15/03/1439. Lecture 4: Linear Time Invariant (LTI) systems

Transcription

1 Lecre 4: Liner Time Invrin LTI sysems 2. Liner sysems, Convolion 3 lecres: Implse response, inp signls s coninm of implses. Convolion, discree-ime nd coninos-ime. LTI sysems nd convolion Specific objecives for ody: We re looking discree ime signls nd sysems Undersnd sysem s implse response properies Sow ow ny inp signl cn be decomposed ino coninm of implses DT Convolion for ime vrying nd ime invrin sysems LTI sysems Two imporn bsic Properies of sysems: Lineriy. Time-invrince sperposiion propery. Plys n imporn role in signls nd sysems nlysis. Mny of pysicl processes possess ese properies. Tey re modeled s LTI sysems Any sysem possess ese wo properies is clled liner ime- invrin LTI sysem. EE-227 SS, L4: /7 EE-227 SS, L4: 2/7 LTI sysems Properies Represenion of DTS in Terms of Implses A DTS [n] cn be viewed s seqence of individl implses or s liner combinion of ime-sifed implses: Develop complee crcerizion of LTI sysem in erms o is implse response sing convolion sm for DTS nd convolion inegrl for CTS. 3/7 4/7

2 Represenion of DTS in Terms of Implses Represenion of DTS in Terms of Implses 5/7 6/7 Discree Implses & Time Sifs Represenion of DTS in Terms of Implses Bsic ide: se infinie se of of discree ime implses o represen ny signl. Consider ny discree inp signl [n]. Tis cn be wrien s e liner sm of se of ni implse signls: [ ] n [ ] [ n ] n [ ] [ n ] [] n [] [ n] n [] n [] [ n ] cl vle Implse, ime n sifed signl Terefore, e signl cn be epressed s: [ n] [ 2] [ n 2] [ ] [ n ] [] [ n] [] [ n ] In generl, ny discree signl cn be represened s: k [ n] [ k] [ n k] Te sifing propery EE-227 SS, L4: 7/7 8/7 2

3 Te discree signl [n] Is decomposed ino e following ddiive componens [-4][n+4] + Emple [-3][n+3] + [-2][n+2] + [-][n+] + DT LTI sysems: e convolion sm Is e memicl relionsip links e inp nd op signls in ny LTI discree-ime sysem. Given n: LTI sysem, Inp signl [n]. Implse Response H[n]: e response o one of e bsic signls sc s implse signl; Te convolion sm will llow s o compe e corresponding op signl y[n] of e sysem. Compe y[n]?????????? 9/7 /7 Inrodcion o Convolion Definiion Convolion is n operor kes n inp signl nd rerns n op signl, bsed on knowledge bo e sysem s ni implse response [n]. Response of LTI s liner combinion of implse response [n] = [n] [n] Sysem Sysem: [n] y[n] = [n] y[n] Te bsic ide beind convolion is o se e sysem s response o simple inp signl o clcle e response o more comple signls Tis is possible for LTI sysems becse ey possess e sperposiion propery lecre 3: n] k k [ k k y n] [ n] [ n] [ n] [ n] [ k k y [ n] y [ n] y [ n] y [ n] EE-227 SS, L4: /7 2/7 3

4 Response of LTI s liner combinion of implse response Response of LTI s liner combinion of implse response 3/7 4/7 Response of LTI s liner combinion of implse response Convolion Sm 5/7 6/7 4

5 Convolion Sm Response of LTI s combinion of H[n] 7/7 8/7 Convolion sm Convolion sm 9/7 2/7 5

6 Emple : LTI Convolion Emple 2: LTI Convolion A LTI sysem wi e following ni implse response: [n] = [ ] For e inp seqence: [n] = [.5 2 ] Te resl is: y[n] = + [][n] + [][n-] + = +.5*[ ] + 2.*[ ] + = [ ] Consider e problem described for emple Skec [k] nd [n-k] for ny priclr vle of n, en mliply e wo signls nd sm over ll vles of k. For n<, we see [k][n-k] = for ll k, since e nonzero vles of e wo signls do no overlp. y[] = S k [k][-k] =.5 y[] = S k [k][-k] =.5+2 y[2] = S k [k][2-k] =.5+2 y[3] = S k [k][3-k] = 2 As fond in Emple EE-227 SS, L4: 2/7 EE-227 SS, L4: 22/7 Convolion sm Emple 5: Compe y[] for e inp signl nd implse response of n LTI sysem sown in e following Figre. Convolion sm Emple 5: Compe y[] for e inp signl nd implse response of n LTI sysem sown in e following Figre. 23/7 24/7 6

