信号与系统奥本海默原版PPT第二章.ppt
,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 Impulse 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 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),So,or yn=xn*hn,(3)Calculation of Convolution Sum,Time Inversal: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 Systems,2.2 Continuous-time LTI system:The convolution integral,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 Representation 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 Systems,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-Invariant Systems,2.3.1 The Commutative Property,Discrete time:xn*hn=hn*xnContinuous 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*h2nContinuous time:x(t)*h1(t)+h2(t)=x(t)*h1(t)+x(t)*h2(t),Example 2.10,2 Linear Time-Invariant Systems,2.3.3 The Associative Property,Discrete time:xn*h1n*h2n=xn*h1n*h2nContinuous 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 that: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 system satisfy the condition:hn=0 for n0Continuous 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 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 Differential and Difference Equation,Discrete time system:Differential EquationContinuous 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: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,2.4.3 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-Invariant Systems,2.5 Singularity Functions,2.5.1 The unit impulse as idealized short pulse,(1),(2),2 Linear Time-Invariant Systems,Several important formula:,Problems:2.1 2.3 2.5 2.7 2.10 2.11 2.12 2.18 2.19 2.20 2.23 2.24 2.40 2.47,
