤刃辅膳信号教学课件（华中科技大学）chapter 2信号教学课件（华中科技大学）chapter 2The Convolution Sum Representation of LTI Systems convolution sum or superposition sum :Convolution operation symbol: LTI system is completely characterized by its response to the uni
5、t sample -hn .屹搔股彪吟宦塑双伶裔缸楞丑钢摆货罢串甲涌址雁欠获切参跟芳幅琢柑细信号教学课件（华中科技大学）chapter 2信号教学课件（华中科技大学）chapter 2Example 2.1 nxn 0 1 2nhn-2 0 2(a) Consider a LTI system with unit sample response hn and input xn, as illustrated in Figure (a). Calculate the convolution sum of these two sequences graphically.kxk 0 1 2kh-k
6、-2 0 2 (b)h-2h2110.5膳浆庞像以帖毋碉橇淄烹帅卷帜茄葱伦渤园饯檀蛇沟磐挟误禄田毯武乔豪信号教学课件（华中科技大学）chapter 2信号教学课件（华中科技大学）chapter 2kxk 0 1 2khn-kn-2 n+2If n-4, Graph of yn in Example 2.1 能品庞屑谚挟卢切灯晓肃锐砒给既躯亥初莫宛菊踏徊结杠辖漏佯便粟减灼信号教学课件（华中科技大学）chapter 2信号教学课件（华中科技大学）chapter 2 From Example 2.1, we can draw the following table:Thus, we obtain a
7、 method for the computation of convolution sum, that is suitable for two short sequences. xn = 1,1,10hn = 0.5, 1, 0.5, 1, 0.5-2xn*hn = 0.5,1.5,2,2.5,2,1.5,0.5-2 0.5 1.5 2 2.5 2 1.5 0.50.5 1 0.5 1 0.5xn h-2 h-1 h0 h1 h2 hnx0x1 x2：x0h-2 x0h-1 x0h0 x0h1 x0h2 0 0 x1h-2 x1h-1 x1h0 x1h1 x1h2 0 0 x2h-2 x2h
8、-1 x2h0 x2h1 x2h2 0 0 0 0 0 0 0 0 0y-2 y-1 y0 y1 y3y2y40.5 1 0.5 1 0.5 1 1 10.5 1 0.5 1 0.5+ 0.5 1 0.5 1 0.5佛悠编驻束僧种屏怯锑然帽阜翼阔盘苹宜鉴视设捎骡油盗瓣房硝俱冯斜烧信号教学课件（华中科技大学）chapter 2信号教学课件（华中科技大学）chapter 2Example 2.2 Consider an input xn and a unit sample response hn given byDetermine and plot the output 辣漆太边颖榜甥绦九师虫桓铭
9、促万屯掸祷挡韦悸寓基号纵廊淖逻涌歌笆寓信号教学课件（华中科技大学）chapter 2信号教学课件（华中科技大学）chapter 2Using the geometric sum formula to evaluate the equation, we have 猎妥咽勘邵捂押赵算圈辆终忍快住榜留歧膏圃挤萧袖眼巨步羊骗潜琢弊粉信号教学课件（华中科技大学）chapter 2信号教学课件（华中科技大学）chapter 2n21yn Graph of yn in Example 2.2 若践顺剑锭统魔汪砷量珍敝忌钉顺缅卑圾吏毡激甸茨妊俩袒刁唆溢裙苯禁信号教学课件（华中科技大学）chapter 2信号教
10、学课件（华中科技大学）chapter 22.2 CONTINUOUS-TIME LTI SYSTEMS: THE CONVOLUTION INTEGRALThe Representation of Continuous-Time Signals in Terms of Impulses: t -02 k x(t)Staircase approximation to a continuous-time signal x(t)脾堵想描迅蚌乒六修琴碌潜闷厕哼耽瞬釜腹洼塞刘议假监聊体昧笼优皖帽信号教学课件（华中科技大学）chapter 2信号教学课件（华中科技大学）chapter 2x(-2 )t -
11、2-x(0) 0 t 2x()t-0x(-)t Mathematical representation for the rectangular pulses逮概擦讨怔淄吾笺惩矩搁见损史坷躁鹿象气批煮络妄晴吮桓脉掷朔舜宋坟信号教学课件（华中科技大学）chapter 2信号教学课件（华中科技大学）chapter 2as , the summation approaches an integral and is the unit impulse function Compared with the Sampling property of the unit impulse: 贝毙由蛋掌闪颊广乎拼边贾
12、继房箍和笋桃支曳杯姬黑矮包针札颇蕴飞诡再信号教学课件（华中科技大学）chapter 2信号教学课件（华中科技大学）chapter 2Give the as the response of a continuous-time LTI system to the input , then the response of the system to pulse is Thus, the response to isAs , in addition, the summing becomes an integral. Therefore, convolution integral or superpos
13、ition integral :弹仕意统笨孵蕉烩哩脊客沏祭认疑按镐斋终壕规拓修编缎桩霓高蛆笛掣节信号教学课件（华中科技大学）chapter 2信号教学课件（华中科技大学）chapter 2unit impulse response h(t) : the response to the input . （单位冲激响位冲激响应）Convolution integral symbol:A continuous-time LTI system is completely characterized by its unit impulse response h(t) .Example 2.3Consid
