课件：信号与系统Chapter-2

1、 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-Invariant SystemsIf xn=un, then Linear combinations of delayed impulse (Sifting property) 2 Linear Time-Invariant Systems2.1.2 The Dis

2、crete-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 SystemsSifting property represents xn as a superposition of scaled versions of shifted unit impulse n-k (2) Convoluti

3、on Sum of LTI System LTIxnyn= ?Solution:Question: n hnn-k hn-kxkn-k xkhn-k 2 Linear Time-Invariant SystemsTime-invariantAdditivityScaling 2 Linear Time-Invariant SystemsHere, hkn denotes the response of the linear system to the shifted unit impulse n-k. LTIhkn=hn-k 2 Linear Time-Invariant Systems( C

4、onvolution Sum )Soor yn = xn * hn(3) Calculation of Convolution SumTime Reversal: hk h-kTime Shift: h-k hn-kMultiplication: xkhn-kSumming: Example 2.1 2.2 2.3 2.4 2.5 2 Linear Time-Invariant SystemsNote: A discrete-time LTI system is completely characterized by its unit impulse response hnTo visuali

5、ze the calculationExample2.3 Consider the two sequences xn = un，hn = anun，0a1，please calculate the convolution of the two signals yn = xn*hnh - k k1.Viewed as functions of k2.Time reversaln 0, No overlap between xk and hn-kyn=0Example1 Consider the two sequences xn = un，hn = anun，0a1，please calculat

6、e the convolution of the two signals yn = xn*hn3. Time shift4. Multiplication & SummingExample1 Consider the two sequences xn = un，hn = anun，0a1，please calculate the convolution of the two signals yn = xn*hnn 0, xk overlaps hn-kOver all values of kn 0, xk overlaps hn-kn 0, No overlap between xk and

7、hn-kyn=0Example1 Consider the two sequences xn = un，hn = anun，0a1，please calculate the convolution of the two signals yn = xn*hnExample 2 Calculate the convolution yn = RNn* RNn whereyn = 0 n 0, No overlap between RN k and RN n-k 0 n N -1, RN k overlaps RN n-k in 0，n Example 2 Calculate the convolut

8、ion yn = RNn* RNn where N-1 2N-2，No overlap again yn = 0Example 2 Calculate the convolution yn = RNn* RNn where n 0, No overlap between RN k and RN n-kyn = 0 0 n N -1，Overlaps in 0，n N-1 2N-2，No overlap againyn = 0Example 2 Calculate the convolution yn = RNn* RNn where解:Example 3 Calculate the convo

9、lution yn = xn*hn where 2.2 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(Sifting property)Pulse approximation(Sifting property) 2 Linear Time-Invar

10、iant SystemsPulse approximation2.2.2 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 Systems (Unit Impulse Response)A. LTI(t)h(t)x

11、(t)y(t)= ?Because of So,we can get ( Convolution Integral ) or y(t) = x(t) * h(t) 2 Linear Time-Invariant SystemsTime-invariantAdditivityScalingNote: A continuous-time LTI system is completely characterized by its unit impulse response h(t).(3) Computation of Convolution Integral Time Reversal: h()

12、h(-)Time Shift: h(-) h(t-)Multiplication: x()h(t-)Integrating: Example 2.6 2.8 2 Linear Time-Invariant SystemsTo visualize the calculationRegarded as a function of : x(t) x(), h(t) h()ExampleReflection h() h(-) For t 0，ExampleCalculateExample y(t) = p1(t) * p1(t)。a) - t -1b) -1 t 0y (t) = 0 c) 0 1y

13、(t) = 0Example y(t) = p1(t) * p1(t)。c) 0 1y (t) = 0a) - t -1b) -1 t 0y (t) = 0Example y(t) = p1(t) * p1(t)。 Exercise 1: u(t) * u(t) Exercise 2: y (t) = x(t) * h(t)= r(t)trapezoid2.3 Properties of Linear Time Invariant SystemConvolution formula:h(t)x(t)y(t)=x(t)*h(t)hnxnyn=xn*hn 2 Linear Time-Invaria

