28
DSP (Spring, 2007) Sampling of Continuous-time Signals NCTU EE 1 Sampling of Continuous-time Signals Advantages of digital signal processing, e.g., audio/video CD. Things to look at: Continuous-to-discrete (C/D) Discrete-to-continuous (D/C) – perfect reconstruction Frequency-domain analysis of sampling process Sampling rate conversion Periodic Sampling Ideal continuous-to-discrete-time (C/D) converter Continuous-time signal: ) (t x c Discrete-time signal: < < = n nT x n x c ), ( ] [ , T: sampling period In theory, we break the C/D operation into two steps: (1) Ideal sampling using “analog delta function (impulse)” (2) Conversion from impulse train to discrete-time sequence Step (1) can be modeled by mathematical equation. Step (2) is a “concept”, no mathematical model. In reality, the electronic analog-to-digital (A/D) circuits can approximate the ideal C/D operation. This circuitry is one piece; it cannot be split into two steps. ) ( t x c ] [ n x C/D ) (t x c ) (t s ) (t x s Conversion from impulse train to discrete-time sequence ] [ n x

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Page 1: Sampling of Continuous-time Signalstwins.ee.nctu.edu.tw/courses/dsp_08/dsp4.pdf · Discrete-time Processing of Continuous-time Sig- ... Sampling of Continuous-time Signals NCTU EE

DSP (Spring, 2007) Sampling of Continuous-time Signals

NCTU EE 1

Sampling of Continuous-time Signals Advantages of digital signal processing, e.g., audio/video CD.

Things to look at:

Continuous-to-discrete (C/D)

Discrete-to-continuous (D/C) – perfect reconstruction

Frequency-domain analysis of sampling process

Sampling rate conversion

Periodic Sampling Ideal continuous-to-discrete-time (C/D) converter

Continuous-time signal: )(txc

Discrete-time signal: ∞<<∞−= nnTxnx c ),(][ , T: sampling period

In theory, we break the C/D operation into two steps:

(1) Ideal sampling using “analog delta function (impulse)”

(2) Conversion from impulse train to discrete-time sequence

Step (1) can be modeled by mathematical equation.

Step (2) is a “concept”, no mathematical model.

In reality, the electronic analog-to-digital (A/D) circuits can approximate the ideal C/D

operation. This circuitry is one piece; it cannot be split into two steps.

)(txc ][nx C/D

)(txc

)(ts

)(txs

Conversion from impulse train

to discrete-time sequence ][nx

Page 2: Sampling of Continuous-time Signalstwins.ee.nctu.edu.tw/courses/dsp_08/dsp4.pdf · Discrete-time Processing of Continuous-time Sig- ... Sampling of Continuous-time Signals NCTU EE

DSP (Spring, 2007) Sampling of Continuous-time Signals

NCTU EE 2

Ideal sampling

Ideal sampling signal: impulse train (an analog signal)

( ) ( )∑∞

−∞=−=

nnTtts δ , T: sampling period

Analog (continuous-time) signal: )(txc

Sampled (continuous-time) signal: )(txs

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )∑∑

∑∞

−∞=

−∞=

−∞=

−=−=

−==

nc

nc

nccs

nTtnTxnTttx

nTttxtstxtx

δδ

δ

Frequency-domain Representation of Sampling

( ) ( ) ( )∑∞

−∞=Ω−Ω=Ω↔

ksk

TjSts δπ2 , where Ts /2π=Ω

Remark: Ω : analog frequency (radians/s)

ω : discrete (normalized) frequency (radians/sample)

Tω=Ω ; πωπ ≤<− , TTππ

≤Ω<−

Step 1: Ideal Sampling (all in analog domain)

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )( )∑ ∑

∑∞

−∞=

−∞=

−∞=

Ω−Ω=Ω−Ω∗Ω=

Ω−Ω∗Ω=Ω∗Ω=Ω

k kscc

ksccs

kjXT

kjXT

kjXT

jSjXjX

11

121

δ

δπ

The sampled signal spectrum is the sum of shifted copies of the original.

