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Sampling Theorem
主講者:虞台文
Content Periodic Sampling Sampling of Band-Limited Signals Aliasing --- Nyquist rate CFT vs. DFT Reconstruction of Band-limited Signals Discrete-Time Processing of Continuous-Time Signals Continuous-Time Processing of Discrete-Time Signals Changing Sampling Rate Realistic Model for Digital Processing
Sampling Theorem
Periodic Sampling
Continuous to Discrete-Time Signal Converter
C/DC/D
T
xc(t) x(n)= xc(nT)
Sampling rate
C/D System
Conversion from impulse train to
discrete-time sequence
Conversion from impulse train to
discrete-time sequence
xc(t)x(n)= xc(nT)
s(t)
xs(t)
Sampling with Periodic Impulse train
t
xc(t)
0 T 2T 3T 4TT2T3T
n
x(n)
0 1 2 3 4123
t
xc(t)
0 2T 4T 8T 10T2T4T8T
n
x(n)
0 2 4 6 8246
Sampling with Periodic Impulse train
t
xc(t)
0 T 2T 3T 4TT2T3T
n
x(n)
0 1 2 3 4123
t
xc(t)
0 2T 4T 8T 10T2T4T8T
n
x(n)
0 2 4 6 8246
What condition has to be placed on the sampling rate?
What condition has to be placed on the sampling rate?
We want to restore xc(t) from x(n). We want to restore xc(t) from x(n).
C/D System
Conversion from impulse train to
discrete-time sequence
Conversion from impulse train to
discrete-time sequence
xc(t)x(n)= xc(nT)
s(t)
xs(t)
n
n
nTtts )()(
)()()( tstxtx cs
n
nc nTttx )()(
n
nc nTtnTx )()(
C/D System
Conversion from impulse train to
discrete-time sequence
Conversion from impulse train to
discrete-time sequence
xc(t)x(n)= xc(nT)
s(t)
xs(t)
n
n
nTtts )()(
)()()( tstxtx cs
n
nc nTttx )()(
n
nc nTtnTx )()(
)(*)(2
1)(
jSjXjX cs
)(*)(2
1)(
jSjXjX cs
2 2( ) ( ), s s
k
S j kT T
2 2
( ) ( ), s sk
S j kT T
C/D System
)(*)(2
1)(
jSjXjX cs
)(*)(2
1)(
jSjXjX cs
Tk
TjS s
ks
2 ,)(
2)(
Tk
TjS s
ks
2 ,)(
2)(
s:Sampling Frequency
kscs k
TjXjX )(
2*)(
2
1)(
C/D System
kscs k
TjXjX )(
2*)(
2
1)(
k
sc kjXT
)(*)(1
k
sc kjjXT
)(1
k
scs kjjXT
jX )(1
)(
k
scs kjjXT
jX )(1
)(
Sampling Theorem
Sampling of
Band-Limited Signals
Band-Limited Signals
Yc(j)
Band-Limited
Band-Unlimited
Xc(j)
NN
1
Sampling of Band-Limited Signals
Band-Limited
TkjjX
TjX s
kscs
2 ),(
1)(
TkjjX
TjX s
kscs
2 ),(
1)(
Xc(j)
NN
1
ss 2s 3s2s3s
S(j)2/T
4s4s 2s 6s2s6s
S(j)2/T
Sampling withHigher Frequency
Sampling withLower Frequency
Sampling Theorem
Aliasing ---
Nyquist Rate
Recoverability
Band-Limited
TkjjX
TjX s
kscs
2 ),(
1)(
TkjjX
TjX s
kscs
2 ),(
1)(
Xc(j)
NN
1
ss 2s 3s2s3s
S(j)2/T
4s4s 2s 6s2s6s
S(j)2/T
Sampling withHigher Frequency
Sampling withLower Frequency
s > 2N s > 2N
s < 2N s < 2N
Case 1: s > 2N T
kjjXT
jX sk
scs
2 ),(
1)(
TkjjX
TjX s
kscs
2 ),(
1)(
Xc(j)
NN
1
ss 2s 3s2s3s
S(j)2/T
1/T
ss 2s 3s2s3s
Xs(j)
Case 1: s > 2N T
kjjXT
jX sk
scs
2 ),(
1)(
TkjjX
TjX s
kscs
2 ),(
1)(
Xc(j)
NN
1
ss 2s 3s2s3s
S(j)2/T
1/T
ss 2s 3s2s3s
Xs(j)
Passing Xs(j) through a low-pass filter with cutoff frequency N < c< s N , the original signal can be recovered.
