Bogazici%20University

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Laplace Transforms

• Very useful in the analysis and design of LTI
  systems.
• Operations of differentiation and integration are
  converted to algebraic equations of s.
• We can solve differential equations using Laplace
  transforms.

Definition For a time function
F(s) is
s has unit 1/Time
Note that the time is integrated out.
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For the transform to exist, the integral should
be finite. It means that
The inverse transform is given by
This equation is rarely used. L-1 is usually
determined using Partial Fraction Expansion
(PFE).
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Example
Step function
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A TABLE OF LAPLACE TRANSFORMS
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Properties
Linearity
Transform of a derivative
Repeated application of the rule above
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Final Value Theorem
Use the property of the derivative
Very useful result
States that the steady state behavior of f(t) is
the same as the behavior of F(s) around s0.
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Other Properties
Integration
Frequency shift
Time shift