Remark: In analog domain,

)(*)(

21

)(*)()()(

ΩΩ=

jYjX

fYfXtytx

π

)(txc )(txsSampling

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DSP (Spring, 2007) Sampling of Continuous-time Signals

NCTU EE 3

Step 2: Analog Impulses to Sequence (analog to discrete-time)

No mathematical model. The spectrum of )(txs , )( ΩjX s , is the same as the

spectrum of ][nx , )( TjeX Ω . (See the Appendix at the end.)

Now, ( )( )sk

cTj kjX

TeX Ω−Ω= ∑

−∞=

Ω 1)(

Thus, ⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛ −= ∑

−∞= Tk

TjX

TeX

kc

j πωω 21)(

Remark: In time domain, )(txs and ][nx are two very different signals but they

have the “same” spectra in frequency domain.

Two Cases:

(1) no aliasing: Ns Ω>Ω 2 , and

(2) aliasing: Ns Ω<Ω 2 , where NΩ is the highest nonzero frequency compo-

nent of )( ΩjX c .

After sampling, the replicas of )( ΩjX c overlap (in frequency domain). That is, the

higher frequency components of )( ΩjX c overlap with the lower frequency com-

ponents of ( ))( sc jX Ω−Ω .

⇒ t t

xc(t) xs(t)

T

⇓ ⇓Xc(jΩ)

Ω

ΩN

Xs(jΩ)

ΩΩS

FT FT

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DSP (Spring, 2007) Sampling of Continuous-time Signals

NCTU EE 4

Nyquist Sampling Theorem:

Let x(t) be a bandlimited signal with 0)( =ΩjX c for NΩ≥Ω || . (i.e., no com-

ponents at frequencies greater than NΩ ) Then )(txc is uniquely determined by its

samples K,2,1,0),(][ ±±== nnTxnx c , if Ns TΩ≥=Ω 22π . (Nyquist,

Shannon)

-- Nyquist frequency = NΩ , the bandwidth of signal.

-- Nyquist rate = 2 NΩ , the minimum sampling rate without distortion. (In some books,

Nyquist frequency = Nyquist rate.)

-- Undersampling: Ns Ω<Ω 2

-- Overdampling: Ns Ω>Ω 2

Fourier Series, Fourier Transform, Discrete-Time

Fourier Series & Discrete-Time Fourier Transform Fourier Series x(t): periodic continuous-time signal with period T0

Power: ∑∫

−∞=

==k

kTx XdttxT

P 22

0 0

)(1

T0

t

x(t)

k

Xk

complex valued

⎪⎪⎩

⎪⎪⎨

=Ω=

==

∑∑Ω−

−∞=

Ω∞

−∞=

00

0

2

2 ,)(1

)(

0

0

00

Tdtetx

TX

eXeXtx

T

tjkk

k

tjkk

k

Ttkj

k

π

π

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DSP (Spring, 2007) Sampling of Continuous-time Signals

NCTU EE 5

Fourier Transform x(t): continuous-time signal

Energy: ∫∫∞

∞−

∞−ΩΩ== djXdttxPx

22 )(21)(π

Remark: (1) Other Notations

(2) Relationships between F.S. & F.T.

t

x(t)

Ω

X(jΩ)

complex valued

⎪⎩

⎪⎨

ΩΩ=

∫∫∞

∞−

Ω−

∞−

Ω

)()(

)(21)(

dtetxjX

dejXtx

tj

tj

π

⎪⎩

⎪⎨

=

=

∫∫

∞−

∞−

)()(

)(21)(

dtetxwX

dwewXtx

jwt

jwt

π⎪⎩

⎪⎨⎧

=

=

∫∫

∞−

∞−

)()(

)()(2

2

dtetxfX

dfefXtxftj

ftj

π

π

k

Xk ⇓ F.S.

t T0

T0 → ∞

⇒ t

Ω

X(jΩ)⇓ F.T.

