Unit7_LT - EE 3210 Signals and Systems Unit 7 Laplace...

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EE 3210 Signals and Systems Unit 7 Laplace Transform
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Unit 7.1 What is Laplace Transform? After studying this unit, you will be able to 1. describe the definition of Laplace transform and its relationship with Fourier transform 2. find the Laplace transform of some simple signals 3. describe the properties of Laplace transform
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EE3210 Signals and Systems Unit 7 3 Non-Existence of Fourier Transform Fourier transform is extremely useful for studying signals & LTI systems. However, not all signals have F.T. e.g. some infinite-energy signals. Example: ) ( ) ( t u e t x t =
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EE3210 Signals and Systems Unit 7 4 Example 1 = = = = = 0 ) 1 ( ) 1 ( 0 ) 1 ( ) ( ) ( ) ( ) ( ω j t j t j t j t t e dt e dt e t u e j X t u e t x Consider the real part: It does not converge!
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EE3210 Signals and Systems Unit 7 5 Laplace Transform Laplace Transform : dt e t x s X st ) ( ) ( What is its relationship with Fourier Transform? where s is a complex number.
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EE3210 Signals and Systems Unit 7 6 Fourier Transform as a Special Case Fourier Transform is a special case of Laplace Transform. When s is purely imaginary (e st e j ω t ), L. T. reduces to F. T. dt e t x s X st ) ( ) ( dt e t x j X t j ω ) ( ) ( j s = Put
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EE3210 Signals and Systems Unit 7 7 Example 1 Laplace Transform : dt e t x s X st ) ( ) ( X(s) exists for Re{s} > 1. [ ] 1 1 1 1 ) ( 0 ) 1 ( 0 = = = s e s dt e e s X t s st t ) ( ) ( t u e t x t = Consider the previous example:
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EE3210 Signals and Systems Unit 7 8 Damping Effect What is the difference between Fourier Transform and Laplace Transform? [] {} ) ( ) ( ) ( ) ( ) ( ) ( t t j t t j st e t x F dt e e t x dt e t x dt e t x s X σ ωσ + = = = damping function
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EE3210 Signals and Systems Unit 7 9 Damping Effect t j t t e e e ω σ ] [ The term e - σ t converts infinite-energy signals to finite- energy ones. 1 >
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EE3210 Signals and Systems Unit 7 10 Example 2 (Example 9.1 in textbook) Find the Laplace Transform of ) ( ) ( t u e t x at = t a>0 t a<0 x(t) x(t)
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EE3210 Signals and Systems Unit 7 11 a s a s a j a > + = > + + + = } { Re , 1 0 , ) ( 1 σ ωσ a s a s t u e L at > + ⎯→ } Re{ , 1 ) ( ( ) sj ω = + () 0 0 at st a j t aj t s eu t ed t e d t e σω −− + + −∞ −+ + Χ= = ⎡⎤ = ⎢⎥ ++ ⎣⎦ ∫∫ Example 2 When a<0, F {x(t)} does not converge, but L{x(t)} does for Re{s}>-a.
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EE3210 Signals and Systems Unit 7 12 Example 3 (Example 9.2 in textbook) Consider the following function: How to sketch it? Find the Laplace Transform.
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Unit7_LT - EE 3210 Signals and Systems Unit 7 Laplace...

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