Passing Xs(j) through a low-pass filter with cutoff frequency N < c< s N , the original signal can be recovered.
Xs(j) is a periodic function with period s.
Xs(j) is a periodic function with period s.
Case 2: s < 2N T
kjjXT
jX sk
scs
2 ),(
1)(
TkjjX
TjX s
kscs
2 ),(
1)(
Xc(j)
NN
1
1/T
2s2s 4s 6s4s6s
S(j)2/T
2s2s 4s 6s4s6s
Xs(j)
Case 2: s < 2N T
kjjXT
jX sk
scs
2 ),(
1)(
TkjjX
TjX s
kscs
2 ),(
1)(
Xc(j)
NN
1
1/T
2s2s 4s 6s4s6s
S(j)2/T
2s2s 4s 6s4s6s
Xs(j)Aliasi
ngAliasi
ng
No way to recover the original signal.No way to recover the original signal.
Xs(j) is a periodic function with period s.
Xs(j) is a periodic function with period s.
Nequist Rate
Xc(j)
NN
1Band-Limited
Nequist frequency (N) The highest frequency of a band-limited signal
Nequist rate = 2N
Nequist Sampling Theorem
Xc(j)
NN
1Band-Limited
s > 2N
s < 2N
Recoverable
Aliasing
Sampling Theorem
CFT vs. DFT
C/D System
Conversion from impulse train to
discrete-time sequence
Conversion from impulse train to
discrete-time sequence
xc(t)x(n)= xc(nT)
s(t)
xs(t)
k
scs kjjXT
jX )(1
)(
k
scs kjjXT
jX )(1
)(
n
n
nTtts )()(
)()()( tstxtx cs
n
nc nTttx )()(
n
nc nTtnTx )()(
Continuous-Time Fourier Transform
Conversion from impulse train to
discrete-time sequence
Conversion from impulse train to
discrete-time sequence
xc(t)x(n)= xc(nT)
s(t)
xs(t)
n
n
nTtts )()(
)()()( tstxtx cs
n
nc nTttx )()(
n
nc nTtnTx )()(
k
scs kjjXT
jX )(1
)(
k
scs kjjXT
jX )(1
)(
n
Tnjcs enTXjX )()(
n
Tnjcs enTXjX )()(
CFT vs. DFT
Conversion from impulse train to
discrete-time sequence
Conversion from impulse train to
discrete-time sequence
xc(t)x(n)= xc(nT)
s(t)
xs(t)
n
Tnjcs enTXjX )()(
n
Tnjcs enTXjX )()(
n
njj enxeX )()(
n
njj enxeX )()(
k
scs kjjXT
jX )(1
)(
k
scs kjjXT
jX )(1
)(
x(n)
CFT vs. DFT
Conversion from impulse train to
discrete-time sequence
Conversion from impulse train to
discrete-time sequence
xc(t)x(n)= xc(nT)
s(t)
xs(t)
n
Tnjcs enTXjX )()(
n
Tnjcs enTXjX )()(
n
njj enxeX )()(
n
njj enxeX )()(
k
scs kjjXT
jX )(1
)(
k
