signals and systems_introduction

1、Signals and Systems - One of the most important courses for us! 陈布雨技术中心B区217室图书馆720第一章 信号与系统简介 （Introduction）介绍信号与系统的基本概念； 信号分类及基本信号；系统分类和特性。第二章 线性非时变系统的时域分析 （Time-Domain Representations for Linear Time- Invariant Systems）单位冲激响应和单位脉冲响应，卷积积分和卷积和；系统的互联；系统的响应求解；系统的方框图表示, 系统的状态空间分析。第三章 信号与线性非时变系统的傅里叶分析

2、（Fourier Representations for signals and Linear Time- Invariant Systems ）离散时间周期与非周期信号、连续时间周期与非周期信号的傅里叶分析；傅里叶分析的性质；LTI系统的频域分析。Contents Contents 第四章 混合信号中的傅立叶分析的应用 （Applications of Fourier Representations to mixed signals）使用傅里叶分析方法对线性非时变系统进行分析；取样定理第五章 傅里叶分析方法在通信系统的应用 （Application to communication syst

3、ems）第六章 连续时间信号与系统的复频域分析拉普拉斯变换 （Representation of Signals by Using Continuous-Time Complex Exponentials:The Laplace Transform）拉普拉斯变换及其性质，连续系统响应的求解，系统函数及其与零极点关系，系统稳定性分析。 第七章 离散时间信号与系统的Z域分析-Z变换 （Representation of Signals by Using Discrete-Time Complex Exponentials: The z-Transform）Z变换及其性质，离散系统响应的求解，系统函

4、数及其与零极点关系，系统稳定性分析。 Contents 第八章 信号与系统分析方法在滤波器和均衡器中的应用 （Applications to Filters and Equalizers） 第九章 信号与系统分析方法在反馈系统中的应用 （Applications to Feedback systems） 第十章 尾声 （Epilogue）简要论述了非平稳信号的时频分析方法等先进分析工具以及非线性和自适应系统等的概念。Learning method 理论与实际相结合，物理概念、数学概念和工程概念并重。 掌握信号与系统分析的基本思想和方法；注意问题的提出、分析问题和解决问题的方法。 讲、练、做相结

5、合；增加适当实践环节，学生可自行学用 MATLAB进行信号与系统的分析。My requirement(要求要求) for the class: 1. Dont be late for class. 2. No late homework will be accepted. 3. You can work together on homework, but you cant copy from others4. Questions in class, raise your hand please. 5. Turn off or mute your cell-phone6. Suggestions

6、 or comments are welcome. The course teaching and learning arrangements1. 课堂教学：讲解重点内容和学生学习中遇到的疑难问题。 2. 作业： 书面作业，每周一交，12：30前交3. 期中和期末考试：闭卷形式。主要考察学生对本门课的基本理论基本原理及重点内容的掌握程度。4.课程成绩的组成： 总评=平时成绩（作业、测验、实验）*30%+期末考试*70。5.答疑时间： 周一下午14：00_16:00 B楼一楼教师休息室主要参考书主要参考书1 Simon H.,Barry V.V. Signals and Systems. Joh

7、n Wiley & Sons,Inc.19992 Edward W.K.,Bonnie S.H. Fundamentals of Signals and Systems Using MATLAB. Prentice-Hall International,Inc. 19973 A.V.Oppenheim. Signals and Systems 或中译本（第2版）. 西安交通大学出版社.4 郑君里，应启珩等. 信号与系统. 第2版. 高等教育出版社，2000. 主要参考书主要参考书5 吴湘淇等. 信号、系统与信号处理(上). 第2版. 电子工业出版社，20016 吴湘淇,郝晓莉等. 信号

8、、系统与信号处理软硬件 实现.电子工业出版社，20027 陈后金等. 信号与系统. 清华大学出版社， 20038 陈后金等. 信号与系统学习指导与习题精解. 清华大学出版社，2004First Big Picture of the Class: 1. Signals Easy, ha! 2. Systems Input a signal, get an output. General description of its response to any input signal System （系统）（系统）Input signal x(t)output signal y(t)Second B

9、ig Picture of the Class: 1. Time and Frequency And they are equivalent !And they are both important to us ! 2. How are they related? Fourier Transform (傅立叶变换傅立叶变换)Third Big Picture of the Class: 1. Continuous(连续连续) and Discrete(离散离散) 2. Periodic(周期周期) or Aperiodic(非周期非周期) Periodicity allows easy des

