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1、2023/5/24,1,Signals and Systems,He Chun(SCIE,UESTC);Room:KB244A,2023/5/24,2,Course Description,Lectures:68+4 Week 1-17 Mon.（5,6）:Room A-412 Wed.（3,4）:Room A-412 Fri.（3,4;even）:Room A-302,Experiment:4+4+16 Software Experiment:4 Hardware Experiment:4 Project based on software platform:16,2023/5/24,3,C
2、ourse Description,2023/5/24,4,Grading,2023/5/24,5,Textbook,Second Edition,Alan V.Oppenheim,Alan S.Willsky(M.I.T),S.Hamid Nawab(B.U.),Signals and Systems,1 Simon Haykin，Barry Van Veen，Signals&Systems,Second Edition,Publishing House of Electronics Industry，20032 郑君里，应启珩，杨为理，信号与系统(第二版)，北京：高等教育出版社，2000年
3、 3 闵大镒，朱学勇，信号与系统分析，电子科技大学出版社，2000年 4 吕幼新，张明友，信号与系统，电子工业出版社，2003年,References,Exploration in Signals and Systems Using MATLAB John R.Buck,Michael M.Daniel,Signals and Systems,2023/5/24,6,2023/5/24,7,Foreword,What shall we study?,-The concepts、theory and techniques of signals and systems analysis,Why s
4、hall we study?,-Extraordinarily wide variety of applications,How shall we study?,-Understand with mathematical concept,physic concept and engineering concept,-Do exercises and activities,2023/5/24,8,ForewordWhat is Signal?,A function of one or more independent variables that contain information abou
5、t the behavior or nature of some phenomenon.We encounter many types of signals in various applicationsElectrical signals:voltage,current,magnetic and electric fields,Mechanical signals:velocity,force,displacement,Acoustic signals:sound,vibration,Other signals:pressure,temperature,2023/5/24,9,Respond
6、 to particular signals by producing other signals or some desired behaviors.e.g.,Filters，Parameter estimation.,ForewordWhat is System?,2023/5/24,10,ForewordHow they work?,2023/5/24,11,Foreword,Filter the noise,2023/5/24,12,Foreword,Original information,After lowpass filtering,After highpass filterin
7、g,Image Processing,2023/5/24,13,Foreword,Example.Communication,Radio signal,intermediate,inputsignal,Received signal,outputsignal,Text,Audio,Video,Electric,Electromagnetic,Text,Audio,Video,With Information,2023/5/24,14,Outline,1 Signals and Systems2 Linear Time-Invariant Systems3 Fourier Series Repr
8、esentation of Periodic Signals 3.03.5 3.6 3.7 3.8 3.9 3.10 3.114 The Continuous-Time Fourier Transform5 The Discrete-Time Fourier Transform 5.05.5 5.6 5.8 6 Time and Frequency Characterization of Signals and Systems 6.1 6.4 6.56.77 Sampling 7.1 7.3 8 Communication System 8.1 8.2 9 The Laplace Transf
9、orm 10 The Z-Transform,Note:Contents-Keypoints.Contents-Read by yourselves.,2023/5/24,15,Signals and Systems,Continuous-time and discrete-time signalsTransformations of the Independent VariableExponential and Sinusoidal signalThe Unit Impulse and Unit Step FunctionsContinuous-time and Discrete-time
10、SystemBasic System Properties,2023/5/24,16,1.1 Continuous-time and discrete-time signals,A.Examples,(1)A simple RC circuit,Source voltage Vs and Capacitor voltage Vc,(2)An automobile,Force f from engineRetarding frictional force VVelocity V(t),2023/5/24,17,1.1 Continuous-time and discrete-time signa
11、ls,(4)A image signal,(3)A speech signal-should we chase,2023/5/24,18,1.1 Continuous-time and discrete-time signals,B.Types of Signals,(1)Continuous-time Signal,(2)Discrete-time Signal,-the independent variable is continuous,-the independent variable is discrete,n is integer number,Continuous-time di
12、screte-time signals,1.1 Continuous-time and discrete-time signals,2023/5/24,20,1.1 Continuous-time and discrete-time signals,C.Representation of Signal,(2)Graphical Representation,(1)Function Representation,Example:x(t)=cos0t x(t)=ej 0t,n=0,2023/5/24,21,1 Signals and Systems 1.1.2 Signal Energy and
