Leo Lam © 2010-2011

Signals and Systems

EE235

Leo Lam © 2010-2011

Today’s menu

• Yesterday: Exponentials• Today: Linear, Constant-Coefficient Differential

Equation

Leo Lam © 2010-2011

LCCDE, what will we do

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 est

Leo Lam © 2010-2011

Circuit example

4

• Want to know the current i(t) around the circuit• Resistor

• Capacitor

• Inductor

R L

C

E(t) = E 0 s in t

RIER

C

QEC

dt

dQI

dt

dILEL

Leo Lam © 2010-2011

Circuit example

5

• Kirchhoff’s Voltage Law (KVL)

R L

C

E(t) = E 0 s in t

RIER

C

QEC

dt

dILEL

tEC

QRI

dt

dIL sin0

tEdt

dQ

Cdt

dIR

dt

IdL cos1

02

2

tEICdt

dIR

dt

IdL cos1

02

2

output

input

Leo Lam © 2010-2011

Differential Eq as LTI system

6

• Inputs and outputs to system T have a relationship defined by the LTI system:

• Let “D” mean d()/dt

Tx(t) y(t)

(a2D2+a1D+a0)y(t)=(b2D2+b1D+b0)x(t)Defining

Q(D)Defining

P(D)

Leo Lam © 2010-2011

Differential Eq as LTI system (example)

7

• Inputs and outputs to system T have a relationship defined by the LTI system:

• Let “D” mean d()/dt

Tx(t) y(t)

Leo Lam © 2010-2011

Differential Equation: Linearity

8

• Define:

• Can we show that:

• What do we need to prove?

dt

tytydbtxktxkaty

))()(()()()( 21

2211

Leo Lam © 2010-2011

Differential Equation: Time Invariance

9

• System works the same whenever you use it• Shift input/output – Proof• Example:

• Time shifted system:• Time invariance?• Yes: substitute t for t (time shift the input)

dt

tdxty

)()(

dt

ttdxtty

)()( 00

d

dxy

)()(

Leo Lam © 2010-2011

Differential Equation: Time Invariance

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, time-invariant

Linear, not TI

Leo Lam © 2010-2011

LTI System response

11

• A little conceptual thinking• Time: t=0

• Linear system: Zero-input response and Zero-state output do not affect each other

TUnknown past Initial conditionzero-input response (t)

TInput x(t) zero-state output (t)

Total response(t)=Zero-input response (t)+Zero-state output(t)

Leo Lam © 2010-2011

Zero input response

12

• General nth-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

Leo Lam © 2010-2011

Zero input response (example)

13

• Using the first example:

• Zero-input response: x(t)=0

• Need to solve:

• Solve (challenge)n for “natural response”

Leo Lam © 2010-2011

Zero input response (example)

14

• Solve

• Guess solution:

• Substitute:

• One term must be 0:

Characteristic Equation

Leo Lam © 2010-2011

Zero input response (example)

15

• Solve

• Guess solution:

• Substitute:

• We found:

• Solution:

Characteristic roots = natural frequencies/

eigenvalues

Unknown constants:Need initial conditions

Leo Lam © 2010-2011

Summary

• Differential equation as LTI system• Complete example tomorrow