Laplace Examples, Inverse, Rational Form

Size: px

Start display at page:

Download "Laplace Examples, Inverse, Rational Form"

Katherine Ryan
5 years ago
Views:

1 Lecture 20 Outine: Lapace Eampe, Invere, Rationa Form Announcement: HW 5 poted, due Friday ore Lapace Tranform Eampe Invere Lapace Tranform Rationa Lapace Tranform ROC for Right/Left/Toided Signa agnitude/phae of Fourier Tranform from Lapace

2 Revie of Lat Lecture Lapace tranform generaize Fourier Tranform Aay eit ithin a Region of Convergence Ued to tudy ytem/igna /out Fourier Tranform Defn: L Region of Convergence (ROC): A vaue of + uch that L[(t)] eit Depend ony on Eampe t t [ ] e dt, + ; eit if e dt < [ ] F Reation ith Fourier Tranform: L ( ) F[ ] e at u [ e t ] L + a, Re ( ) > a Smaet : () eit trip aong ai () () defined in ROC Potted on pane

3 Lapace Tranform Eampe Rect function: æ t ö ì Pç í è 2t ø î0 Famiiar friend; ha a Fourier tranform t t t t e e Lapace Tranform: 2 e dt t d t t ( e e ) d ROC: Finite everyhere ecept poiy 0: im 2 im 0 0 d d So ROC i entire pane: ROC{a }: true for any finite duration aoutey integrae function Fourier tranform (ÎROC): 2 t t t > t 2 e t e t t in t t 2 0 t t ç è p æ ö 2t 2t inc ø 2t t LeftSided Rea Eponentia: Lapace: 4 0 ( ) e +a t dt 4 + a, Re < a at e u( t), converge if Re()<a Same a for rightided cae ut ith a different ROC (Re()>a) Doe not have a Fourier tranform if a<0, ee a t

4 Another Eampe: ToSided Rea Eponentia e t, rea 5 Can rite a um of to term: 5 t t e u + e u( t) «, Re ( ) > e t u ( t) e t u «+, < Re ( )( + ) { Re( ) > } Ç{ < } ROC Re

$through partia fraction epanion Ao anayi through poezero pot 2p t t e F [ ( + ) ] ( + ) e d ( + t e ) t$

5 Invere Lapace Tranform Ue Fourier Reation: Can appy invere Fourier: utipy oth ide y e t : ( + ) F Ho to integrate over pane: change of variae +; fied, dd: 2p + t e d Compe integration part of compe anayi (ath 6) We need a different approach Rationa Lapace Tranform: t t t [ e ] [ e ] e dt Invere otained through partia fraction epanion Ao anayi through poezero pot 2p t t e F [ ( + ) ] ( + ) e d ( + t e ) t + d 2 p

6 Rationa Lapace Tranform umerator and Denominator are poynomia Can factor a product of monomia are zero (here ()0), g are poe (here ()) ROC cannot incude any poe If () rea, a and are rea a zero of () are rea or occur in compeconugate pair. Same for the poe. m /a n, zero, poe, and ROC fuy pecify () Eampe: Toided Eponentia ( )( + )

7 ore on Lapace ROC for Right, Left, and ToSided Signa Rightided: (t)0 for t<a for ome a ROC i to the right of the rightmot poe, e.g. RH eponentia Leftided: (t)0 for t>a for ome a ROC i to the eft of the eftmot poe, e.g. LH eponentia Toided: neither right or eft ided ROC i a vertica trip eteen to poe, e.g. 2ided eponentia agnitude/phae of Fourier from Lapace Given Lapace in rationa form ith in ROC Can otain magnitude and phae from individua component uing geometry, formua, or ata Õ Õ a H g Õ Õ a H g Õ Õ a H g å å Ð Ð + ø ö ç ç è æ Ð Ð a H g

8 OW pg. 69, Reader pp (Simiar to CTFT propertie Repace ith, chec ROC) Ued for LTI Sytem Anayi Ued to ove ODE 0 Appie to caua igna; Uefu to chec initia and fina vaue of (t) ithout inverting ()

9 ain Point Lapace tranform incude the ROC; different function can have ame Lapace tranform ith different ROC Invere Lapace tranform require compe anayi to compute: need a different approach Epreing the Lapace tranform in rationa form ao invere via partia fraction epanion Ao ao Lapace characterization via poezero pot OneSided igna have ided ROC. To ided igna have trip a ROC Can determine magnitude and phae of Fourier tranform eaiy from rationa form of Lapace Propertie of Lapace tranform imiar to thoe of Fourier tranform, ith imiar proof. ut chec ROC.

BSM510 Numerical Analysis

BSM510 Numerical Analysis BSM510 Numerica Anaysis Roots: Bracketing methods : Open methods Prof. Manar Mohaisen Department of EEC Engineering Lecture Content v Introduction v Bracketing methods v Open methods v MATLAB hints 2 Introduction