1、-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 Systemskknkxnxnxnxnxnxnx 2 2 1 1 0 1 1 2 2If xn=un,then 0kknnx-2 2 Linear Time-Invariant Systems-3 2 Linear Time-Invariant S
2、ystems2.1.2 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-4 2 Linear Time-Invariant Systems(2)Convolution Sum of LTI System LTIxnyn=?Solution:Question:n hnn-k hn-kxkn-k xk hn-kkkknhkx
3、nyknkxnx-5 2 Linear Time-Invariant Systems-6 2 Linear Time-Invariant Systems-7 2 Linear Time-Invariant Systems(Convolution Sum)Sokknhkxnyor yn=xn*hn(3)Calculation of Convolution SumTime Inversal:hk h-kTime Shift:h-k hn-kMultiplication:xkhn-kSumming:kknhkxnyExample 2.1 2.2 2.3 2.4 2.5-8 2 Linear Time
4、-Invariant Systems2.2 Continuous-time LTI system:The convolution integral2.2.1 The Representation of Continuous-time Signals in Terms of Impulsesotherwisett,00,1)(Define We have the expression:kktkxtx)()()(Therefore:kktkxtx)()(lim)(0-9 2 Linear Time-Invariant Systems-10 2 Linear Time-Invariant Syste
5、msor dtxtx)()()(-11 2 Linear Time-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)=?-12 2 Linear Time-Invariant SystemsA.LTI(t)h(t)x(t)y
6、(t)=?dtxtx)()()(Because of dthxty)()()(So,we can get(Convolution Integral)or y(t)=x(t)*h(t)-13 2 Linear Time-Invariant SystemsB.)(lim)(0ttor y(t)=x(t)*h(t)(lim)(0ththLTI(t)h(t)(t)h(t)()(tht)()(kthkt)()()()(kthkxktkxkkkthkxktkx)()(lim)()(lim00dtxtx)()()(dthxty)()()(Convolution Integral)-14 2 Linear T
7、ime-Invariant Systems-15 2 Linear Time-Invariant Systems(3)Computation of Convolution Integral Time Inversal:h()h(-)Time Shift:h(-)h(t-)Multiplication:x()h(t-)Integrating:dthxty)()()(Example 2.6 2.8-16 2 Linear Time-Invariant Systems2.3 Properties of Linear Time Invariant SystemConvolution formula:d
8、thxthtxty)()()(*)()(kknhkxnhnxny*h(t)x(t)y(t)=x(t)*h(t)hnxnyn=xn*hn-17 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(t)*h(t)x(t)h(t)y(t)=h(t)*x(t)-18 2 Linear Time-Invariant Systems2.3.2 The Distributive Proper
9、tyDiscrete 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)Example 2.10-19 2 Linear Time-Invariant Systems2.3.3 The Associative PropertyDiscrete time:xn*h1n*h2n=xn*h1n*h2nContinuous time:x(
10、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)-20 2 Linear Time-Invariant Systems2.3.4 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=x
11、n*knImply that:x(t)*(t)=x(t)and xn*n=xn-21 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)or hn*h1n=nh(t)x(t)x(t)h1(t)Example 2.11 2.12-22 2 Linear Time-Invariant Systems2.3.6 Caus
12、ality for LTI systemDiscrete time system satisfy the condition:hn=0 for n0Continuous time system satisfy the condition:h(t)=0 for t0-23 2 Linear Time-Invariant Systems2.3.7 Stability for LTI system Definition of stability:Every bounded input produces a bounded output.Discrete time system:kkkhknxknhk
13、xny或If|xn|B,the condition for|yn|A iskkh|AnythenkhifkhBkhknxnykkk|,|-24 2 Linear Time-Invariant SystemsContinuous time system:If|x(t)|B,the condition for|y(t)|A isdhtxdthxty)()()()()(或dh|)(|AtythendhifdhBdhtxty|)(|,|)(|)(|)(|)(|)(|Example 2.13-25 2 Linear Time-Invariant Systems2.3.8 The Unit Step Re
14、sponse of LTI systemDiscrete time system:1)1(nhnhnsnhnhkhnsnk或hn nhnunsn=un*hnContinuous time system:h(t)(t)h(t)u(t)s(t)=u(t)*h(t)()()()()()1(tsththdhtst或-26 2 Linear Time-Invariant Systems2.4 Causal LTI Systems Described by Differential and Difference EquationDiscrete time system:Differential Equat
15、ionContinuous time system:Difference Equation-27 2 Linear Time-Invariant Systems2.4.1 Linear Constant-Coefficient Differential EquationA general Nth-order linear constant-coefficient differential equation:MkkkkNkkkkdttdxbdttyda00)()(or)()()()()()()()(01)1(1)(01)1(1)(txbtxbtxbtxbtyatyatyatyaMMMMNNNNa
16、nd initial condition:y(t0),y(t0),y(N-1)(t0)(N values)-28 2 Linear Time-Invariant Systems2.4.2 Linear Constant-Coefficient Difference EquationA general Nth-order linear constant-coefficient difference equation:MkkNkkknxbknya00or 1)1(1)1(011011nxbnxbMnxbMnxbnyanyaNnyaNnyaMMNNand initial condition:y0,y
17、-1,y-(N-1)(N values)Example 2.15-29 2 Linear Time-Invariant Systems2.4.3 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-30 2 Linear Time-Invariant
18、 SystemsBasic elements:-31 2 Linear Time-Invariant SystemsExample:yn+ayn-1=bxn -32 2 Linear Time-Invariant Systems(2)Continuous time system Basic elements:A.An adder B.Multiplication by a coefficient C.An(differentiator)integrator-33 2 Linear Time-Invariant SystemsBasic elements:-34 2 Linear Time-In
19、variant SystemsExample:y(t)+ay(t)=bx(t)-35 2 Linear Time-Invariant Systems2.5 Singularity Functions2.5.1 The unit impulse as idealized short pulseotherwisett,00,1)()(lim)(0tt(1)1(2)(*)()(tttr)(lim)(0trt-36 2 Linear Time-Invariant Systemstdxtxtutxtxttxtttxttttxttxtttxtxttxtxttx)()()(*)()6()()(*)()5()()(*)()4()()(*)()3()()0()()()2()()(*)()1()1(212100Several 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
