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Laplace Transform: Definition and Region of Convergence: Yao Wang Polytechnic University

The document defines the Laplace transform and discusses its region of convergence (ROC). The Laplace transform generalizes the Fourier transform to allow analysis of a broader class of signals and unstable systems. It relates the Laplace transform of a signal to the eigenfunctions of linear time-invariant systems. Examples are provided to illustrate the ROC for different types of signals, which is important because the same Laplace transform may correspond to different original signals depending on the ROC. In general, the ROC is a vertical half-plane or stripe bounded by poles.

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Chandrakeshwar Nayak
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110 views19 pages

Laplace Transform: Definition and Region of Convergence: Yao Wang Polytechnic University

The document defines the Laplace transform and discusses its region of convergence (ROC). The Laplace transform generalizes the Fourier transform to allow analysis of a broader class of signals and unstable systems. It relates the Laplace transform of a signal to the eigenfunctions of linear time-invariant systems. Examples are provided to illustrate the ROC for different types of signals, which is important because the same Laplace transform may correspond to different original signals depending on the ROC. In general, the ROC is a vertical half-plane or stripe bounded by poles.

Uploaded by

Chandrakeshwar Nayak
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Laplace Transform: Definition

and Region of Convergence


Yao Wang
Polytechnic University

Some slides included are extracted from lecture notes from MIT open courseware
http://ocw.mit.edu/OcwWeb/Electrical-Engineering-and-Computer-Science/6-003Fall-
2003/CourseHome/
Why do we need another
transform?

 Fourier transform cannot handle large (and important)


classes of signals and unstable systems, i.e. when

 Laplace Transform can be viewed as an extension of the


Fourier transform to allow analysis of broader class of
signals and systems (including unstable systems!)

EE3054, S08 Yao Wang, Polytechnic University 2


Eigen Function of LTI System

 est is an eigenfunction of any LTI system


 s= σ+ jω can be complex in general

 Show on the board


 H(s) is the Laplace transform of h(t)!
EE3054, S08 Yao Wang, Polytechnic University 3
The (Bilateral) Laplace
Transform

EE3054, S08 Yao Wang, Polytechnic University 4


Relation with Fourier
Transform

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Example 1

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 Derive result on board, sketch ROC for
both a>0 and a<0

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Example 2

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 Derive result on board

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Note: same X(s) may correspond to different x(t) depending on ROC!
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Example 3

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EE3054, S08 Yao Wang, Polytechnic University 12
General trend of ROC
 ROCs are always vertical half planes or stripes, bounded by poles
 Right side signals -> ROC in right half plane
 Left side signals -> ROC in left half plane
 Double sided signals -> ROC in a central stripe, or does not exist

EE3054, S08 Yao Wang, Polytechnic University 13


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 Finite duration signals that are absolutely integrable ->
ROC contains entire S-plane

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Importance of ROC

 X(s) cannot uniquely define x(t)


 Need ROC and X(s)!

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jω in the integral limit should be replaced by j∞

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EE3054, S08 Yao Wang, Polytechnic University 19

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