10、cription of the signal. 3. Four different forms of Fourier Transform: Continuous and Periodic Continuous and Aperiodic Discrete and Periodic Discrete and Aperiodic 4. Use the right one !Ch1 Introduction (绪论绪论)本章重点本章重点: What is a signal? 什么是信号? What is a system? 什么是系统? Classifications of signals 信号的分

11、类 Basic operations of signals 信号的基本运算 Basic signals 基本信号 Properties of systems 系统的特性Ch1.1 What is a Signal (信号信号)? A Speech signal Its amplitude varies with time, depending on the spoken word and who speaks it. Its a one-dimensional signal.x(t)What is a Signal? Electrocardiogram (ECG) Signal Represe

12、nt the electrical activity of the heart. Its a one-dimensional signal.x(t)What is a Signal?I(x,y) An image is a function of two spatial coordinates. Its a two-dimensional signal.What is a Signal (信号信号)? Definitions A Signal is formally defined as a function of one or more variables that conveys info

13、rmation on the nature of a physical phenomenon.信号信号是一个或多个变量的函数，携带着某个物理现象的信息。Ch1.2 What is a System(系统系统)?A System is formally defined as an entity that manipulates one or more signals to accomplish a function, thereby yielding new signals. Definitions Ch1.3 Overview of Specific SystemsElements of a

14、communication system. The transmitter changes the message signal into a form suitable for transmission over the channel. The receiver processes the channel output (i.e., the received signal) to produce an estimate of the message signal.Examples: Communication Systems(通信系统通信系统)(a) Snapshot of Pathfin

15、der exploring the surface of Mars.(b) The 70-meter (230-foot) diameter antenna located at Canberra, Australia. The surface of the 70-meter reflector must remain accurate within a fraction of the signals wavelength. (Courtesy of Jet Propulsion Laboratory.)Examples: Control Systems(控制系统控制系统)Block diag

16、ram of a feedback control system. The controller drives the plant, whose disturbed output drives the sensor(s). The resulting feedback signal is subtracted from the reference input to produce an error signal e(t), which, in turn, drives the controller. The feedback loop is thereby closed.Examples: B

17、iomedical Signal Processing(生物信号处理生物信号处理)The traces shown in (a), (b), and (c) are three examples of EEG signals recorded from the hippocampus of a rat. Neurobiological studies suggest that the hippocampus plays a key role in certain aspects of learning and memory. (a) In this diagram, the basilar m

18、embrane in the cochlea is depicted as if it were uncoiled and stretched out flat; the “base” and “apex” refer to the cochlea, but the remarks “stiff region” and “flexible region” refer to the basilar membrane. Examples: Auditory Systems(听觉系统听觉系统)(b) This diagram illustrates the traveling waves along

19、 the basilar membrane, showing their envelopes induced by incoming sound at three different frequencies.Ch1.4 classifications of signals (信号的分类信号的分类) 1. continuous-time and discrete-time signals 连续时间信号和离散时间信号2. periodic and non-periodic signals 周期信号和非周期信号3. deterministic and random signals 确定信号和随机信号

20、4. Energy and power signals 能量信号和功率信号Continuous-time and Discrete-time signals 连续时间信号和离散时间信号连续时间信号和离散时间信号(a) Continuous-time signal x(t). (b) Representation of x(t) as a discrete-time signal xn. Continuous-time and Discrete-time signals 连续时间信号和离散时间信号连续时间信号和离散时间信号 Discrete-time signal: a signal if it

21、 is defined only at discrete instants of time. 离散时间信号：离散时间信号：若信号若信号仅在某些离散时刻处仅在某些离散时刻处有定有定义义, 用用xn表示。表示。Continuous-time signal: a signal if it is defined for all time t. 连续时间信号：连续时间信号：若信号若信号在所有时间在所有时间t 处处都有定义都有定义, 用用x(t)表示。表示。Definitions Continuous-time and Discrete-time signals 连续时间信号和离散时间信号连续时间信号和离

