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Bài giảng Xử lý tín hiệu số - Chapter 2: Quantization - Hà Hoàng Kha

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Chapter 2  
Quantization  
Ha Hoang Kha, Ph.D.  
Ho Chi Minh City University of Technology  
Email: [email protected]  
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.  
Ha H. Kha  
2
Quantization  
1. Quantization process  
Fig: Signal quantization  
™ Quantizer resolution or quantization width  
R
2B  
Q =  
R
R
− ≤ xQ (nT) <  
™ A bipolar ADC  
2
2
0 ≤ x
Q
(nT) < R  
™ A unipolar ADC  
Ha H. Kha  
3
Quantization  
1. Quantization process –Quantization error  
™ Quantization by rounding: replace each value x(nT) by the nearest  
quantization level.  
™ Quantization by truncation: replace each value x(nT) by its below  
quantization level.  
e(nT) = xQ (nT) − x(nT)  
™ Quantization error:  
Q
Q
− ≤ e ≤  
™ Consider rounding quantization:  
2
2
Fig: Uniform probability density of quantization error  
Ha H. Kha  
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Quantization  
1. Quantization process –Quantization error  
Q/2  
Q/2  
1
™ The mean value of quantization error  
e = ep(e)de = e de =0  
∫
∫
Q
−Q/2  
−Q/2  
Q/2  
Q/2  
1
Q2  
2
2
2
2
™
The
mean-square error
(power)  
™
Root-mean-square (rms) error:  
σ
=
e
=
e p
(
e
)
de
=
e de
 
=  
∫
∫
Q
12  
−Q/2  
−Q/2  
™ 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  
⎛ ⎞  
which is referred to as 6 dB bit rule.  
Ha H. Kha  
5
Quantization  
1. Quantization process –Example  
™ 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 quantinzation error mush be kept below 50 microvolts. Then,  
determine the actual rms error and the bit rate in bits per second.  
Ha H. Kha  
6
Quantization  
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).  
Ha H. Kha  
7
Quantization  
2. DAC-Example DAC Circuit  
Rf  
™ Full scale R=VREF, B=4 bit  
I
∑
i
xQ=V  
out  
2Rf  
4Rf  
8Rf 16Rf  
MSB  
bB  
b1  
LSB  
Fig: DAC using binary weighted resistor  
-VREF  
⎛
⎞
b
b2  
b3  
b4  
1
I =V  
+
+
+
⎜
⎜
⎟
⎟
∑
REF  
2Rf 4Rf 8Rf 16Rf  
⎝
⎠
2
b3  
b b  
b4  
⎛
⎞
⎠
1
x =V = I ⋅ R =V  
+ + +  
∑
Q
OUT  
f
REF  
⎜
⎝
⎟
2 4 8 16  
x
=
R2−4 b 2−3 + b 2−2 + b 2−1 + b 20 = Q b 2−3 + b 2−2 + b 2−1 + b 20  
(
)
(
)
Q
1
2
3
4
1
2
3
4
Ha H. Kha  
8
Quantization  
2. D/A Converters  
™ Unipolar natural binary  
where m is the integer whose binary representation is b=[b1, b2,…,bB].  
m = b 2+ b2+...+ b2
™ Bipolar offset binary: obtained by shifting the xQ of unipolar natural  
binary converter by half-scale R/2:  
Q 2 B
1
™ Two’s complement code: obtained from the offset binary code by  
complementing the most significant bit, i.e., replacing b by  
.
b =1−b  
1
1
1
2
Ha H. Kha  
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Quantization  
2. D/A Converters-Example  
™ 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] ?  
Ha H. Kha  
10  
Quantization  
3. A/D converter  
™ 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  
Ha H. Kha  
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Quantization  
3. A/D converter  
™ 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.  
Ha H. Kha  
12  
Quantization  
3. A/D converter  
™ Successive approximation algorithm  
1 if x ≥ 0  
0 if x < 0  
⎧
⎨
⎩
where the unit-step function is defined by  
u(x) =  
This algorithm is applied for the natural and offset binary with  
truncation quantization.  
Ha H. Kha  
13  
Quantization  
3. A/D converter-Example  
™ 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.  
Ha H. Kha  
14  
Quantization  
3. A/D converter  
™ For rounding quantization, we  
™ For the two’s complement  
code, the sign bit b1 is treated  
separately.  
shift x by Q/2:  
Ha H. Kha  
15  
Quantization  
3. A/D converter-Example  
™ 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 .  
Ha H. Kha  
16  
Quantization  
Homework  
™ Problems 2.1, 2.2, 2.3, 2.5, 2.6  
Ha H. Kha  
17  
Quantization  
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