信号与系统 吕幼新张明友

1、 1 Signal and System1.1 Continuous-time and discrete-time signals1.1.1 Examples and Mathematical RepresentationA. Examples(1) A simple RC circuitSource voltage Vs and Capacitor voltage Vc1. Signals and Systems 1 Signal and System(2) An automobileForce f from engineRetarding frictional force VVelocit

2、y V 1 Signal and System(3) A Speech Signal 1 Signal and System(4) A Picture 1 Signal and System(5) Vertical Wind Profile 1 Signal and SystemB. Types of Signals(1) Continuous-time Signal 1 Signal and System(2) Discrete-time Signal 1 Signal and SystemC. Representation(1) Function Representation Exampl

3、e: x(t) = cos0t x(t) = ej 0t(2) Graphical Representation Example: ( See page before ) 1 Signal and System1.1.2 Signal Energy and PowerA. Energy (Continuous-time)Instantaneous power:Let R=1, so p(t)=i2(t)=v2(t)=x2(t) 1 Signal and SystemEnergy over t1 t t2:Total Energy:Average Power: 1 Signal and Syst

4、emB. Energy (Discrete-time)Instantaneous power:Energy over n1 n n2:Total Energy :Average Power: 1 Signal and SystemC. Finite Energy and Finite Power SignalFinite Energy Signal :Finite Power Signal :( P 0 )( E ) 1 Signal and System1.2 Transformations of the Independent Variable1.2.1 Examples of Trans

5、formationsA. Time ShiftRight shift : x(t-t0) xn-n0 (Delay)Left shift : x(t+t0) xn+n0 (Advance) 1 Signal and SystemExamples 1 Signal and SystemB. Time Reversalx(-t) or x-n : Reflection of x(t) or xn 1 Signal and SystemC. Time Scalingx(at) ( a0 ) Stretch if a0 1 Signal and System1.2.2 Periodic Signals

6、Definition: There is a posotive value of T which : x(t)=x(t+T) , for all t x(t) is periodic with period T . T Fundamental Period For Discrete-time period signal: xn=xn+N for all n N Fundamental Period 1 Signal and SystemExamples of periodic signal 1 Signal and System1.2.3 Even and Odd Signals Even s

7、ignal: x(-t) = x(t) or x-n= xn Odd signal : x(-t)= -x(t) or x-n= -xnEven-Odd Decomposition:or: 1 Signal and SystemExamples 1 Signal and System1.3 Exponential and Sinusoidal signal1.3.1 Continuous-time Complex Exponential and Sinusoidal SignalsA. Real Exponential Signals x(t)= C eat ( C, a are real v

8、alue) 1 Signal and SystemB. Periodic Complex Exponential and Sinusoidal Signals (1) x(t) = e j0t (2) x(t) = Acos(0t+) (3) x(t) = e jk0t All x(t) satisfy for x(t) = x(t+T) , and T=2/ 0 So x(t) is periodic. 1 Signal and SystemEulers Relation: e j0t = cos0t + sin 0t and cos0t = (e j0t + e -j0t ) / 2 si

9、n0t = (e j0t - e -j0t ) / 2 We also have 1 Signal and SystemC. General Complex Exponential Signals x(t) = C e jat , in which C = |C| ej , a = r + j 0 So x(t) = |C| ej eat ej0t = |C| eat ej(0t+ ) = |C| eat cos(0t+ ) + j |C| eat sin(0t+ ) 1 Signal and SystemSignal waves 1 Signal and System1.3.2 Discre

10、te-time Complex Exponential and Sinusoidal SignalsComplex Exponential Signal (sequence) : xn = C n or xn = C en 1 Signal and SystemA. Real Exponential Signal Real Exponential Signal xn = C n (a) 1 (b) 01 (c) -10 (d) -1 1 Signal and SystemB. Sinusoidal Signals Complex exponential:xn = e j0n = cos 0n

11、+ jsin0n Sinusoidal signal: xn = cos(0n+) 1 Signal and SystemC. General Complex Exponential Signals Complex Exponential Signal: xn = C n in which C = |C| ej , = |ej0 (polar form)then xn=|C| |ncos(0n + )+j|C| |nsin(0n + ) 1 Signal and SystemReal or Imaginary of Signal 1 Signal and System1.3.3 Periodi

