The Laplace Transform - PowerPoint PPT Presentation

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The Laplace Transform

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The Laplace Transform Montek Singh Thurs., Feb. 19, 2002 3:30-4:45 pm, SN115 What we will learn The notion of a complex frequency Representing a signal in the ... – PowerPoint PPT presentation

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Title: The Laplace Transform


1
The Laplace Transform
  • Montek Singh
  • Thurs., Feb. 19, 2002330-445 pm, SN115

2
What we will learn
  • The notion of a complex frequency
  • Representing a signal in the frequency domain
  • Manipulating signals in the frequency domain

3
Complex Exponential Functions
  • Complex exponential est, where s ? j?
  • Examples

?lt0, ?0
?gt0, ?0
?0, ?0
Re(est)
?0
?lt0
?gt0
4
Some Useful Equalities
5
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

6
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

7
Example 1 The Unit Impulse Function
  • F(s) 1 everywhere!

8
Example 2 The Unit Step Function
9
Some Useful Transform Pairs
10
Properties of the Laplace Transform (1)
11
Properties of the Laplace Transform (2)
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