ECE 4371, 2009 Class 9 Zhu Han Department of Electrical and Computer Engineering Class 9 Sep. 22 nd, 2009.

ECE 4371, 2009

Class 9

Zhu Han

Department of Electrical and Computer Engineering

Class 9

Sep. 22nd, 2009

QuantizationQuantization Scalar Quantizer Block Diagram

Mid-tread

Mid-rise

EquationsEquations

function. staircase a is which stic,characteriquantizer thecalled is

(3.22) )g( mapping The

size. step theis , levelstion reconstrucor tion representa theare

L..,.1,2, , where isoutput quantizer then the )( If

3.9 Figin shown as )(

amplitude discrete a into )( amplitude sample

theing transformof process The:onquantizati Amplitude

hreshold.decision t or the leveldecision theis Where

(3.21) ,...,2,1 , :

cellpartition Define

1

1

m

mm

tm

nT

nTm

m

Lkmmm

kk

s

s

k

kk

kννJ

J

kkk

k

Quantization NoiseQuantization Noise

Quantization Noise Level Quantization Noise Level

(3.28) 12

1

)(][

(3.26) otherwise

2

2 ,0

,1

)(

levels ofnumber total: ,

(3.25) 2

is size-step the

typemidrise theofquantizer uniforma Assuming

(3.24) )0][( ,

(3.23)

valuesample of variable

random by the denoted beerror onquantizati Let the

2

2

2

22

2

222

max max

max

dqqdqqfqQE

qqf

LmmmL

m

MEVMQ

mq

qQ

QQ

Q

Quantization SNRQuantization SNR

).(bandwidth increasing lly withexponentia increases (SNR)

(3.33) )23

(

)(

)( ofpower average thedenote Let

(3.32) 23

1

(3.31) 2

2

(3.30) log

sampleper bits ofnumber theis where

(3.29) 2

form,binary in expressed is sample quatized theWhen

o

22max

2o

22max

2

2

max

R

m

P

PSNR

tmP

m

m

LR

R

L

R

Q

RQ

R

R

, 6dB per bit

ExampleExample SNR for varying number of representation levels for sinusoidal

modulation 1.8+6 X dB, example 3.1

Number of representation level L

Number of Bits per Sample, R

SNR (dB)

32 5 31.8

64 6 37.8

128 7 43.8

256 8 49.8

Conditions for Optimality of Scalar Quantizers

Let m(t) be a message signal drawn from a stationary process M(t)

-A m A

m1= -A mL+1=A

mk mk+1 for k=1,2,…., L

The kth partition cell is defined as

Jk: mk< m mk+1 for k=1,2,…., L

d(m,vk): distortion measure for using vk to represent values inside Jk.

Condition for Optimal QuantizerCondition for Optimal Quantizer

kk

kk

M

M

L

km k

L

kkL

kk

mmd

mf

dmmfmdDk

by zedcharacteridecoder aandby zedcharacteri

encoderan : components twoof consistsquantizer

owever thesolution.H form closed havenot may

whichproblemnonlinear a ison optimizati The

(3.38) )( ),(

commonly used is distortion square-mean The

known. is which pdf, theis )( where

(3.37) )(),(

distortion average the

minimize that , and sets two theFind

,

2

1

11

J

J

J

Condition OneCondition One

)(2

1

0)()()()(

,)()(

distortion square-meanFor

,,1,2 ,)g(

mappingnonlinear by the definedencoder thefind toisThat

. D minimizes that set thefind , set Given the

decodergiven afor encoder theof Optimality . 1Condition

,,1opt ,

221

1k

2

11

(3.40)

optkoptkk

kMkkkMkkk

M

L

m k

k

L

kkL

kk

ffD

dmmfmD

Lkm

k

J

JJJJJ

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Condition TwoCondition Two

minimum a reaches D untiliteration, Using

)(

)(

0)()(2

)()(

distortion square-meanFor

. minimized that set thefind ,set Given the

encodergiven afor decoder theofy .Optimalit 2Condition

(3.47)1

opt ,

1k

2

1k

2

11

kk

m M

Mmk

M

L

m kk

M

L

m k

L

kkL

kk

mmmME

dmmf

dmmfm

dmmfmD

dmmfmD

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k

k

k

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Vector QuantizationVector Quantization

Vector QuantizationVector Quantization

image and voice compression, voice recognition

statistical pattern recognitionvolume rendering

Rate Distortion CurveRate Distortion Curve Rate: How many codewords

(bits) are used?– Example: 16-bit audio vs. 8-

bit PCM speech

Distortion: How much distortion is introduced?– Example: mean absolute

difference(L1), mean square error (L2)

Vector Quantizer often performs better than Scalar Quantizer with the cost of complexity

Rate (bps)

Distortion

SQ

VQ

Non-uniform QuantizationNon-uniform Quantization

Motivation– Speech signals have the

characteristic that small-amplitude samples occur more frequently than large-amplitude ones

– Human auditory system exhibits a logarithmic sensitivity

More sensitive at small-amplitude range (e.g., 0 might sound different from 0.1)

Less sensitive at large-amplitude range (e.g., 0.7 might not sound different much from 0.8)

histogram of typical speech signals

Non-uniform QuantizerNon-uniform Quantizer

x Q xF F-1

Example F: y=log(x) F-1: x=exp(x)

y y

F: nonlinear compressing functionF-1: nonlinear expanding function

F and F-1: nonlinear compander

We will study nonuniform quantization by PCM example next

A law and law

Law/A LawLaw/A Law

The -law algorithm (μ-law) is a companding algorithm, primarily used in the digital telecommunication systems of North America and Japan. Its purpose is to reduce the dynamic range of an audio signal. In the analog domain, this can increase the signal to noise ratio achieved during transmission, and in the digital domain, it can reduce the quantization error (hence increasing signal to quantization noise ratio).

