1. Leo Lam © 2010-2011 Signals and Systems EE235

2. Leo Lam © 2010-2011 Today’s menu Yesterday: Exponentials Today: Linear, Constant-Coefficient Differential Equation

3. LCCDE, what will we do Leo Lam © 2010-2011 3 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

4. Circuit example Leo Lam © 2010-2011 4 Want to know the current i(t) around the circuit Resistor Capacitor Inductor

5. Circuit example Leo Lam © 2010-2011 5 Kirchhoff’s Voltage Law (KVL) output input

6. Differential Eq as LTI system Leo Lam © 2010-2011 6 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)

7. Differential Eq as LTI system (example) Leo Lam © 2010-2011 7 Inputs and outputs to system T have a relationship defined by the LTI system: Let “D” mean d()/dt T x(t)y(t)

8. Differential Equation: Linearity Leo Lam © 2010-2011 8 Define: Can we show that: What do we need to prove?

9. Differential Equation: Time Invariance Leo Lam © 2010-2011 9 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)

10. Differential Equation: Time Invariance Leo Lam © 2010-2011 10 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

11. LTI System response Leo Lam © 2010-2011 11 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)

12. Zero input response Leo Lam © 2010-2011 12 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

13. Zero input response (example) Leo Lam © 2010-2011 13 Using the first example: Zero-input response: x(t)=0 Need to solve: Solve (challenge) n for “natural response”

14. Zero input response (example) Leo Lam © 2010-2011 14 Solve Guess solution: Substitute: One term must be 0: Characteristic Equation

15. Zero input response (example) Leo Lam © 2010-2011 15 Solve Guess solution: Substitute: We found: Solution: Characteristic roots = natural frequencies/ eigenvalues Unknown constants: Need initial conditions

16. Leo Lam © 2010-2011 Summary Differential equation as LTI system Complete example tomorrow