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1、信号与系统 Signals and Systems,郝晓莉 陈后金 北京交通大学电子信息工程学院,Ch1 Introduction (绪论),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)?,A Speech signal Its amplitu
2、de 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 c
3、oordinates. Its a two-dimensional signal.,What is a Signal (信号)?,Definitions,A Signal is formally defined as a function of one or more variables that conveys information on the nature of a physical phenomenon. 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)?,A System is formally defined as an
4、 entity that manipulates one or more signals to accomplish a function, thereby yielding new signals.,Definitions,Ch1.3 Overview of Specific Systems,Elements of a communication system. The transmitter changes the message signal into a form suitable for transmission over the channel. The receiver proc
5、esses the channel output (i.e., the received signal) to produce an estimate of the message signal.,Examples: Communication Systems(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
6、reflector must remain accurate within a fraction of the signals wavelength. (Courtesy of Jet Propulsion Laboratory.),Examples: Control Systems(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
7、 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),The traces shown in (a), (b), and (c) are three examples of EEG signals recorded from the hippocampus o
8、f 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 “s
9、tiff region” and “flexible region” refer to the basilar membrane.,Examples: Auditory Systems(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,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
10、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,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,1 Simon H.,Barry V.V. Signals and Systems. John Wiley & Sons,I
11、nc.1999 2 Edward W.K.,Bonnie S.H. Fundamentals of Signals and Systems Using MATLAB. Prentice-Hall International,Inc. 1997 3 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,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，2001 6 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
12、版社，2002 7 陈后金等. 信号与系统. 清华大学出版社， 2003 8 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，2004,Ch1.4 classifications of signals (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
13、和功率信号,Continuous-time and Discrete-time signals 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,Discrete-time signal: a signal if it is defined only at discrete instants of time. 离散时间信号：若信
14、号仅在某些离散时刻处有定义, 用xn表示。,Continuous-time signal: a signal if it is defined for all time t. 连续时间信号：若信号在所有时间t 处都有定义, 用x(t)表示。,Definitions,Continuous-time and Discrete-time signals 连续时间信号和离散时间信号,离散信号可以由连续信号取样(sampling)得来： xn=x(t)|t=nT =x(nT) T称为取样间隔,periodic and non-periodic signals 周期信号和非周期信号,(a) Square
15、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),A Periodic signals is a function of time that satisfies the condition: x(t+T) = x(t) for all t.,Definition,Fundamental Period: T0， the smallest value of
16、 T that satisfies the definition.,Period：,Periodic and Non-Periodic Signals (周期信号与非周期信号),Definition,Frequency,Fundamental Frequency,Angular Frequency,Periodic and Non-Periodic Signals (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 dura
17、tion T1.,Problem: Triangular wave alternative between 1 and +1,Periodic and Non-Periodic Signals (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， the smallest integer that satisfies the definition.
18、,Periodic and Non-Periodic Signals (周期信号与非周期信号),Definition,Period,Fundamental 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),Deterministic and Random Signa
19、ls 确定性信号和随机信号(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：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：在任意时刻都有确定的值的信号。,Deterministic and Random Signals 确
20、定性信号和随机信号,Deterministic and Random Signals 确定性信号和随机信号,t,t,t,X(t),Energy and Power signals (能量信号和功率信号),Definitions:,Energy,Average Power,Energy and Power signals (能量信号和功率信号),Definitions:,Energy Signal,Power Signal,Energy and Power signals 能量信号和功率信号,Problem: Determine the total energy of the discrete-
21、time signal.,Energy and Power signals (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.,Energy and Power signals (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,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
22、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),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),2. Addition (信
23、号相加) y(t) = x1(t)+x2(t) y(n) = x1n+x2n,3. Multiplication (信号相乘) y(t) = x1(t) x2(t) y(n) = x1n x2n,operations performed on dependent variables (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),Example：电感两端的电压与其电流为微分关系： Example： 电容两端的
24、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),(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01， 则x(at)是x(t)压缩a倍。,Time scaling（尺度展缩）,(a) discrete-time signa
