DSP-Lec 02-Quantization.pdf

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Quantization

Chapter 2

Department of Telecommunications Engineering

Ho Chi Minh City University of Technology


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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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Homework

17 Quantization

Problems 2.1, 2.2, 2.3, 2.5, 2.6

DSP-Lec 02-Quantization.pdf

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