14、nt SystemsNote: The characteristics of an LTI system are completely determined by its impulse response. (Holds only for LTI system)2.3.1 The Commutative PropertyDiscrete time: xn*hn=hn*xnh(t)x(t)y(t)=x(t)*h(t)x(t)h(t)y(t)=h(t)*x(t) 2 Linear Time-Invariant SystemsContinuous time: x(t)*h(t)=h(t)*x(t)P

15、roof:Proof:Note: The output of an LTI system with input x(t) and unit impulse response h(t) is identical to the output of an LTI system with input h(t) and unit impulse response x(t). 2.3.2 The Distributive PropertyDiscrete time: xn*h1n+h2n=xn*h1n+xn*h2nContinuous time: x(t)*h1(t)+h2(t)=x(t)*h1(t)+x

16、(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)Example 2.10 2 Linear Time-Invariant SystemsNote: A parallel combination of LTI systems can be replaced by a single LTI system whose unit impulse response is the sum of the individual unit impulse responses in the p

17、arallel combination.2.3.3 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 SystemsNote: The unit impulse response of a cascade of two LTI syste

18、ms does not depend on the order in which they are cascaded. However, the order in which nonlinear systems are cascaded can not be changed.If ，then 2 Linear Time-Invariant SystemsThe Time Shift PropertyThe Derivation PropertyIf ，thenNote: These properties can be use to simplify the calculation.Exampl

19、e:From the derivation property we know* 2 Linear Time-Invariant Systemsx(t)1*= ? 2 Linear Time-Invariant SystemsProperties of LTI systemFrom the properties of linear and time-invariant, we know 1）Differential or Difference property：If T x(t)=y(t)thenIf Txk= ykthen T xk -xk-1= yk - yk-1 2）Integral or

20、 Sum property：If Tx(t)=y(t)thenIf Txk= ykthen 2 Linear Time-Invariant SystemsExample Consider an LTI system, we know that the input x1(t) leads to the output y1(t) ，please determine the response of this system to the input x2(t)。解：The relation between x1(t) and x2(t) is as follows: From the properti

21、es of linearity and time-invariance ，we get the same relation between y2(t) and y1(t) 2.3.4 LTI system with and without MemoryMemoryless system: Discrete time: yn=kxn, hn=kn 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

22、 2 Linear Time-Invariant Systems2.3.5 Invertibility of LTI systemOriginal system: h(t)inverse system: h1(t)(t) x(t)x(t)*(t)=x(t)So, for the invertible system: h(t)*h1(t)=(t) or hn*h1n=nh(t) x(t)x(t)h1(t) Example 2.11 2.12 2 Linear Time-Invariant Systems2.3.6 Causality for LTI systemDiscrete time sys

23、tem satisfy the condition: hn=0 for n0Continuous time system satisfy the condition: h(t)=0 for t0 2 Linear Time-Invariant SystemsCausal Signal: xn=0 for n0 or x(t)=0 for t02.3.7 Stability for LTI system Definition of stability: Every bounded input produces a bounded output. Discrete time system:If |

24、xn| B, the condition for |yn| A is 2 Linear Time-Invariant SystemsSufficient & necessaryContinuous time system:If |x(t)|B, the condition for |y(t)|0 (2) To determine the particular solution yp(t) to y(t)+6y(t)+8y(t) = x(t)yp(t) has the similar form of the input signal x(t)yp(t) = Ce-tSubstitute yp(t

25、) to the original system function, we gett0Example The LTI system is given bywith initial conditions y(0)=1, y (0)=2. Please give the response y(t) of this system to the input signal x(t)=e-t u(t).C=1/3 (3) To get the complete solution A=5/2，B= -11/6Example The LTI system is given bywith initial con

26、ditions y(0)=1, y (0)=2. Please give the response y(t) of this system to the input signal x(t)=e-t u(t).Using the initial conditions we knowThen we have1) With the same initial conditions, but different input signal x(t) = sin t u(t)，then y(t) = ？2) Using the same input signal，but different initial