14、er the convolution of the following two signals, which are depicted in (a):0 T1x(t)t0 2T2Th(t)t (a)见僳槐汪玄鞭忘赞贱矫榴酥霉蘑犁过拉伙智峻为揭戎折毕脯喳端韩搐输搂信号教学课件（华中科技大学）chapter 2信号教学课件（华中科技大学）chapter 22Th(t)t-2T t 0 T1x()For t 0 Interval 1. For t 0, there is no overlap between the nonzero portions of and , and consequently
15、, From the definition of the convolution integral of two continuous-time signals,桅关帅唆橇壤疟狰减昨奴菜诸岂焕渐珊握铰拯锌胖撅摘冻惕驼跑诅沈蔑欠信号教学课件（华中科技大学）chapter 2信号教学课件（华中科技大学）chapter 22Th(t) t-2T 0 t T1x()For 0 t T .Thus, for 0 t T .Interval 2. For 0 t T,罢坦颗赛帧创葡超油霜烧绩铜怯苔命徒齐颠屉擎湘辆蚕搜郑迪人很墓效忽信号教学课件（华中科技大学）chapter 2信号教学课件（华中科技大学）c
16、hapter 22Th(t) t-2T T t1x()For T t 2T Thus, for T t T but t-2T 0, i.e. T t 0, but t-2T T, i.e. 2T t 3T Thus, for 2T t 3T.For 2T t T, or equivalently, t 3T, there is no overlap between the nonzero portions of and hence, x()2Th(t) 0 T t-2T t1For t 3TSummarizing, 亿撕垢绍民爬痹仓环溶据弛堰寂篆家犹娱胸逆骤陋溺森庸宅科康抹林辗柿信号教学课件（
17、华中科技大学）chapter 2信号教学课件（华中科技大学）chapter 22.3 PROPERTIES OF CONVOLUTION OPERATION 2.3.1 The Commutative Property(交换律交换律) in discrete time : in continuous time: 2.3.2 The Distributive Property (分配律分配律) in discrete time : in continuous time: 啤林始吾婉授服穿背匿覆滴策哄儿绣传判姬仑酶坑完币至佩网得欲宁堡年信号教学课件（华中科技大学）chapter 2信号教学课件（华
18、中科技大学）chapter 2y(t)y2(t)y1(t) h2(t)x(t) h1(t)x(t) h1(t)+h2(t)y(t) Two equivalent systems: having same impulse responses 兵魔临消亡果县图慑雾扩装直壤仆扑挖庙蝴衬浦邦瘫羌勾揪汇菌绽春兹拦信号教学课件（华中科技大学）chapter 2信号教学课件（华中科技大学）chapter 22.3.3 The Associative Property (结合律结合律) in discrete time : in continuous time: xnh1nynh2nxnhn=h1n*h2ny
19、nxnhn=h2n*h1nynxnh2nynh1nFour equivalent systems鹃共厨班誓吼潮码壤绰锤盯乖御翼揩剁螟张芳封术烈诣揖瘦渐丢堑沼抱酗信号教学课件（华中科技大学）chapter 2信号教学课件（华中科技大学）chapter 22.3.4 Convolving with Impulse2.3.5 Differentiation and Integration of Convolution IntegralCombining the two properties, we have 桩死擞汛淀肚呛倾爽吃电维敲藉赁含宇约乙怖滤难渔目煞吟赘朔空涂番挝信号教学课件（华中科技大学）
20、chapter 2信号教学课件（华中科技大学）chapter 22.3.6 First Difference and Accumulation of Convolution Sum皮砰净野茬翅构柜摇吵侠蹿颠约湘盗耍竹哥致库腋服艾紧虏敝硕憎舍菊靠信号教学课件（华中科技大学）chapter 2信号教学课件（华中科技大学）chapter 22.4.1 LTI Systems with and without Memoryfor a discrete-time LTI system without memory: for a continuous-time LTI system without mem
21、ory:2.4.2 Invertibility of LTI SystemsThe impulse responses of a system and its inverse system satisfy the following condition: in discrete-time :in continuous-time: Since2.4 PROPERTIES OF LTI SYSTEMS如按济瑟辨歇抿镰孰瞩葫棘藉占椿纂家缮阻宣姑都呜亭眯弊俏策留易谅足信号教学课件（华中科技大学）chapter 2信号教学课件（华中科技大学）chapter 22.4.3 Causality for LT
22、I Systems for a causal discrete-time LTI system: for a causal continuous-time LTI system: 2.4.4 Stability for LTI Systemsfor a stable discrete-time LTI system: for a stable continuous-time LTI system: absolutely summable absolutely integrable 吸销埂雕住术希剧遵匈呵癸曼转辗朋科烹徐型景秦寻仪奴纫排昧往霉埃斟信号教学课件（华中科技大学）chapter 2信号
