信号教学课件华中科技大学chapter2.ppt
CHAPTER 2 LINEAR TIME-INVARIANT SYSTEMS 2.0 INTRODUCTION,Representation of signals as linear combination of delayed impulses.Convolution sum(卷积和)or convolution integral(卷积积分)representation of LTI systems.Impulse response and systems properties Solutions to linear constant-coefficient difference and differential equations(线性常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 2,2.1 DISCRETE-TIME SYSTEMS:THE CONVOLUTION SUM,Derivation steps:Step 1:Representing discrete-time signals in 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 delayed 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 2,The Convolution Sum Representation of LTI Systems,convolution sum or superposition sum:,Convolution operation symbol:,LTI system is completely characterized by its response to the unit 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 2,Example 2.1,1,1,0.5,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 2,链谋渗杖苞狭叮旬合扑投北曳密磊录占袱堂洋噶已寐梳邻漏鸯谴独计两扒信号教学课件（华中科技大学）chapter 2信号教学课件（华中科技大学）chapter 2,孟框想瘤梳浩疽搜祈缝赌庐挞颐彩瘤绞炳朱虞对怜撇演孰衍炒韩备蛾鸳训信号教学课件（华中科技大学）chapter 2信号教学课件（华中科技大学）chapter 2,毫缠冕畦照哨占且贩毯软鞋扶浙瑟酮舜源傣蝶恭淘戚剁受柔咀仁霹拓宛搭信号教学课件（华中科技大学）chapter 2信号教学课件（华中科技大学）chapter 2,言藻磕拦崇币垂圃统雹盲同翘拾吩钙煮蹋褪净噶勤农胶坎以溪疥须跌谈秤信号教学课件（华中科技大学）chapter 2信号教学课件（华中科技大学）chapter 2,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 method for the computation of convolution sum,that is suitable for two short sequences.,xn=1,1,10,hn=0.5,1,0.5,1,0.5-2,xn*hn=0.5,1.5,2,2.5,2,1.5,0.5-2,0.5 1.5 2 2.5 2 1.5 0.5,0.5 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 2,Example 2.2,Consider an input xn and a unit sample response hn given by,Determine and plot the output,辣漆太边颖榜甥绦九师虫桓铭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 2,Using 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 2,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信号教学课件（华中科技大学）chapter 2,2.2 CONTINUOUS-TIME LTI SYSTEMS:THE CONVOLUTION INTEGRAL,The Representation of Continuous-Time Signals in Terms of Impulses:,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 2,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 2,as,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 2,Give 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 is,As,in addition,the summing becomes an integral.Therefore,convolution integral or superposition 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 2,unit impulse response h(t):the response to the input.（单位冲激响应）,Convolution integral symbol:,A continuous-time LTI system is completely characterized by its unit impulse response h(t).,Example 2.3,Consider the convolution of the following two signals,which are depicted in(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 2,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 2,2T,h(t),t-2T 0 t T,1,x(),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）chapter 2,2T,h(t),t-2T T t,1,x(),For T t 2T,Thus,for T t 2T,Interval 3.For t T but t-2T 0,i.e.T t 2T,漠霉站椎熙捏沃扇波殉乘蟹双判凳紫敏附鸯颇沛焊揍蒜砖峰弗学挤祈贤忱信号教学课件（华中科技大学）chapter 2信号教学课件（华中科技大学）chapter 2,2T,h(t),t-2T t,1,x(),Interval 4.For t-2T 0,but t-2T T,i.e.2T t 3T,Thus,for 2T t 3T,.,For 2T t 3T,札矿纠呛奶柿分蝴溜磨垢际伯渍圭铅树健届睫镊哥课溢字窥证尺塌写峻辫信号教学课件（华中科技大学）chapter 2信号教学课件（华中科技大学）chapter 2,Interval 4.For t-2T T,or equivalently,t 3T,there is no overlap between the nonzero portions of and,hence,Summarizing,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（华中科技大学）chapter 2信号教学课件（华中科技大学）chapter 2,2.3 PROPERTIES OF CONVOLUTION OPERATION,2.3.1 The Commutative Property(交换律),2.3.2 The Distributive Property(分配律),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（华中科技大学）chapter 2,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 2,2.3.3 The Associative Property(结合律),Four 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 2,2.3.4 Convolving with Impulse,2.3.5 Differentiation and Integration of Convolution Integral,Combining 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）chapter 2信号教学课件（华中科技大学）chapter 2,2.3.6 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 2,2.4.1 LTI Systems with and without Memory,2.4.2 Invertibility of LTI Systems,Since,2.4 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 2,2.4.3 Causality for LTI Systems,2.4.4 Stability for LTI Systems,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信号教学课件（华中科技大学）chapter 2,Suppose,Proof:,Then,If,Then,Therefore,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 2,To show that the absolutely summable is also a necessary condition for the