Leo Lam © 2010-2012

Signals and Systems

EE235Lecture 30

Leo Lam © 2010-2012

Today’s menu

• Laplace Transform!

Leo Lam © 2010-2012

Laplace Transform

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

Leo Lam © 2010-2012

Laplace Transform

• Definition:

• Where

• Inverse:

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

Leo Lam © 2010-2012

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

Leo Lam © 2010-2012

Region of Convergence

• Example: Find the Laplace Transform of:

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

Leo Lam © 2010-2012

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 © 2010-2012

Region of Convergence

• A very similar example: Find Laplace Transform of:

• For what value does:

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

Leo Lam © 2010-2012

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 © 2010-2012

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 © 2010-2012

Laplace and Fourier

• Very similar (Fourier for Signal Analysis, Laplace for Control 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 © 2010-2012

Laplace and Fourier

• No Fourier Transform Example:

• ROC exists: Laplace ok• ROC does not include jw-axis, no Fourier Transform

)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 © 2010-2012

Laplace and Fourier

• 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 © 2010-2012

Summary

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