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