stability of a discrete-time LTI system,Let,where,is conjugate.,Then,xn is bounded by 1,that is,However,If,Then,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 2,2.5 The Unit Step Response(单位阶跃响应)of an LTI System,The unit step response,sn or s(t),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 2,The 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 2,2.6 CAUSAL LTI SYSTEMS DESCRIBED BY DIFFERENTIAL AND DIFFERENCE EQUATIONS,Linear constant-coefficient differential equation,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,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 differential(difference)equation.,Initial rest(初始静止):for a causal LTI system,if x(t)=0 for tt0,then y(t)must also equal 0 for tt0.,General Nth-order linear constant-coefficient differential equation:,谐俊虐寝玉毡演巧八耀旱契令治鹃敲妄蹭炼羚里弃憋亡呼靡踞奄赏棋藩猾信号教学课件（华中科技大学）chapter 2信号教学课件（华中科技大学）chapter 2,N auxiliary conditions:,褐溜求诅矮倘寿亿肠竿户抓堤滁抒凭潘颂栗脚糟墓灼栅腮森语殉悄爽稀淳信号教学课件（华中科技大学）chapter 2信号教学课件（华中科技大学）chapter 2,If R=4,C=1/2,and the input signal is,Then we obtain,From the eigenvalue of the homogeneous equation,we can write,Since for t0,so let,Taking x(t)and for t 0 into the original equation yields,Thus,So 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 2,Taking 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 2,Example 2.5,Jack saves money every month.It is known that at the beginning of the nth month the amount 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 the 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 month,So the difference equation is,also,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）chapter 2,Difference:,For sequence xn,its First forward difference(一阶前向差分)is defined as xn=xn+1 xn,its First backward difference(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 2,General 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（华中科技大学）chapter 2信号教学课件（华中科技大学）chapter 2,Example 2.6,Solve the difference equation and the initial condition is y0=1.,The eigen equation is,So the eigenvalue is a=2,We 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 2,The solution for the given equation is,From 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 2,Example 2.7,Consider the difference equation,Determine the output recursively with the condition of initial rest and,Rewrite the given difference equation as,Starting 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 2,Considering 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）chapter 2,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 2,Example 2.8,Consider a continuous-time LTI system described by the following differential equationwith initial conditions of and input,From the definition of the zero-input response,we have,In the case of zero input,Thus we can write,Equivalently,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 2,Next,solve for h(t).,From the definition of the unit impulse response,we have,And for t0,it becomes,h(t)is the 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 are,But,Let(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 2,Taking equations(2)and(3)into equation(1),we have,Comparing the coefficients of the corresponding terms on each side,Compute the integral in the interval of 0-,0+on both sides of equation(2)to obtain,Analogously to equation(3)to obtain,Consequently,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 2,Then from,We obtain,So,Then,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 2,Example 2.9,Consider a causal LTI system described by,Determine the unit sample response hn.,For n0,hn satisfies the difference equation,And there should be,Substituting hn for yn and n for xn in the original difference equation,and let n=0,we obtain,Its obvious that,Taking 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 2,In fact,for n=0,h0 also satisfy,Thus,we can write,You may also try the recursive method to obtain the hn for 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 2,2.7 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: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.,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）chapter 2,Block 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,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 2,2.6 SUMMARY,A 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 integral 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 2,Homework,2.21(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,
