The unit step response of an LTI system
22
16.362 Signal and System I representation of discrete-time signals in terms of impulse Example ] 0 [ ] [ ] 0 [ x n x k k n k x n x ] [ ] [ ] [ ] [ ] [ ] [ k x k n k x ] 1 [ ] 1 [ ] 1 [ x n x 0 ] [ ] [ ] [ ] [ k k k n k n k u n u
description
The unit step response of an LTI system. The unit step response of an LTI system. The unit step response of an LTI system. Linear constant-coefficient difference equations. +. delay. depends on x[n]. We don’t know y[n] unless x[n] is given. - PowerPoint PPT Presentation
Transcript of The unit step response of an LTI system
16.362 Signal and System I • The representation of discrete-time signals in terms of impulse
]0[][]0[ xnx
k
knkxnx ][][][
][][][ kxknkx
]1[]1[]1[ xnx
0
][
][][][
k
k
kn
knkunu
Example
16.362 Signal and System I • The representation of discrete-time signals in terms of impulse
k
knkxnx ][][][
][n ][nh ][][ nhny
k
knkxnx ][][][
k
k
knxkh
knhkxny
][][
][][][
][][][ nxnhny
][][][ nhnxny
Convolution
16.362 Signal and System I • The representation of continuous-time signals in terms of impulse
')'()'()( dttttxtx
')'()'()( dttthtxty
)()()( txthty
• Properties of LIT systems
Commutative property
)()()( txthty
)()()( thtxty
Distributive property
)()()()(
)()()()(
21
21
txthtxth
txththty
16.362 Signal and System I • Properties of LIT systems
Associative property
)()()(
)()()(
)()()()(
21
21
21
txthth
txthth
txththty
Causality
,0)( th for t<0.
,0][ nh for n<0.
Stability
dtth )(
n
nh ][
16.362 Signal and System I • The unit step response of an LTI system
][n ][nh ][ny
][nu ][nh ][ns
k
knhkny ][][][
][
][][][
nh
knkhnyk
n
k
k
kh
knukhns
][
][][][
16.362 Signal and System I • The unit step response of an LTI system
][nu ][nh ][1 ns
n
k
khns ][][1
]1[ nu ][nh ][2 ns
1
2
][
]1[][][
n
k
k
kh
knukhns
]1[][][ 1
1
2
nskhnsn
k
][]1[][ 11 nhnsns
16.362 Signal and System I • The unit step response of an LTI system
][n ][nh ][nh
][nu ][nh ][ns ][]1[][ nhnsns
16.362 Signal and System I • Linear constant-coefficient difference equations
][nx
][]1[2
1][ nxnyny
?][ ny ?][ nh
][ny depends on x[n]. We don’t know y[n] unless x[n] is given.
But h[n] doesn’t depend on x[n]. We should be able to obtain h[n] without x[n].
How?• Discrete Fourier transform, --- Ch. 5.
• LTI system response properties, this chapter.
][nh
][ny
2
1
+
delay
16.362 Signal and System I • Linear constant-coefficient difference equations
][]1[2
1][ nxnyny
][]1[2
1][ nnhnh
]1[2
1][ nhnh
][]1[2
1][ nnhnh
When n 1, 2
1
]1[
][
nhnh
n
Anh
2
1][
][2
1][ nuAnh
n
Causality
][n
][nh
][nh
2
1
+
delay
16.362 Signal and System I • Linear constant-coefficient difference equations
][n
][nh
][nh
2
1
][]1[2
1][ nxnyny +
][]1[2
1][ nnhnh delay
][]1[2
1][ nnhnh
][2
1][ nuAnh
n
Determine A by initial condition:
When n = 0, 1]0[]0[ h
]0[2
1]0[
0
uAh
A = 1
16.362 Signal and System I • Linear constant-coefficient difference equations
]1[ n ][]1[2
1][ nxnyny
][]1[2
1][ nnhnh
][2
1][ nunh
n
?][ ny
Two ways:
(1) Repeat the procedure
(2) ][][][ nhnxny
]1[2
1
]1[
][]1[][
1
nu
nh
nhnny
n
][nh
][nh
2
1
+
delay
16.362 Signal and System I • The unit step response of an LTI system, continuous time
)(t )(th )(ty
)(tu )(th )(ts
)(
)()()(
th
dnhty
)()(
thdt
tds
t
dh
dtuhts
)(
)()()(
16.362 Signal and System I • Linear constant-coefficient difference equations
)(tx)(
2
1
2
1)( tx
dt
dyty
?)( ty ?)( th
)(ty depends on x(t). We don’t know y(t) unless x(t) is given.
