1. Leo Lam © 2010-2012 Signals and Systems EE235 Lecture 30

2. Leo Lam © 2010-2012 Today’s menu Laplace Transform!

3. Laplace Transform Leo Lam © 2010-2012 Focus on: –Doing (Definitions and properties) –Understanding its possibilities (ROC) –Poles and zeroes (overlap with EE233)

4. Laplace Transform Leo Lam © 2010-2012 Definition: Where Inverse: Good news: We don’t need to do this, just use the tables.

5. Laplace Transform Definition: Where Inverse: Leo Lam © 2010-2012 Good news: We don’t need to do this, just use the tables. Inverse Laplace expresses f(t) as sum of exponentials with fixed   has specific requirements

6. Region of Convergence Example: Find the Laplace Transform of: Leo Lam © 2010-2012 We have a problem: the first term for t=∞ doesn’t always vanish!

7. Region of Convergence Example: Continuing… In general: for In our case if: then Leo Lam © 2010-2012 For what value of s does: Pole at s=-3. Remember this result for now!

8. 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

9. Region of Convergence Comparing the two: Leo Lam © 2010-2012 ROC -3 ROC -3 s-plane Laplace transform not uniquely invertible without region of convergence Casual, Right-sided Non-casual, Left-sided

10. Finding ROC Example Example: Find the Laplace Transform of: From table: Leo Lam © 2010-2012 ROC: Re(s)>-6 ROC: Re(s)>-2 Combined: ROC: Re(s)>-2

11. Laplace and Fourier Very similar (Fourier for Signal Analysis, Laplace for Control and System Designs) ROC includes the j-axis, then Fourier Transform = Laplace Transform (with s=j) If ROC does NOT include j-axis but with poles on the j-axis, FT can still exist! Example: But Fourier Transform still exists: No Fourier Transform if ROC is Re(s)<0 (left of j-axis) Leo Lam © 2010-2012 ROC: Re(s) > 0 Not including jw-axis

12. Laplace and Fourier No Fourier Transform Example: ROC exists: Laplace ok ROC does not include j-axis, no Fourier Transform Leo Lam © 2010-2012 ROC: Re(s)>-3 ROC: Re(s)<-1 Combined: -3<ROC<-1

13. Laplace and Fourier No Laplace Example: Leo Lam © 2010-2012 ROC: Re(s)>-1 ROC: Re(s)<-3 Combined: ROC: None!

14. Summary Laplace intro Region of Convergence Causality Existence of Fourier Transform Leo Lam © 2010-2012