Leo Lam © 2010-2011

Signals and Systems

EE235

Leo Lam © 2010-2011

Futile

Q: What did the monserous voltage source say to the chunk of wire?

A: "YOUR RESISTANCE IS FUTILE!"

Leo Lam © 2010-2011

Today’s menu

• Laplace Transform

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

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

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

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

Leo Lam © 2010-2011

Another Inverse Example

• Example, find h(t) (assuming causal):

• Using linearity and partial fraction:

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Another Inverse Example

• Here is the reason:

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Laplace & LTI Systems

• If:

• Then

LTI

LTI

Laplace of the zero-state (zero initialconditions) response

Laplace of the input

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

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.

Leo Lam © 2010-2011

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 © 2010-2011

Laplace Stability Example:

• Is this stable?

Leo Lam © 2010-2011

Laplace Stability Example:

• Is this stable?

• Mathematically stable (all poles cancelled)• In reality…explosive

3

5

1

1)(

s

ssH

Woohoo!!!

Leo Lam © 2010-2011