23、教学课件（华中科技大学）chapter 2SupposeProof:ThenIfThenTherefore, the absolutely summable is a sufficient condition to guarantee the stability of a discrete-time LTI system.麻呐谜直沉供休敖池琼散浸圆柴荚铝跪触晌渣度莱镀浓毡度莲乘劣媒茅以信号教学课件（华中科技大学）chapter 2信号教学课件（华中科技大学）chapter 2To show that the absolutely summable is also a necessary con
24、dition for the stability of a discrete-time LTI system, Letwhere, is conjugate .Then, xn is bounded by 1, that isHowever, IfThen郁淫柠售授实钱研升援郡袁芜泉棕莽企脊瀑泣挠物麦壹她市躲娥揍恶街趣信号教学课件（华中科技大学）chapter 2信号教学课件（华中科技大学）chapter 22.5 The Unit Step Response(单位阶跃响应单位阶跃响应) of an LTI System The unit step response, sn or s(t),
25、is the output of an LTI system when input xn=un or x(t)=u(t). The unit step response of a discrete-time LTI system is the running sum of its unit sample response: The unit sample response of a discrete-time LTI system is the first difference of its unit step response: 底射凶芝六绅户术锌愧贮玉掏活菱声凿腆巩毛辟捕簿砌埠并顿钞穗傲画
26、儒信号教学课件（华中科技大学）chapter 2信号教学课件（华中科技大学）chapter 2The unit step response of a continuous-time LTI system is the running integral of its unit impulse response: The unit impulse response of a continuous-time LTI system is the first derivative of the unit step response :笺照倦兽鸭桌赃掀阅委售暇烧飞淡絮卓悦竣藕瞧恶蠕梗摆流就翅在造确你信号教
27、学课件（华中科技大学）chapter 2信号教学课件（华中科技大学）chapter 22.6 CAUSAL LTI SYSTEMS DESCRIBED BY DIFFERENTIAL AND DIFFERENCE EQUATIONS Linear constant-coefficient differential equation: input signal; : output signal. Ci (t)VsR + Example 2.4漏恐蜕磅盐乒馒索及劣迢叛悍陵存拭铀坑竭蛔怯丑灶过札檬旋罢乞夹椎溪信号教学课件（华中科技大学）chapter 2信号教学课件（华中科技大学）chapter 2
28、Linear constant-coefficient difference equation is the mathematical representation of a discrete-time LTI system. Linear constant-coefficient differential equation is the mathematical representation of a continuous-time LTI system. We must specify one or more auxiliary conditions to solve a differen
29、tial (difference) equation . Initial rest(初始静止初始静止): for a causal LTI system, if x(t)=0 for tt0, then y(t) must also equal 0 for t0, so let Taking x(t) and for t 0 into the original equation yields ThusSo the solution of the differential equation for t0 is In Example 2.4,排哟耘覆皖谱棋蹭磨炬采楞认樱反囚浅特佩吹趟喀焊神箭如誓公
30、直侨某仔信号教学课件（华中科技大学）chapter 2信号教学课件（华中科技大学）chapter 2Taking use of the condition initial rest, we obtain Consequently, or for t0灌琢拔峨窄摆泵聊遥痰丘勾仲让点桃买胸汉秧许醒聚操痛仆瞄掌岭狼异啥信号教学课件（华中科技大学）chapter 2信号教学课件（华中科技大学）chapter 2Example 2.5Jack saves money every month. It is known that at the beginning of the nth month the a
31、mount he saves into the bank is RMB xn yuan, and the rate of interest is per month. Suppose Jack wouldnt withdraw his bank deposits in whatever situation, try to give the difference equation relating xn and yn, which is the deposits of Jack at the end of the nth month. (before the bank calculates th
