1. Leo Lam © 2010-2013 Signals and Systems EE235

2. Today’s menu Leo Lam © 2010-2013 Laplace Transform

3. Laplace Stability Conditions Leo Lam © 2010-2013 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 j-axis LTIC system is if marginally stable : poles in the LHP + unrepeated poles on the jaxis.

4. Laplace Stability Conditions Leo Lam © 2010-2013 Generally: system H(s) stability conditions: The system’s ROC includes the jaxis Stable? Causal? σ jωjω x x x Stable+CausalUnstable+Causal σ jωjω x x x x σ jωjω x x x Stable+Noncausal

5. Laplace: Poles and Zeroes Leo Lam © 2010-2013 Given: Roots are poles: Roots are zeroes: Only poles affect stability Example:

6. Laplace Stability Example: Leo Lam © 2010-2013 Is this stable?

7. Laplace Stability Example: Leo Lam © 2010-2013 Is this stable?

8. Standard Laplace question Find the Laplace Transform, stating the ROC. So: Leo Lam © 2010-2013 ROC extends from to the right of the most right pole ROC xxo

9. Inverse Laplace Example (2 methods!) Find z(t) given the Laplace Transform: So: Leo Lam © 2010-2013

10. 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: Leo Lam © 2010-2013

11. Inverse Laplace Example (Diffy-Q) Find the differential equation relating y(t) to x(t), given: Leo Lam © 2010-2013

12. Laplace for Circuits! Don’t worry, it’s actually still the same routine! Leo Lam © 2010-2013 Time domain inductor resistor capacitor Laplace domain Impedance!

13. Laplace for Circuits! Find the output current i(t) of this ugly circuit! Then KVL: Solve for I(s): Partial Fractions: Invert: Leo Lam © 2010-2013 R L +-+- Given: input voltage And i(0)=0 Step 1: represent the whole circuit in Laplace domain.

14. 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 © 2010-2013 LTIC

15. A “strange signal” example Find the Laplace transform of this signal: What is x(t)? We know these pairs: So: Leo Lam © 2010-2013 x(t) 1 2 3 2 1

16. One last bit: Parseval’s Theorem Leo Lam © 2010-2013

17. And that’s it! House cleaning –Sample Final –Review sessions –Class review (now online) Leo Lam © 2010-2013