16.362 Signal and System I The representation of discrete-time signals in terms of impulse Example.
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Transcript of 16.362 Signal and System I The representation of discrete-time signals in terms of impulse Example.
![Page 1: 16.362 Signal and System I The representation of discrete-time signals in terms of impulse Example.](https://reader035.fdocuments.net/reader035/viewer/2022070605/5a4d1b167f8b9ab059991b51/html5/thumbnails/1.jpg)
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
![Page 2: 16.362 Signal and System I The representation of discrete-time signals in terms of impulse Example.](https://reader035.fdocuments.net/reader035/viewer/2022070605/5a4d1b167f8b9ab059991b51/html5/thumbnails/2.jpg)
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
![Page 3: 16.362 Signal and System I The representation of discrete-time signals in terms of impulse Example.](https://reader035.fdocuments.net/reader035/viewer/2022070605/5a4d1b167f8b9ab059991b51/html5/thumbnails/3.jpg)
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
txthtxthtxththty
![Page 4: 16.362 Signal and System I The representation of discrete-time signals in terms of impulse Example.](https://reader035.fdocuments.net/reader035/viewer/2022070605/5a4d1b167f8b9ab059991b51/html5/thumbnails/4.jpg)
16.362 Signal and System I • Properties of LIT systems
Associative property
)()()(
)()()()()()()(
21
21
21
txththtxththtxththty
Causality
,0)( th for t<0.
,0][ nh for n<0.
Stability
dtth )(
n
nh ][
![Page 5: 16.362 Signal and System I The representation of discrete-time signals in terms of impulse Example.](https://reader035.fdocuments.net/reader035/viewer/2022070605/5a4d1b167f8b9ab059991b51/html5/thumbnails/5.jpg)
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
][
][][][
![Page 6: 16.362 Signal and System I The representation of discrete-time signals in terms of impulse Example.](https://reader035.fdocuments.net/reader035/viewer/2022070605/5a4d1b167f8b9ab059991b51/html5/thumbnails/6.jpg)
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
![Page 7: 16.362 Signal and System I The representation of discrete-time signals in terms of impulse Example.](https://reader035.fdocuments.net/reader035/viewer/2022070605/5a4d1b167f8b9ab059991b51/html5/thumbnails/7.jpg)
16.362 Signal and System I • The unit step response of an LTI system
][n ][nh ][nh
][nu ][nh ][ns ][]1[][ nhnsns
![Page 8: 16.362 Signal and System I The representation of discrete-time signals in terms of impulse Example.](https://reader035.fdocuments.net/reader035/viewer/2022070605/5a4d1b167f8b9ab059991b51/html5/thumbnails/8.jpg)
16.362 Signal and System I • Linear constant-coefficient difference equations
][nx
][]1[21][ 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
21
+
delay
![Page 9: 16.362 Signal and System I The representation of discrete-time signals in terms of impulse Example.](https://reader035.fdocuments.net/reader035/viewer/2022070605/5a4d1b167f8b9ab059991b51/html5/thumbnails/9.jpg)
16.362 Signal and System I • Linear constant-coefficient difference equations
][]1[21][ nxnyny
][]1[21][ nnhnh
]1[21][ nhnh
][]1[21][ nnhnh
When n 1, 21
]1[][
nhnh
n
Anh
21][
][21][ nuAnh
n
Causality
][n
][nh
][nh
21
+
delay
![Page 10: 16.362 Signal and System I The representation of discrete-time signals in terms of impulse Example.](https://reader035.fdocuments.net/reader035/viewer/2022070605/5a4d1b167f8b9ab059991b51/html5/thumbnails/10.jpg)
16.362 Signal and System I • Linear constant-coefficient difference equations
][n
][nh
][nh
21
][]1[21][ nxnyny +
][]1[21][ nnhnh delay
][]1[21][ nnhnh
][21][ nuAnh
n
Determine A by initial condition:
When n = 0, 1]0[]0[ h
]0[21]0[
0
uAh
A = 1
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16.362 Signal and System I • Linear constant-coefficient difference equations
]1[ n ][]1[21][ nxnyny
][]1[21][ nnhnh
][21][ nunh
n
?][ ny
Two ways:
(1) Repeat the procedure
(2) ][][][ nhnxny
]1[21
]1[][]1[][
1
nu
nhnhnny
n
][nh
][nh
21
+
delay
![Page 12: 16.362 Signal and System I The representation of discrete-time signals in terms of impulse Example.](https://reader035.fdocuments.net/reader035/viewer/2022070605/5a4d1b167f8b9ab059991b51/html5/thumbnails/12.jpg)
