Sampling Theorem 主講者：虞台文. Content Periodic Sampling Sampling of Band-Limited Signals...

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



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





xc(t)x(n)= xc(nT)

s(t)

xs(t)

n

n

nTtts )()(

)()()( tstxtx cs

n

nc nTttx )()(

n

nc nTtnTx )()(

C/D System





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.


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.



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





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





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





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





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)


FilterHr(j)

T


impulse train


impulse train

T


FilterHr(j)


FilterHr(j)

T


/T/T

Hr(j)

T

Obtained from sampling xc

(t) using an ideal C/D system.

)()()( nTtnxtxn

s


impulse train


impulse train

T


FilterHr(j)


FilterHr(j)

T


)()( Tjs eXjX

))(/(

))(/sin()()(

nTtT

nTtTnxtx

nr

))(/(

))(/sin()()(

nTtT

nTtTnxtx

nr

)()()( Tjrr eXjHjX )()()( Tj

rr eXjHjX


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


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)



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)()(


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

/||)()()(




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



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



xc(t) )()( txdt

dty cc

T

TjjH eff /||0

/||)( jjH )(




xc(t) )()( txdt

dty cc

T

TjjH eff /||0

/||)( jjH )(

|Heff(j)|

T

T


|| ,/)( TjeH j



xc(t) )()( txdt

dty cc

|Heff(j)|

T

T

Impulse Invariance

Continuous-TimeLTI system

hc(t), Hc(j)


hc(t), Hc(j)xc(t) yc(t)

y(n) yc(t)D/CD/C

T

Discrete-TimeLTI System

h(n)H(ej)


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


hc(t), Hc(j)


hc(t), Hc(j)xc(t) yc(t)

y(n) yc(t)D/CD/C

T


h(n)H(ej)


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


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



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



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


)/()( TjHeH cj )/()( TjHeH c

j

The Model

Discrete-TimeSystem

Discrete-TimeSystem


)/()( TjHeH cj )/()( TjHeH c

j

yc(t) y(n)C/DC/D

T



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


MTT ' )()()( nMTxnMxnx cd )()()( nMTxnMxnx cd


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


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


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


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


Cutoff = /M MM)(nx )(~ nx

)(~)(~ nMxnxd

Increasing Sampling Rate By an Integer Factor

UpSampling

UpSampling


LTT /'

T4/' TT

Increasing Sampling Rate By an Integer Factor

UpSampling

UpSampling


LTT /'

)( jeX )( jeX )(' jeX )(' jeX

X (ej)

1/T

X’ (ej)

L/T

Interpolator

Lowpass filterGain = L

Cutoff = /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


L

TMT '

Changing the Sampling Rate By a Noninteger Factor

)(nx


Cutoff = /M


Cutoff = /M MMLowpass filter

Gain = LCutoff = /L


Cutoff = /LLL )(nxe )(nxi )(~ nxi )(~ nxd

Sampling Periods:Sampling Periods:

TT LT /LT / LTM /LTM /

MMLowpass filter

Gain = LCutoff = min(/L, /M)


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



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



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


Sample and Hold

n


n


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


)(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)


)(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


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


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

Sampling Theorem 主講者：虞台文. Content Periodic Sampling Sampling of Band-Limited Signals...

Documents

Transcript of Sampling Theorem 主講者：虞台文. Content Periodic Sampling Sampling of Band-Limited Signals...