Laplace Examples, Inverse, Rational Form

Transcription

1 Lecure 3 Ouline: Lplce Exple, Invere, Rionl For Announceen: Rein: 6: Lplce Trnfor pp. 3-33, , 7 HW 8 poe, ue nex We. Free -y exenion OcenOne Roo Tour will e fer cl y 7 (:3-:) Lunch provie ferwr. Cn rrne epre our for hoe w/conflic Lplce Trnfor Pir/Properie (pp only) LTI Sye Anlyi uin hee Properie ROC for Rih/Lef/Two-ie Sinl niue n Phe of FT fro Lplce Boe Plo Invere of Rionl Lplce Trnfor

2 Review of L Lecure Lplce Exple Rec funcion: x ROC e, Lef-Sie Rel Exponenil: Two-Sie Rel Exponenil e < e u ) 4, Re( 4 x 5 e, rel inc Invere Lplce rnfor ue coplex nlyi Cn oin invere for rionl Lplce rnfor uin iple h (pril frcion expnion) 5 ROC Re Re

3 Review Con Rionl Lplce Trnfor Zero n Pole: re zero (where ()=), re pole (where ()=) ROC cnno inclue ny pole If () rel, ll zero of () re rel or occur in coplexconue pir. Se for he pole. / n, zero, pole, n ROC fully pecify () Exple: Two-ie Exponenil Typo in We lecure: no zero in 5 () A B 5

4 OW p. 69, Reer pp. -7 (Siilr o CTFT properie Replce wih, chec ROC) Ue for LTI Sye Anlyi Ue o olve ODE Applie o cul inl; Ueful o chec iniil n finl vlue of x() wihou inverin ()

5 LTI Sye Anlyi uin hee Properie LTI Anlyi uin convoluion propery x H h y y( ) x( )* h( ) Y ( ) H ( ) ( ) H() clle he rnfer funcion of he ye ROCROC x ROC h ROC Equivlence of Sye (explore in HW) x G h H y x * h G H y x G h H S Follow fro lineriy of convoluion n of he Lplce Trnfor y x G h H y Follow fro couiviy of convoluion n he convoluion propery of he Lplce Trnfor

6 (Siilr o CTFT pir Replce wih, chec ROC) OW p. 69

7 ROC for Sie Sinl Rih-ie: x()= for < for oe ROC i o he rih of he riho pole Exple: RH exponenil L x x e Lef-ie: x()= for > for oe ROC i o he lef of he lefo pole Exple: LH exponenil Lx x e Two-ie: neiher rih or lef ie Cn e wrien u of RH n LH ie inl ROC i vericl rip eween wo pole, -ie exponenil

8 niue/phe of Fourier fro Lplce n Boe Plo niue/phe of Fourier fro Lplce Given Lplce in rionl for wih in ROC Cn oin niue n phe fro iniviul coponen uin eoery, forul, or l Boe Plo: how niue/phe on B cle Plo lo H() =lo H() v. (B) Cn plo excly or vi rih-line pproxiion Pole le o B per ece ecree in Boe plo, Zero le o B per ece incree Frequency in r/ i ploe on lo cle for only H H H H

9 Exple: Fir-orer LPF RC= H h e u, rel H ROC { / } ROC H H 4 4 H ω ω ω

10 lo H() (B) H() (r) Boe Plo for Orer LPF B for. r for B/ece for 4 r for Boe Exc.. Boe Exc. r for

11 Inverion of Rionl Lplce Trnfor Exrc he Sricly Proper Pr of () If <, i ricly proper, procee o nex ep If, perfor lon iviion o e, where Inver D() o e ie inl: Follow fro n The econ er i ricly proper Perfor pril frcion expnion: Inver pril frcion expnion er-y-er For rih-ie inl: A B, D D, n n n Z z n n n A B / B A B p p B p p A A Oin coefficien vi reiue eho p p u e A u e A x!

12 in Poin Properie of Lplce rnfor iilr o hoe of Fourier rnfor, wih iilr proof. u chec ROC. Convoluion n oher properie of Lplce llow u o uy inpu/oupu relionhip LTI ye Cn lo uy LTI ye in erie or prllel One-Sie inl hve one-ie ROC. Two ie inl hve rip ROC Cn eerine niue n phe of Fourier rnfor eily fro rionl for of Lplce Boe plo how niue/phe on B cle Pole chne lope y -B/ece, zero chne lope y B per ece Inver rionl Lplce rnfor uin pril frcion expnion