22、散时间信号离散信号可以由连续信号离散信号可以由连续信号取样取样(sampling)得来：得来： xn=x(t)|t=nT =x(nT) T称为称为取样间隔取样间隔 periodic and non-periodic signals 周期信号和非周期信号周期信号和非周期信号(a) Square wave with amplitude A = 1 and period T = 0.2s.(b) Rectangular pulse of amplitude A and duration T1.periodic and nonperiodic signals 周期信号和非周期信号周期信号和非周期信号P

23、eriodic signals are such that x(t+T) = x(t) for all t. The smallest value of T that satisfies the definition is called the period(周期).In general, for CT signals for DT signalsWhere r, N are integers)()(txrTtx)()(nxrNnxProblem: Triangular wave alternative between 1 and +1 (a) Discrete-time square wav

24、e alternative between 1 and +1. (b) Non-periodic discrete-time signal consisting of three nonzero samples.periodic and non-periodic signals 周期信号和非周期信号周期信号和非周期信号Energy and Power signals 能量信号和功率信号能量信号和功率信号Definitions 能量信号能量信号TTTdttxE2)(lim功率信号功率信号TTTdttxTP2)(21limNNnnnxE2)(limNNnnnxNP2)(21limEnergy an

25、d Power signals 能量信号和功率信号能量信号和功率信号te2te2分别指出功率信号和能量信号Energy and Power signals 能量信号和功率信号能量信号和功率信号Problem: Determine the average power of the triangular wave. Energy and Power signals 能量信号和功率信号能量信号和功率信号Problem: Determine the total energy of the discrete-time signal.Deterministic and Random Signals确定性信

26、号和随机信号确定性信号和随机信号(非确定性信号非确定性信号)Definitions A random signal is a signal about which there is uncertainty before it occurs.随机信号：随机信号：再出现之前具有不确定性的信号再出现之前具有不确定性的信号。A deterministic signal is a signal about which there is no uncertainty with respect to its value at any time.确定性信号：确定性信号：在任意时刻都有确定的值的信号在任意时刻都

27、有确定的值的信号。Deterministic and Random Signals确定性信号和随机信号确定性信号和随机信号xn0NnDeterministic and Random Signals确定性信号和随机信号确定性信号和随机信号tttX(t)(1tx)(2tx)(3txCh1.5 basic operations on signals信号的基本运算信号的基本运算 1. 基于从变量（信号本身或信号之间）的运算 幅度变化； 相加和相乘； 连续信号的微积分，离散信号的差分与累加 2. 基于自变量的运算 连续信号的翻转、展缩和平移 离散信号的翻转、展缩和平移operations perform

28、ed on dependent variables (基于从变量的运算基于从变量的运算) 1. Amplitude scaling (幅度比例变化) x(t) cx(t) xn cxn (c为常数) 波形不变，幅度成比例放大或缩小。 Example：x(t)=sin(210t) ; y(t)=5x(t)=5sin(210t) ; operations performed on dependent variables (基于从变量的运算基于从变量的运算) 2. Addition (信号相加) y(t) = x1(t)+x2(t) y(n) = x1n+x2n3. Multiplication (

29、信号相乘) y(t) = x1(t) x2(t) y(n) = x1n x2noperations performed on dependent variables (基于从变量的运算基于从变量的运算)tdxty)()( 4. Differentiation(连续信号的微分) 5. integration（连续信号的积分）dttdxty)()(operations performed on dependent variables (基于从变量的运算基于从变量的运算)tdiCtv)(1)( Example：电感两端的电压与其电流为微分关系： Example： 电容两端的电压与其电流为积分关系：d

30、ttdiLtv)()(operations performed on independent variables (基于信号自变量的运算基于信号自变量的运算)101 (a) continuous-time signal x(t)(b) version of x(t) compressed by a factor of 2,(c) version of x(t) expanded by a factor of 2.1.Time scaling（尺度展缩）: y(t)=x(at) a0 若0a1，则x(at)是x(t)扩展1/a倍。 若a1， 则x(at)是x(t)压缩a倍。 Time scali

31、ng（尺度展缩）（尺度展缩） (a) discrete-time signal xn (b) version of xn compressed by a factor of 2, with some values of the original xn lost as a result of the compression.yn= xknoperations performed on independent variables (基于信号自变量的运算基于信号自变量的运算) (a) continuous-time signal x(t)(b) reflected version of x(t) a