Energy SpectrumPower Spectrum

∫ Ω−=0

0)(1

0T

tjkk dtetx

TX ∫

∞−

Ω−=Ω dtetxjX tj)()(

)(lim)( 000

00

kT

kT

k XTjXXT∞→

Ω→Ω∞→

=Ω⎯⎯⎯ →⎯

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DSP (Spring, 2007) Sampling of Continuous-time Signals

NCTU EE 6

(3) Periodic Signal

Discrete-Time Fourier Series x[n]: periodic discrete-time signal with period N.

Power: ∑∑−

=

=

==1

0

21

0

2][1 N

kk

N

nx Xnx

NP

k

Xk

F.S.

t T0

Ω

X(jΩ)

F.T.

2π/T0

k

Xk

complex valued

D.F.S.

⇔ n

x[n]

N N

Ω

X(jΩ)

2π/T

t

x(t)

T

F.T.

⎪⎪⎩

⎪⎪⎨

=

=

∑−

=

=

][

1][

1

0

2

1

0

2

N

n

Nnkj

k

N

k

Nnkj

k

enxX

eXN

nxπ

π

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DSP (Spring, 2007) Sampling of Continuous-time Signals

NCTU EE 7

Discrete-Time Fourier Transform x[n]: discrete-time signal

Energy: ∫∑ −

−∞=

==π

ππdweXnxE jw

nx

22 )(21][

Remark: X(ejw) v.s. X(jΩ)

X(ejw) is a frequency-scaled version of X(jΩ)

Tw

jweXjXΩ=

=Ω )()(

w

X(ejw)

DTFT

⇔ n

x[n]

Ω

X(jΩ)

2π/T

t

x(t)

T

F.T.

⎪⎪⎩

⎪⎪⎨

=

=

∑∫

−∞=

][)(

)(21][

n

jwnjw

jwnjw

enxeX

dweeXnxπ

ππ

Page 8: Sampling of Continuous-time Signalstwins.ee.nctu.edu.tw/courses/dsp_08/dsp4.pdf · Discrete-time Processing of Continuous-time Sig- ... Sampling of Continuous-time Signals NCTU EE

DSP (Spring, 2007) Sampling of Continuous-time Signals

NCTU EE 8

Reconstruction of a Band-limited Signal from Its

Samples -- Perfect reconstruction: recover the original continuous-time signal without distortion,

e.g., ideal lowpass (bandpass) filter

Based on the frequency-domain analysis, if we can “clip” one copy of the original spectrum,

)( ΩjX c , without distortion, we can achieve the perfect reconstruction. For example, we

use the ideal low-pass filter as the reconstruction filter.

Remark: Note that )(txs is an analog signal (impulses).

∑ →−→−=→→ ][..)()()()( nxconvrseqnTtnTxtxsamplingtx sc δ

)(.)(][)(.][ txreconnTtnxtxconvrimpulsenx rs ∑ →→−=→−→ δ

{ } ∑∑ ∫

∫ ∑∞

−∞=

−∞=

∞−

∞−

−∞=

−=−−=

⎭⎬⎫

⎩⎨⎧ −−=∗=

nr

nr

nrrsr

nTthnxdthnTnx

dthnTnxthtxtx

)(][)()(][

)()(][)()()(

λλλδ

λλλδ

)()()()()()(

}][){()(][)(

ΩΩ=Ω=Ω=

Ω=Ω=Ω

Ω

Ω=

−∞=

Ω−∞

−∞=

Ω− ∑∑jXjHeXjHeXjH

enxjHejHnxjX

rTj

rTw

jwr

n

Tnjr

n

Tnjrr

Ideal low-pass reconstruction filter:

⎩⎨⎧ ≤Ω<−

=Ωotherwise

TTTjH r 0

)(ππ

)()sin()(

TtTtthr π

π=

Reconstruction filter

)(txs )(txrConvert from Sequence to Impulse train

][nx

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DSP (Spring, 2007) Sampling of Continuous-time Signals

NCTU EE 9

T 2T t

1

hr(t)

t

xs(t)

