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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 1 2 Linear Time-Invariant Systems2 2 Linear Time-Invariant Systems2.1.2 The Discrete-time Unit Impulse
2、 Response and the Convolution Sum Representation of LTI Systems(1) Unit Impulse(Sample) Response LTIxn=nyn=hn Unit Impulse Response: hn 3 2 Linear Time-Invariant Systems(2) Convolution Sum of LTI System LTIxnyn=?Solution:Question: n hnn-k hn-kxkn-k xk hn-k4 2 Linear Time-Invariant Systems5 2 Linear
3、Time-Invariant Systems6 2 Linear Time-Invariant Systems( Convolution Sum )Soor yn = xn * hn(3) Calculation of Convolution SumTime Inversal: hk h-kTime Shift: h-k hn-kMultiplication: xkhn-kSumming: Example 2.1 2.2 2.3 2.4 2.57 2 Linear Time-Invariant Systems2.2 Continuous-time LTI system: The convolu
4、tion integral2.2.1 The Representation of Continuous-time Signals in Terms of ImpulsesDefine We have the expression: Therefore: 8 2 Linear Time-Invariant Systems9 2 Linear Time-Invariant Systemsor 10 2 Linear Time-Invariant Systems2.2.2 The Continuous-time Unit impulse Response and the convolution In
5、tegral 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)=?11 2 Linear Time-Invariant SystemsA. LTI(t)h(t)x(t)y(t)=?Because of So,we can get ( Convolution Integral ) or y(t) = x(t) * h(t) 12 2 Linear Time-Invariant SystemsB. or y(
6、t) = x(t) * h(t) LTI(t)h(t)(t) h(t)( Convolution Integral ) 13 2 Linear Time-Invariant Systems14 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 2.815 2 Linear Time-Invariant Syste
7、ms2.3 Properties of Linear Time Invariant SystemConvolution formula:h(t)x(t)y(t)=x(t)*h(t)hnxnyn=xn*hn16 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)17 2 Linear Time-Invariant S
8、ystems2.3.2 The Distributive PropertyDiscrete time: xn*h1n+h2n=xn*h1n+xn*h2nContinuous time: x(t)*h1(t)+h2(t)=x(t)*h1(t)+x(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 2.1018 2 Linear Time-Invariant Systems2.3.3 The Associative PropertyDiscrete time: xn
9、*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)19 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
10、)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=xn20 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) Exampl
11、e 2.11 2.1221 2 Linear Time-Invariant Systems2.3.6 Causality for LTI systemDiscrete time system satisfy the condition: hn=0 for n0Continuous time system satisfy the condition: h(t)=0 for t022 2 Linear Time-Invariant Systems2.3.7 Stability for LTI system Definition of stability: Every bounded input p
12、roduces a bounded output. Discrete time system:If |xn|B, the condition for |yn|A is23 2 Linear Time-Invariant SystemsContinuous time system:If |x(t)|B, the condition for |y(t)|A isExample 2.1324 2 Linear Time-Invariant Systems2.3.8 The Unit Step Response of LTI systemDiscrete time system:hn nhnunsn=
13、un*hnContinuous time system:h(t) (t)h(t)u(t)s(t)=u(t)*h(t)25 2 Linear Time-Invariant Systems2.4 Causal LTI Systems Described by Differential and Difference EquationDiscrete time system: Differential EquationContinuous time system: Difference Equation26 2 Linear Time-Invariant Systems2.4.1 Linear Con
14、stant-Coefficient Differential EquationA general Nth-order linear constant-coefficient differential equation:orand initial condition: y(t0), y(t0), , y(N-1)(t0) ( N values )27 2 Linear Time-Invariant Systems2.4.2 Linear Constant-Coefficient Difference EquationA general Nth-order linear constant-coef
15、ficient difference equation:orand initial condition: y0, y-1, , y-(N-1) ( N values )Example 2.1528 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. Multipl
16、ication by a coefficient C. An unit delay29 2 Linear Time-Invariant SystemsBasic elements: 30 2 Linear Time-Invariant SystemsExample: yn+ayn-1=bxn 31 2 Linear Time-Invariant Systems(2) Continuous time system Basic elements: A. An adder B. Multiplication by a coefficient C. An (differentiator) integr
17、ator32 2 Linear Time-Invariant SystemsBasic elements: 33 2 Linear Time-Invariant SystemsExample: y(t)+ay(t)=bx(t) 34 2 Linear Time-Invariant Systems2.5 Singularity Functions2.5.1 The unit impulse as idealized short pulse(1)(2)35 2 Linear Time-Invariant SystemsSeveral 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 2.4736 刚才的发言，如刚才的发言，如有不当之处请多指有不当之处请多指正。谢谢大家！正。谢谢大家！37