scs kjjXT
jX )(1
)(
x(n)
Ts
j jXeX )()(
Tsj jXeX )()(
T
js eXjX
)()(T
js eXjX
)()(
CFT vs. DFT
Tsj jXeX )()(
Tsj jXeX )()(
k
scs kjjXT
jX )(1
)(
k
scs kjjXT
jX )(1
)(
kc
j
T
kj
TjX
TeX )
2(
1)(
kc
j
T
kj
TjX
TeX )
2(
1)(
CFT vs. DFT
kc
j
T
kj
TjX
TeX )
2(
1)(
kc
j
T
kj
TjX
TeX )
2(
1)(
Xs(j)
ss
Ts
2Ts
21/T
X(ej)
2
1/T
2 44
Xc(j)1
CFT vs. DFT
Xs(j)
ss
Ts
2Ts
21/T
X(ej)
2
1/T
2 44
Xc(j)1
Amplitude scaling&
Repeating
Frequency scaling
s2
kc
j
T
kj
TjX
TeX )
2(
1)(
kc
j
T
kj
TjX
TeX )
2(
1)(
Sampling Theorem
Reconstruction of Band-limited Signals
Key Concepts
t
xc(t)
0 T 2T 3T 4TT2T3T
n
x(n)
0 1 2 3 4123
X(ej)
FT
IFT
Xc(j)
/T/T
SamplingC/D
RetrieveOne period
ICFT
CFT
TTjXT
eX cTj //
1)( TTjX
TeX c
Tj // 1
)(
Interpolation
T
T
tjcc dejXtx
/
/2
1)(
T
T
tjTj deeTX/
/2
1
T
T
tj
n
Tnjc deenTx
T /
/)(
2
T
T
tjTnj
nc dee
TnTx
/
/2)(
T
T
Tntj
nc de
TnTx
/
/
)(
2)(
TnTt
TnTtnTx
nc /)(
]/)(sin[)(
Interpolation
TnTt
TnTtnTxtx
ncc /)(
]/)(sin[)()(
x(n) n(t)
)()()( tnxtx nn
c
)()()( tnxtx nn
c
Ideal D/C Reconstruction System
x(n) xs(t) xr(t)Covert from sequence to
impulse train
Covert from sequence to
impulse train
T
Ideal Reconstruction
FilterHr(j)
Ideal Reconstruction
FilterHr(j)
T
x(n) xs(t) xr(t)Covert from sequence to
impulse train
Covert from sequence to
impulse train
T
Ideal Reconstruction
FilterHr(j)
Ideal Reconstruction
FilterHr(j)
T
Ideal D/C Reconstruction System
/T/T
Hr(j)
T
Obtained from sampling xc
(t) using an ideal C/D system.
)()()( nTtnxtxn
s
x(n) xs(t) xr(t)Covert from sequence to
impulse train
Covert from sequence to
impulse train
T
Ideal Reconstruction
FilterHr(j)
Ideal Reconstruction
FilterHr(j)
T
Ideal D/C Reconstruction System
)()( Tjs eXjX
))(/(
))(/sin()()(
nTtT
nTtTnxtx
nr
))(/(
))(/sin()()(
nTtT
nTtTnxtx
nr
)()()( Tjrr eXjHjX )()()( Tj
rr eXjHjX
Ideal D/C Reconstruction System
x(n) xr(t)D/CD/C
T
))(/(
))(/sin()()(
nTtT
nTtTnxtx
nr
))(/(
))(/sin()()(
nTtT
nTtTnxtx
nr
xc(t)C/DC/D
T
In what condition xr(t) = xc(t)?In what condition xr(t) = xc(t)?