T

n

x[n]

t

xr(t)

T

Tπ− Tπ Ω

T

Page 10: Sampling of Continuous-time Signalstwins.ee.nctu.edu.tw/courses/dsp_08/dsp4.pdf · Discrete-time Processing of Continuous-time Sig- ... Sampling of Continuous-time Signals NCTU EE

DSP (Spring, 2007) Sampling of Continuous-time Signals

NCTU EE 10

Discrete-time Processing of Continuous-time Sig-

nals

If this is an LTI system,

(1) ][][ nynx → : )()()( ωωω jjj eXeHeY =

(2) ][)( nxtxc → : ⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ −= ∑

−∞= Tk

TjX

TeX

kc

j πωω 21)(

(3) )(][ tyny r→ : )()()( Tjrr eYjHjY ΩΩ=Ω

Note that “T” is included in the expression of ),( ωjeY TΩ←ω . This means

“physical” frequency (not normalized).

(4) )()( tytx rc →→L :

∑∞

−∞=

Ω

ΩΩ

⎟⎠⎞

⎜⎝⎛ −ΩΩ=

Ω=Ω

kc

Tjr

TjTjrr

TkjjX

TeHjH

eXeHjHjY

π21)()(

)()()()(

If )( ΩjHr is an ideal low-pass reconstruction filter, then

⎩⎨⎧ <ΩΩ=Ω

Ω

otherwiseTjXeHjY c

Tj

r ,0/||),()()( π

In other words, if )(txc is band-limited and is ideally sampled at a rate above the

Nyquist rate, and the reconstruction filter is the ideal low-pass filter, then the

equivalent analog filter has the same spectrum shape of the discrete-time filter.

⎩⎨⎧ <Ω=Ω

Ω

otherwiseTeHjH

Tj

eff ,0/||),()( π

][nx ][ny)(txc )(tyr

C/D Discrete-time

system D/C

T T

)( ωjeH

)( ΩjHeff

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DSP (Spring, 2007) Sampling of Continuous-time Signals

NCTU EE 11

Remark: In order to have the above equivalent relation between )( ωjeH and

)( ΩjHeff , we need

(i) The system is LTI;

(ii) The input is bandlimited;

(iii) The input is sampled without aliasing and the ideal impulse train is

used in sampling;

(iv) The ideal reconstruction filter is used to produce the analog output.

In practice, the above conditions are only approximately valid at best.

However, there are methods in designing the sampling and the reconstruction

processes to make the approximation better.

)()( Ω=Ω jHjH ceff

⇒ )(][ nTThnh c=

The impulse response of the discrete-time system is a scaled, sampled version of )(thc .

TjHT

TwjHeH

c

cjw

π

π

≥Ω=Ω

<=

for ,0)( s.t. chosen is

w )()(

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DSP (Spring, 2007) Sampling of Continuous-time Signals

NCTU EE 12

Continuous-time Processing of Discrete-time Signals

⇒ π<== weXTwjH

TwjX

TwjH

TeY jw

cccjw )()()()(1)(

⇒ π<= wTwjHeH c

jw )()(

or, equivalently, T

jHeH cTj π

<ΩΩ=Ω )()(

Example: Noninteger Delay

π<= Δ− w )( jwjw eeH

][nx ][ny)(txc )(tycD/C

Continu.-time system C/D

T T

)( ωjeH

)( ΩjHc

π

π

π

<=

<ΩΩΩ=Ω

<Ω=Ω Ω

wTwjY

TeY

TjXjHjY

TeTXjX

cjw

ccc

Tjc

),(1)(

),()()(

),()(

n

x[n]

n

y[n]

yc(t) = xc(t-ΔT)

xc(t)

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DSP (Spring, 2007) Sampling of Continuous-time Signals

NCTU EE 13

Change the Sampling Rate Using Discrete-time

Processing

⎩⎨⎧

=→→=→→

)'(]['')(][

)(nTxnxTnTxnxT

txc

cc

Original sampling period: T

New sampling period: 'T 'TT ≠

• Sampling rate reduction by an integer factor Sampling rate compressor:

