The Laplace Transform

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The Laplace Transform

• Montek Singh
• Thurs., Feb. 19, 2002330-445 pm, SN115

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What we will learn

• The notion of a complex frequency
• Representing a signal in the frequency domain
• Manipulating signals in the frequency domain

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Complex Exponential Functions

• Complex exponential est, where s ? j?
• Examples

?lt0, ?0
?gt0, ?0
?0, ?0
Re(est)
?0
?lt0
?gt0
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Some Useful Equalities
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The Laplace Transform Overview

• Key Idea
• Represent signals as sum of complex exponentials
• since all exponentials have the form Aest, it
  suffices to know the value of A for each s, to
  completely represent the original signal
• i.e., representation transformed from t to s
  domain
• Benefits
• Complex operations in the time domain get
  transformed into simpler operations in the
  s-domain
• e.g., convolution, differentiation and
  integration in time ? algebraic operations in
  the s-domain!
• Even fairly complex differential equations can be
  transformed into algebraic equations

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The Laplace Transform

• F(s) Laplace Transform of f(t)

• 1-to-1 correspondence between a signal and its
  Laplace Transform
• Frequently, only need to consider time t gt 0

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Example 1 The Unit Impulse Function

• F(s) 1 everywhere!

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Example 2 The Unit Step Function
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Some Useful Transform Pairs
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Properties of the Laplace Transform (1)
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Properties of the Laplace Transform (2)