Sampling Theorem
Discrete-Time Processing of Continuous-Time Signals
The Model
y(n) yr(t)D/CD/C
T
Discrete-TimeSystem
Discrete-TimeSystem
T
xc(t)C/DC/D
x(n)
Continuous-TimeSystem
Continuous-TimeSystem
xc(t) yr(t)
The Model
y(n) yr(t)D/CD/C
T
Discrete-TimeSystem
Discrete-TimeSystem
T
xc(t)C/DC/D
x(n)
Continuous-TimeSystem
Continuous-TimeSystem
xc(t) yr(t)Heff(j)Heff(j)
H (ej)H (ej)
LTI Discrete-Time Systems
y(n) yr(t)D/CD/C
T
Discrete-TimeSystem
Discrete-TimeSystem
T
xc(t)C/DC/D
x(n)H (ej)H (ej)
)( jX c)( jX c )( jeX )( jeX )( jeY )( jeY )( jYr
)( jYr
Hr (j)Hr (j)
kc
j
T
kj
TjX
TeX )
2(
1)(
)()()( Tjrr eYjHjY )()()( TjTj
r eXeHjH
kc
Tjr T
kjjX
TeHjH )
2(
1)()(
LTI Discrete-Time Systems
y(n) yr(t)D/CD/C
T
Discrete-TimeSystem
Discrete-TimeSystem
T
xc(t)C/DC/D
x(n)H (ej)H (ej)
)( jX c)( jX c )( jeX )( jeX )( jeY )( jeY )( jYr
)( jYr
Hr (j)Hr (j)
kc
Tjrr T
kjjX
TeHjHjY )
2(
1)()()(
T
TjXeHjY c
Tj
r/||0
/||)()()(
LTI Discrete-Time Systems
Continuous-TimeSystem
Continuous-TimeSystem
xc(t) yr(t)Heff(j)Heff(j)
T
TjXeHjY c
Tj
r/||0
/||)()()(
)( jX c)( jX c )()()( jXjHjY reffr
)()()( jXjHjY reffr
T
TeHjH
Tj
eff/||0
/||)()(
T
TeHjH
Tj
eff/||0
/||)()(
Example:Ideal Lowpass Filter
y(n) yr(t)D/CD/C
T
Discrete-TimeSystem
Discrete-TimeSystem
T
xc(t)C/DC/D
x(n)
)( jX c)( jX c )( jYr
)( jYr
1
cc
H(ej)
T
TjH
c
ceff /||0
/||1)(
T
TjH
c
ceff /||0
/||1)(
T
TeHjH
Tj
eff/||0
/||)()(
T
TeHjH
Tj
eff/||0
/||)()(
Example:Ideal Lowpass Filter
Continuous-TimeSystem
Continuous-TimeSystem
xc(t) yr(t)1
cc
Heff(j)
2||0
||1)(
T
TeH
c
cj
T
TeHjH
Tj
eff/||0
/||)()(
T
TeHjH
Tj
eff/||0
/||)()(
Example: Ideal Bandlimited Differentiator
Continuous-TimeSystem
Continuous-TimeSystem
xc(t) )()( txdt
dty cc
T
TjjH eff /||0
/||)( jjH )(
Example: Ideal Bandlimited Differentiator
Continuous-TimeSystem
Continuous-TimeSystem
xc(t) )()( txdt
dty cc
T
TjjH eff /||0
/||)( jjH )(
|Heff(j)|
T
T
Example: Ideal Bandlimited Differentiator
|| ,/)( TjeH j
Continuous-TimeSystem
Continuous-TimeSystem
xc(t) )()( txdt
dty cc
|Heff(j)|
T
T
Impulse Invariance
Continuous-TimeLTI system
hc(t), Hc(j)
Continuous-TimeLTI system
hc(t), Hc(j)xc(t) yc(t)
y(n) yc(t)D/CD/C
T
Discrete-TimeLTI System
h(n)H(ej)
Discrete-TimeLTI System
h(n)H(ej)
T
xc(t)C/DC/D
x(n)
What is the relation between hc(t) and h(n)?What is the relation between hc(t) and h(n)?
|| ),/()( TjHeH cj || ),/()( TjHeH c
j
Impulse Invariance || ),/()( TjHeH c
j || ),/()( TjHeH cj
)()( jeXnx )()( jeXnx )()( jXtx cc)()( jXtx cc
)()( nTxnx c
k
cj
T
kj
TjX
TeX
21)(
|| , 1
)(T
jXT
eX cj
Impulse Invariance || ),/()( TjHeH c
j || ),/()( TjHeH cj
)()( jeHnh )()( jeHnh )()( jHth cc)()( jHth cc
)()( nThnh c
|| , 1
)(T
jHT
eH cj
)()( nTThnh c
|| , )(T
jHeH cj
Impulse Invariance
Continuous-TimeLTI system
hc(t), Hc(j)
Continuous-TimeLTI system
hc(t), Hc(j)xc(t) yc(t)
y(n) yc(t)D/CD/C
T
Discrete-TimeLTI System
h(n)H(ej)
Discrete-TimeLTI System
h(n)H(ej)
T
xc(t)C/DC/D
x(n)
What is the relation between hc(t) and h(n)?What is the relation between hc(t) and h(n)?
