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Ve216LectureNotesChapter6Part1

# Ve216LectureNotesChapter6Part1 - Ve216 Lecture Notes...

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Ve216 Lecture Notes Dianguang Ma Spring 2009

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Chapter 6 (Part I) Representing Signals by Using Continuous-Time Complex Exponentials: The Laplace Transform
6.1 Introduction The Laplace transform is a generalization of the continuous-time Fourier transform. It provides a broader characterization of continuous-time LTI systems and their interaction with signals than is possible with Fourier methods. Pierre-Simon, marquis de Laplace (March 23, 1749 – March 5, 1827) was a French mathematician who invented the Laplace transform which appears in many branches of mathematical physics, a field that he took leading role in forming.

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6.2 The Laplace Transform Eigenfunction property of e st : Let e st be a complex exponential with complex frequency s=σ+jω. Consider applying an input of the form x(t)= e st to an LTI system with impulse response h(t). The system output is given by ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) : the transfer function of the system s t st s st s y t h t x t h x t d h e d e h e d e H s H s h e d τ -∞ - - -∞ -∞ - -∞ = = - = = = =
6.2 The Laplace Transform ( ) ( ) st y t e H s = Recall that an eigenfunction is a signal that passes through the system without being modified except for multiplication by a scalar. Hence, we identify e st as an eigenfunction of the LTI system and H(s) as the corresponding eigenvalue. Given the simplicity of describing the action of the system on inputs of the form e st , we now seek a representation of arbitrary signals as a weighted superposition of eigenfunction e st .

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6.2 The Laplace Transform The transfer function H(s) is the Lapace transform of h(t), or the Fourier transform of h(t)e -σt . ( ) ( ) ( ) ( ) ( ) ( ) [ ( ) ] s st j t t j t H s h e d h t e dt H j h t e dt h t e e dt τ σ ϖ - - -∞ -∞ - + - - -∞ -∞ = = + = = We may recover h(t) by the inverse Laplace transform of H(s). ( ) 1 1 ( ) ( ) ( ) ( ) 2 2 1 1 ( ) ( ) 2 2 t j t t j t j j t st j h t e H j e d h t e H j e d H j e d H s e ds j π - -∞ -∞ + ∞ + -∞ - ∞ = + = + = + =
6.2 The Laplace Transform For an arbitrary signal x(t), the Lapace transform of x(t) is ( ) ( ) st X s x t e dt - -∞ = And the inverse Laplace transform of X(s) is 1 ( ) ( ) 2 j st j x t X s e ds j σ π + ∞ - ∞ = We express this relationship with the notation ( ) ( ) L x t X s ←→

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6.2 The Laplace Transform [ ] ϖ σ j s t t j t s X j X e t x F dt e e t x s X = - - - - = = = ) ( ) ( } ) ( { ) ( ) ( The Laplace transform of x(t) can be interpreted as the Fourier transform of x(t) after multiplication by a real exponential signal e -σt . Hence, a necessary condition for convergence of the
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Ve216LectureNotesChapter6Part1 - Ve216 Lecture Notes...

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