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102_1_lecture15_student - UCLA Fall 2011 Systems and...

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Unformatted text preview: UCLA Fall 2011 Systems and Signals Lecture 15: Inversion of Laplace Transform November 21, 2011 EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 1 Review Last class we introduced Laplace transform: • Generalizes Fourier transform • Allows handling of growing signals, unstable systems • Simplifies analysis of LCCDEs (converts differential equations into algebraic equations) • Important for systems with feedback, control EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 2 We defined a complex frequency s = σ + jω • Oscillation component jω • A decay/growth component σ Also defined the corresponding complex exponential e st For which s does f ( t ) e st → as t → ∞ ? Defined bilateral Laplace transform: F ( s ) = Z ∞-∞ f ( t ) e- st dt. with the inverse: f ( t ) = 1 2 πj Z c + j ∞ c- j ∞ F ( s ) e st ds EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 3 Notice that Fourier transform is just the special case of this: F ( jω ) = F ( s ) | s = jω ℜ ℑ ℜ ℑ Fourier Transform Laplace Transform s = j ω s = σ + j ω σ f ( t ) = 1 2 π j Z c + j ∞ c- j ∞ F ( s ) e st ds f ( t ) = 1 2 π Z ∞- ∞ F ( j ω ) e j ω t d ω Important: Laplace transform is not unique! F ( s ) is usually specified along with region of convergence (region in complex plane for which F ( s ) does not blow up) EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 4 We are primarily interested in causal signals (for which f ( t ) = f ( t ) u ( t ) ), so we defined a unilateral Laplace transform: F ( s ) = Z ∞- f ( t ) e- st dt The lower limit- indicates that we include impulses at the origin. A bilateral Laplace transform can correspond to different signals (causal, anti-causal, or infinite extent) depending on the region of convergence. The e σt factor that makes the integral converge for a causal signal can make the integral for an anti-causal signal blow up. If we restrict ourselves to the unilateral transform the Laplace transform is (almost) unique, and we can ignore the region of convergence. EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 5 Example. Consider the Laplace transform of f ( t ) = e- at u ( t ) : F ( s ) = Z ∞ e- at e- st dt = Z ∞ e- ( a + s ) t dt = 1 s + a provided we can say e- ( s + a ) t → as t → ∞ . If < ( s + a ) = σ + a > : e- ( s + a ) t = e- ( σ + jω + a ) t = e- jωt | {z } =1 e- ( σ + a ) t = e- ( σ + a ) t The region of convergence is then σ >- a , or < s >- a . The Laplace transform pair is e- at ⇔ 1 s + a EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 6 This is very similar to the Fourier transform relationship: e- at ⇔ 1 jω + a EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 7 Example : • What is the Laplace transform of unit step signal?...
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This note was uploaded on 03/30/2012 for the course ELEC ENGR 102 taught by Professor Lee during the Fall '11 term at UCLA.

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102_1_lecture15_student - UCLA Fall 2011 Systems and...

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