)()( nTThnh c )()( nTThnh c
Sampling Theorem
Continuous-Time Processing of Discrete-Time Signals
The Model
yc(t) y(n)C/DC/D
T
Continous-TimeSystem
Continous-TimeSystem
T
x(n)D/CD/C
xc(t)
Discrete-TimeSystem
Discrete-TimeSystem
x(n) y(n)
The Model
yc(t) y(n)C/DC/D
T
Continous-TimeSystem
Continous-TimeSystem
T
x(n)D/CD/C
xc(t)
Discrete-TimeSystem
Discrete-TimeSystem
x(n) y(n)H (ej)H (ej)
Hc(j)Hc(j)
The Model
yc(t) y(n)C/DC/D
T
Continous-TimeSystem
Continous-TimeSystem
T
x(n)D/CD/C
xc(t)Hc(j)Hc(j)
)( jX c)( jX c)( jeX )( jeX )( jeY )( jeY)( jYc
)( jYc
TnTt
TnTtnxtx
nc /)(
]/)(sin[)()(
TnTt
TnTtnxtx
nc /)(
]/)(sin[)()(
TnTt
TnTtnyty
nc /)(
]/)(sin[)()(
TnTt
TnTtnyty
nc /)(
]/)(sin[)()(
TeTXjX Tjc /|| ),()(
TjXjHjY ccc /|| ),()()(
|| ),/(1
)( TjYT
eY j
The Model
TeTXjX Tjc /|| ),()(
TjXjHjY ccc /|| ),()()(
|| ),/(1
)( TjYT
eY cj
)/()/(1
)( TjXTjHT
eY ccj
)()/(1 j
c eTXTjHT
)()/( jc eXTjH
The Model
)/()/(1
)( TjXTjHT
eY ccj
)()/(1 j
c eTXTjHT
)()/( jc eXTjH
Discrete-TimeSystem
Discrete-TimeSystem
x(n) y(n)H (ej)H (ej)
)/()( TjHeH cj )/()( TjHeH c
j
The Model
Discrete-TimeSystem
Discrete-TimeSystem
x(n) y(n)H (ej)H (ej)
)/()( TjHeH cj )/()( TjHeH c
j
yc(t) y(n)C/DC/D
T
Continous-TimeSystem
Continous-TimeSystem
T
x(n)D/CD/C
xc(t)Hc(j)Hc(j)
Sampling Theorem
Changing Sampling Rate Using
Discrete-Time Processing
The Goal
Down/UpSamplingDown/UpSampling
)()( nTxnx c )'()(' nTxnx c
Sampling Rate Reduction By an Integer Factor
DownSampling
DownSampling
)()( nTxnx c )'()(' nTxnx c
MTT ' )()()( nMTxnMxnx cd )()()( nMTxnMxnx cd
Sampling Rate Reduction By an Integer Factor
k
cj
T
kj
TjX
TeX
21)(
MTT ' )()()( nMTxnMxnx cd )()()( nMTxnMxnx cd
r
cj
d T
rj
TjX
TeX
'
2
''
1)(
r
c MT
rj
MTjX
MT
21
Sampling Rate Reduction By an Integer Factor
k
cj
T
kj
TjX
TeX
21)(
r
cj
d MT
rj
MTjX
MTeX
21)( Let r = kM + i
1
0
2211 M
i kc MT
ij
T
kj
MTjX
TM
1
0
2211)(
M
i kc
jd T
kj
MT
ijX
TMeX
1
0
2211)(
M
i kc
jd T
kj
MT
ijX
TMeX
Sampling Rate Reduction By an Integer Factor
1
0
2211)(
M
i kc
jd T
kj
MT
ijX
TMeX
1
0
2211)(
M
i kc
jd T
kj
MT
ijX
TMeX
1
0
/)2( )(1
)(M
i
Mijjd eX
MeX
1
0
/)2( )(1
)(M
i
Mijjd eX
MeX
NN
Xc(j)
NN
Xs(j), X (ejT)
2/T2/T
1/T
N=NTN
X (ej)
22
1/T
Sampling Rate Reduction By an Integer Factor
Xd (ej)
22
1/MTM=2M=2
Xd (ejT)1/T’
2/T’2/T’ 4/T’4/T’
1
0
/)2( )(1
)(M
i
Mijjd eX
MeX
1
0
/)2( )(1
)(M
i
Mijjd eX
MeX
NN
Xc(j)
NN
Xs(j), X (ejT)
2/T2/T
1/T
N=NTN
X (ej)
22
1/T
N < : no aliasingN < : no aliasing
Antialiasing
NN
X (ej)
22
1/T
M=3M=3
Xd (ej)1/MT
22
AliasingAliasing
Antialiasing
NN
X (ej)
22
1/T
22
)(~ j
d eX
/3
Hd (ej)
22
1
/3
22
)()()( jjd
jd eXeHeX
/3/3
However, xd(n) x(nT’)However, xd(n) x(nT’)
