信号与系统英文版教学课件：ch2 Linear Time-invariant Systems

1、1INTRODUCTIONLTI systems possesses the superposition property.Represent signals as linear combinations of delayed impulses .Convolution sum or convolution integral.linear constant-coefficient difference or differential equations. 21) The Representation of Discrete-time Signals in Terms of ImpulsesIf

2、 xn=un,then Sifting Property of Unit Sample: 1. Discrete-Time LTI: Convolution Sum32) The Discrete-time Unit Impulse Response of LTI SystemsLTIxn=nyn=hn Unit Impulse Response hn :response of the LTI system to the unit sample n. n hn Why do we need it? 43) The Discrete-time Response of LTI Systems to

3、 any Input Signal: Convolution SumLTIxnyn=?Solution:Question: n hnn-k hn-kxkn-k xk hn-kThe response yn to xn is the weighted linear combination of delayed unit sample responses.5Convolution SumSoRepresenting the convolution operation symbolically as: yn = xn * hn- Convolution Sum That is, the unit i

4、mpulse response -hn can fully characterize an LTI system.Summary on calculating convolution sumTime Inversal: hk h-kTime Shift: h-k hn-kMultiplication: xkhn-kSumming: 6Example 2.1 Consider a LTI system with unit sample response hn and input xn, as illustrated in Figure (a). Calculate the convolution

5、 sum (convolution) of these two sequences graphically. nxn 0 1 2nhn-2 0 2(a) 122kxk 0 1 2kh-k -2 0 2 (b)2217kxk 0 1 22kh-k -2 0 221n=0kh-1-k -3 -2 0 121n=-1kh1-k -1 0 1 2 321n=18Example 2.2 Consider an input xn and a unit sample response hn given byDetermine and plot the output Using the geometrical

6、 sum formula to evaluate last equation, we have 9n21yn101) The Representation of Continuous-time Signals in Terms of ImpulsesDiscrete-time:Continuous-time:Why?t -02 k x(t)Staircase approximation to a continuous-time signal x(t)2. Continuous-Time LTI: Convolution Integral11Therefore: What is this? De

7、fine We have the expression: as , the summing approaches an integral and is the unit impulse function 122) The Continuous-time Unit Impulse Response of LTI SystemsLTIx(t)=(t)y(t)=h(t)Unit Impulse Response h(t) : the response of the LTI system to the input . 3) The Continuous-time Response of LTI Sys

8、tems to any Input Signal: Convolution IntegralLTIx(t)y(t)=?13Give the as the response of a continuous-time LTI system to the input , then the response of the system to pulse is Thus, the response to isAs , in addition, the summing becomes an integral. Therefore, - Convolution Integral 14Represent co

9、nvolution integral of two signals x(t) and h(t) symbolically as:Convolution IntegralA continuous-time LTI system is completely characterized by its unit impulse response h(t) .Computation of Convolution Integral: Time Inversal: h() h(- )Time Shift: h(-) h(t- )Multiplication: x()h(t- )Integrating: 15

10、Example 2.3 Consider the convolution of the following two signals, which are depicted in (a): 2 x(t) 1 h(t) 0 1 2 t 0 1 2 3 t -1 (a) x() h(-) -2 0 1 2 3 t=0 x() h(t-) -2 0 1 2 3 t 0t1 When t1 : x()h(t-) = 0So 16 x() h(t-) -2 0 1 t 2 3 1t2 x() h(t-)-2 0 1 2 t 3 2t3 x() h(t-) -2 0 1 2 3 t 4 3t4 17 x()

11、 h(t-) -2 0 1 2 3 4 t 5 4t5 y(t) 0 1 3 5 -2 x() h(t-) -2 0 1 2 3 4 5 t t 5 When t5 : x()h(t-) = 0So 18 h(t) or hn completely characterizes an LTI system What property should h(t) or hn have for the LTI system to be stable, causal, memoryless and invertible? 3. Properties of LTI Systems191) The Commu

12、tative PropertyDiscrete time: xn*hn=hn*xnContinuous time: x(t)*h(t)=h(t)*x(t)h(t)x(t)y(t)=x(t)*h(t)x(t)h(t)y(t)=h(t)*x(t)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)3. Properties of LTI Systems20h1(t)+h2(t)x(t)y(t)=x(t)*h1

