signals and systems chapter 2 signals and systems analysis in time domain

Signals And

Systems

Chapter 2Signals and systems analysis in time

domain

Part I

Review

Systems Representation?

Differential Equation

LTI Systems representation?

2

0 1 2 0 12( ) ( ) ( ) ( ) ( )

d d da r t a r t a r t b e t b e tdt dt dt

Linear Constant-coefficient Differential Equation

Example of second-order continuous-time systems

2

2

1 1( ) ( ) ( ) ( )

d R d di t i t i t e t

dt L dt LC L dt

C: capacitor

L: inductor

R: resistor

general response

)(tr

zero-input response and

zero-state response

)()()( trtrtr zszi

superposition theorem

r(t) is the sum of

zero-input response

)(trzi0)( te

Initial condition 0- state

zero-state response

)(trzs

Conditions:

Initial state is “0”

Conditions:)(te 0- state 0+ state

0+ state =

0- state(0)

Part II

Zero-input Response

)(trzi

?

0)( te

Ai: Initial condition

zero-input Response

2

2

1 1( ) ( ) ( ) ( )

d R d di t i t i t e t

dt L dt LC L dt

0)(1

)()(2

2

tiLC

tidt

d

L

Rti

dt

dHomogeneous equation

Characteristic equation 012 LCL

R

1 21 2( ) t t

zii t A e A e

Example1:

Suppose that:

)()(2)(3)(2

2

tetrtrdt

dtr

dt

d

0)0(',1)0( rr

determine r(t)

and)(2)( 2 tuete t

Example2:

Consider the differential equation of a second-

order system

)(8)(12)(2)(10)(7)( '2

2

tutttrtrdt

dtr

dt

d

0)0(',5

4)0( rr

determine rzi(t)

if

Exercise

Page 83 #2-6(zero-input response)