12、city Properties of Discrete-time Complex ExponentialsContinuous-time: e j0t , T=2/0Discrete-time: e j0n , N=?Calculate period: By definition: e j0n = e j0(n+N) thus e j0N = 1 or 0N = 2 m So N = 2m/0 Condition of periodicity: 2/0 is rational 1 Signal and SystemPeriodicity Properties 1 Signal and Syst

13、em1.4 The Unit Impulse and Unit Step Functions1.4.1 The Discrete-time Unit Impulse and Unit Step Sequences(1) Unit Sample(Impulse): 1 Signal and SystemUnit Step Function:(2) Relation Between Unit Sample and Unit Stepor 1 Signal and System(3) Sampling Property of Unit Sample 1 Signal and SystemIllust

14、ration of Sampling 1 Signal and System1.4.2 The Continuous-time Unit Step and Unit Impulse Functions(1) Unit Step Function: 1 Signal and SystemUnit Impulse Function: 1 Signal and System(2) Relation Between Unit Impulse and Unit Step 1 Signal and System(3) Sampling Property of (t) 1 Signal and System

15、1.5 Continuous-time and Discrete-time SystemDefinition: (1) Interconnection of Component,device, subsystem. (Broadest sense) (2) A process in which signals can be transformed. (Narrow sense)Representation of System: (1) Relation by the notation 1 Signal and System(2) Pictorial Representation Continu

16、ous-time system x(t)y(t)Discrete-time system xnyn 1 Signal and System1.5.1 Simple Example of systemsExample 1.8: RC Circuit in Figure 1.1 : Vc(t) Vs(t)RC Circuit (system)vs(t)vc(t) 1 Signal and SystemExample 1.10: Balance in a bank account from month to month: balance yn net deposit xn interest 1% s

17、o yn=yn-1+1%yn-1+xn or yn-1.01yn-1=xnBalance in bank (system)xnyn 1 Signal and System1.5.2 Interconnections of System(1) Series(cascade) interconnection 1 Signal and System(2) Parallel interconnection Series-Parallel interconnection 1 Signal and System(3) Feed-back interconnection 1 Signal and Syste

18、mExample of Feed-back interconnection 1 Signal and System1.6 Basic System Properties1.6.1 Systems with and without MemoryMemoryless system: Its output is dependent only on the input at the same time.Features: No capacitor, no conductor, no delayer.Examples of memoryless system: y(t) = C x(t) or yn =

19、 C xnExamples of memory system:or yn-0.5yn-1=2xn 1 Signal and System1.6.1 Invertibility and Inverse SystemsDefinition:(1) If system is invertibility,then an inverse system exists.(2) An inverse system cascaded with the original system,yields an output equal to the input. 1 Signal and System 1 Signal

20、 and System1.6.3 CausalityDefinition: 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, there must be y(t)=0 for tt0. ( nonanticipative ) Memoryless systems are causal. 1 Signal and Systemx(t)y(

21、t)t1t2 1 Signal and System1.6.4 StabilityDefinition: Small inputs lead to responses that don not diverge. Finite input lead to finite output: if |x(t)|M, then |y(t)|N . Examples: Stable pendulum Motion of automobile 1 Signal and System 1 Signal and System1.6.5 Time InvarianceDefinition: Characterist

22、ics of the system are fixed over time. Time invariant system: If x(t) y(t), then x(t-t0) y(t-t0) .Example 1.14 1.15 1.16 1 Signal and Systemx(t)y(t)x(t-t0)y(t-t0) 1 Signal and System1.6.6 LinearityDefinition: The system possesses the important property of superposition: (1) Additivity property: The

23、response to x1(t)+x2(t) is y1(t)+y2(t) . (2) Scaling or homogeneity property: The response to ax1(t) is ay1(t) . (where a is any complex constant, a0 .) 1 Signal and SystemLx1(t)x2(t)y1(t)y2(t)a x1(t)x1(t) +x2(t)ax1(t) +bx2(t)a y1(t)y1(t) +y2(t)ay1(t) +by2(t)Represented in block-diagram:Example 1.17

