大二下信号与系统第02章

1、Chapter2 Linear Time-Invariant System Representation of Signal by Impulse Signal Time Domain Analysis of LTI system Convolution Integral / Sum Basic Idea for Analysis of LTI Systems Response of LTI System to Signals Response of LTI System to Impulse Signal5, 7, 10, 11, 12, 19, 20, 23, 40, 46, 47Prob

2、lems:2.0 IntroductionA. Fundamental Role of LTI SystemMany physical systemLTI SystemAnalized in detailpowerful toolscoreSignal and SystemB. Basic Idea for Analysis of LTI SystemsBasic signalSuper-position( )( )iix ty t LResponse of basic signal( )( )iiiia x ta y t L?( )( )x ty t L positioncompositio

3、nComplicate input signalResponse of complicate input signalMathematic Tools ?Mathematic Tools ?Ch.2/3/4/5/9/10Ch.2/3/4/5/9/10nkh nk L( )nnzH z z ( )ststeH s e C.Basic signal and their response to LTI system()()th t LL000()jtjteH je L000()jnjneH je LL0jtte e0jtte e0jnne e0jnne eUnite impulseD.Mathema

4、tic tools for position and composition 0sj0jzre0je eComplex exponentialDecaying complex exponentialCh.2The convolution sum/integrateFourier TransformLaplas & Z- Transform Ch.3/4/5Ch.9/10Ch.2Ch.3/4/5Ch.9/10& The convolution sum/integrate& Fourier Transform& Laplas & Z- Transform p

5、osition2.1 Discrete-Time LTI System: the Convolution SumBasic signalsnkh nk LResponse of basic signals2.1.1 Discrete-Time signal represented by unit impulse signal Complicate signalA set of basic signals x n kx knki.e kkx nank x k ku nu knkcompositionL kh nkx k Time-invariant 2.1.2 The Response of L

6、TI System to InputA. The response of LTI System to n nImpulse signalImpulse response h n LB. Response to an InputL nh n Lnkh nk knkx kLinearitya set of basic signals and its response x n y nLhnLinearity？ x nL y nC. Convolution Sum x n y nLConvolution SumD. Impulse response representation of LTI Syst

7、emResponse to xnAny input signalImpulse response of LTI System y nx nh n L x n h n y n x nh n x nnLkkL x k x kLnkh nk？!0 1 21k x k0 1421n y n1 2 2 11410131 11 1 11 11 11 1不进位不进位E.Computation of Convolution Sum kx nh nx k h nk01kh nkn kx k h nk21n0 141n x n1 200 1 21n h n ; ,01nx nu nh nu n kku k u n

8、k11 y n0n10k kx ku k1 kku k u nk ky nx nh nx k h nk0,nkk11,010,nnotherwise11 1nu n0k1u nkn0n otherwise0,tShifted impulse at : a set of basic signal 2.2 Continuous-Time LTI System: The Convolution integration 2.2.1 Continuous-Time Signal Represented by Impulse Signal( )( ) ()x txtd Any signalScaling

9、factor for impulse at tt0( ) tttt 偶0( ) ()x ttt dt0( )x tB. The Response of LTI System to Input Signal2.2.2 The Response of LTI System to input( )( )th t LImpulse input Impulse response L( )x t( )h t( )y t( ) tA. The Response of LTI System to L( )( )th t ()xtd ( )y t( )x tL?LinearityLinearity L( ) (

10、)txhd Time-invariant L()()th t : a set of basic signals and its response D. Time-Domain Analysis of LTI System( )( )( )y tx th tResponse of input signalAny input signalImpulse response of LTI SystemC. Convolution IntegralConvolution Integral L( )x t( )h t( )y t( )x tL( )y tL()t()h tLdd( )( )x tt( )(

11、 )x th tL( )x( )x!E. Computation of convolution integral( )( )(0),( )( )atx teu tah tu t1( )x0ae0t()yt1a( ) ()xt htd 1at001t( )h tt1( )x t0atet1()h t00,taed1(1),00,atetaotherwise1(1)( )atau taotherwise0t ( )( )( )y tx th t( ) ()xh td02( )();( )(3)tx te uth tu t1( )x02et()h t03t11t( )h t031t( )x t02t

12、e()ytt0123( )( )( )( ) ()y tx th txh td2(3)1;321;32tett32;te d30t 30t 2(3)1(3)2teut 1(3)2u t 02;e d2.3 Properties of Linear Time-Invariant System and Convolution Sum/Integral 2.3.1 The Commutative Property( )( )( )( )( )x th ty th tx tOutput response to inputImpulse response L2( )h t( )x t( )y t( )h

13、 tOutput response toinputImpulse response L1( )x t( )h t( )y t( )x t y n x nh n h nx n2.3.2 The Distributive Property12 ( )x nh nh n1212( ) ( ( )( )( )( )( )( )x th th tx th tx th t1( )h t2( )h t( )x t( )y t12( )( )h tht( )x t( )y t12 x nh nx nh n2.3.3 The Association Property1212() ( )x nh nh nx nh

14、 nh n1212( )( )( ) ( )( )( )x th th tx th th t数学公式数学公式物理意义物理意义12( )( )h tht( )x t( )y t( )h t21( )( )hth t( )x t( )y t( )h tAssociation propertyCommutative property1( )h t2( )h t( )x t( )y t2( )h t( )x t( )y t2( )h tAssociative property2.3.12 Time-Delaying Property( )( )( )y tx th t00()()( )y ttx tt

