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1、Contents2.1 Discrete LTI systems: the convolution sum2.2 Continuous LTI systems: the convolution integral2. 3 Properties of LTI systems2.4 Causal LTI systems described by differential or difference equations 2.5 Singularity functionsDiscrete LTI systems: the convolution sumConsider signal xn: An arb

2、itrary sequence can be represented as a linearcombination of shifted unit impulses n-k, wherethe weights in this linear combination are xk.Write as :-4-3 -2-10 12 34nxnDiscrete LTI systems: the convolution sum ( sifting property 篩選性篩選性 ) kknkxnx or.2-n x21-n x1 0 1 1 2 2.+nxnxnxnxDiscrete LTI system

3、s: the convolution sumLet hn denote the response of linear system to n. i.e. hn the unit impulse response.then, each n of xn response: . .kkxnxk-n nh0 xn0 x 1nh1x1n1x knhkxknkx Discrete LTI systems: the convolution sum . . This result is referred to as the convolution sum, and the operation on the r

4、ight-hand side of equation is known as the convolution sum of xn and hn.knhkxknkx k(2.6) knhkxyn Discrete LTI systems: the convolution sumWe represent the operation as yn =xn hn (2.7)The same, is referred to as the convolution sum of xn and n.Some notes:1) An LTI system is completely characterized b

5、y its hn. kk-n kxnxnnx Discrete LTI systems: the convolution sum2) The graph of convolution sum.a)Transform independent variable: xn, hn xk, hk and hk h-kb)Shift h-k n steps hn-k.c)For any n, xk multiplied by hn-k and xkhn-k . k knhkxyn Discrete LTI systems: the convolution sum Example 2.2 determine

6、 yn=xn hn.Answer:(a) kknhkxnyDiscrete LTI systems: the convolution sum (b) Shift h-k to the right (n0) or to the left (n0).n0 toward right ; otherwise toward left Multiply together Integral )()( xtx)()( hth)()( hh)()( thh)()( thx dthx)()( Continuous LTI systems: the convolution integral CompareExamp

7、le 2.6:Answer: )t ()t (xd)t ()(x)t (x)1()1()( txttx )3()2()1( ttt u(t) h(t) 0a ),t(u ex(t) whereat - d )t (u)(ue)t ( y aContinuous LTI systems: the convolution integralAnswer:For t00)()( thx0)( ty tdthxty0)()()( tade0 )1 (110attateaea 01)(h 01)(x t 0)t (h t0t10t0h(t- )1Continuous LTI systems: the co

8、nvolution integral for all tExample:DetermineAnswer:)()1(a1)y(tuetat 01e2 e-(t- )t 01e2 te-(t- )a1 0y(t)t)()()(2tuetuetytt 0 1e2 x( )=h( )=e- deedeetyttt)(0)( 2 )(+)(31)(312tuetuett + + (t0)Continuous LTI systems: the convolution integral 01e2 e-(t- )t 01e2 te-(t- )0y(t)tte32te31 31)(31)(31)(2tuetue

9、tytt + + Continuous LTI systems: the convolution integralExample 2.7 consider:And then determine x(t)= 1, 0tT 0, otherwiseh(t)=t, 0t2T0, otherwise)()()(thtxty 1T0 x(t)tT 2T2Th(t)0tContinuous LTI systems: the convolution integralAnswer:(1) t3T (2) 0tT(3)Tt2T dtdthxtytt 00)()()()(2222121ttt 2021 )()(T

10、tTdttyT 0)( ty1T0 x( )t-2T 0tT1Tt2T)t(h )(x T 2T2Th( )0t-2T 0 tT10tT)t (h )(x Continuous LTI systems: the convolution integral(4) 2Tt3Ty(t)=0, t3TAssignments (P139) 2.10 , 2.11 0Tt dttyTTt)()(2 222321TTtt-+ + + t-2T0tT1)t (h )(x ,t21t21t222 2TtT ,21 2TtT ,232122TTtt+ + + 32TtT tT2T 3T. y(t)0234T221T

11、Properties of convolution The commutative propertynxnhnhnx )()()()(txththtx x(t)y(t)h(t)x(t)y(t)h(t)Properties of convolution The distributive propertyAlso, )(2121nhnxnhnxnhnhnx + + + + ）（）（）（）（thtxthtxththtx2121()( + + + + x(t)）t (h1）t (h2y(t)）t(h1）t(h2x(t)y(t)(2121nhnxnhnxnhnxnx + + + +Properties