32、e interest)Solution:yn is consists of the sum of the following three parts: xn saved at the beginning of the nth month yn-1 interest at the end of the (n-1)th month yn-1 deposit of the (n-1)th monthSo the difference equation isalso牢侗篷簇孵离卞牙泞声捣测杖锻沂铡蹲臭迈掺雏蛔井俱吱荫枫巢撂宫溯顽信号教学课件（华中科技大学）chapter 2信号教学课件（华中科技大学）
33、chapter 2Difference:For sequence xn, its First forward difference(一阶前一阶前向差分向差分) is defined as xn = xn+1 xnSecond backward difference as 2x n = xn xn-1 = xn-2xn-1+xn-2 Analogously, Second forward difference can be constructed as 2xn = xn = xn+1 xn = xn+2-2xn+1+xn its First backward difference (一阶后向差分
34、一阶后向差分) is defined as xn = xn xn-1涨仁声逸昼巡胀诫菜碾譬阉啮垄束匪褂生镊并付由音国邓哺爹否前斟吗祝信号教学课件（华中科技大学）chapter 2信号教学课件（华中科技大学）chapter 2General Nth-order linear constant-coefficient difference equation:First resolution:N auxiliary conditions: Second resolution: (recursive method(迭代法)破智涯寒肯评放勤恃映淫技娟韭卓琶贵彝疹磋拂臆攫什强产蛛钝尺拨遵钉信号教学课件（华
35、中科技大学）chapter 2信号教学课件（华中科技大学）chapter 2Example 2.6Solve the difference equation and the initial condition is y0=1.The eigen equation is So the eigenvalue is a = 2We can write Let Taking into the original equation yields Thus殖晨默损垃谤册招床陷回浪谈亢违位物税卫仕蚀起系破驼枕笛兔宪群推陈信号教学课件（华中科技大学）chapter 2信号教学课件（华中科技大学）chapter
36、2The solution for the given equation isFrom the initial condition of y0=1, we have Consequently, 惧贩拌完目拽闭讼塔糯代斗尘粳团透党氓本奎携混泥椒脂清因骨区夺扣箔信号教学课件（华中科技大学）chapter 2信号教学课件（华中科技大学）chapter 2Example 2.7Consider the difference equationDetermine the output recursively with the condition of initial rest and Rewrite th
37、e given difference equation asStarting from initial condition, we can solve for successive values of yn for n1:擎诊驰履戎窥事糖赚囤小亢支陋朵浅朝戈臀独食导录蔚哥斯滤钾洁印读隔信号教学课件（华中科技大学）chapter 2信号教学课件（华中科技大学）chapter 2Considering yn =0 for n0, the solution is 撞倚爹诵诸遮要限几忧翱洼掩力钳房锥己粮补膊卉匿谈叶缩马疫麓视嫩皂信号教学课件（华中科技大学）chapter 2信号教学课件（华中科技大学）
38、chapter 2What are the relationships between yp, yh, yzi and yzs ?Question:邵绍弦恭奋斤豹惮硬访禄野据械怯褪晃尊畅亮佰填陇陕苍世劣矫吞函獭幂信号教学课件（华中科技大学）chapter 2信号教学课件（华中科技大学）chapter 2Example 2.8Consider a continuous-time LTI system described by the following differential equationwith initial conditions of and input From the defin
39、ition of the zero-input response, we haveIn the case of zero input, Thus we can writeEquivalently,Consequently,遇旁锤彪宛攫匹汤八叮忧扮帝帝鸭叁掸照茅鬼鸟秦断镶西岛姨瓮与老辞质信号教学课件（华中科技大学）chapter 2信号教学课件（华中科技大学）chapter 2Next, solve for h(t).From the definition of the unit impulse response, we haveAnd for t0, it becomes h(t) is th
40、e solution for the homogeneous equation. Thus,And because the system is a causal one, there should be The initial conditions used to determine A1 and A2 areButLet (2) Then (3)该滔串仰咐模腥禽咽姻德掂纪殊绥酞盈眷辖勤撤惭宙释歌枢痞或袱奋卯柑信号教学课件（华中科技大学）chapter 2信号教学课件（华中科技大学）chapter 2Taking equations (2) and (3) into equation (1),
41、 we haveComparing the coefficients of the corresponding terms on each sideCompute the integral in the interval of 0-, 0+ on both sides of equation (2) to obtainAnalogously to equation (3) to obtainConsequently,汕缔碴蜒逊仰慧套胁明长棚倚氰碟爹百釉取郎援芯鼓瀑假屑山控壹显珐眶信号教学课件（华中科技大学）chapter 2信号教学课件（华中科技大学）chapter 2Then fromWe
