Lecture Digital signal processing: Chapter 2 - Nguyen Thanh Tuan

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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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
Q /2

 The mean value of quantization error e 



Q /2

ep(e)de 

 Q /2

 The mean-square error  q 2
(power)



 Q /2

Q /2

e

1
de 0
Q

Q /2

2
1
Q
 e 2   ( e  e ) 2 p (e)de   e 2 de 
Q
12
 Q /2
 Q /2

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

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
  x2 

R
SNR dB  10log10  2   20log10    20log10 (2 B )  6 B dB
 
Q
 q

which is referred to as 6 dB bit rule.
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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 must 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

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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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Mid-riser and mid-tread quantization

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Bonus 2.1
 Write a program to simulate DAC.
b1
b2

MSB

b3

DAC

bB

xQ

LSB
R (full-scale range)

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Bonus 2.2
 Write a program to simulate ADC.
MSB

x(n)

b1
b2
b3

ADC

bB
LSB
R (full-scale range)
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Review
Các thông số cơ bản của quá trình lượng tử hóa?
Quan hệ giữa các nguyên tắc lượng tử?
Quan hệ giữa các nguyên tắc mã hóa?
Tính chất của sai số lượng tử?
Hiệu quả của lấy mẫu dư và định dạng nhiễu?
Hiệu quả của dither?
Giải thuật test bit?
Xác định mức lượng tử và các bit lượng tử?
Xác định dung lượng cần lưu trữ?
Xác định tốc độ xử lý yêu cầu của chip DSP?
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Homework 1
 Cho bộ lượng tử và mã hóa nhị phân tự nhiên B = 5 bit hoạt động
theo nguyên tắc làm tròn gần nhất (rounding) với khoảng lượng tử
đều Q = 1.1@ (biết 0 là giá trị lượng tử nhỏ nhất).
a) Xác định giá trị lượng tử lớn nhất?
b) Kiểm tra xem liệu giá trị 20.10 có là giá trị lượng tử hay không?
c) Xác định giá trị lượng tử tương ứng với từ mã 10011?
d) Xác định từ mã của mẫu tín hiệu ngõ vào 20.10?
e) Làm lại câu d trong trường hợp B = 8 bit?

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