15、h t y nx nh n y nkx nkh n(Time-Invariant Property)orL( )x t( )h t( )y tL( )h t0()y ttdelay0t0()x ttdelay0torL x n h n y nL h n0y nndelay0t0 x nndelay0t2.3.9 Difference and Differential Properties of ConvolutionDenote :( )( )dx tx tdt 1x nx nx n( )( )( )y tx th t( )( )( )( )( )y tx th tx th t y nx nh

16、 n y nx nh nx nh n orL( )x t( )h t( )y t( )x tL( )h t( )y t2( )( )teu tu tor x nLL x n h n y n h n y n(补充，不证补充，不证) 2( )( )teu tt2( )( )teu tu t2( )teu t2.3.10 Running Sum and Running Integral Properties of Convolution( )( )( )y tx th t( )( )( )( )( )tttydxdh tx thdorL( )x t( )h t( )y t( )txdL( )h tt

17、( )tydt y nx nh n nnnkkky kx kh nx nh korL x n h n y n nkx kL h nnknk nky k 2.3.11 Running Sum/Difference and Running Integral /Differential Properties( )( )( )y tx th t( )( )( )( )( )tty tx thdxdh t y nx nh n nnkky nx nh kx kh n1( )x tt112310t0( )y tt0( )x tL1111200000( )( )( )Lh tx ty t ?011( )( )

18、( )Lh tx ty t 0(2)x t0(2)y t00)(1)(xx tt0( )y t0(1)y tLt2311101( )y t1000( )( )(1)(2)y ty ty ty t1000( )( )(1)(2)y ty ty ty t000( )( )( )x th ty t(given)1000000( )( )( )(1)( )(2)( )y tx th tx th tx th tdelaydistributivecommutative10000( )( )(1)(2)( )y tx tx tx th t110( )( )( )y tx th t1000( )( )(1)(

19、2)x tx tx tx t解：解：L0( )x tL0(1)x t L0(2)x t 1( )x tLinearL000( )( )( )Lh tx ty t 0100( )( )(2)?Lh tx tx t 002(1)?(2)Lh tx t030( )( )?Lh txt t110( )y t0000( )( )(2)( )y tx tx th t00( )(2)(1)y tx th t解：解：t0( )yt110t0( )yt110100( )( )( )y txtht00()(2yytt t1123100(1)y t t11200000( )( )(2)( )x th tx th t

20、00( )(2)y ty t0(1)y t00( )( )x th t0( )ytt110( )y t02.3.4 System With and Without MemoryMemoryless LTI System ( )( )h tkt h nkn2.3.6 Causality for LTI SystemCausal LTI System ( )00h tfor t 00h nfor n( )( ) ()y txh td( )xt(0)h( )h ( )h t()h tfuture input( )0h Causal System 2.3.5 Inevitability of LTI

21、SystemInevitability LTI System 1( )( )( )h th tt1 h nh nnImpulse response of the LTI SystemImpulse response of the inverse LTI System2.3.7 Stability for LTI SystemStability LTI System h k ( )h t dt bookAbsolutely summable Absolutely integrable 2.3.8 The Unit Step Response of LTI SystemA. Unit step r

22、esponse ( )& ( );( )& ( );th tnot easyto getu ts teasyto getL u n s nL n h nL( )u t( )s tL( ) t( )h tnkB. Relations Between Unit Step Response and Unit Impulse Response ( )( )tu td tL( )u t( )s tL( ) t( )h tt( )( )ts thdnk nku nkL u n s nL n h n nks nh k(Proof see the book)( )( )tu t nu n( )

23、( )h ts t h ns n2.4 Causal LTI System Described by Differential and Difference Equations 2.4.1 Linear Constant-Coefficient Equation and Continuous- Time LTI SystemDenote: ( )00 x tfortLinear Constant Differential Equation ( )( )00( )( )NMkkkkkka ytb yt( )00y tfort( )( )( )kkkdyty tdtContinuous-Time

24、LTI SystemCausalImplicit specification explicit specificationgiven( )(0)? (0 1)kykN( ) ( )(0)y tf x tt(附加条件，初始状态)隐式表示显式表示( )( )( )(0)phy tytyttparticular solution (特解)homogeneous solution (齐次解)More powerful solution see chapter 9particular (特解)3( )5tKy te(0)0y5KA 3( )2 ( )( ) ;( )( ) ;(0)0ty ty tx t

25、x tKe u ty( )( )( )phy tytythomogeneous(高等数学知识)( )2 ( )( )y ty tx tsolution any function satisfy 3( )(0)tpytYeta solution( )(0)sthytAetcomplex exponential solution( )2 ( )0y ty tsolutionsolution(齐次解)(自然响应)naturalresponsesimilar to input5KY 2s 333032ttttYeYeKe020ststtAseAe32( )() ( )5ttKy teeu t2tAe,

26、0t 2.4.2 Linear Constant-Coefficient Difference EquationsLinearConstant-CoefficientDifference Equations 00NMkkkka y nkb x nkdescribeCausal 00 x nforn 00y nfornDiscrete-Time LTI Systemgiven ? (1)y kkN Implicit expression 隐式表示显式表示explicit expression ( )(0)y nf x nforn (0)phy nyny nnparticular solution

27、 (特解)homogeneous solution (齐次解)especially for Difference Equations 001 MNkkkky nabx nkay nk, 10;1,01;NN 2,12;,1;NnNnn 递推法求解More powerful tools see chapter 10A. Basic Idea Getting Output Signal (Time-Domain Analysis)( )( ) ()f tftd 信号分解 position BasicSignals响应合成 compositionL( )( ) ()y tfh tdLLL ItsResponseL kf nf knk信号分解 position响应合成 compositionLLLL ky nf k h nk2.5 Singularity Functions (See the Book) 2.6 Summary (Time Domain Solution to LTI Systems) -Convolution I