32、bout the origin. 2. reflection（ 翻转）: y(t)=x(-t) x(-t)表示将x(t)以纵轴为中心作180翻转。Reflection（ 翻转）翻转）Problem: Find the reflected version of xn and yn2. reflection（ 翻转）: xn x-n xn以纵轴为中心作180翻转operations performed on independent variables (基于信号自变量的运算基于信号自变量的运算) (a) continuous-time signal in the form of a rectang

33、ular pulse of amplitude 1.0 and duration 1.0, symmetric about the origin;(b) time-shifted version of x(t) by 2 time shifts. 3.Time shifting (时移 ): y(t) =x(tt0) x(tt0)表示信号 x(t)右移t0单位； x(tt0)表示信号x(t)左移t0单位。Time shifting (时移时移 )n10nx-12 34-21-12Problem: Find the time-shifted signal yn= xn+3 yn=xn k ，k0

34、 xn+k，左移k单位； xn-k， 右移k单位。operations performed on independent variables (基于信号自变量的运算基于信号自变量的运算)abtaxbatx)()()()()(abtaxatxtxtx时移展缩翻转 总结公式：operations performed on independent variables (基于基于信号自变量的运算信号自变量的运算)1t30 x(t)2)3(2(3)2(2)()( txtxtxtx右移缩翻转1t1x(2t)1.5x(2t+6)1.54t11t2x(t)3Example: 已知x(t)的波形如图所示，试画出

35、x(2t)、 x(t/3)、 x(t+6) 、x(-t)、 x(62t)的波形。operations performed on independent variables (基于基于信号自变量的运算信号自变量的运算)n10nx-12 34-21-12Example: 已知x(n)的波形如图所示，求xn+2、 x-n 、x-n-2、x-n/3、 x2n 的波形。Ch1.6 Basic Signals基本信号基本信号1.Exponential Signals 指数信号指数信号2.Sinusoidal Signals 正弦信号正弦信号3.Exponential Damped Sinusoidal S

36、ignals 按指数按指数衰减的正弦信号衰减的正弦信号4.Step Signals 阶跃信号阶跃信号5.Impulse Signals 冲激信号冲激信号6.Derivatives of The Impulse 冲激信号的导数冲激信号的导数7.Ramp Function 斜坡函数斜坡函数Exponential Signals指数信号指数信号 (a) Decaying exponential form of continuous-time signal.(b) Growing exponential form of continuous-time signal.atBtxe)(Examples o

37、f Exponential Signals指数信号指数信号Lossy capacitor, with the loss represented by shunt resistance R. Exponential Signals(指数信号指数信号) (a) Decaying exponential form of discrete-time signal.(b) Growing exponential form of discrete-time signal. nBrtxSinusoidal signal(正弦信号正弦信号) (a) Sinusoidal signal Acos(t +) wi

38、th phase=+/6 radians. (b) Sinusoidal signal Asin(t +) with phase=+/6 radians.Sinusoidal signal： x (t)=A cos(t + )Examples of Sinusoidal signal正弦信号正弦信号Parallel LC circuit, assuming that the inductor L and capacitor C are both ideal. Sinusoidal signal正弦信号正弦信号Discrete-time sinusoidal signal.Relation be

39、tween Sinusoidal and Complex Exponential Signals正弦信号和复指数信号的关系正弦信号和复指数信号的关系Complex plane, showing eight points uniformly distributed on the unit circle.Exponentially damped sinusoidal signal按指数衰减的正弦信号按指数衰减的正弦信号Exponentially damped sinusoidal signal Ae-at sin(t), with A = 60 and = 6.x (t)= Ae-at sin(t

40、)， 0Exponential(指数指数) and sinusoidal(正弦正弦) signals1). Continuous-time Complex Exponential and Sinusoidal Signals 连续时间复指数和正弦信号连续时间复指数和正弦信号2). Discrete-time Complex Exponential and Sinusoidal Signals 离散时间复指数和正弦信号离散时间复指数和正弦信号3). Periodicity Properties of Discrete-time Complex Exponentials 离散时间复指数序列的周期性

41、质离散时间复指数序列的周期性质1). Continuous-time Complex Exponential and Sinusoidal Signals 连续时间复指数信号：连续时间复指数信号： x(t)= Cex(t)= CeatatA. 实指数信号实指数信号(Real Exponential Signals) x(t)= C eat ( C, a are real value)1). Continuous-time Complex Exponential and Sinusoidal SignalsB. 周期复指数和正弦信号周期复指数和正弦信号 （Periodic Complex Exp