42、obtainSoThen呆遭捅组桅傍阜任正烈叼汇附私阁诡硒咐皿噎取咆凰扣私鹊神魏遣蒙眠囚信号教学课件（华中科技大学）chapter 2信号教学课件（华中科技大学）chapter 2Example 2.9Consider a causal LTI system described byDetermine the unit sample response hn. For n0, hn satisfies the difference equationAnd there should be Substituting hn for yn and n for xn in the original dif
43、ference equation, and let n=0, we obtainIts obvious thatTaking use of h0, we make out the coefficient in hn:So斜坊被哎淳原尖篱鸳慎海孵幅图朗竖垒维邯瞧徽忱舟瞩香那贬入坠穿配舌信号教学课件（华中科技大学）chapter 2信号教学课件（华中科技大学）chapter 2In fact, for n=0, h0 also satisfy Thus, we can write You may also try the recursive method to obtain the hn for
44、this system! 竖踢占妙枚茹责赵纂拟辗藻包叠淄易匀省曰治膏枕谗耶皆粹跟腿炳甥断妖信号教学课件（华中科技大学）chapter 2信号教学课件（华中科技大学）chapter 22.7 Block Diagram Representations of First-Order Systems Described by Differential and Difference Equations First-order difference equation : addition delay multiplication Three basic elements in block diagram
45、: adder, multiplier and delayer. (方框图) (加法器)(乘法器) (延时器) Basic elements for the block diagram representation of causal discrete-time systems. (a) an adder (b) a multiplier (c) a delayer. x1n+x2n + x2nx1n(a)axnxna(b) Dxnxn-1(c)茎闲幅赦遥酶吉湿除仕堪窥斯主伞间淫立疵制郧斜孪带荚也涯瘸曙痒皱豺信号教学课件（华中科技大学）chapter 2信号教学课件（华中科技大学）chapte
46、r 2 + Dxnyn b -ayn-1Block diagram representation for the causal discrete-time system described by the first-order difference equation (yn+ayn-1=bxn). First-order differential equation :differentiation 惕烂兽揍肮劈酱貌单签董油礁炯钮帘菌裸崇亨骗蹲衍姨据削蝶套峪扒烩符信号教学课件（华中科技大学）chapter 2信号教学课件（华中科技大学）chapter 2 x(t)Block diagram re
47、presentation of an integrator + x(t)y(t)b-aBlock diagram representation for the system described by the first-order differential equation Three basic elements in block diagram: adder, multiplier and integrator(积分器) . 涯扰烷称削牛纸射眶挣盒迸跪携婉框粘吞巴套颇档玲蓉冀扒渭丝椒刁著树信号教学课件（华中科技大学）chapter 2信号教学课件（华中科技大学）chapter 22.6 S
48、UMMARYA representation of an arbitrary discrete-time signal as weighted sums of shifted unit samples;Convolution sum representation for the response of a discrete-time LTI systems;A representation of an arbitrary continuous-time signal a weighted integrals of shifted unit impulses; Convolution integ
49、ral representation for continuous-time LTI systems; Relating LTI system properties, including causality, stability, to corresponding properties of the unit impulse (sample) response; Some properties of systems described by linear constant-coefficient differential (difference) equations; Understanding of the condition of initial rest. 呸宠虐眺堵锦卷赵狗胰邀摧纳撮斤邵痰岁锐杯仟蛊侦忠回射掣匙攀含兵抢信号教学课件（华中科技大学）chapter 2信号教学课件（华中科技大学）chapter 2HomeworkHomework2.21 (a) (c) 2.22 (a) (d) 2.28 (b) (e) (g) 2.29 (b) (e) (f)2.32湾樊蛋张蜂彦弯算沦面个盘椽拱撒尺孽实软楷煞拿烂翌驮适硫错峰埃贞匣信号教学课件（华中科技大学）chapter 2信号教学课件（华中科技大学）chapter 2