But h(t) doesn’t depend on x(t). We should be able to obtain h(t) without x(t).
How?• Continuous time Fourier transform.
• LTI system response properties, this chapter.
)()(2 txtydt
dy)(th
)(ty
2
1
+
dt
d
2
1
16.362 Signal and System I • Linear constant-coefficient difference equations
)(t
When t>0,dt
dyty
2
1)( tAety 2)(
Determine A by initial condition:
)()( 2 tuAeth t
Causality
)(2
1
2
1)( tx
dt
dyty
)(2
1
2
1)( t
dt
dyty
)(2
1
2
1)( t
dt
dyty
)(th
)(ty
2
1
+
dt
d
2
1
16.362 Signal and System I • Linear constant-coefficient difference equations
Determine A by initial condition:
)()( 2 tuAeth t
)(2
1)()
2
1()()2(
2
1)( 222 ttAetuAetuAe ttt
A = 1 )()( 2 tueth t
)(t)(
2
1
2
1)( tx
dt
dyty
)(2
1
2
1)( t
dt
dyty
)(th
)(ty
2
1
+
dt
d
2
1
16.362 Signal and System I • Linear constant-coefficient difference equations
)()( 3 tuKetx t
][5
][
)]()[(
)()(
)()()(
23
52
)(23
)(23
tt
t
o
t
t
o
t
t
eeK
deKe
deKe
dtueuKe
dthx
thtxty
)(th
)(ty
2
1
)(2
1
2
1)( tx
dt
dyty +
dt
d
2
1
)()( 2 tueth t
)(][5
)( 23 tueeK
ty tt
16.362 Signal and System I • Singularity functions
)()(0 ttu Define:
dt
tdtu
)()(1
n
n
n dt
tdtu
)()(
)()()(1 tudtut
du
dtuu
tututu
t
)(
)()(
)()()( 112
du
tutututut
n
n
)(
)()()()(
)1(
16.362 Signal and System I • Singularity functions
)()()()()( 0 txttxtutx
dt
dx
tudt
dx
dtd
tdxu
tdxuutx
dutx
dtxututx
)(
)(
)()(
)()(|)()(
)()(
)()()()(
0
0
00
0
11
n
n
n dt
txdtutx
)()()(
dt
tdxtutx
)()()( 1
)()()( 0 txtutx
16.362 Signal and System I • Singularity functions
n
n
n dt
txdtutx
)()()(
dt
tdxtutx
)()()( 1
)()()( 0 txtutx )()(
)()( 21
11 tudt
tdututu
)()()()( 111 tutututu k
k terms
16.362 Signal and System I • Singularity functions
dx
dtux
tutxtutx
t
)(
)()(
)()()()( 1
n
n
n dt
txdtutx
)()()(
t
dxtutx )()()( 1
)()()( 0 txtutx
ddx
tudx
tututxtutx
t
t
')'(
)()(
)()()()()( 2
)()()(2 tututu
16.362 Signal and System I • Singularity functions --- discrete time
]1[][][1 nnnu Define:
]1[][][ 11 nununu kkk
]1[][
]1[][][][][][ 1
nxnx
nnxnnxnunx
]1[][][][ 1111 nunununu
]1[][][][ 1 nxnxnunx
16.362 Signal and System I • Singularity functions --- discrete time
][][1 nunu
Define:
1][][][
][][][ 112
n
kk
nkuknuku
nununu
n
kx
knukx
nunxnunx
][
][][
][][][][ 1
