Leo Lam © 2010-2012

Signals and Systems

EE235

Leo Lam © 2010-2012

Today’s menu

• Almost done!• Laplace Transform

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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 © 2010-2012

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 © 2010-2012

Laplace & Differential Equations

• Example (causal LTIC):

• Cross Multiply and inverse Laplace:

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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 © 2010-2012

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

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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?

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Laplace Stability Example:

• Is this stable?

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Standard Laplace question

• Find the Laplace Transform, stating the ROC.

• So:

ROC extends from to the right of the most right pole

ROCxxo

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Inverse Laplace Example (2 methods!)

• Find z(t) given the Laplace Transform:

• So:

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Inverse Laplace Example (2 methods!)

• Find z(t) given the Laplace Transform (alternative method):

• Re-write it as:

• Then:

• Substituting back in to z(t) and you get the same answer as before:

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Inverse Laplace Example (Diffy-Q)

• Find the differential equation relating y(t) to x(t), given:

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Laplace for Circuits!

• Don’t worry, it’s actually still the same routine!

Time domain

inductor

resistor

capacitor

Laplace domain

Impedance!

Laplace for Circuits!

• Find the output current i(t) of this ugly circuit!

• Then KVL: • Solve for I(s):

• Partial Fractions:• Invert:

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RL

+-

Given: input voltage

And i(0)=0

Step 1: representthe whole circuit inLaplace domain.

Step response example

• Find the transfer function H(s) of this system:

• We know that:

• We just need to convert both the input and the output and divide!

Leo Lam © 2010-2012

LTIC

)(

)()(

sX

sYsH

LTIC

A “strange signal” example

• Find the Laplace transform of this signal:

• What is x(t)?

• We know these pairs:

• So:

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x(t)

1 2 3

2

1