信号与系统 Signal and System

1、2022-2-221Signals and Systems He Chun (SCIE,UESTC);Room:KB244Ahttp:/ Description n Lectures: 68 + 4 Week 1-17 Mon. （5,6） : Room A-412 Wed. （3,4） : Room A-412 Fri.（3,4;even） : Room A-302nExperiment: 4 + 4 +16 Software Experiment : 4 Hardware Experiment : 4 Project based on software platform :162022-2

2、-223Course Description 2022-2-224n Grading ACTIVITIESPERCENTAGESMiddle Exam20%Final Exam50%Exercises10%Experiments20%2022-2-225TextbookSecond EditionAlan V. Oppenheim , Alan S. Willsky (M.I.T), S. Hamid Nawab (B.U.)Signals and Systems1 Simon Haykin，Barry Van Veen，Signals & Systems,Second Edition

3、, Publishing House of Electronics Industry，20032 郑君里，应启珩，杨为理，信号与系统郑君里，应启珩，杨为理，信号与系统(第二版第二版)，北京：高等教育出版社，北京：高等教育出版社，2000年年 3 闵大镒，朱学勇，信号与系统分析，电子科技大学出版社，闵大镒，朱学勇，信号与系统分析，电子科技大学出版社，2000年年 4 吕幼新，张明友，信号与系统，电子工业出版社，吕幼新，张明友，信号与系统，电子工业出版社，2003年年 References Exploration in Signals and Systems Using MATLAB John R

4、.Buck , Michael M. DanielSignals and Systems2022-2-2262022-2-227ForewordWhat shall we study?- The concepts、theory and techniques of signals and systems analysisWhy shall we study?-Extraordinarily wide variety of applicationsHow shall we study? - Understand with mathematical concept, physic concept a

5、nd engineering concept - Do exercises and activities2022-2-228ForewordWhat is Signal?A function of one or more independent variables that contain information about the behavior or nature of some phenomenon. We encounter many types of signals in various applicationsqElectrical signals: voltage, curre

6、nt, magnetic and electric fields,qMechanical signals: velocity, force, displacement,qAcoustic signals: sound, vibration,qOther signals: pressure, temperature,2022-2-229n Respond to particular signals by producing other signals or some desired behaviors. e.g., Filters， Parameter estimation. ForewordW

7、hat is System?2022-2-2210ForewordHow they work?2022-2-2211ForewordFilter the noise2022-2-2212ForewordOriginal informationAfter lowpass filteringAfter highpass filteringImage Processing2022-2-2213ForewordExample. CommunicationExample. CommunicationRadio Radio signalsignalTransmitterTransmitterReceive

8、rReceiverintermediateintermediateinputinputsignalsignalReceived Received signalsignaloutputoutputsignalsignalText,Text,Audio,Audio,Video Video Electric,Electric,ElectromagneticElectromagnetic Text,Text,Audio,Audio,Video Video With Information2022-2-2214 Outline1 Signals and Systems2 Linear Time-Inva

9、riant Systems3 Fourier Series Representation 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 7.4-7.5 8 Communi

10、cation System 8.1 8.2 8.3-8.8 9 The Laplace Transform 10 The Z-Transform Note: Note: Contents Contents - Keypoints.- Keypoints. Contents - Contents - Read by yourselvesRead by yourselves. .2022-2-22151.Signals and Systems nContinuous-time and discrete-time signalsnTransformations of the Independent

11、VariablenExponential and Sinusoidal signalnThe Unit Impulse and Unit Step FunctionsnContinuous-time and Discrete-time SystemnBasic System Properties2022-2-22161.1 Continuous-time and discrete-time signalsA. Examples(1) A simple RC circuitSource voltage Vs and Capacitor voltage Vc(2) An automobile70(

12、 )(/ )v t m s010(sec)t1090Force f from engineRetarding frictional force VVelocity V(t)2022-2-22171.1 Continuous-time and discrete-time signals(4) A image signal(3) A speech signal- should we chase2022-2-22181.1 Continuous-time and discrete-time signalsnB. Types of Signals(1) Continuous-time Signal(2

13、) Discrete-time Signal- the independent variable is continuous- the independent variable is discreten is integer number RttAtx),sin()(0124 . 08 . 01 2 3n4 5 6 7111098nx0124 . 08 . 0123t)(txSamplingContinuous-time discrete-time signals1.1 Continuous-time and discrete-time signals2022-2-22201.1 Contin

