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