27、conditions y(0) = 0, y (0) = 1, then y(t) = ？QuestionsTotal response = zero-input response + zero-state responseSolve the homogeneous differential equation to get yzi (t)Calculate the convolution x(t)*h(t) to get yzs (t)(2) The Convolution Method:Methods to solve the response of continuous-time LTI

28、system 2 Linear Time-Invariant Systems The Homogeneous Solution yh(t)(1) Different real characteristic roots s1, s2, , sn(2) Multiple real characteristic roots s1=s2=sn =s(3) Complex conjugate roots 2 Linear Time-Invariant SystemsZero-Input Response Definition: The zero-input response results only f

29、rom the initial state of the system and not from any external drive. Example The LTI system is given by y (t)+5y (t) +6y (t) =4x(t), t0 with initial conditions y(0-) = 1，y (0-) = 3. Please determine the zero-input response yzi(t) of this system. y(0-)=yzi(0-)=K1+K2=1 y (0-)= yzi(0-)= - 2K1-3K2 =3K1=

30、 6，K2= -5The characteristic equation isThe characteristic roots are（different roots ） （multiple roots） y(0-)=yzi(0-)=K1=1;y(0-)= y zi(0-)= -2K1+K2 =3 K1 = 2, K2= 3Example The LTI system is given by y (t)+4y (t) +4y (t) =4x(t), t0 with initial conditions y(0-) = 1，y (0-) = 3. Please determine the zer

31、o-input response yzi(t) of this system.The characteristic equation isThe characteristic roots arey(0-)=yzi(0-)=K1=1y (0-)= y zi(0-)= -K1+2K2 =3 K1= 1，K2= 2Example The LTI system is given by y (t)+2y (t) +5y (t) =4x(t), t0 with initial conditions y(0-) = 1，y (0-) = 3. Please determine the zero-input

32、response yzi(t) of this system.The characteristic equation isThe characteristic roots are(conjugate roots) Methods to solve the zero-state response yzs (t): 1) Solve the differential equation with initial state of zero. 2) The convolution method ：Calculate the convolution yzs (t)= x(t)*h(t) Definiti

33、on：The zero-state response is the behavior or response of a system with initial state of zero. It results only from the external inputs or driving functions of the system and not from the initial state. 2 Linear Time-Invariant SystemsZero-State Response Example The LTI system is given by ：y(t) + 3y(

34、t) = 2x(t) with the impulse response h(t) = 2e-3t u(t) and input signal x(t) =3u(t). Please determine the zero-state response yzs(t) of this system.Example The LTI system is given by Please determine the impulse response h(t) of this system. x(t) = d (t) y(t) = h(t)，A=2Method to Determine the Impuls

35、e Response h(t)The characteristic root is s = -3. And as nm, then we knowExample The LTI system is given by Please determine the impulse response h(t) of this system.A= -16, B =3The characteristic root is s = -6. And as n=m, then we knowMethod to Determine the Impulse Response h(t)Method to Determin

36、e the Impulse Response h(t)(冲激平衡法)x(t)=(t) (nm ) 将h(t)代入微分方程，使方程两边平衡，确定系数Ki ， Aj (nm)Methods to solve the response of discrete-time LTI system Using the initial values y-1, y-2, y-2, y-n and the input signal, the output can be iteratively given by1. Iterative Method 2 Linear Time-Invariant SystemsEx

37、ample The LTI system is given by yk-0.5yk-1=uk,with the initial condition of y-1 =1 . Please use the iterative method to determine the output of this system.The difference equation can be written asSubstitute the initial condition, we get: IterativelyShortcoming：Can not get the closed solution.2. Th

38、e Classical Approach 2 Linear Time-Invariant SystemsComplete solution = homogeneous solution+ particular solutionyh k is determined by the characteristic roots of the homogeneous difference equation. yp k has the same form as the input signalMethods to solve the response of discrete-time LTI system

