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Leo Lam © 2010-2011
Signals and Systems
EE235
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Leo Lam © 2010-2011
Futile
Q: What did the monserous voltage source say to the chunk of wire?
A: "YOUR RESISTANCE IS FUTILE!"
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Leo Lam © 2010-2011
Today’s menu
• Laplace Transform
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Leo Lam © 2010-2011
Ambiguous? Define it away!
• Bilateral Laplace Transform:
• Unilateral Laplace Transform (for causal system/signal):
• For EE, it’s mostly unilateral Laplace (any signal with u(t) is causal)
• Not all functions have a Laplace Transform (no ROC)
0
)( dteetf tjt
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Leo Lam © 2010-2011
Inverse Laplace
• Example, find f(t) (assuming causal):
• Table:
• What if the exact expression is not in the table? – Hire a mathematician– Make it look like something in the table (partial fraction etc.)
)()sin( tubt
)()5sin()( tuttf
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Leo Lam © 2010-2011
Laplace properties (unilateral)
Linearity: f(t) + g(t) F(s) + G(s)
Time-shifting:
FrequencyShifting:
Differentiation:
and
Time-scaling
a
sFa
1
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Leo Lam © 2010-2011
Laplace properties (unilateral)
Multiplication in time Convolution in Laplace
Convolution in time Multiplication in Laplace
Initial value
Final value
Final value resultOnly works ifAll poles of sF(s) in LHP
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Leo Lam © 2010-2011
Another Inverse Example
• Example, find h(t) (assuming causal):
• Using linearity and partial fraction:
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Leo Lam © 2010-2011
Another Inverse Example
• Here is the reason:
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Leo Lam © 2010-2011
Laplace & LTI Systems
• If:
• Then
LTI
LTI
Laplace of the zero-state (zero initialconditions) response
Laplace of the input
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Leo Lam © 2010-2011
Laplace & Differential Equations
• Given:
• In Laplace:– where
• So:
• Characteristic Eq:– The roots are the poles in s-domain, the “power” in time domain.
012
2
012
2
)(
)(
bsbsbsP
asasasQ
0)( sQ
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Leo Lam © 2010-2011
Laplace Stability Conditions
• LTI – Causal system H(s) stability conditions:• LTIC system is stable : all poles are in the LHP• LTIC system is unstable : one of its poles is in the RHP• LTIC system is unstable : repeated poles on the jw-axis• LTIC system is if marginally stable : poles in the LHP +
unrepeated poles on the j -w axis.
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Leo Lam © 2010-2011
Laplace: Poles and Zeroes
• Given:
• Roots are poles:
• Roots are zeroes:
• Only poles affect stability
• Example:
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Leo Lam © 2010-2011
Laplace Stability Example:
• Is this stable?
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Leo Lam © 2010-2011
Laplace Stability Example:
• Is this stable?
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Leo Lam © 2010-2011
Laplace Stability Example:
• Is this stable?
• Mathematically stable (all poles cancelled)• In reality…explosive
3
5
1
1)(
s
ssH
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Woohoo!!!
Leo Lam © 2010-2011