1 signals and systems lecture 22 aliasing. 2 chapter 7 sampling sampling theorem: let be a...
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Signals and SystemsLecture 22
•Aliasing
2
Chapter 7 Sampling
Sampling Theorem:
Let be a band-limited signal with
then is uniquely determined by its samples
if where
MjX , 0 tx tx ,1,0, nnTx
Ms 2Ts
2
RateNyquisyM :2
FrequencyNyquistM :
MsT
3
Chapter 7 Sampling
§7.3 The Effect of Undersampling: Aliasing
欠采样 混叠
2s M rx t x t
sin cr
n
T t nTx t x nT
t nT
2s
c T
Let
r an
x t x nT S t nTT
rx nT x nT
P565 7.25
4
Chapter 7 Sampling
1. Consider a sinusoidal signal ttx 0cos2s
c
Fix , change s 0
0 0a 66s
s
0cosrx t x t t
rX j X j
00
0
jH
2s
2s
s 0 s
Figure (a)00
5
Chapter 7 Sampling
0a 6s
6
Chapter 7 Sampling
0 0b 33s
s
0cosrx t x t t s s 0
Figure (b)
00
jH
2s
2s
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Chapter 7 Sampling
0 0
2 3c
3 2s
s
s s
Figure (c)
0 00
jH2s
2s
0 ss 0
0cosr sx t t x t
0cos2rx t t
8
0 0
5 6d
6 5s
s
0cos r sx t t x t s
s0
Figure (d)
00
jH2s
2s
0 ss 0
0cos5rx t t
Chapter 7 Sampling
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Chapter 7 Sampling
0 0reconstructed1 cos cos st t
When aliasing occurs, 02 s
0 0 02 2 , s s s
0 03 , 0 constants s rx t ，
2. Consider a sinusoidal signal
ttx 0cos
jtjjtj eeeetx 00
2
1
2
100
je
0
je
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Chapter 7 Sampling
00 33
a ss
ttxtxr 0cos
00 0s s
jH
2s
2s
00 2
3
3
2 b s
s
00 0s s
0 ss 0
0 ss 0
jH2s
2s
0cos r sx t t
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Chapter 7 Sampling
3. 02s
ttx 0cos
20s
c
00
je
0
je
00
je
0
je jeje
00 coscos jX r
ttxr 0coscos
txtxr 1,0,k,2k ①
0 2
txr②
0sinx t t 02
2sin
00
nnTx
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Anti-Alias Filtering (AAF)• Therefore, the first step in sampling is anti-alias
filtering (AAF) –– a LPF to assure thatMs 2
Chapter 7 Sampling
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The effect of AAF
• The AAF low-pass filter will rid of the information in x(t) beyond
, but will avoid the much more serious aliasing problem.
2s
Chapter 7 Sampling
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Chapter 7 Sampling
Excise 7.8 (P558)
5M 102 Ms2.0
10
2
sT
4
4
4
0
)sin(2
1)(
k
tjkk
k
k
eatktg
4
142
1
12
1
00
1
1
kj
kj
k
a
k
k
k
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Chapter 7 Sampling
Excise 7.9 (P558)
100M 150s Ms 2
k
skjXT
jG ))((1
)(
50 Ms 50)( Ms
75
12
s
T
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Chapter 7 Sampling
Excise 7.10 (P558)
a. x(t) is not a periodic signal.
b. |ω|< ω0
c.
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Chapter 7 Sampling
4. Strobe Effect （频闪效应）
Figure 7.18
2 0 s①
The speed of rotation of the disc is perceived correctly.
2 0 s②
The rotation appears to be at a lower frequency than is
actually the case.
Furthermore, because of phase reversal, the disc will appearto be rotating in the wrong direction.
Rotatingdisc
0Strobe
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Excise 7.38(P570)
Chapter 7 Sampling
otherwise
TjH0
2
11
)(
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Chapter 7 Sampling
T
20
Ts
2
))()(()(2)( 00 jj eeBAjX
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Chapter 7 Sampling
))cos((
))(cos(
)()(
0
0
taBA
atBA
atxty
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Chapter 7 Sampling
00
2
sAssume:
otherwiseTjH c
s
042
11
)(
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Chapter 7 Sampling
00 24
ss
cs
0014
4
s
nn s00
1
Consider the general case:
00 )1( scs nn
nn s0
014
4
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Chapter 7 Sampling
4
1 sc T
TnTn )4
11()1(
T
nTna s 1
0
0
nn s0
014
4
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Summary
Analysis of aliasing phenomenon
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Readlist
• Signals and Systems:– 7.4~7.5
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Chapter 6 Time and Frequency Characterization
Homework:P564 7.26