25、l 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= xkn,operations performed on independent variables (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(-
26、t) x(-t)表示将x(t)以纵轴为中心作180翻转。,Reflection（ 翻转）,Problem: Find the reflected version of xn and yn,2. 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),(a) continuous-time signal in the form of a rectangular pulse of amplitude 1.0 and duration 1.0, symmetri
27、c 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 ),Problem: Find the time-shifted signal yn= xn+3,yn=xn k ，k0 xn+k，左移k单位； xn-k， 右移k单位。,operations performed on independent varia
28、bles (基于信号自变量的运算),总结公式：,operations performed on independent variables (基于信号自变量的运算),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,Exponent
29、ial Signals 指数信号 Sinusoidal Signals 正弦信号 Exponential Damped Sinusoidal Signals 按指数衰减的正弦信号 Step Signals 阶跃信号 Impulse Signals 冲激信号 Derivatives of The Impulse 冲激信号的导数 Ramp Function 斜坡函数,Exponential Signals 指数信号,(a) Decaying exponential form of continuous-time signal. (b) Growing exponential form of con
30、tinuous-time signal.,实指数信号：,Examples of Exponential Signals 指数信号,Lossy capacitor, with the loss represented by shunt resistance R.,Exponential Signals(指数信号),(a) Decaying exponential form of discrete-time signal. (b) Growing exponential form of discrete-time signal.,实指数信号：,Sinusoidal signal(正弦信号),(a)
31、 Sinusoidal signal Acos(t +) with 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,Parallel LC circuit, assuming that the inductor L and capacitor C are both ideal.,Sinusoidal signal 正弦信号,Discrete-time
32、sinusoidal signal.,Relation between Sinusoidal and Complex Exponential Signals 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,Exponentially damped sinusoidal signal Ae-at sin(t), with A = 60 and = 6.,x (t)
33、= Ae-at sin(t)， 0,Step function 阶跃函数,Continuous-time version of the unit-step function of unit amplitude.,Definitions:,Step function(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
34、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,(a) Series RC circuit with a switch that is closed at time t = 0, ther
35、eby energizing the voltage source. (b) Equivalent circuit, using a step function to replace the action of the switch.,(t)=0 , t0,Definitions:,Unit Impulse 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 imp
36、ulse. (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,(a) Series circuit consisting of a capacitor, a dc voltage source, and a switch; the switch is closed at time t = 0. (b) Equiv
37、alent circuit, replacing the action of the switch with a step function u(t).,Examples of Unit Impulse 冲激信号, 筛选特性,Properties of Unit Impulse 冲激信号的性质, 取样特性,证明：,利用筛选特性,Properties of Unit Impulse 冲激信号, Time Scaling(展缩特性),推论：冲激信号是偶函数。,根据d(t)泛函定义证明,取 a = -1 , 可得 d(t) = d(-t),Properties of Unit Impulse 冲激信
38、号的性质,The Time-scaling Property of Unit Impulse 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
39、, restoring it to unit area., 冲激信号与阶跃信号的关系,Properties of Unit Impulse 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）,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）,Ramp function(斜坡函数),Definitions:,Relation Between Unit Funct
40、ion and Ramp function 阶跃函数与斜坡函数的关系,(1),(1),(2),(2),解：,Problems,Relations of Impulse Function, Unit Function and Ramp function 冲激函数、阶跃函数与斜坡函数的关系,Ch1.7 Systems Viewed as Interconnections of Operations,Block diagram representation of operator H for (a) continuous time and (b) discrete time.,Discrete-ti
41、me-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) implementations of the moving-average system: (a) cascade form of implementation and (b) parallel form of implementation.,Ex: Mov
42、ing-Average Systems 滑动平均系统：y(n)=x(n)+x(n-1)+x(n-2)/3,Ch1.8 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）,Ex: Mov
43、ing-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=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) , r1,Dramatic photographs showing the collapse of the Tacoma Narrows suspens
44、ion 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
45、 as it crashed in Puget Sound, Washington. Note the car in the top right-hand corner of the photograph.,An Unstable System,2.Causality （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-Average Systems (滑动平均系统) y(n)=x(n)+x(n-1)+x(n-2)/3 Is this system causal?,Causality （因果性）,Series RC circui
46、t 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),The notion of system invertibility. The second operator Hinv is the inverse of the first operator H. Hence, the input x(t) is passed t
47、hrough the cascade correction of H and H-1 completely unchanged.,4. Time Invariance(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
48、 Invariance(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),线性系统：具有均匀性 与 叠加性的系统。,(2) Homogeneity (均匀性),(1) Superposition (叠加性),5. Linearity (线性),线性：同时