part6 - Part 6: Laplace Transform Laplace Transform l l l...

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Part 6: Laplace Transform Laplace Transform l Fourier Transform of some signals cannot be derived from the definition. l Is it possible to have a more general transform? l Bilateral Laplace Transform or where l In some cases, bilateral is improper; eg. the above integral does not converge. l Unilateral Laplace Transform ( 29 [ ] ( 29 - - - = dt e e t x t x t j t w s L ( 29 [ ] ( 29 - - = e t x t x st L w s j s + = ( 29 ( 29 t f A t x 0 2 cos p = ( 29 ( 29 s X t x → L Used in this course Unless specified. ( 29 [ ] - - = 0 e t x t x L
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EE3118 Linear System & Signal Analysis (KST) Example: () [] t δ L 1 1 lim 1 lim 2 / 2 / 0 2 / 2 / 0 0 = = = = dt dt e dt e t t st st L Example: t u L s s e dt e t u st st 1 1 0 0 = = × = + L with the assumption that 0 > σ (Keep in mind that t j t t s e e e ω = ) Example: [ ] at e t u L a s a s e dt e e e t u t a s st at at + = + = = + 1 0 0 L with the assumption that 0 > + a (Note: - a , called the pole of the system, is excluded).
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Region of Convergence (ROC) l Refer to the examples, the Laplace Transform is only convergent in certain range of l Consider to be finite, sufficient condition: where A is a real positive number, and are real. X(s) converges as long as , which is known as ROC of X(s) s ( 29 - - - - - + = 0 0 dt e t x e t x e t x t t t s s s < < 0 for 0 t Ae t t x t t b a b s a < < a b Properties of ROC l ROC consists of strips parallel to the –axis in the s-plane (i.e. it depends only on the real part of s) l If x(t) is a) finite duration and there is at least one value of s for which the Laplace Transform converges, then the ROC is the entire s-plane . b) right-sided and if the line is in the ROC, then all values of s for which will also be in the ROC c) left-sided and if the line is in the ROC, then all values of s for which will also be in the ROC d) two-sided and if the line is in the ROC, then the ROC will consist of a strip in the s-plane including the line { } 0 s = s e { } 0 s = s e { } 0 s s e { } 0 s < s e w j { } 0 s = s e T 1 time T 2 T 1 T 2 (a) (b) (c) (d)
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Properties of Laplace Transform l Linearity l Time Shifting l S-Shifting l Time Domain Differentiation l Time Domain Integration ( 29 ( 29 [ ] ( 29 0 0 0 st e s X t t u t t x - = - - L ( 29 ( 29 [ ] ( 29 ( 29 s X a s X a t x a t x a 2 2 1 1 2 2 1 1 + = + L ( 29 [ ] ( 29 a a + = - s X t x e t L ( 29 ( 29 ( 29 ( 29 ( 29 - - - - - - - = 0 0 1 1 n n n n n x x s s X s dt t x d L L ( 29 ( 29 ( 29 ( 29 s x s s X d x t - - - + = 0 1 l l L Properties of Laplace Transform l Time Scaling where a > 0 l Frequency Scaling where a > 0 l Convolution l Product l Initial Value Theorem l Final Value Theorem ( 29 [ ] = - a s X a at x 1 L ( 29 s a X a t x a = 1 L ( 29 ( 29 [ ] ( 29 ( 29 s X s X t x t x 2 1 2 1 = L ( 29 ( 29 [ ] ( 29 ( 29 + - - = j j d s X X j t x t x s s l l l p 2 1 2 1 2 1 L ( 29 ( 29 s X s t x s t = + lim 0 ( 29 ( 29 s X s t x s t 0 =
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EE3118 Linear System & Signal Analysis (KST) 1. Linearity () () [] s X a s X a dt e t x a dt e t x a dt e t x a t x a t x a t x a t s t s t s 2 2 1 1 0 2 2 0 1 1 0 2 2 1 1 2 2 1 1 + = + = + = +
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This note was uploaded on 01/11/2011 for the course EE 3118 taught by Professor Kitsangtsang during the Spring '08 term at City University of Hong Kong.

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part6 - Part 6: Laplace Transform Laplace Transform l l l...

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