13、Power,A.Energy(Continuous-time),Instantaneous power:,Example:,Let R=1 and,so,2023/5/24,22,1.1.2 Signal Energy and Power A.Energy(Continuous-time),Energy over t1 t t2:,Total Energy of signal:,Time_Average Power:,2023/5/24,23,1.1.2 Signal Energy and Power B.Energy(Discrete-time),Instantaneous power:,T
14、otal Energy:,Time_Average Power:,Energy over n1 n n2:,2023/5/24,24,1.1.2 Signal Energy and Power C.Finite Energy and Finite Power Signal,(Finite Total)Energy Signal:,(Finite Average)Power Signal:,Infinite Total Energy,Infinite Average Power Signal:,HW1-1,Read textbook P71:MATHEMATICAL REVIEWHW1_1：P5
15、7-1.2 1.3(b)(c)(e)(f),2023/5/24,25,2023/5/24,26,1 Signals and Systems 1.2 Transformations of the Independent Variable,Time shiftTime reversalTime-scaling,2023/5/24,27,1.2 Transformations of the Independent Variable 1.2.1 Examples of Transformations,Examples,t00；Advance,Identical shape,A.Time Shift,n
16、00;Delay,A.Time Shift,2023/5/24,28,t00；Advance,t00；Delay,are identical in shape.,2023/5/24,29,1.2 Transformations of the Independent Variable B.Time Reversal,x(-t)or x-n:Reflection of x(t)or xn,Examples,x(-t)is a reflection of x(t)about t=0X-n is a reflection of xn about n=0,B.Time Reversal,2023/5/2
17、4,30,2023/5/24,31,1.2 Transformations of the Independent Variable C.Time-Scaling,Continues-Time Signals X(t),1.2 Transformations of the Independent Variable C.Time-Scaling,Discrete Time Signals xn,2023/5/24,32,Decimation,Solution 1：,Solution 2：,1/2 3/2,1/6 1/2,0 1/3,1/6 1/2,x(3t-1/2)?,2023/5/24,34,1
18、.2 Transformations of the Independent Variable Example 1.1 f(1-3t)is wanted.,shift,reversal,Scaling,reversal,reversal,shift,shift,Scaling,Scaling,Always to“t”,2023/5/24,35,1.2 Transformations of the Independent Variable1.2.2 Periodic Signals,For Continues time period signal:There is a positive value
19、 of T which:x(t)=x(t+T),for all t x(t)is periodic with period T.,The smallest T Fundamental Period T0,For Discrete-time period signal:There is a positive value of N which xn=xn+N for all n,The smallest N Fundamental Period N0,Note:T,T0 positive real N,N0 positive integer,If a signal is not periodic,
20、it is called aperiodic signal.,2023/5/24,36,1.2.2 Periodic SignalsExamples of periodic signal,T0=T,N0=3,CT:,DT:,Note:x(t)=C is a periodic signal,but its fundamental period is undefined.,Examples:,1,It is periodic signal.Its period is T0=16/3.,2,It is not periodic.,3,x(t)is periodic.Its period is,The
21、 smallest multiples of T1 and T2 in common,4,It is aperiodic.,There is no the smallest multiples of T1 and T2 in common,5,x(t)is aperiodic.,6,It is periodic with period N0=16.,CostCos2tcost+cos2t,2023/5/24,39,1.2 Transformations of the Independent Variable1.2.3 Even and Odd Signals,Even signal:x(-t)
22、=x(t)or x-n=xn Odd signal:x(-t)=-x(t)or x-n=-xn,2023/5/24,40,1.2 Transformations of the Independent Variable1.2.3 Even and Odd Signals,Even-Odd Decomposition,Any signal can be expressed as a sum of Even and Odd signals.,x(t)=xeven(t)+xodd(t),xn=xevenn+xoddn,Example of the even-odd decomposition,Exam
23、ple of the even-odd decomposition,2023/5/24,43,HW1-2P57:1.9,1.10,1.21(a)(b)(c)(d),1.22(a)(b)(c)(g),1.23,1.24,2023/5/24,44,1 Signals and Systems 1.3 Exponential and Sinusoidal signal,1.3.1 Continuous-time Complex Exponential and Sinusoidal Signals,The continuous-time complex exponential signal has th
24、e General form as,where C and s are,in general,complex number.,2023/5/24,45,A.Real Exponential Signals(C,a are real value),a0,a0,1 Signal and System 1.3 Exponential and Sinusoidal signal,1.3.1 Continuous-time Complex Exponential and Sinusoidal Signals,growing,decaying,B.Periodic Complex Exponential(
25、Purely Imaginary Exponential Singnal)and Sinusoidal Signals,(1),Periodic,Energy and Power,(2)Sinusoidal Signals A-magnitude f-frequency(Hz)0=2 f-angular frequency(Rad/s)-initial phase(Rad),2023/5/24,47,1.3.1 Continuous-time Complex Exponential and Sinusoidal Signals,1 2 3,2,1,3,y=cos2t,y=cos5t,y=cos