12、of convolution The associative property)()(2121nhnhnxnhnhnx 21nhnhnxny (c)ynxnnh=*nh1nh2(a)xnynnh1nh2(d)xnynnh1nh2(b)ynxnnh=*nh1nh2Properties of LTI systems LTI systems with and without memoryFor discrete system without memory: hn=0 for n0In this casewhere k=h0 is a constant, and convolution sum isn

13、knh kknhkxnhnxny )()()()()( thxthtxtynkxny Properties of LTI systemsWith memory: hn 0 for n0.For continuous system without memory: h(t)=0 for t0 k a constantIf k=1, systems become identity systems, and)()()()(tkxtytkth nnxnx )()()(ttxtx Properties of LTI systems Invertibility of LTI systemIf then th

14、e system with is the inverse of the system with .)()()(10tthth )(0th)(1thx(t)h0(t)w(t)=x(t)(a)h1(t)x(t)identity system h(t)= (t)x(t)(b)Properties of LTI systemsSimilarly, if then system of is the inverse system of Example: consider determine10nnhnh 0nh1nh0nunh 1nh110nnhnunhnhhold 11 nnnh )1( 1 nnnun

15、unun Properties of LTI systems Causality for LTI systemsIf hn=0, for n0 or h(t)=0, for t0 then the system is causal. In this case0 knkorknxkhknhkxny0)()( )()()(dtxhthxtyortProperties of LTI systems Stability for LTI systems absolutely summable. absolutely integrable.Example: if then is stable if isn

16、t stable Assignments: 2.28 (a)(d),2.29(a)(d)()(0ttth kkh| dh| )(|1|)(| dt)()()(tudtht 0)( dduProperties of LTI systems The unit step response of an LTI system The unit step response, s(t) or sn,corresponding to the output when xn=un or x(t)=u(t) nkkkhknukhnhnuns 1 nsnsnh tdhdtuhthtuts )()()()()()(Pr

17、operties of LTI systems Assignments: P139 2.12, 2.23(t) sdt)t (ds)t (h Causal LTI systems described by equations Linear constant-coefficient differential equationsA first-order a general Nth - order (2.109)()(2)(txtydttdy + + mkkkkNkkkkdttxdbdttyda00)()(Causal LTI systems described by equations(1) t

18、he response to an input x(t) will generally consist of the sum of a particular solution and a homogeneous solution, i.e.(2) In order to determine y(t), we must specify auxiliary conditions.(3) We will use the condition of initial rest as auxiliary condition. That is, if x(t)=0 for , we assume that)(

19、)()(tytytyhp+ + 0tt 0)(.)( 10100 NNdttyddttdy)y(t(4) Under the condition of initial rest, the system described by Eq. (2.109) is causal and LTI. determine h(t) from eq(2. 109), see problem 2.56. Causal LTI systems described by equations Linear constant coefficient difference equationsThe nth-order(1

20、) In a manner exactly analogous to that for differential(2) Auxiliary condition also is initial rest, i.e if xn=0 for nn0, then yn=0 for n 0, integer0k ),t (u . ),t (u ),t (uk21 dt)t (d)t ()t (u1 kkkdttxdtutx)()(*)( Singularity functions Examples k timesfor unit doublet (t)= u1(t), if x(t)=1, then)(

21、*)()(*)(dt(t)d1ttxtutxx)t (u)t (u*)t (x)t (u*)t (xdt)t (xd11222 )t (u)t (u)t (u e . i112 )t (u.)t (u)t (u11k )t (u*)t (xdt)t (dx1 0d )(ud )t (x)u11（Singularity functions(2) If kth derivatives of x(t) exist and continuous at t=0, then)t (u)t ( of0 area 1 1 t)t ( 0)t (u)t ()t (lim10 0t)t( )0() 1()()()

22、(kkkxdttutx0tkk(k)dt)t(xd)0(x here Singularity functions Show: apply kth derivative to two sides for t0 -00)t (xdt)t-(tx(t) )t (xdt)tt ()t (x)1(0)k(0)k(k- ) 0 ()()( 1 xdtttx kif then , 0 t0let0(k)()0(tkkdttxdxSingularity functions 3. Denominatethen k0, integerExample an integrator),(),.( ),( ,)d( ,d)(21tututuask timesk-)(.)(u*(t)ktttdxtx t-)dx(y(t) d )t ()t (u)t (h itst d )(x)t (u*)t (x t t2)t (tu)t (ud )(u)t (u*)t (uSingularity functions tu(t) unit ramp function.h(t) of k-times integrator cascade is For any integers k and r101tu -2(t)=tu(