Laplace Transform: Definition and Region of Convergence

Yao Wang

Polytechnic University

Some slides included are extracted from lecture notes from MIT open coursewarehttp://ocw.mit.edu/OcwWeb/Electrical-Engineering-and-Computer-Science/6-003Fall-

2003/CourseHome/

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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!)

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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)!

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The (Bilateral) Laplace

Transform

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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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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

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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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∞jj by replaced be shouldlimit integral in the ω

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