MTT =' , where M is an integer

)(][][ nMTxnMxnx cd ==

Compressor

Aliasing: If the original signal BW is not small enough to meet the Nyquist rate requirement,

t

T1

Ω

1

2Tπ

Downsampling

Ω

)(1

1

ΩjXT c

2

2Tπ

)(1

2

ΩjXT c

FT

t

T2 = MT1

FT

M↓][nx )(][][ nMTxnMxnx cd ==

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DSP (Spring, 2007) Sampling of Continuous-time Signals

NCTU EE 14

prefiltering is needed.

The Original

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛ −= ∑

−∞= Tk

TjX

TeX

kc

j πωω 21)(

The Downsampled

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛ −= ∑

−∞= '2

''1)(

Tr

TjX

TeX

rc

jd

πωω

LLLLLLLLL

1100001121)1(210)1()2(

)1()1(21012

−−−−−

+−−−

kMMMi

MMMr

Old and new index: kMir += ∞−−−∞= ,,2,1,0,1,2,,, LLkr , 1,,2,1,0 −= Mi L

∑ ∑

−∞=

=

−∞=

−∞=

⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ −−=

⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ −=

⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ −=

k

M

ic

rc

rc

jd

MTi

MTkM

MTjX

MT

MTr

MTjX

MT

Tr

TjX

TeX

1

0

221

21

'2

''1)(

ππω

πω

πωω

∑ ∑

=

⎟⎠⎞

⎜⎝⎛ −

=

−∞=

⎟⎟

⎜⎜

⎛=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

−=

1

0

2

1

0

1

2211)(

M

i

Mi

Mj

M

i kc

jd

eXM

Tkj

MTijX

TMeX

πω

ω ππω

The down-sampled spectrum = sum of shifted replica of the original

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DSP (Spring, 2007) Sampling of Continuous-time Signals

NCTU EE 15

Downsampling with aliasing

To avoid aliasing

⇒ π<MwN

General System for Sampling Rate Reduction by M

Lowpass filter Cutoff= Mπ

M↓][nx ][~ nx ][~][ nMxnxd =

T T T’=MT

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DSP (Spring, 2007) Sampling of Continuous-time Signals

NCTU EE 16

• Increasing sampling rate by an integer factor

Sampling rate expander

LTT /'= , where L is an integer

⎟⎠⎞

⎜⎝⎛=

LTnxnx ci ][

(1) shape is compressed; (2) replicas are removed

Lowpass filter Cutoff= Lπ

L↑ ][nx ][nxe ][nxi

T T’ T’=T/L

t

T1

1

2Tπ

Ω

Downsampling

Ω

2

2Tπ

FT

t

T2 = T1/L

FT

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DSP (Spring, 2007) Sampling of Continuous-time Signals

NCTU EE 17

(1) Increase samples

<Time-domain>

∑∞

−∞=−=

⎩⎨⎧ ±±==

k

e

kLnkx

otherwiseLLnL

nxnx

][][

,0,2,,0],[][

δ

L

<Frequency-domain>

)(][][

][][)(

Lj

k n

nj

nj

n k

je

eXekLnkx

ekLnkxeX

ωω

ωω

δ

δ

=⎟⎠⎞

⎜⎝⎛ −=

⎟⎠

⎞⎜⎝

⎛ −=

∑ ∑

∑ ∑∞

−∞=

−∞=

−∞

−∞=

−∞=

Note that Lkj

n

nj eekLn ωωδ −∞

−∞=

− =−∑ ][

Remark: Essentially, the horizontal frequency axis is compressed.

The shape of the spectrum is not changed.

Told Ω=ω_ , LTTnew Ω=Ω= '_ω , Lnewold ⋅= ωω __

Remark: At this point, we only insert zeros into the original signal. In time domain,

this signal doesn’t look like the original.