24、 1.18 1 Signal and SystemLTI SystemLTIx(t)y(t)x(t-t0)ax(t) +bx(t-t0)y(t-t0)ay(t) +by(t-t0)Linear and Time-invariant systemProblems: 2 Linear Time-Invariant Systems2.1 Discrete-time LTI system: The convolution sum2.1.1 The Representation of Discrete-time Signals in Terms of Impulses2. Linear Time-Inv

25、ariant SystemsIf xn=un, then 2 Linear Time-Invariant Systems 2 Linear Time-Invariant Systems2.1.2 The Discrete-time Unit Impulse Response and the Convolution Sum Representation of LTI Systems(1) Unit Impulse(Sample) Response LTIxn=nyn=hn Unit Impulse Response: hn 2 Linear Time-Invariant Systems(2) C

26、onvolution Sum of LTI System LTIxnyn=?Solution:Question: n hnn-k hn-kxkn-k xk hn-k 2 Linear Time-Invariant Systems 2 Linear Time-Invariant Systems 2 Linear Time-Invariant Systems( Convolution Sum )Soor yn = xn * hn(3) Calculation of Convolution SumTime Inversal: hk h-kTime Shift: h-k hn-kMultiplicat

27、ion: xkhn-kSumming: 2 Linear Time-Invariant Systems2.2 Continuous-time LTI system: The convolution integral2.2.1 The Representation of Continuous-time Signals in Terms of ImpulsesDefine We have the expression: Therefore: 2 Linear Time-Invariant Systems 2 Linear Time-Invariant Systemsor 2 Linear Time

28、-Invariant Systems2.2.2 The Continuous-time Unit impulse Response and the convolution Integral Representation of LTI Systems(1) Unit Impulse Response LTIx(t)=(t)y(t)=h(t)(2) The Convolution of LTI System LTIx(t)y(t)=? 2 Linear Time-Invariant SystemsA. LTI(t)h(t)x(t)y(t)=?Because of So,we can get ( C

29、onvolution Integral ) or y(t) = x(t) * h(t) 2 Linear Time-Invariant SystemsB. or y(t) = x(t) * h(t) LTI(t)h(t)(t) h(t)( Convolution Integral ) 2 Linear Time-Invariant Systems 2 Linear Time-Invariant Systems(3) Computation of Convolution Integral Time Inversal: h() h(- )Time Shift: h(-) h(t- )Multipl

30、ication: x()h(t- )Integrating: 2 Linear Time-Invariant Systems2.3 Properties of Linear Time Invariant SystemConvolution formula:h(t)x(t)y(t)=x(t)*h(t)hnxnyn=xn*hn 2 Linear Time-Invariant Systems2.3.1 The Commutative PropertyDiscrete time: xn*hn=hn*xnContinuous time: x(t)*h(t)=h(t)*x(t)h(t)x(t)y(t)=x

31、(t)*h(t)x(t)h(t)y(t)=h(t)*x(t) 2 Linear Time-Invariant Systems2.3.2 The Distributive PropertyDiscrete time: xn*h1n+h2n=xn*h1n+xn*h2nContinuous time: x(t)*h1(t)+h2(t)=x(t)*h1(t)+x(t)*h2(t)h1(t)+h2(t)x(t)y(t)=x(t)*h1(t)+h2(t)h1(t)x(t)y(t)=x(t)*h1(t)+x(t)*h2(t)h2(t) 2 Linear Time-Invariant Systems2.3.3

32、 The Associative PropertyDiscrete time: xn*h1n*h2n=xn*h1n*h2nContinuous time: x(t)*h1(t)*h2(t)=x(t)*h1(t)*h2(t)h1(t)*h2(t)x(t)y(t)=x(t)*h1(t)*h2(t)h1(t)x(t)y(t)=x(t)*h1(t)*h2(t)h2(t) 2 Linear Time-Invariant Systems2.3.4 LTI system with and without MemoryMemoryless system: Discrete time: yn=kxn, hn=k

