Chapter 5 The Laplace Transform. x(t) input y(t) output Dynamic System
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1 EE 422G No: Chapr 5 Inrucor: Chung Chapr 5 Th Laplac Tranform 5- Inroducion () Sym analyi inpu oupu Dynamic Sym Linar Dynamic ym: A procor which proc h inpu ignal o produc h oupu dy ( n) ( n dy ( n) + a n ( a n b dx ( m) m ( b m Quion: Can w rmin for a givn? Anwr: U Fourir ranform o convr h ODE ino algbraic quaion! X(f) Y(f ) Dynamic Sym H(f ) Y ( f ) H ( f ) X ( f ) F ( Y ( f )) (2) Two Problm. Som common ignal do no hav a Fourir Tranform! Exampl: Wha i h Fourir ranform of? X j 2πf j 2πf ( f ) blow up whn. do no xi for any f a Evn hough x ( grow unboundd a, i may ill xi a an inrmdia p in a largr ym. Conidr h oupu whn x ( i fd 2 ino a ym wih impul rpon h( ( 3 2 ). Pag 5- PDF Crad wih dkpdf PDF Wrir - Trial :: hp://
2 EE 422G No: Chapr 5 Inrucor: Chung which crainly dcay o a τ 2τ τ τ )( 3 2 ) τ 2τ τ ( 3 2 ) dτ 2 ( ). τ ) dτ 2. Iniial Condiion Problm Say w know h oupu of h abov dynamic ym i 5 a. Nowhr in h Fourir ym quaion blow w could inr hi informaion: Y ( f ) H ( f ) X ( f ) F ( Y ( f )) (3) Soluion: Laplac Tranform Evn hough do no go o zro (whn nough α. ), bu σ may for larg W will aum all ignal ar caual : x ( for < Fourir ranform of ( σ ) jω σ : ( σ + jω) whr σ + jω Laplac Tranform of W will how Laplac ranform ak car of iniial condiion lar. Evn hough invr Laplac Tranform xi, i involv mor ophiicad concp from complx numbr hory. Ju lik Invr Fourir Tranform, w will ju u abl (and Malab). Laplac ranform i ju a nic a Fourir Tranform: Y ( ) H ( ) X ( ) + iniial condiion L ( Y ( )) Fourir Tranform of x ( can obaind by ubiuing jω (i.. ing σ ) in X ().. If h Rgion of Convrg of X() do no includ h imaginary axi ( ω ), hn i Fourir Tranform do no xi. (Mor lar) j Pag 5-2 PDF Crad wih dkpdf PDF Wrir - Trial :: hp://
3 EE 422G No: Chapr 5 Inrucor: Chung 5-2 Exampl of Evaluaing Laplac Tranform uing h dfiniion () Sp funcion L[ ] d( lim ( ) + R( ) j Im( ) lim ( ) + / if R( ) > a if R( ) < a no ur if R( ) a lim R( ) R( ) j Im( )? Whn R(), dfin ωim(). Th ingral bcom jω u jω ( ) which i h Fourir ranform of. From chapr 4, w know ha F [ ] 2πδ ( ω) jω + No ha for ω, h Fourir Tranform can b valuad by ubiuing jω in h xprion /. α (2) Exponnial L[ α ] α ( α ) ( α ) + + d ( α) α ( α ) lim + + α + α lim + α /( + α) if if /( + α) + 2πδ (Im( )) if R( α ) j Im( α ) ( ) + + α R( ) > R( α) a R( ) < R( α) a R( ) R( α ) a lim R( α ) R( α ) j Im( α )? Pag 5-3 PDF Crad wih dkpdf PDF Wrir - Trial :: hp://
4 EE 422G No: Chapr 5 Inrucor: Chung (3) δ ( Rcall ha h impul funcion i rprnd a a limi of convnion funcion raddling h origin. To incorpora h full dla funcion, w dfin h lowr limi of our Laplac ingral o b -. L[ δ ( ] No conrain on. σ δ ( σ (coω 5-2B Rgion of Convrgnc (ROC) jω jinω ROC: Picorial dcripion of h valu whr h Laplac Tranform of a funcion xi. Anaomy of ROC Im() Rgion whr X() i infini. Im() R() Rgion whr X() i fini, bu no abolu convrgn. α R(-α) R() ROC of / Rgion whr X() i fini. In fac, i convrg aboluly ( blow). ROC of /(α) Propri of ROC. I i bordrd by h RIGHTMOST POLES of X(). A pol i dfind by h complx valu uch ha h algbraic X( ) i infini. 2. If i in h inrior of h ROC, hn all wih R()R( ) ar inid h ROC. 3. I xnd o poiiv infiniy Pag 5-4 PDF Crad wih dkpdf PDF Wrir - Trial :: hp://
5 EE 422G No: Chapr 5 Inrucor: Chung In h inrior of ROC (no boundary), no only do h Laplac ingral convrg, i convrg ABSOLUTELY. Normal convrgn (or X() i fini): Abolu convrgn: < X ( ) < W canno prov ha wihou vnuring ino a branch of mahmaic calld h complx variabl hory. Howvr, knowing ha X() convrg aboluly inid h ROC, w can xplain om of h propri: For xampl: 2. If i in h inrior of h ROC, hn all wih R()R( ) ar inid h ROC. Why? xp( R( j Im( ) xp( R( ) xp( j Im( ) 2 2 No ha xp( j Im( ) co (Im( ) + in (Im( ), w hav xp( R( ) which do no dpnd on Im(). 3. ROC xnd o poiiv infiniy. Aum p i in h ROC. L q b a complx numbr wih R( q ) > R( p). q xp( R( q) xp( R( p) <. Th cond la inqualiy i du o h fac ha xp( R( q) xp( R( p). Thu q mu alo b in h ROC. Exampl: 2 ( 4) X ( ) ha wo pol a - and -3 a wll a a doubl zro a 4. I ROC ( + )( + 3) i {: >-}. Pag 5-5 PDF Crad wih dkpdf PDF Wrir - Trial :: hp://
6 EE 422G No: Chapr 5 Inrucor: Chung Final no: Fourir Tranform: if h imaginary axi ( jω ) i nirly inid h ROC of X (), hn h Fourir Tranform X ( f ) xi. If jω i nirly ouid, X ( f ) do no xi. If jω i h boundary of h ROC, h Fourir ranform mu b valuad by ohr man du o h prnc of pol (.g. Try ) Pag 5-6 PDF Crad wih dkpdf PDF Wrir - Trial :: hp://
7 EE 422G No: Chapr 5 Inrucor: Chung No: Uing malab o find Laplac Tranform >> x ym( co(omga* ) x co(omga* >> X laplac(x) % Laplac Tranform X /(^2+omga^2) >> prx) omga >> X2 ym( omga/(^2+omga^2) ) X2 omga/(^2+omga^2) >> x2 ilaplac(x2) % Invr Laplac x2 in(omga* >> dirac() % Dla Funcion an inf >> dirac(.45) an >> laplac(ym( dirac( )) an >> haviid() % Sp Funcion an NaN >> haviid(-3) an >> haviid() an >> laplac(ym( haviid( )) an / Pag 5-7 PDF Crad wih dkpdf PDF Wrir - Trial :: hp://
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