Lesson_27 - Lesson 27 Lesson 27 Challenge 26 Lesson 27 -...

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Unformatted text preview: Lesson 27 Lesson 27 Challenge 26 Lesson 27 - Laplace Transform and continuous-time signals and systems Challenge 27 Lesson 27 Challenge 26 ) ( = ) ( + ) ( t v dt t dv RC t v ) ( 1 = ) ( /- t u e RC t h RC t RC=1 y(t=0.5)=0.4 ) ( )- 1 ( = ) (- t u e t y t Lesson 27 Lesson 27 Again, convolution is the issue. How do you normally perform this convolution operation? Response: Laplace transform ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 d h t x d t h x t x t h t y - -- =- = = Lesson 27 Lesson 27 Laplace assumed that the solution to an ODE was: and postulated a mapping: What would happen if x(t)=e st ? ( 29 = = n i t i s i e a t x 1 ( 29 ( 29 -- = st e t x s X Lesson 27 Lesson 27 Example x(t)=e t u(t). ( 29 ( 29 ) ( 1 - = = = -- s dt e dt e e s X t s st t for Re(s)> . What happens when s= ? Lesson 27 Lesson 27 Table 1: Common Laplace Transforms x(t) Laplace Transform (t) 1 u(t) 1/s tu(t) 1/s 2 t n u(t) n!/s n+1 e at u(t) 1/(s-a) t e at u(t) 1/(s-a) 2 t n e at u(t) n!/(s-a) n+1 sin( t) u(t) /(s 2 + 2 ) cos( t) u(t) s/(s 2 + 2 ) e at sin( t) u(t) /((s-a) 2 + 2 ) e at cos( t) u(t) (s-a)/((s-a) 2 + 2 ) Lesson 27 Lesson 27 Table 1: Laplace Properties x(t) Laplace Transform x(t) X(s) y(t) Y(s) ax(t) +by(t) aX(s) +bY(s) x(t-t ) e-st o X(s) x(at) (1/a)X(s/a) e at x(t) X(s-a) t n x(t) (-1) n d n X(s)/ds n x(0) lim s sX(s) if x(t) is causal and has no impulses orhigher order singularites x(...
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This note was uploaded on 08/21/2010 for the course EEL 3135 taught by Professor ? during the Spring '08 term at University of Florida.

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Lesson_27 - Lesson 27 Lesson 27 Challenge 26 Lesson 27 -...

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