14、uous-time and discrete-time signalsC. Representation of Signal(2) Graphical Representation(1) Function RepresentationExample: x(t) = cos 0t x(t) = ej 0tothersttG001)( .,0,1,2, 1.5,2,0,1,0,.xn n=02022-2-22211 Signals and Systems 1.1.2 Signal Energy and PowerA. Energy (Continuous-time)Instantaneous po

15、wer:)()(1)()()(22tiRtvRtitvtp222( )( )( )( )( )( )p tv ti tvtitxtExample :( )( )i tx t2*() | ()|() ()ptxtxt x t+R_)(tv)(ti2022-2-22221.1.2 Signal Energy and Power A. Energy (Continuous-time)Energy over t1 t t2:222111221( )| ( ) |( )ttttttEp t dtx tdtvt dtRTotal Energy of signal:221|( ) |( )Ex tdtvt

16、dtRTime_Average Power:21|( ) |2limTTTPv td tT R 2022-2-22231.1.2 Signal Energy and Power B. Energy (Discrete-time)Instantaneous power:Total Energy :Time_Average Power:2 | |p nx n212| |nnnEx n2| |nEx n21| |21limNNnNPx nN Energy over n1 n n2:2022-2-22241.1.2 Signal Energy and Power C. Finite Energy an

17、d Finite Power Signal (Finite Total) Energy Signal :,02limTEEPT (Finite Average) Power Signal :(0,)limTTTIf Pthen EP dt P Infinite Total Energy, Infinite Average Power Signal:,EP HW1-1nRead textbook P71: MATHEMATICAL REVIEWnHW1_1： P57-1.2 1.3(b)(c)(e)(f)2022-2-22252022-2-22261 Signals and Systems 1.

18、2 Transformations of the Independent Variablen Time shiftn Time reversaln Time- scaling2022-2-22271.2 Transformations of the Independent Variable 1.2.1 Examples of TransformationsExamplest00; DelayA. Time Shift2022-2-222800tttnnn 00)()( x tx t tx nx n nt00；Delay are identical in shape.0)( )(x tand x

19、 tt2022-2-22291.2 Transformations of the Independent Variable B. Time Reversalx(-t) or x-n : Reflection of x(t) or xnExamplesx(-t) is a reflection of x(t) about t = 0X-n is a reflection of xn about n= 0B. Time Reversal2022-2-2230-ttnn)()(- -x tx tx nx n2022-2-22311.2 Transformations of the Independe

20、nt Variable C. Time-Scaling Continues-Time Signals X(t)( )() ,0ta tx tx a ta1/2set a ()01,1,.For x atastreched acompressed：；2set a 1.2 Transformations of the Independent Variable C. Time-ScalingDiscrete Time Signals xn 2022-2-2232,x ann anN2:set a 1/2:set a DecimationSolution 1：Solution 2：0 11tx(t)0

21、1tx(t-1/2)1/2 3/20 1tx(3t-1/2)1/6 1/20 11tx(t)0 1/31tx(3t)0 1tx(3t-1/2)1/6 1/2x(3t-1/2)?2022-2-22341.2 Transformations of the Independent Variable Example 1.1 f(1-3t) is wanted.( )f t12121 0t122 1t0(1)f t )1 (tf 01 122t)31 (tf t0131 232)( tf t01121 2 t1 12012)1 (tf )3( tf12t031 32)31 (tf 012t3231shi

22、ftreversalScalingreversalreversalshiftshiftScalingScalingAlways to “t”2022-2-22351.2 Transformations of the Independent Variable1.2.2 Periodic SignalsFor Continues time period signal : There is a positive value of T which : x(t)=x(t+T) , for all t x(t) is periodic with period T .The smallest T Funda

23、mental Period T0For Discrete-time period signal: There is a positive value of N which xn=xn+N for all nThe smallest N Fundamental Period N0Note: T, T0 positive real N,N0 positive integer If a signal is not periodic, it is called aperiodic signal.2022-2-22361.2.2 Periodic SignalsExamples of periodic

24、signalT0=TN0=3CT:DT:Note: x(t)=C is a periodic signal, but its fundamental period is undefined.Examples:1 3( )sin8x tAtIt is periodic signal. Its period is T0=16/3.2 0, 00,cos)(ttttxIt is not periodic.3 tBtAtx41sin31cos)(8,621TTx(t) is periodic. Its period is024TThe smallest multiples of T1 and T2 i