7 Convolion sm Convolion sm 25/7 26/7 Emple 3: LTI Convolion Convolion sm Consider LTI sysem s sep response [n] = [n] o e ni implse inp signl W is e response wen n inp signl of e form [n] = n [n] were <<, is pplied? For n: n k y[ n] Terefore, k n n y[ n] [ n] EE-227 SS, L4: 27/7 28/7 7

8 Convolion sm Convolion sm 29/7 3/7 Convolion sm Discree, Uni Implse Sysem Response A very imporn wy o nlyse sysem is o sdy e op signl wen ni implse signl is sed s n inp [n] Sysem: q [n] Loosely speking, is corresponds o giving e sysem kick n=, nd en seeing w ppens Tis is so common, specific noion, [n], is sed o denoe e op signl, rer n e more generl y[n]. Te op signl cn be sed o infer properies bo e sysem s srcre nd is prmeers q. 3/7 EE-227 SS, L4: 32/7 8

9 Types of Uni Implse Response Looking ni implse responses, llows yo o deermine cerin sysem properies Csl, sble, infinie implse response y[n] = [n] +.7y[n-] Csl, sble, finie implse response y[n] = [n] +.5[n-] +.25[n-2] Liner, Time Vrying Sysems If e sysem is ime vrying, le k [n] denoe e response o e implse signl [n-k] becse i is ime vrying, e implse responses differen imes will cnge. Ten from e sperposiion propery Lecre 3 of liner sysems, e sysem s response o more generl inp signl [n] cn be wrien s: Inp signl [ n] [ k] [ n k] k Sysem op signl is given by e convolion sm k y [ n] [ k] [ n] k i.e. i is e scled sm of implse responses Csl, nsble, infinie implse response y[n] = [n] +.3y[n-] EE-227 SS, L4: 34/7 EE-227 SS, L4: 33/7 Emple: Time Vrying Convolion [n] = [.5 ] - [n] = [ ] [n] = [ ] y[n] = [ ] Liner Time Invrin Sysems Wen sysem is liner, ime invrin, e ni implse responses re ll ime-sifed versions of ec oer: k [ n] n k I is sl o drop e sbscrip nd simply define e ni implse response [n] s: [ n] n In is cse, e convolion sm for LTI sysems is: k y [ n] [ k] [ n k] I is clled e convolion sm or sperposiion sm becse i involves e convolion of wo signls [n] nd [n], nd is someimes wrien s: y[ n] [ n]* [ n] EE-227 SS, L4: 35/7 EE-227 SS, L4: 36/7 9

10 Sysem Idenificion nd Predicion Noe e sysem s response o n rbirry inp signl is compleely deermined by is response o e ni implse. Terefore, if we need o idenify priclr LTI sysem, we cn pply ni implse signl nd mesre e sysem s response. T d cn en be sed o predic e sysem s response o ny inp signl [n] Sysem: [n] Noe describing n LTI sysem sing [n], is eqivlen o descripion sing difference eqion. Tere is direc mpping beween [n] nd e prmeers/order of difference eqion sc s: y[n] = [n] +.5[n-] +.25[n-2] y[n] Discree LTI Convolion in Mlb In Mlb o find o bo commnd, yo cn serc e elp files or ype: >> lookfor convolion e Mlb commnd line. Tis rerns ll Mlb fncions conin e erm convolion in e bsic descripion Tese inclde: conv To see ow is works nd oer fncions my be pproprie, ype: >> elp conv e Mlb commnd line Emple: >> = [ ]; >> = [.5 2 ]; >> y = conv, >> y = [ ] EE-227 SS, L4: 37/7 DT Uni-Implse Response Consider e DT SISO sysem: Generl Response n [ ] yn [ ] Sysem If e inp signl is nd e sysem s no energy, e op is clled e implse response [ n] [ n] of e sysem n y[ n] [ n] Emple Consider e DT sysem described by y[ n] y[ n ] b[ n] Is implse response cn be fond o be n b, n,,2, n [ ], n, 2, 3, [ n] Sysem Implse Response n [ ]