M=3M=3
Decimator
Lowpass filterGain = 1
Cutoff = /M
Lowpass filterGain = 1
Cutoff = /M MM)(nx )(~ nx
)(~)(~ nMxnxd
Increasing Sampling Rate By an Integer Factor
UpSampling
UpSampling
)()( nTxnx c )'()(' nTxnx c
LTT /'
T4/' TT
Increasing Sampling Rate By an Integer Factor
UpSampling
UpSampling
)()( nTxnx c )'()(' nTxnx c
LTT /'
)( jeX )( jeX )(' jeX )(' jeX
X (ej)
1/T
X’ (ej)
L/T
Interpolator
Lowpass filterGain = L
Cutoff = /L
Lowpass filterGain = L
Cutoff = /LLL)(nx )(nxe)(nxi
otherwise
LLLnxnxe 0
,2,,0)/()(
k
e kLnkxnx )()()(
k
e kLnkxnx )()()(
Interpolator
k
e kLnkxnx )()()(
k
e kLnkxnx )()()()()( Ljje eXeX )()( Ljj
e eXeX
nj
n k
je ekLnkxeX
)()()(
k n
njekLnkx )()(
k
Lkjekx )(
Interpolator
k
e kLnkxnx )()()(
k
e kLnkxnx )()()()()( Ljje eXeX )()( Ljj
e eXeX
X (ej)
1/T
Xe(ej)
1/T
Xi(ej)
L/T
L=3L=3
Hi(ej)
L
/3/3
Changing the Sampling Rate By a Noninteger Factor
ResamplingResampling
)()( nTxnx c )'()(' nTxnx c
L
TMT '
Changing the Sampling Rate By a Noninteger Factor
)(nx
Lowpass filterGain = 1
Cutoff = /M
Lowpass filterGain = 1
Cutoff = /M MMLowpass filter
Gain = LCutoff = /L
Lowpass filterGain = L
Cutoff = /LLL )(nxe )(nxi )(~ nxi )(~ nxd
Sampling Periods:Sampling Periods:
TT LT /LT / LTM /LTM /
MMLowpass filter
Gain = LCutoff = min(/L, /M)
Lowpass filterGain = L
Cutoff = min(/L, /M)LL)(nx )(nxe )(~ nxi
)(~ nxd
Sampling Theorem
Realistic Model for
Digital Processing
Ideal Discrete-Time Signal Processing Model
y(n) yc(t)D/CD/C
T
Discrete-TimeLTI System
Discrete-TimeLTI System
T
xc(t)C/DC/D
x(n)
Real world signal usually is not
bandlimited
Ideal continuous-to-discrete converter is
not realizable
Ideal discrete-to-continuous
converter is not realizable
More Realistic Model
y(n) yc(t)D/CD/C
T
Discrete-TimeLTI System
Discrete-TimeLTI System
T
xc(t)C/DC/D
x(n)
)(ˆ nx )(ˆ ny)(txc )(txa
)( jH aa
)(txo )(tyDA )(ˆ tyr
)(~ jH r
Anti-aliasing
filter
Sample and
Hold
A/D converter
Discrete-time system
D/A converter
Compensated reconstruction
filter
T TT
Analog-to-Digital Conversion
T
Sample and
Hold
Sample and
Hold
A/D converter
A/D converter
T T
)(txa )(txo )(ˆ nxB
)(txo
)(txa
Sample and Hold
T )(txo
)(txa
otherwise
Ttth
0
01)(0
tT
ho(t)
n
nTt )(t
Sample and Hold
T )(txo
)(txa
otherwise
Ttth
0
01)(0
tT
ho(t)
n
nTtnx )()(
t
xo(t)
n
a nTtnTx )()(
Sample and Hold
otherwise
Ttth
0
01)(0
tT
ho(t)
n
nTtnx )()(
t
xo(t)
n
a nTtnTx )()(
n
aoo nTtnTxthtx )()(*)()(
n
aoo nTtnTxthtx )()(*)()(
Sample and Hold
n
aoo nTtnTxthtx )()(*)()(
n
aoo nTtnTxthtx )()(*)()(
Zero-OrderHoldho(t)
Zero-OrderHoldho(t)
n
nTtts )()(
)(txc )(txo)(txs
Goal: To hold constant sample value for A/D converter.
