1、,2 Linear Time-Invariant Systems,2.1 Discrete-time LTI system: The convolution sum,2.1.1 The Representation of Discrete-time Signals in Terms of Impulses,2. Linear Time-Invariant Systems,If xn=un, then,2 Linear Time-Invariant Systems,2 Linear Time-Invariant Systems,2.1.2 The Discrete-time Unit Impul

2、se Response and the Convolution Sum Representation of LTI Systems,(1) Unit Impulse(Sample) Response,Unit Impulse Response: hn,2 Linear Time-Invariant Systems,(2) Convolution Sum of LTI System,Solution:,Question:,n hn n-k hn-k xkn-k xk hn-k,2 Linear Time-Invariant Systems,2 Linear Time-Invariant Syst

3、ems,2 Linear Time-Invariant Systems,( Convolution Sum ),So,or yn = xn * hn,(3) Calculation of Convolution Sum,Time Inversal: hk h-k Time Shift: h-k hn-k Multiplication: xkhn-k Summing:,Example 2.1 2.2 2.3 2.4 2.5,2 Linear Time-Invariant Systems,2.2 Continuous-time LTI system: The convolution integra

4、l,2.2.1 The Representation of Continuous-time Signals in Terms of Impulses,Define,We have the expression:,Therefore:,2 Linear Time-Invariant Systems,2 Linear Time-Invariant Systems,or,2 Linear Time-Invariant Systems,2.2.2 The Continuous-time Unit impulse Response and the convolution Integral Represe

5、ntation of LTI Systems,(1) Unit Impulse Response,(2) The Convolution of LTI System,2 Linear Time-Invariant Systems,A.,Because of,So,we can get,( Convolution Integral ),or y(t) = x(t) * h(t),2 Linear Time-Invariant Systems,B.,or y(t) = x(t) * h(t),( Convolution Integral ),2 Linear Time-Invariant Syst

6、ems,2 Linear Time-Invariant Systems,(3) Computation of Convolution Integral,Time Inversal: h() h(- ) Time Shift: h(-) h(t- ) Multiplication: x()h(t- ) Integrating:,Example 2.6 2.8,2 Linear Time-Invariant Systems,2.3 Properties of Linear Time Invariant System,Convolution formula:,2 Linear Time-Invari

7、ant Systems,2.3.1 The Commutative Property,Discrete time: xn*hn=hn*xn Continuous time: x(t)*h(t)=h(t)*x(t),2 Linear Time-Invariant Systems,2.3.2 The Distributive Property,Discrete time: xn*h1n+h2n=xn*h1n+xn*h2n Continuous time: x(t)*h1(t)+h2(t)=x(t)*h1(t)+x(t)*h2(t),Example 2.10,2 Linear Time-Invari

8、ant Systems,2.3.3 The Associative Property,Discrete time: xn*h1n*h2n=xn*h1n*h2n Continuous time: x(t)*h1(t)*h2(t)=x(t)*h1(t)*h2(t),2 Linear Time-Invariant Systems,2.3.4 LTI system with and without Memory,Memoryless system: Discrete time: yn=kxn, hn=kn Continuous time: y(t)=kx(t), h(t)=k (t),Imply th

9、at: x(t)* (t)=x(t) and xn* n=xn,2 Linear Time-Invariant Systems,2.3.5 Invertibility of LTI system,Original system: h(t) Reverse system: h1(t),So, for the invertible system: h(t)*h1(t)=(t) or hn*h1n=n,Example 2.11 2.12,2 Linear Time-Invariant Systems,2.3.6 Causality for LTI system,Discrete time syste

10、m satisfy the condition: hn=0 for n0 Continuous time system satisfy the condition: h(t)=0 for t0,2 Linear Time-Invariant Systems,2.3.7 Stability for LTI system,Definition of stability: Every bounded input produces a bounded output. Discrete time system:,If |xn|B, the condition for |yn|A is,2 Linear

11、Time-Invariant Systems,Continuous time system:,If |x(t)|B, the condition for |y(t)|A is,Example 2.13,2 Linear Time-Invariant Systems,2.3.8 The Unit Step Response of LTI system,Discrete time system:,Continuous time system:,2 Linear Time-Invariant Systems,2.4 Causal LTI Systems Described by Differenti

12、al and Difference Equation,Discrete time system: Differential Equation Continuous time system: Difference Equation,2 Linear Time-Invariant Systems,2.4.1 Linear Constant-Coefficient Differential Equation,A general Nth-order linear constant-coefficient differential equation:,or,and initial condition:

13、y(t0), y(t0), , y(N-1)(t0) ( N values ),2 Linear Time-Invariant Systems,2.4.2 Linear Constant-Coefficient Difference Equation,A general Nth-order linear constant-coefficient difference equation:,or,and initial condition: y0, y-1, , y-(N-1) ( N values ),Example 2.15,2 Linear Time-Invariant Systems,2.

14、4.3 Block Diagram Representations of First-order Systems Described by Differential and Difference Equation,(1) Dicrete time system Basic elements: A. An adder B. Multiplication by a coefficient C. An unit delay,2 Linear Time-Invariant Systems,Basic elements:,2 Linear Time-Invariant Systems,Example: yn+ayn-1=bxn,2 Linear Time-Invariant Systems,(2) Continuous time system Basic elements: A. An adder B. Multiplication by a coefficient C. An (differentiator) integrator,2 Linear Time-Invariant Systems,Basic elements:,2 Linear Time-Invariant Systems,Example: y(t)+ay(t)=bx(t),2 Linear Time