13、(t)+h2(t)h1(t)x(t)y(t)=x(t)*h1(t)+x(t)*h2(t)h2(t)3) The Associative PropertyDiscrete time: xn*h1n*h2n=xn*h1n*h2nContinuous time: x(t)*h1(t)*h2(t)=x(t)*h1(t)*h2(t)21h1(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)4) LTI System with and without MemoryMemoryless system: Discrete

14、time: yn=kxn, hn=? Continuous time: y(t)=kx(t), h(t)=?k (t) x(t)y(t)=kx(t)=x(t)*k(t)k n xnyn=kxn=xn*kn225) 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) or hn*h1n=nh(t) x(t)x(t)h1(t) 6) Causality for LTI system

15、Discrete time system satisfy: hn=0 for n0Continuous time system satisfy: h(t)=0 for t0Why?237) Stability for LTI systemDefinition of stability:Every bounded input produces a bounded output. If |xn|B, the sufficient and necessary condition for |yn|A isDiscrete time system:Continuous time system:If |x

16、(t)|B, the condition for |y(t)|A is248) The Unit Step Response of LTI systemThe unit step response, sn or s(t), is the output of an LTI system when input xn=un or x(t)=u(t). A.The step response of a discrete-time LTI system is the running sum of its sample response: B.The impulse response of a discr

17、ete-time LTI system is the first difference of its step response: hn / h(t) n/ (t)hn/h(t)un/u(t)sn=un*hn /s(t)=u(t)*h(t)25C.The unit step response of a continuous-time LTI system is the running integral of its impulse response: D.The unit impulse response of a continuous-time LTI system is the first

18、 derivative of the unit step response :E.Properties of convolution integral:Derivative property: Integral property: Combining the two properties, we have Solution Example 2.3 with Properties E : y(t) = x(t) * h(t) = dx(t)/dt * -t h(x)dx = 2d(t-1)-2d(t-3)*f (t) =2 d(t-1)*f (t)-2d(t-3)*f (t) =2f (t-1)

19、-2f (t-3) x(t) h(t) dx(t)/dt -t h(x)dx=f (t) 0 1 3 t 0 1 2 t y(t) 0 1 3 5 -2 2f (t-1) -2f (t-3)2627: input signal; : output signal. Ci (t)VsR + 1) Continuous-time system: Differential EquationLinear constant-coefficient differential equationLinear constant-coefficient differential(difference) equati

20、on provides an implicit relationship between the input and output rather than an explicit expression for the system output as a function of the input . 4. Causal LTI Systems described by Differential and Difference Equations28How to find the system output given an input signal? natural response Forc

21、ed response We must specify one or more auxiliary conditions to solve a differential (difference) equation . Initial rest: for a causal LTI system, if x(t)=0 for tt0, then y(t) must also equal 0 for t t0. 29A general Nth-order linear constant-coefficient differential equation:orand initial condition

22、: y(t0), y(t0), , y(N-1)(t0) ( N values )For a causal LTI system:302) Discrete-time system: Difference EquationA general Nth-order linear constant-coefficient difference equation:orand initial condition: y0, y-1, , y-(N-1) ( N values )Under initial rest, the system described by linear constant-coeff

23、icient differential(difference) equation is causal and LTI.31General solutions to such difference equations: later in Chapter 5 or 10. Second resolution:(recursive method)First resolution:N auxiliary conditions: 323) Block Diagram Representations(1) Dicrete-time systemBasic elements: A. An adder B.

24、Multiplication by a coefficient C. An unit delayFirst-order difference equation : addition delay multiplication 33Example: yn+ayn-1=bxn (2) Continuous-time system First-order differential equation :differentiation Three basic elements in block diagram: adder, multiplier and integrator . 34Example: y

25、(t)+ay(t)=bx(t) Such block diagrams can also be developed for higher order systems. 355. Singularity Functions 1)The unit impulse as an idealized short pulse(1)(2)Important: for small , they both behaves the same from an LTI system, see Figure 2.34.362)Defining the unit impulse through convolution- Operational definition （运算定义）Or，equivalently, The primary importance of the unit impulse is not what it is at each value of t, but rather what it do