lecture 3

lecture 3 - 1 1 Summary of Lecture#3 • Examples of...

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Unformatted text preview: 1 1 Summary of Lecture #3 8/27/2010 • Examples of Laplace transform • Remarks on Laplace transform • Inverse Laplace transform • Inv. Laplace trans. by partial fraction expansion of rational functions • Why rational functions? • Single poles • Double poles • Triple and higher order poles • Complex poles • Four examples 2 One-sided or unilateral Laplace transform Complex s , s = σ + j ϖ Region of convergence (ROC) Abscissa of convergence (AOC) Pole or poles of F(s) Inverse Laplace transform dt e ) t ( f ) s ( F )) t ( f ( st ∫ ∞-- ≡ ≡ L ) t ( f ds e ) s ( F 2 j 1 )) s ( F ( st 1- = π ≡ ∫ Γ + L σ j ϖ ROC Γ 0- ∞ 3 Ex. 3: Find Laplace transform of Ku(t) . ≥ < = t 1 t ) t ( u L (Ku(t)) exists for exists for exists for exists for Re(s) > 0 . ROC: ROC: ROC: ROC: Re(s) > 0 AOC: σ = 0 Pole: s = 0 st s st t (f (t)) f (t) e dt e dt Ku(t) e K K if Re(s) s s--- ∞-- ∞ ∞- = =- = = ∫ ∫ L s = σ + j ϖ Re (s) Im (s) 4 Ex. 4: Find Laplace transform of e- at u(t). < ≥ =- t t e ) t ( f at L (e (e (e (e-α t u(t) ) exists for Re(s) > ) exists for Re(s) > ) exists for Re(s) > ) exists for Re(s) >- a. a. a. a. ROC: Re(s) > ROC: Re(s) > ROC: Re(s) > ROC: Re(s) >- a AOC: σ = - a Pole : s = - a at ( st st a s)t (f (t)) f ( e e 1 if Re(s) a (a s) t) e dt a e s dt--- ∞ ∞--- ∞- + = = =- + + =- ∫ ∫ L Re (s) Im (s)- a s = σ + j ϖ a : a real number 5 Ex. 6: Find L [u(t+1)- u(t- 1)]. - =-- + = elsewhere 1 t 1 1 ) 1 t ( u ) 1 t ( u ) t ( f L ( u(t+1)- u(t- 1) ) exists for exists for exists for exists for all values of s. all values of s. all values of s. all values of s. ROC: whole complex s ROC: whole complex s ROC: whole complex s ROC: whole complex s-plane plane plane plane AOC : σ = -∞-∞-∞-∞ No pole Re (s) Im (s) 1 st s st st 1 st (u(t 1) (f (t)) f ( u(t 1) e 1 e 1 s s t) e dt e dt e dt---- ∞ ∞----- +-- = = =- = =- ∫ ∫ ∫ L 6 Ex. 7: Find Laplace transform of δ (t) ....
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lecture 3 - 1 1 Summary of Lecture#3 • Examples of...

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