25、n common2/T 4 tttx2coscos)(21, 2 TTIt is aperiodic.There is no the smallest multiples of T1 and T2 in common5 nnx4cosx(t) is aperiodic. 6 nnx83cosIt is periodic with period N0=16. -20-15-10-505101520-101-20-15-10-505101520-101-20-15-10-505101520-202CostCos2tcost+cos2t2022-2-22391.2 Transformations o

26、f the Independent Variable1.2.3 Even and Odd Signals Even signal: x(-t) = x(t) or x-n= xn Odd signal : x(-t)= -x(t) or x-n= -xn2022-2-22401.2 Transformations of the Independent Variable1.2.3 Even and Odd Signals)()(21)()(txtxtxtxEve)()(21)()(txtxtxtxOdo21nxnxnxnxEve21nxnxnxnxOdoEven-Odd Decompositio

27、n Any signal can be expressed as a sum of Even and Odd signals.x(t) = xeven(t) + xodd(t)xn = xevenn + xoddnExample of the even-odd decomposition Example of the even-odd decomposition 2022-2-2243HW1-2P57: 1.9, 1.10, 1.21(a)(b)(c)(d), 1.22(a)(b)(c)(g),1.23 , 1.242022-2-22441 Signals and Systems 1.3 Ex

28、ponential and Sinusoidal signal1.3.1 Continuous-time Complex Exponential and Sinusoidal SignalsThe continuous-time complex exponential signal has the General form as( ),stx tCewhere C and s are, in general, complex number.0,saj jCC e stj(+j)ttj(+t)x t = Ce=|C|e e=|C|e e2022-2-2245A. Real Exponential

29、 Signals (C, a are real value)a0a01 Signal and System 1.3 Exponential and Sinusoidal signal1.3.1 Continuous-time Complex Exponential and Sinusoidal Signalsgrowingdecaying( ),atx tCeB. Periodic Complex Exponential (Purely Imaginary Exponential Singnal) and Sinusoidal Signals0( ),1,stx tCewhentCsjthen

30、 (1)000020001,TTjtTEedtdtT00011TTPEPT 0( )jtx te Periodic,E Energy and Power00()jt Tjtee021,0, 1, 2,.jTjkeek 02,1, 2,.Tkk 002TFundamental Peroid 0- Fundamental Frequency(2) Sinusoidal Signals A - magnitude f - frequency (Hz) 0 = 2 f - angular frequency (Rad/s) - initial phase(Rad)2022-2-22471.3.1 Co

31、ntinuous-time Complex Exponential and Sinusoidal Signals0( )cos()cos(2)x tAtAft-20-15-10-505101520-101-20-15-10-505101520-101-20-15-10-505101520-101 1 2 0a 1(b) 0 1(c) -1 0(d) 1|a|12022-2-22581.3.3 Periodicity Properties of Discrete-time Complex Exponentials Two properties of continuous-time signal

32、ej 0t : (1) ej 0t is periodic for any value of 0， 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 ej 0n ? Periodic? By definition: ej 0n = e j 0(n+N) thus e j 0N = 1, 0N = 2 m, m is a integer So N = m (2 / 0) and N m

33、ust be a positive integer.Conclusion: the condition of periodicity for ej 0n is 2 / 0 is rational. D. Discrete-time Pure imaginary Signals Examples:Periodic? If Yes, determine its fundamental period:(1) nnxttx318cos318cos)(T=31/4(2) 6cosnnxIt is not period. (3) njnjeenx)43()32(N1 =3, N2 =8N=N1N2 =24

34、The smallest multiple of N1 and N2 in commonN=312022-2-2260From these figures, we can conclude:nSignals oscillate rapidly when 0= , 3 ,(high-frequency); nSignals oscillate slowly when 0=0,2 , 4 , (low-frequency)0002on the most occasions we will use the interval(3)Harmonically related complex exponen

35、tialsNote: (2/) ,0, 1, 2,jkN nknek nnkNkNnNjNNnjNnjenenenn/)1(21/42/210, 1Comparison of the signals e j 0t and e j 0n ,see P28 Table 1.1So, Only N distinct periodic exponentials in the setFor2022-2-2263 HW1-3: P61 - 1.26,*1.25(d)(e)(f) P57 - 1.6, 1.7(a)(b), 1.22(e)(f)2022-2-22641.4 The Unit Impulse

36、and Unit Step Functions1.4.1 The Discrete-time Unit Impulse and Unit Step Sequences(1)Unit Sample(Impulse) n1n=00 n 0n01 n(2)Unit Step Function u n 1n 00 n0n2101 21 nu3(3) Relation Between Unit Sample and Unit Step 1 nmnu nu nu nm First Difference Accumulator-2022-2-22651.4.1 The Discrete-time Unit