33、n Continuous time: y(t)=kx(t), h(t)=k (t)k (t) x(t)y(t)=kx(t)=x(t)*k(t)k n xnyn=kxn=xn*knImply that: x(t)* (t)=x(t) and xn* n=xn 2 Linear Time-Invariant Systems2.3.5 Invertibility of LTI systemOriginal system: h(t)Reverse system: h1(t)(t) x(t)x(t)*(t)=x(t)So, for the invertible system: h(t)*h1(t)=(t

34、) or hn*h1n=nh(t) x(t)x(t)h1(t) 2 Linear Time-Invariant Systems2.3.6 Causality for LTI systemDiscrete time system satisfy the condition: hn=0 for n0Continuous time system satisfy the condition: h(t)=0 for t0 2 Linear Time-Invariant Systems2.3.7 Stability for LTI system Definition of stability: Every

35、 bounded input produces a bounded output. Discrete time system:If |xn|B, the condition for |yn|A is 2 Linear Time-Invariant SystemsContinuous time system:If |x(t)|B, the condition for |y(t)| M . Then x(t) is uniquely determined by its samples x(nT),n=0,1,2, if s2 M, where s=2/T . 2M is called Nyquis

36、t Rate. ( Minimum distortionless sampling frequency ) 7 Sampling(3) RecoverySystem for sampling and reconstruction:7 Sampling7 Sampling7.1.2 Sampling with a Zero-order Hold(1) Sampling system construction:7 Sampling7 Sampling(2) Signal Recovery7 Sampling7.3 The Effect of Undersampling: Aliasings=60s

37、=307 Samplings=1.50s=1.207 Sampling 8 Communication systems8.1 Complex Exponential and Sinusoidal Amplitude ModulationModulating system model:8. Communication Systemsx(t)c(t)y(t)=x(t)c(t)x(t) modulating signalc(t) Carrier signal 8 Communication systems8.1.1 Amplitude Modulation with Complex Exponent

38、ial CarrierExponential carrier:For convenience, let c=0, soOutput signal(modulated signal):(1) Modulation Theory 8 Communication systems 8 Communication systems(1) Implementation 8 Communication systems8.1.2 Amplitude Modulation with Sinusoidal signalFor convenience, choose c=0, so 8 Communication s

39、ystems 8 Communication systems 8 Communication systems8.2 Demodulation for Sinusoidal AM8.2.1 Synchronous demodulation(1) Demodulation process 8 Communication systemsIn frequency domain:In time domain:Expected signal: 8 Communication systems 8 Communication systems(2) Synchronous problem 8 Communica

40、tion systemsTime domain:The output of lowpass filter:Ideal output: x(t)When , it is referred to as synchronous demodulation. 8 Communication systems8.2.2 Asynchronous demodulationAmplitude-modulated signal: 8 Communication systems 8 Communication systems9 The Laplace Transform 9. The Laplace Transfo

41、rm 9.1 The Laplace Transform (1) Definition(2) Region of Convergence ( ROC )ROC: Range of for X(s) to convergeRepresentation: A. Inequality B. Region in S-plane9 The Laplace TransformExample for ROCReReS-planeS-planeImIm-a-a9 The Laplace Transform (3) Relationship between Fourier and Laplace transfo

42、rm 9 The Laplace Transform 9.2 The Region of Convergence for Laplace TransformProperty1: The ROC of X(s) consists of strips parallel to j-axis in the s-plane.Property2: For rational Laplace transform, the ROC does not contain any poles.Property3: If x(t) is of finite duration and is absolutely integ

43、rable, then the ROC is the entire s- plane9 The Laplace TransformProperty4: If x(t) is right sided, and if the line Res=0 is in the ROC, then all values of s for which Res0 will also in the ROC. 9 The Laplace TransformProperty5: If x(t) is left sided, and if the line Res=0 is in the ROC, then all va

44、lues of s for which Res0 will also in the ROC. x(t)T2te-0te-1t9 The Laplace TransformProperty6: If x(t) is two sided, and if the line Res=0 is in the ROC, then the ROC will consist of a strip in the s-plane that includes the line Res=0 . 9 The Laplace TransformS-planeReReReImImImRLLR9 The Laplace TransformProperty7: If the Laplace transform X(s) of x(t)