26、10t,2023/5/24,49,1.3.1 Continuous-time Complex Exponential and Sinusoidal Signals Eulers Relation:,We also have,2023/5/24,50,(3)Harmonically related set of complex exponentials,Basic signal,Common Period,Fundamental Frequency,Kth harmonic:,1.3.1 Continuous-time Complex Exponential and Sinusoidal Sig
27、nals,2023/5/24,51,1.3.1 Continuous-time Complex Exponential and Sinusoidal Signals C.General Complex Exponential Signals,So,2023/5/24,52,Signal waves of eat cos(0t+),a 0,a 0,1.3.1 Continuous-time Complex Exponential and Sinusoidal Signals C.General Complex Exponential Signals,2023/5/24,53,1.3 Expone
28、ntial and Sinusoidal signal1.3.2 Discrete-time Complex Exponential and Sinusoidal Signals,Real Exponential Signals,Real,Complex Exponential and Sinusoidal Signals,Purely imaginary,Sampling,2023/5/24,54,1.3.2 Discrete-time Complex Exponential and Sinusoidal Signals,A.Real Exponential Signal,xn=C n(a)
29、1(b)01(c)-10(d)-1,2023/5/24,55,B.Sinusoidal Signals,1.3.2 Discrete-time Complex Exponential and Sinusoidal Signals,Eulers Relation:,2023/5/24,56,1.3.2 Discrete-time Complex Exponential and Sinusoidal Signals,C.General Complex Exponential Signals,in which C=|C|ej,=|ej0(polar form)then,xn=|C|ncos(0n+)
30、+j|C|nsin(0n+),2023/5/24,57,1.3.2 Discrete-time Complex Exponential and Sinusoidal Signals,Real or Imaginary parts of Signal,|a|1,|a|1,2023/5/24,58,1.3.3 Periodicity Properties of Discrete-time Complex Exponentials,Two properties of continuous-time signal ej0t:(1)ej0t is periodic for any value of 0，
31、T=2/0.(2)the larger the magnitude of 0,the higher is the rate of oscillation in the signal.,How about the Discrete-time signal ej0n?Periodic?,By definition:ej0n=e j0(n+N)thus e j0N=1,0N=2m,m is a integer So N=m(2/0)and,N must be a positive integer.,Conclusion:the condition of periodicity for ej0n is
32、 2/0 is rational.,D.Discrete-time Pure imaginary Signals,Examples:,Periodic?If Yes,determine its fundamental period:,(1),T=31/4,(2),It is not period.,(3),N1=3,N2=8N=N1N2=24,The smallest multiple of N1 and N2 in common,N=31,2023/5/24,60,From these figures,we can conclude:Signals oscillate rapidly whe
33、n 0=,3,(high-frequency);Signals oscillate slowly when 0=0,2,4,(low-frequency),on the most occasions we will use the interval,(3)Harmonically related complex exponentials,Note:,Comparison of the signals e j0t and e j0n,see P28 Table 1.1,So,Only N distinct periodic exponentials in the set,For,2023/5/2
34、4,63,HW1-3:P61-1.26,*1.25(d)(e)(f)P57-1.6,1.7(a)(b),1.22(e)(f),2023/5/24,64,1.4 The Unit Impulse and Unit Step Functions,1.4.1 The Discrete-time Unit Impulse and Unit Step Sequences,(3)Relation Between Unit Sample and Unit Step,First Difference,Accumulator,-,2023/5/24,65,1.4.1 The Discrete-time Unit
35、 Impulse and Unit Step Sequences,Running Sum:,or,2023/5/24,66,1.4.1 The Discrete-time Unit Impulse and Unit Step Sequences(3)Sampling Property of Unit Sample,2023/5/24,67,1.4.2 The Continuous-time Unit Step and Unit Impulse Functions,(1)Definition,(A)Unit Step Function,The Even and Odd part of:,2023
36、/5/24,68,1.4.2 The Continuous-time Unit Step and Unit Impulse Functions,It can be represented as:,69,Signal representation using step functions u(t):,2023/5/24,70,The Continuous-time Unit Step and Unit Impulse Functions(B)Unit Impulse Function,Considering the approximation as Following figure:,has n
37、o duration but unit area.,Dirac:,71,(2)Relation Between Unit Impulse and Unit Step,(First derivative),(Running Integral),Example,2023/5/24,73,1.4.2 The Continuous-time Unit Step and Unit Impulse Functions(3)Sampling Property of(t),Another form(Sifting property of(t):,In General:,2023/5/24,74,1.4.2 T