A-law algorithm used in the rest of worlds.

A-law algorithm provides a slightly larger dynamic range than the mu-law at the cost of worse proportional distortion for small signals. By convention, A-law is used for an international connection if at least one country uses it.

LawLaw

A LawA Law

Law/A LawLaw/A Law

Analog to Digital ConverterAnalog to Digital Converter Main characteristics

– Resolution and Dynamic range : how many bits

– Conversion time and Bandwidth: sampling rate

Linearity– Integral

– Differential

Different types– Successive approximation

– Slope integration

– Flash ADC

– Sigma Delta

Successive approximationSuccessive approximation Compare the signal with an n-bit DAC

output

Change the code until – DAC output = ADC input

An n-bit conversion requires n steps

Requires a Start and an End signals

Typical conversion time– 1 to 50 s

Typical resolution– 8 to 12 bits

Cost– 15 to 600 CHF

Single slope integrationSingle slope integration Start to charge a capacitor at constant current

Count clock ticks during this time

Stop when the capacitor voltage reaches the input

Cannot reach high resolution– capacitor

– comparator

-

+IN

C

R

S Enable

N-bit Output

Q

Oscillator Clk

Co

un

ter

StartConversion

StartConversion

02468

101214161820

0 2 4 6 8 10 12 14 16

Time

Vo

lta

ge

acc

ross

th

e c

ap

aci

tor

Vin

Counting time

Flash ADCFlash ADC Direct measurement with 2n-1

comparators

Typical performance:– 4 to 10-12 bits

– 15 to 300 MHz

– High power Half-Flash ADC

– 2-step technique 1st flash conversion with 1/2 the

precision Subtracted with a DAC New flash conversion

Waveform digitizing applications

Sigma-Delta ADCSigma-Delta ADC

Over-sampling ADCOver-sampling ADC

Hence it is possible to increase the resolution by increasing the sampling frequency and filtering

Reason is the noise level reduce by over sampling.

Example :– an 8-bit ADC becomes a 9-bit ADC with an over-sampling factor of

4

– But the 8-bit ADC must meet the linearity requirement of a 9-bit

bitsofnumbereffectivethebeingnnSNR

dBf

fn

ffA

Ax

SNR s

sn

'68.1

log1068.1

212

8log10log1000

2

2

2

2

2

Resolution/Throughput RateResolution/Throughput Rate

<10kHz 10 – 100 kHz 0.1 – 1 MHz 1 – 10 MHz 10 – 100 MHz > 100 MHz

>17 bits

14 – 16 bits

12 – 13 bits

10 – 11 bits

8 – 9 bits

<8 bits

Digital to Analog conversion DACDigital to Analog conversion DAC

DAC

Input code = n0110001010001001001000101011:::

Output voltage = Vout(n) V+ref

V-ref

Digital to Analog ConverterDigital to Analog Converter Pulse Width Modulator DAC

Delta-Sigma DAC

Binary Weighted DAC

R-2R Ladder DAC

Thermometer coded DAC

Segmented DAC

Hybrid DAC

Pulse Code Modulation (PCM)Pulse Code Modulation (PCM)

Pulse code modulation (PCM) is produced by analog-to-digital conversion process. Quantized PAM

As in the case of other pulse modulation techniques, the rate at which samples are taken and encoded must conform to the Nyquist sampling rate.

The sampling rate must be greater than, twice the highest frequency in the analog signal,

fs > 2fA(max) Telegraph time-division multiplex (TDM) was conveyed as early as 1853, by

the American inventor M.B. Farmer. The electrical engineer W.M. Miner, in 1903.

PCM was invented by the British engineer Alec Reeves in 1937 in France.

It was not until about the middle of 1943 that the Bell Labs people became aware of the use of PCM binary coding as already proposed by Alec Reeves.

Figure Figure The basic elements of a PCM system.The basic elements of a PCM system.

Pulse Code Modulation

Encoding

Advantages of PCM

1. Robustness to noise and interference

2. Efficient regeneration

3. Efficient SNR and bandwidth trade-off

4. Uniform format

5. Ease add and drop

6. Secure

DS0: a basic digital signaling rate of 64 kbit/s. To carry a typical phone call, the audio sound is digitized at an 8 kHz sample rate using 8-bit pulse-code modulation. 4K baseband, 8*6+1.8 dB

Virtues, Limitations and Modifications of PCM

0000

1111

1110

1101

1100

1011

1010

1001

0001

0010

0011

0100

0101

0110

0111

0000 0110 0111 0011 1100 1001 1011

Numbers passed from ADC to computer to represent analogue voltage

Resolution=1 part in 2n

PCMPCM