leo lam © 2010-2013 signals and systems ee235. today’s menu leo lam © 2010-2013 laplace...
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Leo Lam © 2010-2013
Signals and Systems
EE235
Leo Lam © 2010-2013
Today’s menu
• Laplace Transform
Leo Lam © 2010-2013
Laplace properties (unilateral)
Linearity: f(t) + g(t) F(s) + G(s)
Time-shifting:
FrequencyShifting:
Differentiation:
and
Time-scaling
a
sFa
1
Leo Lam © 2010-2013
Laplace properties (unilateral)
Multiplication in time Convolution in Laplace
Convolution in time Multiplication in Laplace
Initial value theorem
Final value theorem Final value result
Only works ifAll poles of sF(s) in LHP
Leo Lam © 2010-2013
Laplace transform table
Leo Lam © 2010-2013
Another Inverse Example
• Example, find h(t) (assuming causal):
• Using linearity and partial fraction:
Leo Lam © 2010-2013
Another Inverse Example
• Here is the reason:
Leo Lam © 2010-2013
Another Inverse Example
• Example, find z(t) (assuming causal):
• Same degrees order for P(s) and Q(s)
• From table:
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Inverse Example (Partial Fraction)
• Example, find x(t):
• Partial Fraction
• From table:
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Inverse Example (almost identical!)
• Example, find x(t):
• Partial Fraction (still the same!)
• From table:
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Output
• Example:
• We know:
• From table (with ROC):
Leo Lam © 2010-2013
All tied together
• LTI and Laplace
• So:
LTI
LTIx(t) y(t) = x(t)*h(t)
X(s) Y(s)=X(s)H(s)
Laplace
Multiply
Inverse Laplace
H(s)=X(s)
Y(s)
Leo Lam © 2010-2013
Laplace & LTI Systems
• If:
• Then
LTI
LTI
Laplace of the zero-state (zero initialconditions) response
Laplace of the input
Leo Lam © 2010-2013
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
Leo Lam © 2010-2013
Laplace & Differential Equations
• Example (causal LTIC):
• Cross Multiply and inverse Laplace:
Leo Lam © 2010-2013
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.
Leo Lam © 2010-2013
Laplace Stability Conditions
• Generally: system H(s) stability conditions:• The system’s ROC includes the j -w axis• Stable? Causal?
σ
jω
x
x
x
Stable+Causal Unstable+Causal
σ
jω
x
xx
xσ
jω
x
x
x
Stable+Noncausal
Leo Lam © 2010-2013
Laplace: Poles and Zeroes
• Given:
• Roots are poles:
• Roots are zeroes:
• Only poles affect stability
• Example:
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Laplace Stability Example:
• Is this stable?
Leo Lam © 2010-2013
Laplace Stability Example:
• Is this stable?