11 Represening Signls in Terms of Sifed nd Scled Implses Eploiing Time-Invrince nd Lineriy Le [n] be n rbirry inp signl o DT LTI sysem Sppose for Tis signl cn be represened s n [ ] n, 2, [ n] [] [ n] [] [ n ] [2] [ n 2] [ i] [ n i], n,,2, i y[ n] [ i] [ n i], n i Te Convolion Sm Tis priclr smmion is clled e convolion sm Block Digrm Represenion of DT LTI Sysems Since e implse response [n] provides e complee descripion of DT LTI sysem, we wrie y[ n] [ i] [ n i] Eqion iis clled e convolion represenion of e sysem [ n] [ n] Remrk: DT LTI sysem is compleely described by is implse response [n] y[ n] [ n] [ n] n [ ] n [ ] yn [ ]

12 Emple: Sppose bo [n] nd v[n] re eql Associiviy Commiviy Properies of e Convolion Sm [ n] v[ n] w[ n] [ n] v[ n] w[ n] Disribiviy w.r.. ddiion [ n] v[ n] v[ n] [ n] [ n] v[ n] w[ n] [ n] v[ n] [ n] w[ n] Plo of [ n] v[ n] Properies of e Convolion Sm - Con d q[ n] [ n q] Sif propery: define vq[ n] v[ n q] en w[ n] [ n] v[ n] w[ n q] [ n] v[ n] [ n] v [ n] Convolion wi e ni implse q Convolion wi e sifed ni implse [ n] [ n] [ n] [ n] [ n] [ n q] q q Emple: Comping Convolion wi Mlb n [ ] n [ ] Consider e DT LTI sysem implse response: inp signl: n=:4; =sin.2*n; =sin.5*n; y=conv,; semn,y:lengn [ n] sin.5 n, n [ n] sin.2 n, n yn [ ] 2

13 Convolion sm Convolion sm 49/7 5/7 CT Uni-Implse Response Eploiing Time-Invrince Consider e CT SISO sysem: Sysem If e inp signl is nd e sysem s no energy, e op is clled e implse response of e sysem y y Le [n] be n rbirry inp signl wi for, Using e sifing propery of, we my wrie Eploiing ime-invrince, i is d, Sysem Sysem 3

14 Eploiing Time-Invrince Eploiing Lineriy Eploiing lineriy, i is y d, If e inegrnd does no conin n implse loced, e lower limi of e inegrl cn be ken o be,i.e., y d, Te Convolion Inegrl Tis priclr inegrion is clled e convolion inegrl Block Digrm Represenion of CT LTI Sysems Since e implse response provides e complee descripion of CT LTI sysem, we wrie y d, Eqion is clled e convolion represenion of e sysem Remrk: CT LTI sysem is compleely described by is implse response y y 4

15 Emple: Anlyicl Compion of e Convolion Inegrl were p, p is e recnglr Sppose plse depiced in figre p T In order o compe e convolion inegrl y d, we ve o consider for cses: Emple Con d Cse : Cse 2: T y d T T Cse 3: T T T 2T T y d T T 2T T Cse 4: T T 2T y T T y T T T T y T 2T Properies of e Convolion Inegrl Associiviy v w v w Commiviy Disribiviy w.r.. ddiion v v Properies of e Convolion Inegrl - Con d q q vq v q en w v w q v v Sif propery: define Convolion wi e ni implse q Convolion wi e sifed ni implse q v w v w q q 5

16 6 Convolion Inegrl - Properies ] * [ ] * [ ] *[ ] * *[ ]* * [ * * Commive Associive Disribive Emple Consider CT-LTI sysem. Assme e implse response of e sysem is =e^- for ll > nd > nd inp =. Find e op. =e^- y e e d e d e d y y Drw,, -,ec. ne slide Becse > Te fc > is no n isse! Emple Con. y > < Remember we re ploing i over nd is e vrible U-- U-- e e d e d e d y y y; for =3 64 Emple 2.5: Convolion Inegrl. Given RC circi below RC=s. Use convolion o deermine e volge cross e cpcior y. Inp volge =--2. Solion: y=* - cpcior sr crging = nd discrging =2. b

17 Con d Con d