XỬ LÝ TÍN HIỆU SỐ Dsp chapter2 student 22062015

Chapter 2
Quantization
Nguyen Thanh Tuan, Click
M.Eng.
to edit Master subtitle style
Department of Telecommunications (113B3)
Ho Chi Minh City University of Technology
Email:


1. Quantization process

Fig: Analog to digital conversion

 The quantized sample xQ(nT) is represented by B bit, which can take
2B possible values.
 An A/D is characterized by a full-scale range R which is divided
into 2B quantization levels. Typical values of R in practice are
between 1-10 volts.
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Quantization


1. Quantization process

Fig: Signal quantization
 Quantizer resolution or quantization width (step) Q 
R

R
 A bipolar ADC   xQ (nT ) 
2
2

R
2B

 A unipolar ADC 0  xQ (nT )  R
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1. Quantization process
 Quantization by rounding: replace each value x(nT) by the nearest
quantization level.

 Quantization by truncation: replace each value x(nT) by its below
nearest quantization level.
 Quantization error:

e(nT )  xQ (nT )  x(nT )

 Consider rounding quantization: 

Q
Q

e
2
2

Fig: Uniform probability density of quantization error
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1. Quantization process
 The mean value of quantization error e 

Q /2



Q /2

ep(e)de 

 Q /2



e

 Q /2

Q /2

1
de 0
Q

Q /2

1
Q2
 The mean-square error (power)   e   e p(e)de   e de 
Q
12
 Q /2
 Q /2
2

2

2

 Root-mean-square (rms) error: erms    e2 

2

Q
12

 R and Q are the ranges of the signal and quantization noise, then the

signal to noise ratio (SNR) or dynamic range of the quantizer is
defined as
R
SNR dB  20log10    20log10 (2 B )  B log10 (2)  6 B dB
Q

which is referred to as 6 dB bit rule.
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Quantization


Example 1
 In a digital audio application, the signal is sampled at a rate of 44
KHz and each sample quantized using an A/D converter having a
full-scale range of 10 volts. Determine the number of bits B if the
rms quantization error mush be kept below 50 microvolts. Then,
determine the actual rms error and the bit rate in bits per second.

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2. Digital to Analog Converters (DACs)
 We begin with A/D converters, because they are used as the building

blocks of successive approximation ADCs.

Fig: B-bit D/A converter

 Vector B input bits : b=[b1, b2,…,bB]. Note that bB is the least
significant bit (LSB) while b1 is the most significant bit (MSB).

 For unipolar signal, xQ є [0, R); for bipolar xQ є [-R/2, R/2).
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2. DACs
Rf

 Full scale R=VREF, B=4 bit
2Rf

4Rf

I
8Rf

MSB

i

xQ=Vout

16Rf
bB

b1

LSB
-VREF

Fig: DAC using binary weighted resistor
 b1
b3
b2
b4
I

V




 REF  2R 4R 8R 16R
f
f
f
 f






 b1 b2 b3 b4 
xQ  VOUT   I  R f  VREF     
 2 4 8 16 
xQ  R24  b1 23  b2 22  b3 21  b4 20   Q  b1 23  b2 22  b3 21  b4 20 
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2. DACs
 Unipolar natural binary xQ  R(b1 21  b2 22  ...  bB 2 B )  Qm
where m is the integer whose binary representation is b=[b1, b2,…,bB].
m  b1 2B1  b2 2B2  ...  bB 20

 Bipolar offset binary: obtained by shifting the xQ of unipolar natural
binary converter by half-scale R/2:
R
R
xQ  R(b1 2  b2 2  ...  bB 2 )   Qm 
2
2
1

2

B

 Two’s complement code: obtained from the offset binary code by
complementing the most significant bit, i.e., replacing b1 by b1  1  b1 .
R
xQ  R(b1 2  b2 2  ...  bB 2 ) 
2
1

Digital Signal Processing

2

9

B

Quantization


Example 2
 A 4-bit D/A converter has a full-scale R=10 volts. Find the quantized
analog values for the following cases ?

a) Natural binary with the input bits b=[1001] ?
b) Offset binary with the input bits b=[1011] ?
c) Two’s complement binary with the input bits b=[1101] ?

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3. A/D converters
 A/D converters quantize an analog value x so that is is represented
by B bits b=[b1, b2,…,bB].

Fig: B-bit A/D converter

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3. A/D converters
 One of the most popular converters is the successive approximation
A/D converter

Fig: Successive approximation A/D converter

 After B tests, the successive approximation register (SAR) will hold
the correct bit vector b.
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3. A/D converters
 Successive approximation algorithm

1 if x  0
where the unit-step function is defined by u ( x)  
0 if x  0

This algorithm is applied for the natural and offset binary with
truncation quantization.
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Example 3
 Consider a 4-bit ADC with the full-scale R=10 volts. Using the
successive approximation algorithm to find offset binary of
truncation quantization for the analog values x=3.5 volts and x=-1.5
volts.
Test b1b2b3b4
b1
b2
b3
b4

1000

1100
1110
1101
1101

Digital Signal Processing

xQ

C = u(x – xQ)

0,000
2,500
3,750
3,125
3,125

1
1
0
1

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3. A/D converter
 For rounding quantization, we
shift x by Q/2:

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 For the two’s complement
code, the sign bit b1 is treated
separately.

Quantization


Example 4
 Consider a 4-bit ADC with the full-scale R=10 volts. Using the
successive approximation algorithm to find offset and two’s
complement of rounding quantization for the analog values x=3.5
volts.

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Oversampling noise shaping
 e2
fs

Pee(f)

 e'2
f s'
e(n)

-f’s/2

-fs/2

0

fs/2

f’s/2

'2
 e2  e'2

 '   e2  f s e'
fs
fs
fs

Digital Signal Processing

HNS(f)

f
x(n)

17

ε(n)

xQ(n)

Quantization


Oversampling noise shaping

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Dither

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Uniform and non-uniform quantization

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