Leo Lam © 2010-2013

Signals and Systems

EE235

Leo Lam © 2010-2013

Today’s menu

• Fourier Transform table posted• Laplace Transform

Leo Lam © 2010-2013

Laplace Transform

• Focus on:– Doing (Definitions and properties)– Understanding its possibilities (ROC)– Poles and zeroes (overlap with EE233)

Leo Lam © 2010-2013

j

j

stdsesFj

tf

)(

2

1)(

Laplace Transform

• Definition:

• Where

• Inverse:

Good news: We don’t need to do this, just use the tables.

Inverse Laplace expresses f(t) as sum of exponentials with fixed s has specific requirements

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Region of Convergence

• Example: Find the Laplace Transform of:

We have a problem: the first term for t=∞ doesn’t always vanish!

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Region of Convergence

• Example: Continuing…

• In general: for

• In our case if: then

For what value of s does:

Pole at s=-3. Remember this result for now!

Leo Lam © 2010-2013

Region of Convergence

• A very similar example: Find Laplace Transform of:

• For what value does:

• This time: if then• Same result as before!

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Region of Convergence

• Comparing the two:

ROC

-3

ROC

-3s-plane

Laplace transform not uniquely invertible without region of convergence

Casual,Right-sided

Non-casual,Left-sided

Leo Lam © 2010-2013

Finding ROC Example

• Example: Find the Laplace Transform of:

• From table:

)()(3)( 26 tuetuetx tt

)2)(6(

)3(4)(

2

1

6

3)(

ss

ssX

sssX

ROC: Re(s)>-6

ROC: Re(s)>-2

Combined:ROC: Re(s)>-2

Leo Lam © 2010-2013

Laplace Example

• No Laplace Example:

)3)(1(

2)(

3

1

1

1)(

)()()( 3

sssX

sssX

tuetuetx tt

ROC: Re(s)>-1

ROC: Re(s)<-3

Combined:ROC: None!

Leo Lam © 2010-2013

Laplace and Fourier

• If the ROC includes the j w –axis, then the signal has a Fourier Transform where s= jw

• Caution: If the ROC doesn't quite include the jw-axis (if poles lie on the jw-axis), then it might still have a Fourier transform, but it is not given by s=jw.

σ

jwROC

–a

Leo Lam © 2010-2013

Laplace and Fourier

• No Fourier Transform Example:

• ROC exists: Laplace ok• ROC does not include jw-axis, Fourier Transform is not

F(jw). (In fact, here it does not exist).

)3)(1(

2)(

1

1

3

1)(

)()()( 3

sssX

sssX

tuetuetx tt

ROC: Re(s)>-3

ROC: Re(s)<-1

Combined:-3<ROC<-1

Leo Lam © 2010-2013

Finding ROC Example

• Example: Find the Laplace Transform of:

• From table:

• Thus:

• With ROC:

ROC: Re(s)<-2

ROC: Re(s)>-3

Combined:ROC: -3<Re(s)<-2

xxo

Leo Lam © 2010-2013

Poles and Zeros (the X’s and O’s)

• H(s) is almost always rational for a physical system:

• Rational = Can be expressed as a polynomial• ZEROs = where H(s)=0, which is • POLES = where H(s)=∞, which is • Example:

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Plotting Poles and Zeros

• H(s) is almost always rational for a physical system:

• Plot is in the s-plane (complex plane)

σ

jω

x xo

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Plotting Poles and Zeros

• What does it look like?

Leo Lam © 2010-2013

ROC Properties (Summary)

• All ROCs are parallel to the j w –axis• Casual signal right-sided ROC and vice versa• Two-sided signals appear either as a strip or no ROC

exist (no Laplace).• For a rational Laplace Transform, the ROC is bounded by

poles or ∞.• If ROC includes the jw-axis, Fourier Transform of the

signal exists = F(jw).

Leo Lam © 2010-2013

Laplace and Fourier

• Very similar (Fourier for Signal Analysis, Laplace for Control, Circuits and System Designs)

• ROC includes the jw-axis, then Fourier Transform = Laplace Transform (with s=jw)

• If ROC does NOT include jw-axis but with poles on the jw-axis, FT can still exist!

• Example:

• But Fourier Transform still exists:

• No Fourier Transform if ROC is Re(s)<0 (left of jw-axis)

)()( tutx ROC: Re(s) > 0Not including jw-axis

Leo Lam © 2010-2013

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 © 2010-2013

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 © 2010-2013

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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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 © 2010-2013

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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Summary

• Laplace intro• Region of Convergence• Causality• Existence of Fourier Transform & relationships