信号与系统课件.pptx

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 Represent 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 Signalisformallydefinedasafunctionofoneormorevariablesthatconveysinformationonthenatureofaphysicalphenomenon.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 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 Pathfinder 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 diagram 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:Biomedical 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 membrane 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 the basilar membrane,showing their envelopes induced by incoming sound at three different frequencies.学习方法学习方法理论与实际相结合，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、做相结合；加强计算机实践环节，用 MATLAB进行信号与系统的分析。本课程教学与学习的安排本课程教学与学习的安排1.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（理论）+MATLAB上机作业（实践）。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：由书面作业、MATLAB作业、期中考试和期末考试4部分组成。主要参考书主要参考书1 Simon H.,Barry V.V.Signals and Systems.John 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、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，2004Ch1.4 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 is defined only at discrete instants of time.离离散散时时间间信信号号：若若信信号号仅仅在在某某些些离离散散时时刻刻处处有有定定义义,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 Non-Periodic Signals (周期信号与非周期信号周期信号与非周期信号)A Periodic signals is a function of time that satisfies the condition:x(t+T)=x(t)for all t.Definition Fundamental Period:T0，thesmallestvalueofTthatsatisfiesthedefinition.Period：Periodic and Non-Periodic Signals (周期信号与非周期信号周期信号与非周期信号)Definition FrequencyFundamental FrequencyAngular FrequencyPeriodic and Non-Periodic Signals (周期信号与非周期信号周期信号与非周期信号)(a)Square wave with amplitude A=1 and period T=0.2s.(b)Rectangular pulse of amplitude A and duration T1.Problem:Triangular wave alternative between 1 and+1Periodic and Non-Periodic Signals (周期信号与非周期信号周期信号与非周期信号)A discrete Periodic signals is a function of time that satisfies the condition:xn+N=xn for integer n.Definition Fundamental Period:N，thesmallestintegerthatsatisfiesthedefinition.Periodic and Non-Periodic Signals (周期信号与非周期信号周期信号与非周期信号)Definition PeriodFundamental Frequency(a)Discrete-time square wave 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)Deterministic and Random Signals确定性信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.确确定性信号：定性信号：在任意时刻都有确定的值的信号在任意时刻都有确定的值的信号。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)Energy and Power signals (能量信号和功率信号能量信号和功率信号)Definitions:EnergyAverage PowerEnergy and Power signals (能量信号和功率信号能量信号和功率信号)Definitions:Energy SignalPower SignalEnergy 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.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 square wave.amplitude A=1 and period T=4s,T0=1s.)(txtT0T-0TT-10Energy and Power signals(能量信号和功率信号能量信号和功率信号)Energy signal?Power signal?Ch1.5 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 performed 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(信号相乘)y(t)=x1(t)x2(t)y(n)=x1nx2noperations performed on dependent variables(基于从变量的运算基于从变量的运算)4.Differentiation(连续信号的微分)5.integration（连续信号的积分）operations performed on dependent variables(基于从变量的运算基于从变量的运算)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：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 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)压缩a倍。Time scaling（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)about 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 ofxnand yn2.reflection（翻转）:xn x-nxn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 rectangular 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(t-t0)x(t-t0)表示信号x(t)右移t0单位；x(t+t0)表示信号x(t)左移t0单位。Time shifting(时移时移)Problem:Find the time-shifted signalyn=xn+3yn=xn k，k0 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)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)Example:已知x(t)的波形如图所示，试画出x(2t)、x(t/3)、x(t+6)、x(-t)、x(6-2t)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)Example: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 Signals 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.实指数实指数实指数实指数信号：信号：信号：信号：Examples of 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.实指数实指数实指数实指数信号：信号：信号：信号：Sinusoidal signal(正弦信号正弦信号)(a)Sinusoidal signal Acos(t+)with phase=+/6 radians.(b)Sinusoidal signal Asin(t+)with phase=+/6 radians.Sinusoidalsignal：x(t)=Acos(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 between 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-atsin(t)，0Step function 阶跃函数阶跃函数Continuous-time version of the unit-step function of unit amplitude.Definitions:Step function(阶跃函数阶跃函数)(a)Rectangular pulse x(t)of amplitude A and duration 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)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)=0,t0Definitions:Unit Impulse单位冲激信号单位冲激信号(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 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Properties of Unit Impulse冲激信号的性质冲激信号的性质 取样特性取样特性证明：证明：证明：证明：利用筛选特性Properties of Unit Impulse冲激信号冲激信号 Time Scaling(Time Scaling(展缩特性展缩特性)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The Time-scaling Property of Unit 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.冲激信号冲激信号冲激信号冲激信号与与阶跃信号阶跃信号阶跃信号阶跃信号的关系的关系Properties of Unit Impulse冲激信号的性质冲激信号的性质Problems solution Definitions: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）Properties:(取样特性)(筛选特性)(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)Definitions:Relation Between Unit Function and Ramp 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(1 1)(1 1)(2 2)(2 2)解：解：ProblemsRelations 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 Interconnections of Operations BlockdiagramrepresentationofoperatorHfor(a)continuoustimeand(b)discretetime.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)/3Two different(but equivalent)implementations 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.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）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)/3 （Mx+Mx+Mx）/3=Mxy(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)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.Causality（因果性）（因果性）因果：输出不领先于输入，即现时刻的输出仅取决于当前时刻的输入和（或）过去时刻的输入Ex:Moving-AverageSystems(滑动平均系统)y(n)=x(n)+x(n-1)+x(n-2)/3Isthissystemcausal?Causality（因果性（因果性）Series RC circuit driven from an ideal voltage source v1(t),producing output voltage v2(t).Ex:Consider the RC circuit.Is this system causal or non-causal?3.Invertibility(可逆性可逆性)The notion of system invertibility.The second operator Hinv is the inverse of the first operator H.Hence,the input x(t)is passed through the cascade correction of H and H-1 completely unchanged.4.Time Invariance(时不变性时不变性)The notion of time invariance.(a)Time-shift operator St0 preceding operator H.(b)Time-shift operator St0 following operator H.These two situations are equivalent,provided that H is time invariant.4.Time Invariance(时不变性时不变性)A system is time invariant if a time delay or time advance of the input signals lead to an identical time shift in output signal.系统的输入延迟或超前一段时间，其输出也延迟或超前一段时间，就称为时不变系统。即，时不变系统的特性不随时间发生变化，否则就称为时变系统。Ex:Are these systems time-invariant?时不变系统时变系统判断一个系统是否为时不变系统，只需判断当输入激励x(t)变为x(t-t0)时，相应的输出响应y(t)是否也变为y(t-t0)。(1)y(t)=3x(t)(2)y(t)=t x(t)5.Linearity(线性线性)线性系统：具有均匀性 与 叠加性的系统。(2)Homogeneity(2)Homogeneity(均匀性均匀性均匀性均匀性)(1)(1)Superposition(Superposition(叠加性叠加性叠加性叠加性)5.Linearity(线性线性)线性：同时具有均匀性 与 叠加性。线性系统的数学模型是线性微分方程或线性差分方程。5.Linearity(线性线性)The linearity property of a system.(a)The combined operation of amplitude scaling and summation precedes the operator H for multiple inputs.(b)The operator H precedes amplitude scaling for each input;the resulting outputs are summed to produce the overall output y(t).If these tw