Goal: To hold constant sample value for A/D converter.
A/D Converter
C/DC/D
T
QuantizerQuantizer CoderCoder)(ˆ nx)(txa )(nx )(ˆ nxB
)]([)(ˆ nxQnx )]([)(ˆ nxQnx
Typical Quantizer
2Xm (B+1)-bit Binary code(B+1)-bit Binary code
Bm
Bm XX
22
21 B
mB
m XX
22
21
2
23
25
27
29
2-
23-
25-
27-
29-
2
3
2
3
4
2’s complementcode
Offset binarycode
011
010
001
000
111
110
101
100
111
110
101
100
011
010
001
000
)(ˆ xQx
x
Analysis of Quantization Errors
C/DC/D
T
QuantizerQuantizer CoderCoder)(ˆ nx)(txa )(nx )(ˆ nxB
)(ˆ)(ˆ nxXnx Bm )(ˆ)(ˆ nxXnx Bm
QuantizerQ[ ]
QuantizerQ[ ]
)]([)(ˆ nxQnx )(nx
)()()(ˆ nenxnx )(nx
)(ne
Analysis of Quantization Errors
)()()(ˆ nenxnx )(nx
)(ne 2/)(2/ ne 2/)(2/ ne)2/()()2/( mm XnxX )2/()()2/( mm XnxX
The error sequence e(n) is a stationary random process. e(n) and x(n) are uncorrelated. The random variables of the error process are uncorrelated,
i.e., the error is a white-noise process. e(n) is uniform distributed.
SNR (Signal-to-Noise Ratio)
)()()(ˆ nenxnx )(nx
)(ne 2/)(2/ ne 2/)(2/ ne)2/()()2/( mm XnxX )2/()()2/( mm XnxX
12
22 e 12
)2/( 2BmX
12
2 2m
B X
2
2
10log10e
xSNR
2
2
10log10e
xSNR
2
22
10
212log10
m
xB
XSNR
x
mXB 10log208.1002.6
SNR (Signal-to-Noise Ratio)
12
22 e 12
)2/( 2BmX
12
2 2m
B X
2
2
10log10e
xSNR
2
2
10log10e
xSNR
2
22
10
212log10
m
xB
XSNR
x
mXB 10log208.1002.6
每增加一個 bit , SNR 增加約 6dB每增加一個 bit , SNR 增加約 6dB
SNR (Signal-to-Noise Ratio)
12
22 e 12
)2/( 2BmX
12
2 2m
B X
2
2
10log10e
xSNR
2
2
10log10e
xSNR
2
22
10
212log10
m
xB
XSNR
x
mXB 10log208.1002.6
x大較有利,但不得過大 ( 為何? ) x過小不利 x每降低一倍 SNR 少 6dB X~N(0, x
2) P(|X|<4x )0.00064
Let x=Xm / 4 SNR 6B1.25 dBLet x=Xm / 4 SNR 6B1.25 dB