42、onential and Sinusoidal Signals） (1) x(t) = e j0t (2) x(t) = Acos(0t+) 所有所有 x(t) 满足满足 x(t) = x(t+T) , and T=2/ 0 , so x(t) is periodic.1). Continuous-time Complex Exponential and Sinusoidal Signals欧拉关系欧拉关系（Eulers Relation） : e j0t = cos0t + jsin 0t and cos0t = (e j0t + e -j0t ) / 2 sin0t = (e j0t -

43、e -j0t ) / 2j We also havetjjtjjeeAeeAtA0022)cos(01). Continuous-time Complex Exponential and Sinusoidal SignalsC. 一般复指数信号：一般复指数信号： x(t) = C e at , in which C = |C| ej , a = r + j 0 , so )sin(|)cos(|)(00)(00teCjteCeeCeeeCtxrtrttjrttjrtj1). Continuous-time Complex Exponential and Sinusoidal Signals图中

44、的虚线对应于函数图中的虚线对应于函数 ，其为振荡曲线的包，其为振荡曲线的包络（络（envelope）。）。 rteC2). Discrete-time Complex Exponential and Sinusoidal SignalsA. 实指数信号实指数信号Real Exponential Signal 实指数信号：实指数信号： xn = C n (a) 1 (b) 01 (c) -10 (d) 0 00 1)(tttuDefinitions:Step function(阶跃函数阶跃函数)(a) Rectangular pulse x(t) of amplitude A and durat

45、ion of 1 s, symmetric about the origin. (b) Representation of x(t) as the difference of two step functions of amplitude A, with one step function shifted to the left by and the other shifted to the right by ; the two shifted signals are denoted by x1(t) and x2(t), respectively. Note that x(t) = x1(t

46、) x2(t).Examples of Step function 阶跃函数阶跃函数(a) Series RC circuit with a switch that is closed at time t = 0, thereby energizing the voltage source. (b) Equivalent circuit, using a step function to replace the action of the switch.t)(t)1 (01=d )( tt(t)=0 , t0Definitions:Unit Impulse单位冲激信号单位冲激信号000 1kk

47、k011k2k2(a) Evolution of a rectangular pulse of unit area into an impulse of unit strength (i.e., unit impulse). (b) Graphical symbol for unit impulse. (c) Representation of an impulse of strength a that results from allowing the duration of a rectangular pulse of area a to approach zero.Examples of

48、 Unit Impulse冲激信号冲激信号(a) Series circuit consisting of a capacitor, a dc voltage source, and a switch; the switch is closed at time t = 0. (b) Equivalent circuit, replacing the action of the switch with a step function u(t).Examples of Unit Impulse冲激信号冲激信号筛选特性筛选特性)()()()(000tttxtttxProperties of Unit

49、 Impulse冲激信号的性质冲激信号的性质取样特性取样特性)(d)()(00txttttxttttxd)()(0ttttxd)()(00ttttxd)()(00)(0tx利用筛选特性Properties of Unit Impulse冲激信号冲激信号201展缩特性展缩特性)0( )(1)(tt推论：冲激信号冲激信号是偶函数。根据(t)泛函定义证明取 a = 1 , 可得 (t) = (t)Properties of Unit Impulse冲激信号的性质冲激信号的性质dttadttaatdatadtat)(|1)(|1)()(1)(The Time-scaling Property of U

50、nit Impulse冲激信号的时间展缩冲激信号的时间展缩Steps involved in proving the time-scaling property of the unit impulse. (a) Rectangular pulse x(t) of amplitude 1/ and duration , symmetric about the origin. (b) Pulse x(t) compressed by factor a. (c) Amplitude scaling of the compressed pulse, restoring it to unit area.