37、Impulse and Unit Step Sequences Running Sum: nmu nm0kknnuor2022-2-22661.4.1 The Discrete-time Unit Impulse and Unit Step Sequences (3) Sampling Property of Unit Sample000 0 mx nnxnx nnnx nnnx nx mnm2022-2-22671.4.2 The Continuous-time Unit Step and Unit Impulse Functions(1) Definition0, 10, 0)(tttu(

38、A) Unit Step FunctionThe Even and Odd part of :( )u t1 (0)sign( )1 (0)tttsign(t)1t0-112(dc)1sign( )2tand2022-2-22681.4.2 The Continuous-time Unit Step and Unit Impulse FunctionsIt can be represented as:sign( )2 ( ) 1tu t1 (0)sign( )1 (0)tttsign(t)1t0-169- 0 t tP2- 0 ttu 0 ttu0 1 t 1 ttu1t0 1 t 0 t t

39、u1-1 0 1 t1 tf-1 0 t1 11tut0 1 t 2 ttu20 1 2 t 1 11tutSignal representation using step functions u(t):2022-2-22701.4.2 The Continuous-time Unit Step and Unit Impulse Functions(B) Unit Impulse Function 1t( )ut001( ) tt( )( )dtutdt Considering the approximation as Following figure : t0 t 0 1dtt 100dtt

40、0 ( )( )u tu t0( )( )limtt has no duration but unit area.)(t Dirac:71( )( )du ttdt(2) Relation Between Unit Impulse and Unit Step 0 tt dttu0积分区间积分区间 (First derivative) 0 积分区间积分区间 tu td (Running Integral) Example 1 -1 1 x(t) t(-1) 2 -2 1/2 x1(t) t(-2)(1)?)( 1tx?)(2dxt 1 -1 1 x(t) t?)( tx 2 -2 1 x1(t)

41、 t 2 -2 2 x2(t) t 2 2 x2(t) t12022-2-22731.4.2 The Continuous-time Unit Step and Unit Impulse Functions (3) Sampling Property of (t)( ) ( )(0) ( )x ttxt( ) ( )(0)x tt dtx Another form(Sifting property of (t):00( ) ()( )x tttdtx t000( ) ()( ) ()x tttx tttIn General:2022-2-22741.4.2 The Continuous-tim

42、e Unit Step and Unit Impulse Functions 323(1) (1)(4)?tttdt cos( ) ( )?tt535(21) ( )?ttt dt)(tsin() ( )?tt 0Examples:212022-2-22751.4.2 The Continuous-time Unit Step and Unit Impulse Functions (3) Properties of (t)Proof: So, 1()( )attaObviously 1()( ) ()setatteven )(1)(taat11( / ) ( )(0),0( ) ( )11(

43、/ ) ( )(0),0 xadxaaax tat dtxadxaaa /ta 1 Signals and Systems1.5 Continuous-time and Discrete-time SystemDefinition: (1) Be constituted by some units; (2) Connected with some rules; (3) Have system function. Lx ny n ( )( )Lx ty t (SISO system MIMO system)( ) ( )y tL x t y nL x tExample 1.8 (p39)(1)(

44、1)(tvRCtvRCdttdvsccRC Circuit (system)vs(t)vc(t)Rtvtvtics)()()(From Ohms lawdttdvctic)()(andWe can get1.5.1 Simple Example of systemsLiner constant-coefficient differential equation KVL rule (Kirhhoffs Voltage Law)(physically)( )( )( )dv tf tv tmdtExampleNewton rule (physically)Observation: Very dif

45、ferent physical systems may be modeledmathematically in very similar ways.1.5.1 Simple Example of systems( )1( )( )dv tv tf tdtmmLiner 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-

46、1+xn or yn-1.01yn-1=xnBalance in bankBalance in bank(system)(system)xnyn1.5.1 Simple Example of systems1.5.2 Interconnections of System(1) Series(cascade) interconnection(2) Parallel interconnection Series-Parallel interconnection(3) Feed-back interconnectionExample of Feed-back interconnection 1 Si

47、gnals and Systems1.6 Basic System PropertiesWHY ?A. Important practical/physical implicationsB. They provide us with insight and structure that we can exploit both to analyze and understand systems more deeply.n Memoryn Stabilityn Invertibility n Causalityn Time Invariancen Linearity1.6.1 Systems with and without MemoryMemoryless system: Its output for each value of the independent variable at a given time is dependent only on the input at the same time.Featur