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

3. Leo Lam © 2010-2013 Today’s menu Exponential response of LTI system LCCDE Midterm Tuesday next week

4. LTI system transfer function Leo Lam © 2010-2013 3 Let s=j  LTI systems preserve frequency Complex exponential output has same frequency as the complex exponential input LTI e st H(s)e st LTI

5. Importance of exponentials Leo Lam © 2010-2013 4 Makes life easier Convolving with e st is the same as multiplication Because e st are eigenfunctions of LTI systems cos(t) and sin(t) are real Linked to e st

6. Quick note Leo Lam © 2010-2013 5 LTI e st H(s)e st LTI e st u(t) H(s)e st u(t)

7. Which systems are not LTI? Leo Lam © 2010-2013 6 NOT LTI

8. Leo Lam © 2010-2013 Summary

9. Leo Lam © 2010-2013 Switching gear! Midterm over! Switching gear… Linear, Constant-Coefficient Differential Equation

10. LCCDE, what will we do Leo Lam © 2010-2013 9 Why do we care? Because it is everything! Represents LTI systems Solve it: Homogeneous Solution + Particular Solution Test for system stability (via characteristic equation) Relationship between HS (Natural Response) and Impulse response Using exponentials e st

11. Circuit example Leo Lam © 2010-2013 10 Want to know the current i(t) around the circuit Resistor Capacitor Inductor

12. Circuit example Leo Lam © 2010-2013 11 Kirchhoff’s Voltage Law (KVL) output input

13. Differential Eq as LTI system Leo Lam © 2010-2013 12 Inputs and outputs to system T have a relationship defined by the LTI system: Let “D” mean d()/dt T x(t)y(t) (a 2 D 2 +a 1 D+a 0 )y(t)=(b 2 D 2 +b 1 D+b 0 )x(t) Defining Q(D) Defining P(D)

14. Differential Eq as LTI system (example) Leo Lam © 2010-2013 13 Inputs and outputs to system T have a relationship defined by the LTI system: Let “D” mean d()/dt T x(t)y(t)

15. Differential Equation: Linearity Leo Lam © 2010-2013 14 Define: Can we show that: What do we need to prove?

16. Differential Equation: Time Invariance Leo Lam © 2010-2013 15 System works the same whenever you use it Shift input/output – Proof Example: Time shifted system: Time invariance? Yes: substitute  for t (time shift the input)

17. Differential Equation: Time Invariance Leo Lam © 2010-2013 16 Any pure differential equation is a time- invariant system: Are these linear/time-invariant? Linear, time-invariant Linear, not TI Non-Linear, TI Linear, time-invariant Linear, not TI

18. LTI System response Leo Lam © 2010-2013 17 A little conceptual thinking Time: t=0 Linear system: Zero-input response and Zero-state output do not affect each other T Unknown past Initial condition zero-input response (t) T Input x(t) zero-state output (t) Total response(t)=Zero-input response (t)+Zero-state output(t)

19. Zero input response Leo Lam © 2010-2013 18 General n th -order differential equation Zero-input response: x(t)=0 Solution of the Homogeneous Equation is the natural/general response/solution or complementary function Homogeneous Equation

20. Zero input response (example) Leo Lam © 2010-2013 19 Using the first example: Zero-input response: x(t)=0 Need to solve: Solve (challenge) n for “natural response”

21. Zero input response (example) Leo Lam © 2010-2013 20 Solve Guess solution: Substitute: One term must be 0: Characteristic Equation

22. Zero input response (example) Leo Lam © 2010-2013 21 Solve Guess solution: Substitute: We found: Solution: Characteristic roots = natural frequencies/ eigenvalues Unknown constants: Need initial conditions

23. Zero input response (example) Leo Lam © 2010-2013 22 4 steps to solving Differential Equations: Step 1. Find the zero-input response = natural response y n (t) Step 2. Find the Particular Solution y p (t) Step 3. Combine the two Step 4. Determine the unknown constants using initial conditions