Chapter 2 introduce you to quantization. In this chapter, you will learn to: Quantization process, digital to analog converters (DACs), A/D converters, oversampling noise shaping,...and another contents.
AMBIENT/
Chủ đề:
Nội dung Text: 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: nttbk97@yahoo.com
- 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.
Digital Signal Processing 2 Quantization
- 1. Quantization process
Fig: Signal quantization
R
Quantizer resolution or quantization width (step) Q
2B
R R
A bipolar ADC xQ (nT )
2 2
A unipolar ADC 0 xQ (nT ) R
Digital Signal Processing 3 Quantization
- 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 )
Q Q
Consider rounding quantization: e
2 2
Fig: Uniform probability density of quantization error
Digital Signal Processing 4 Quantization
- 1. Quantization process
Q /2 Q /2
1
The mean value of quantization error e
Q /2
ep(e)de
Q /2
e
Q
de 0
Q /2 Q /2 2
1 Q
The mean-square error q 2 e 2 ( e e ) 2 p (e)de e 2 de
Q 12
(power) Q /2 Q /2
Q
Root-mean-square (rms) error: erms q e2
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.
Digital Signal Processing 5 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 must be kept below 50 microvolts. Then,
determine the actual rms error and the bit rate in bits per second.
Digital Signal Processing 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).
Digital Signal Processing 7 Quantization
- 2. DACs
Rf
Full scale R=VREF, B=4 bit
I i
2Rf 4Rf 8Rf 16Rf xQ=Vout
MSB
bB
b1
LSB
-VREF
Fig: DAC using binary weighted resistor
b1 b2 b3 b4
REF 2R 4R 8R 16R
I V
f f f f
b1 b2 b3 b4
xQ VOUT I R f VREF
2 4 8 16
xQ R24 b1 23 b2 22 b3 21 b4 20 Q b1 23 b2 22 b3 21 b4 20
Digital Signal Processing 8 Quantization
- 2. DACs
Unipolar natural binary xQ R(b1 21 b2 22 ... bB 2 B ) Qm
where m is the integer whose binary representation is b=[b1, b2,…,bB].
m b1 2B1 b2 2B2 ... bB 20
Bipolar offset binary: obtained by shifting the xQ of unipolar natural
binary converter by half-scale R/2:
1 2 BR R
xQ R(b1 2 b2 2 ... bB 2 ) Qm
2 2
Two’s complement code: obtained from the offset binary code by
complementing the most significant bit, i.e., replacing b1 by b1 1 b1 .
1 2 R
B
xQ R(b1 2 b2 2 ... bB 2 )
2
Digital Signal Processing 9 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] ?
Digital Signal Processing 10 Quantization
- 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
Digital Signal Processing 11 Quantization
- 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.
Digital Signal Processing 12 Quantization
- 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.
Digital Signal Processing 13 Quantization
- 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 xQ C = u(x – xQ)
b1
1000 0,000 1
b2
1100 2,500 1
b3
1110 3,750 0
b4
1101 3,125 1
1101 3,125
Digital Signal Processing 14 Quantization
- 3. A/D converter
For rounding quantization, we For the two’s complement
shift x by Q/2: code, the sign bit b1 is treated
separately.
Digital Signal Processing 15 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.
Digital Signal Processing 16 Quantization
- Oversampling noise shaping
Pee(f) e2
fs e'2
f s'
e(n)
HNS(f)
-f’s/2 -fs/2 0 fs/2 f’s/2 f
e2 e'2 '2 x(n) ε(n) xQ(n)
' e2 f s e'
fs fs fs
Digital Signal Processing 17 Quantization
- Oversampling noise shaping
Digital Signal Processing 18 Quantization
- Dither
Digital Signal Processing 19 Quantization
- Uniform and non-uniform quantization
Digital Signal Processing 20 Quantization
Thêm vào BST
Báo xấu
Download
Vui lòng tải xuống để xem tài liệu đầy đủ