38、he Continuous-time Unit Step and Unit Impulse Functions,0,Examples:,2023/5/24,75,1.4.2 The Continuous-time Unit Step and Unit Impulse Functions(3)Properties of(t),Proof:,So,Obviously,1 Signals and Systems,1.5 Continuous-time and Discrete-time System,Definition:(1)Be constituted by some units;(2)Conn
39、ected with some rules;(3)Have system function.,(SISO system MIMO system),Example 1.8(p39),From Ohms law,and,We can get,1.5.1 Simple Example of systems,Liner constant-coefficient differential equation,Example,Newton rule(physically),Observation:Very different physical systems may be modeledmathematic
40、ally in very similar ways.,Simple Example of systems,Liner constant-coefficient differential equation,Example 1.10:Balance in a bank account from month to month:balance-yn net deposit-xn interest-1%so yn=yn-1+1%yn-1+xn or yn-1.01yn-1=xn,1.5.1 Simple Example of systems,1.5.2 Interconnections of Syste
41、m,(1)Series(cascade)interconnection,(2)Parallel interconnection,Series-Parallel interconnection,(3)Feed-back interconnection,Example of Feed-back interconnection,1 Signals and Systems,1.6 Basic System Properties,WHY?,A.Important practical/physical implications,B.They provide us with insight and stru
42、cture that we can exploit both to analyze and understand systems more deeply.,Memory Stability Invertibility Causality Time Invariance Linearity,1.6.1 Systems with and without Memory,Memoryless system:Its output for each value of the independent variable at a given time is dependent only on the inpu
43、t at the same time.Features:No capacitor,no conductor,no delayer.,Memoryless system:Memory system:,1.6.1 Invertibility and Inverse Systems,Definition:(1)If a system is invertible,then an inverse system exists.(2)An inverse system cascaded with the original system,yields an output equal to the input
44、to the original system.,Invertible system distinct inputs lead to distinct outputs.,87,noninvertible systems,Examples:,1.6.3 Causality,Definition:A system is causal If the output at any time depends only on values of the input at the present time and in the past.For causal system,if x(t)=0 for tt0,t
45、here must be y(t)=0 for tt0.(nonanticipative),Systems without memory,Causal systems:,Nocausal systems:,1.6.4 Stability,Definition:Small inputs lead to responses that don not diverge.Bounded Inputs lead to Bounded Output:if|x(t)|M,then|y(t)|N.(BIBO),(unstable system),Examples:,S1：,S2:,S3:,(stable sys
46、tem),(unstable system),a stable pendulum,an unstable inverted pendulum,91,If,Consider a continuous-time system,Time Invariance,Definition:Characteristics of the system are fixed over time.,92,Example:,2023/5/24,93,(Read P52),1.6.6 Linearity,Definition:The system possesses the important property of s
47、uperposition:,Additivity,Scaling,Represented in block-diagram:,1.6.6 Linearity,(linear)(nonlinear)(linear)(nonlinear)(nonlinear),incrementallylinear systems,Example:,Notes:LINEAR AND NONLINEAR SYSTEMS,Many systems are nonlinear.For example:many circuit elements(e.g.,diodes),dynamics of aircraft,econ
48、ometric models,However,in this course we focus exclusively on linear systems.,Why?,Linear models representation of accurate behavior of many systems;linearize models to examine“small signal”;analytically tractable.,1.6.6 Linearity,LINEAR TIME-INVARIANT(LTI)SYSTEMS,LINEAR TIME-INVARIANT(LTI)SYSTEMS,F
49、ocus of most of this course on LTI-Practical importance-The powerful analysis tools associated with LTI systemsA basic fact:If we know the response of an LTI system to some inputs,we actually know the response to many inputs,The general response y(t)of a linear system is the sum of a zero-input resp
50、onse and a zero-condition response,Example:a system when,We want to know y(t),t0,Solution:,Then,Which is the zero-input response of the system,yn-0.5yn-1=2xn,(LTI systems),Example,(Linear,Time-varying system),2023/5/24,103,Example,Consider a discrete-time system with input,and output,related by,is t