(2) Ideal lowpass filtering

<Frequency-domain>

⎩⎨⎧ ≤Ω<−=Ω otherwise

TLTLjHi 0)()(1)( ππ

<Time-domain>

)()sin(][

LnLnnhi π

π= , an interpolator!

[ ]∑∞

−∞= −−

=k

i LkLnLkLnkxnx

/)(/)(sin][][

ππ

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DSP (Spring, 2007) Sampling of Continuous-time Signals

NCTU EE 18

Linear interpolation

2

)2sin()2sin(1)( ⎥⎦

⎤⎢⎣

⎡=

ωωω L

LeH j

lin

∑∞

−∞=−=

klinlin kLnhkxnx ][][][

⎩⎨⎧ ≤−= otherwise

LnLnnhlin ,0||,/||1][

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• Changing sampling rate by a rational factor

Idea: Sampling period TLM

LTT ⎯⎯⎯ →⎯⎯⎯⎯⎯ →⎯ decimationioninterpolat

Remark: In general, if the factor is not rational, go back to the continuous signals.

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• In summary:

-- Sampling

Time-domain Frequency -domain

Prefiltering Limit bandwidth Ns Ω>Ω 2

Analog sampling (impulse train) Duplicate and shift )(Ω

Analog to discrete ][)( nt δδ → ω→Ω

-- Reconstruction

Time-domain Frequency -domain

Discrete to analog )(][ tn δδ → Ω→ω

Interpolation Remove extra copies )(Ω

-- Down-sampling

Time-domain Frequency -domain

Prefiltering Limit bandwidth

Drop samples (rearrange index) Expand (by a factor of M) and duplicate

(insert (M-1) copies)

-- Up-sampling

Time-domain Frequency -domain

Insert zeros Shrink (by a factor of L)

Interpolation Remove extra copies in a π2 period

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Digital Processing of Analog Signals Ideal C/D converter (approximation) analog-to-digital (A/D) converter

Ideal D/C converter (approximation) digital-to-analog (D/A) converter

Prefiltering to Avoid Aliasing Ideal antialiasing filter: Ideal low-pass filter (difficult to implement sharp-cutoff analog

filters).

A solution: simple prefilter and oversampling followed by sharp antialiasing filters in

discrete-time domain.

Remark: Sharp cutoff analog filters are expensive and difficult to implement.

A/D conversion ⇒ the input continuous-time signal is sampled at a very high

sampling rate.

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A/D Conversion Digital: discrete in time and discrete in amplitude

Ideal sample-and-hold: Sample the (input) analog signal and hold its value for T sec-

onds.

∑∞

−∞=−=

nnTthnxtx )(][)( 00

)(})()({)()()( 000 thnTtnTxnTthnTxtxn

an

a ∗−=−= ∑∑∞

−∞=

−∞=

δ

⎩⎨⎧ <<= otherwise

Ttth ,00,1)(0

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Quantization: Transform the input sample ][nx into one of a finite set of prescribed

values.

])[(][ˆ nxQnx = , ][ˆ nx is the quantized sample

Note: Quantization is a non-linear operation.

(i) Uniform quantizer – uniformly spaced quantization levels; very popular (also called

linear quantizer)

(ii) Nonuniform quantizer – may be more efficient for certain applicaitons

Parameters in a quantizer

(1) Decision levels – partition the dynamic range of input signal

(2) Quantization (representation) levels – the output values of a quantizer; a quanti-

zation level represents all samples between two nearby decision levels

(3) Full-scale level – the quantizer input dynamic range

Note: Typically, when the decision levels are first chosen, then the best quantization

levels are decided (for a given input probability distribution). On the other hand,

when the quantization levels are chosen, the best decision levels are decided.

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Quantization error analysis

For a uniform quantizer, there are two key parameters:

(i) step size Δ , and (ii) full-scale level ( mX± )

Assume (B+1) bits are used to represent the quantized values.

Bm

Bm XX

222

1 ==Δ+

Quantization error: ][][ˆ][ nxnxne −= = quanitized value – true value

It is clear that 2

][2

Δ<<

Δ− ne .