16.362 Signal and System I • The unit step response of an LTI system, continuous time
)(t )(th )(ty
)(tu )(th )(ts
)(
)()()(
th
dnhty
)()( thdttds
t
dh
dtuhts
)(
)()()(
![Page 13: 16.362 Signal and System I The representation of discrete-time signals in terms of impulse Example.](https://reader035.fdocuments.net/reader035/viewer/2022070605/5a4d1b167f8b9ab059991b51/html5/thumbnails/13.jpg)
16.362 Signal and System I • Linear constant-coefficient difference equations
)(tx)(
21
21)( txdtdyty
?)( 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 txtydtdy
)(th
)(ty
21
+
dtd
21
![Page 14: 16.362 Signal and System I The representation of discrete-time signals in terms of impulse Example.](https://reader035.fdocuments.net/reader035/viewer/2022070605/5a4d1b167f8b9ab059991b51/html5/thumbnails/14.jpg)
16.362 Signal and System I • Linear constant-coefficient difference equations
)(t
When t>0,dtdyty
21)( tAety 2)(
Determine A by initial condition:
)()( 2 tuAeth t
Causality
)(21
21)( txdtdyty
)(21
21)( tdtdyty
)(21
21)( tdtdyty
)(th
)(ty
21
+
dtd
21
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16.362 Signal and System I • Linear constant-coefficient difference equations
Determine A by initial condition:
)()( 2 tuAeth t
)(21)()
21()()2(
21)( 222 ttAetuAetuAe ttt
A = 1 )()( 2 tueth t
)(t)(
21
21)( txdtdyty
)(21
21)( tdtdyty
)(th
)(ty
21
+
dtd
21
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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
21
)(21
21)( txdtdyty +
dtd
21
)()( 2 tueth t
)(][5
)( 23 tueeKty tt
![Page 17: 16.362 Signal and System I The representation of discrete-time signals in terms of impulse Example.](https://reader035.fdocuments.net/reader035/viewer/2022070605/5a4d1b167f8b9ab059991b51/html5/thumbnails/17.jpg)
16.362 Signal and System I • Singularity functions
)()(0 ttu Define:
dttdtu )()(1
n
n
n dttdtu )()(
)()()(1 tudtut
du
dtuu
tututu
t
)(
)()(
)()()( 112
du
tutututut
n
n
)(
)()()()(
)1(
![Page 18: 16.362 Signal and System I The representation of discrete-time signals in terms of impulse Example.](https://reader035.fdocuments.net/reader035/viewer/2022070605/5a4d1b167f8b9ab059991b51/html5/thumbnails/18.jpg)
16.362 Signal and System I • Singularity functions
)()()()()( 0 txttxtutx
dtdx
tudtdx
dtdtdxu
tdxuutx
dutx
dtxututx
)(
)()()(
)()(|)()(
)()(
)()()()(
0
0
00
0
11
n
n
n dttxdtutx )()()(
dttdxtutx )()()( 1
)()()( 0 txtutx
![Page 19: 16.362 Signal and System I The representation of discrete-time signals in terms of impulse Example.](https://reader035.fdocuments.net/reader035/viewer/2022070605/5a4d1b167f8b9ab059991b51/html5/thumbnails/19.jpg)
16.362 Signal and System I • Singularity functions
n
n
n dttxdtutx )()()(
dttdxtutx )()()( 1
)()()( 0 txtutx )()()()( 21
11 tudttdututu
)()()()( 111 tutututu k
k terms
![Page 20: 16.362 Signal and System I The representation of discrete-time signals in terms of impulse Example.](https://reader035.fdocuments.net/reader035/viewer/2022070605/5a4d1b167f8b9ab059991b51/html5/thumbnails/20.jpg)
16.362 Signal and System I • Singularity functions
dx
dtux
tutxtutx
t
)(
)()(
)()()()( 1
n
n
n dttxdtutx )()()(
t
dxtutx )()()( 1
)()()( 0 txtutx
ddx
tudx
tututxtutx
t
t
')'(
)()(
)()()()()( 2
)()()(2 tututu
![Page 21: 16.362 Signal and System I The representation of discrete-time signals in terms of impulse Example.](https://reader035.fdocuments.net/reader035/viewer/2022070605/5a4d1b167f8b9ab059991b51/html5/thumbnails/21.jpg)
16.362 Signal and System I • Singularity functions --- discrete time
]1[][][1 nnnu Define:
]1[][][ 11 nununu kkk
]1[][]1[][][][][][ 1
nxnxnnxnnxnunx
]1[][][][ 1111 nunununu
]1[][][][ 1 nxnxnunx
![Page 22: 16.362 Signal and System I The representation of discrete-time signals in terms of impulse Example.](https://reader035.fdocuments.net/reader035/viewer/2022070605/5a4d1b167f8b9ab059991b51/html5/thumbnails/22.jpg)
16.362 Signal and System I • Singularity functions --- discrete time
][][1 nunu
Define:
1][][][
][][][ 112
n
kk
nkuknuku
nununu
n
kx
knukx
nunxnunx
][
][][
][][][][ 1