51、与与的关系的关系ttt0 00 1d)()(tuttud)(d)(tProperties of Unit Impulse冲激信号的性质冲激信号的性质The Discrete-time Unit Impulse and Unit Step Sequences0, 10, 0nnn(1)单位冲激单位冲激:序列序列单位阶跃序列单位阶跃序列:0, 10,0nnnuThe Discrete-time Unit Impulse and Unit Step Sequences(2) 单位冲激与单位阶跃序列之间的关系：单位冲激与单位阶跃序列之间的关系：nmmnununun1or0kknnuThe Discre

52、te-time Unit Impulse and Unit Step Sequences(3) 单位冲激序列的抽样性质单位冲激序列的抽样性质0nxnnx000nnnxnnnxkknkxnx- Sifting propertyProblemstttd)4()sin() 1 (325d) 1(e)2(ttt642d)8(e) 3(ttttttd)22(e)4(222d) 13()3()5(tttt)2()32)(6(23ttt)22(e )7(4tt) 1()(e )8(2ttut2/2)4sin(d)4()sin() 1 (ttt51 5325e/1ed) 1(e)2(ttt0d)8(e)3(6

53、42ttte21d) 1(21ed)22(e)4(tttttt0d)3(3)3(d) 13()3()5(222222tttttttt)2(19)2()3222()2()32)(6(2323ttttt) 1(e21) 1(e21) 1(21e)22(e )7(4(-1) 444tttttt0) 1(0) 1() 1(e) 1()(e )8(-1) 22ttuttutsolutionDefinitions:Derivatives of The Impulse function冲激函数的导数（冲激偶）冲激函数的导数（冲激偶）) 1 (t)( t0tttd)(d)( Properties: 0d)(t

54、t)( )( tt)()( )( )()( )(00000tttftttftttf)( d)( )(00tfttttf(取样特性)(筛选特性)0)( )( 1)( tt(展缩特性)Derivatives of The Impulse function冲激函数的导数（冲激偶）冲激函数的导数（冲激偶）Ramp function(斜坡函数斜坡函数)0 00 )(ttttr)()(tuttr或0nknkkukrn123012344rkkDefinitions:t1)(tr1)(d)(dtuttrtutrd)()(0t)(tu1Relation Between Unit Function and Ram

55、p function阶跃函数与斜坡函数的关系阶跃函数与斜坡函数的关系01tf (t)121)2() 1()() 1()(trtrtrtrtf01tf (t)11) 1(2)(2) 1()(trtrtutf解：解：Problemstttd)(d)( ttutd)(d)(ttrtud)(d)(d )()(tutrd )()(ttud )( )(ttRelations of Impulse Function, Unit Function and Ramp function冲激函数、阶跃函数与斜坡函数的关系冲激函数、阶跃函数与斜坡函数的关系Ch1.7 Systems Viewed as Interco

56、nnections of Operations Block diagram representation of operator H for (a) continuous time and (b) discrete time.Discrete-time-shift operator Sk, operating on the discrete-time signal xn to produce xn k.Ex: Moving-Average Systems滑动平均系统：y(n)=x(n)+x(n-1)+x(n-2)/3 Two different (but equivalent) impleme

57、ntations of the moving-average system: (a) cascade form of implementation and (b) parallel form of implementation.Ex: Moving-Average Systems滑动平均系统：y(n)=x(n)+x(n-1)+x(n-2)/3Ch1.8 Properties of Systems 1.Stability (稳定性） 2.Causality （因果性） 3. Invertibility （可逆性）4.Time-Invariance (时不变性)5. linearity (线性)1

58、. Stability (稳定性）稳定性）稳定系统：Bounded Input-Bounded Output（ 有界输入产生 有界输出，BIBO稳定）yxMtyMtx)()(yxMnyMnx)()( )(tx)(ty连续系统)(nx)(ny离散系统不稳定系统：系统的输入有界而输出无界。Stability (稳定性）稳定性） Ex: Moving-Average Systems (滑动平均系统). Show that the System is BIBO stable: y(n)=x(n)+x(n-1)+x(n-2)/3.Solution： y(n) = x(n)+x(n-1)+x(n-2) /

59、3 （Mx+Mx+Mx）/3=Mx y(n)有界，系统稳定。Solution： x(n) Mx1， rn ， y(n)无界，系统不稳定Ex: Unstable System. y(n)=rnx(n) , r1Dramatic photographs showing the collapse of the Tacoma Narrows suspension bridge on November 7, 1940. (a) Photograph showing the twisting motion of the bridges center span just before failure. (b

60、) A few minutes after the first piece of concrete fell, this second photograph shows a 600-ft section of the bridge breaking out of the suspension span and turning upside down as it crashed in Puget Sound, Washington. Note the car in the top right-hand corner of the photograph.An Unstable System2.Causal