Statistical characteristics of ][ne :

(1) ][ne is stationary (probability distribution unchanged)

(2) ][ne is uncorrelated with ][nx

(3) ][ne , ]1[ +ne , … are uncorrelated (white)

(4) ][ne has a uniform distribution

The preceding assumptions are (approximately) valid if the signal is sufficiently com-

plex and the quantization steps are sufficiently small, …

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Mean square error (MSE) of ][ne (= variance if zero mean)

{ }12

1)()(22/

2/222 Δ

=−= ∫Δ

Δ−deeeeEeσ

-- Expressed in terms of B2 and mX

12

2 222 m

B

eX−

-- SNR (signal-to-noise ratio) due to quantization

x

m

m

xB

e

x XB

XSNR

σσ

σσ

102

22

102

2

10 log2002.68.10212

log10log10 −+=⋅

==

Remarks: (1) One bit buys a 6dB SNR improvement.

(2) If the input is Gaussian, a small percentage of the input samples would have an

amplitude greater than xσ4 .

If we choose xmX σ4= , dBBSNR 25.16 −≈

For example, a 96dB SNR requires a 16-bit quantizer.

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D/A Conversion The ideal lowpass filter is replaced by a “practical” filter.

Examples of practical filters: zero-order hold and first-order hold.

Mathematical model:

∑∞

−∞=−=

nDA nTthnxtx )(][ˆ)( 0

= quantized input * impulse response of “practical” interpolation filter

)()(

)(][)(][)(

00

00

tetx

nTthnenTthnxtxnn

DA

+=

−+−= ∑∑∞

−∞=

−∞=

Purpose: Find a compensation filter )(~ thr to compensate for the distortion caused by the

non-ideal )(0 th so that its output )(ˆ txr is close to the analog original

)(txa .

In frequency domain:

)()()(][

)(][)(][)(

00

000

Ω=Ω⎟⎠

⎞⎜⎝

⎛=

Ω=⎭⎬⎫

⎩⎨⎧ −=Ω

Ω∞

−∞=

Ω−

−∞=

Ω−∞

−∞=

∑∑

jHeXjHenx

ejHnxnTthnxFjX

Tj

n

nTj

n

nTj

nt

D/A Converter

Compensated reconstruction filter )(~ ΩjH r

][ˆ nx )(txDA )(ˆ txr

T

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DSP (Spring, 2007) Sampling of Continuous-time Signals

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Because ( ) ∑∞

−∞=⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ −Ω=Ω

ka T

kkjXT

jX π21 ,

( ) ( )Ω⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ −Ω=Ω ∑

−∞=jH

TkkjX

TjX

ka 00

21 π

[The interpolation filter )(0 ΩjH is used to remove the replicas.]

If )(0 ΩjH is not an ideal lowpass filter, we design a compensated reconstruction filter,

)()(

)(~0 Ω

Ω=Ω

jHjH

jH rr

, where )( ΩjH r is the ideal lowpass filter.

(1) Zero-order hold

or 2/0

)2/sin(2)( TjeTjH Ω−

ΩΩ

Thus, the compensated reconstruction filter is

⎪⎩

⎪⎨⎧

<ΩΩ

Ω=Ω

Ω

T

TeT

TjH

Tj

r

/||,0

/||,)2/sin(

2/)(~ 2/

π

π

Remark: A “practical” filter cannot achieve this approximation.

⎩⎨⎧ <<= otherwise

Ttth ,00,1)(0

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Overall system:

Anti-aliasing Processing Zero-order-hold Compensated reconstruction.

)()()()(~)( 0 Ω⋅⋅Ω⋅Ω=Ω Ω jHeHjHjHjH aaTj

reff

22

0 )()()(~)( eTj

re eHjHjHjPa

σΩ⋅Ω⋅Ω=Ω where 12

22 Δ=eσ

)( ΩjH aa )( TjeH Ω )(0 ΩjH )(~ ΩjH r)( ΩjX a )( ΩjYa