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8/10/2019 DSP-Lec 02-Quantization.pdf
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Quantization
Chapter 2
Department of Telecommunications Engineering
Ho Chi Minh City University of Technology
8/10/2019 DSP-Lec 02-Quantization.pdf
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Digital Signal Processing
1. Quantization process
2 Quantization
The quantized sample xQ(nT)is represented by Bbit, which can take
2Bpossible values.
Fig: Analog to digital conversion
An A/D is characterized by a full-scale range Rwhich is dividedinto 2Bquantization levels. Typical values of R in practice are
between 1-10 volts.
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Digital Signal Processing
1. Quantization process
3
Fig: Signal quantization
Quantization
Quantizer resolution or quantization width2B
RQ
A bipolarADC ( )2 2
Q
R Rx nT
A unipolarADC 0 ( )Qx nT R
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Digital Signal Processing
1. Quantization process Quantization error
4 Quantization
Quantization by rounding: replace each value x(nT) by the nearest
quantization level.
( ) ( ) ( )Qe nT x nT x nT
Quantization by truncation: replace each value x(nT) by its below
quantization level.
Quantization error:
Consider rounding quantization:2 2
Q Qe
Fig: Uniform probability density of quantization error
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Digital Signal Processing
1. Quantization process Quantization error
5 Quantization
The mean value of quantization error
The mean-square error (power)
/2 / 2
/2 /2
1
( ) 0
Q Q
Q Qe ep e de e deQ
/2 /2 2
2 2 2 2
/ 2 /2
1( )
12
Q Q
Q Q
Qe e p e de e de
Q
Root-mean-square (rms) error: 212
rmsQe e
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
10 10 1020 log 20 log (2 ) log (2) 6B
dB
RSNR B B dB
Q
which is referred to as 6 dB bit rule.
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Digital Signal Processing
1. Quantization process Example
6 Quantization
In a digital audio application, the signal is sampled at a rate of 44KHz 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.
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Digital Signal Processing
2. Digital to Analog Converters (DACs)
7 Quantization
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 bBis the leastsignificant bit (LSB)while b1is the most significant bit (MSB).
For unipolar signal, xQ[0, R); for bipolar xQ[-R/2, R/2).
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Digital Signal Processing
2. DAC-Example DAC Circuit
8 Quantization
Fig: DAC using binary weighted resistor
Rf
31 2 4
2 4 8 16REF f f f f
bb b bI V
R R R R
31 2 4
2 4 8 16Q OUT f REF
bb b bx V I R V
16Rf8Rf4Rf2RfxQ=Vout
-VREF
iI
LSB
MSB
b1bB
4 3 2 1 0 3 2 1 01 2 3 4 1 2 3 42 2 2 2 2 2 2 2 2Qx R b b b b Q b b b b
Full scale R=VREF, B=4 bit
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Digital Signal Processing
2. D/A Converters
9 Quantization
Unipolar natural binary
where m is the integer whose binary representation is b=[b1, b2,,bB].
1 2 0
1 22 2 ... 2B B
Bm b b b
Bipolar offset binary: obtained by shifting the xQof unipolar naturalbinary converter by half-scale R/2:
1 2
1 2( 2 2 ... 2 )B
Q Bx R b b b Qm
1 2
1 2( 2 2 ... 2 )2 2
B
Q B
R Rx R b b b Qm
Twos complement code: obtained from the offset binary code bycomplementing the most significant bit, i.e., replacing b1by .
1 2
1 2( 2 2 ... 2 )2
B
Q B
Rx R b b b
1 11b b
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Digital Signal Processing
2. D/A Converters-Example
10 Quantization
A 4-bit D/A converter has a full-scale R=10 volts. Find the quantizedanalog values for the following cases ?
a) Natural binary with the input bits b=[1001] ?
b) Offset binary with the input bits b=[1011] ?
c) Twos complement binary with the input bits b=[1101] ?
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Digital Signal Processing
3. A/D converter
11 Quantization
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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Digital Signal Processing
3. A/D converter
12 Quantization
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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Digital Signal Processing
3. A/D converter
13 Quantization
This algorithm is applied for the natural and offset binary with
truncation quantization.
where the unit-step function is defined by
1 0
( ) 0 0
if x
u x if x
Successive approximation algorithm
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Digital Signal Processing
3. A/D converter-Example
14 Quantization
Consider a 4-bit ADC with the full-scale R=10 volts. Using the
successive approximation algorithm to find offset binary oftruncation quantization for the analog values x=3.5 volts and x=-1.5
volts.
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Digital Signal Processing
3. A/D converter
15 Quantization
For rounding quantization, we
shift x by Q/2:
For the twos complement
code, the sign bit b1is treatedseparately.
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Digital Signal Processing
3. A/D converter-Example
16 Quantization
Consider a 4-bit ADC with the full-scale R=10 volts. Using the
successive approximation algorithm to find offset and twos
complement of rounding quantization for the analog values x=3.5
volts .
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Digital Signal Processing
Homework
17 Quantization
Problems 2.1, 2.2, 2.3, 2.5, 2.6