39、Example The LTI system is given by yk-5yk-1+6yk-2 = x k with the initial conditions y0 = 0，y1 = -1, and input xk = 2k ukPlease determine the total response of this system. Then we get the homogeneous solution yhk as follows:Solve the homogeneous difference equation yk-5yk-1+6yk-2 = 0The characterist

40、ic equation isThe characteristic roots areExample The LTI system is given by yk-5yk-1+6yk-2 = x k with the initial conditions y0 = 0，y1 = -1, and input xk = 2k ukPlease determine the total response of this system. (2) To determine the particular solution ypk to yk-5yk-1+6yk-2 =xkyp(t) has the simila

41、r form as the input signal x(t)Substitute ypk to the original system function, we getA= -2(3) To determine the complete solution yk C1= -1，C2= 1Example The LTI system is given by yk-5yk-1+6yk-2 = x k with the initial conditions y0 = 0，y1 = -1, and input xk = 2k ukPlease determine the total response

42、of this system. Using the initial conditions ,we knowThen we get1) With the same initial conditions, but different input signal xk = sin0 k uk ，then yk = ？2) Using the same input signal，but different initial conditions y0=1, y1=1, then yk = ？QuestionsTotal response = zero-input response + zero-state

43、 responseSolve the homogeneous difference equation to get yzi kCalculate the convolution xk*hk to get yzs k(2) The Convolution Method:Methods to solve the response of discrete-time LTI system 2 Linear Time-Invariant Systems The Homogeneous Solution yh k(1) Different real characteristic roots r1, r2,

44、 , rn(2) Multiple real characteristic roots r1=r2=rn(3) Complex conjugate roots 2 Linear Time-Invariant SystemsZero-Input Response Definition: The zero-input response results only from the initial state of the system and not from any external drive. The characteristic roots areThe characteristic equ

45、ation is C1=1，C2= -2Example The LTI system is given by yk+3yk-1+2yk-2=xk with initial conditions y-1=0， y-2= 1/2，. Please determine the zero-input response yzik of this system.（different roots ） C1 = 4, C2= 4The characteristic roots areThe characteristic equation isExample The LTI system is given by

46、 yk+4yk-1+4yk-2=xk with initial conditions y-1=0， y-2= 1/2，. Please determine the zero-input response yzik of this system.（multiple roots ） C1= 1，C2= 0 ，C5= 5Example The LTI system is given by yk-0.5yk-1+yk-2 -0.5yk-3 =xk with initial conditions y-1 = 2， y-2= -1， y-3= 8. Please determine the zero-in

47、put response yzik of this system.The characteristic equation isThe characteristic roots are(conjugate roots) Methods to solve the zero-state response yzs k: 1) Solve the differential equation with initial state of zero. 2) The convolution method ：Calculate the convolution yzs k= xk*hk Definition：The

48、 zero-state response is the behavior or response of a system with initial state of zero. It results only from the external inputs or driving functions of the system and not from the initial state. 2 Linear Time-Invariant SystemsZero-State Response Example The LTI system is given by: with ,Please det

49、ermine the zero state response yzs k of this system解：Example The causal LTI system is given by Please determine the impulse response hk.1) Equivalent initial conditions (等效初始条件法)Causal LTI system h-1 = h-2 = 0， 注意：选择初始条件的基本原则是必须将dk的作用体现在初始条件中二阶系统需要两个初始条件，可以选择h0和h1Method to Determine the Impulse Resp

50、onse hk2) To determine the homogenous solution The characteristic equation isThen we haveUsing the initial values we know C1=-1，C2= 2Example The causal LTI system is given by Please determine the impulse response hk.The characteristic roots areMethod to Determine the Impulse Response hk2.4.3 Block D

51、iagram Representations of First-order Systems Described by Differential and Difference Equation(1) Discrete time system Basic elements: A. An adder B. Multiplication by a coefficient C. A unit delay 2 Linear Time-Invariant SystemsBasic elements: 2 Linear Time-Invariant SystemsAn adderMultiplication by a coefficientA unit delayExample: yn+ayn-1=b xn 2 Linear Time-Invariant S