Lecture 3 Quantization in Signals and Systems

11

Lecture 3Lecture 3Quantization in Signals and Quantization in Signals and

SystemsSystems

byby

Graham C. GoodwinGraham C. Goodwin

University of NewcastleUniversity of NewcastleAustraliaAustralia

Presented at the “Zaborszky Distinguished Lecture Series”December 3rd, 4th and 5th, 2007

22

OverviewOverview

Topics to be covered include:Topics to be covered include:

signal quantization, signal quantization,

predictive and noise shaping quantizers, predictive and noise shaping quantizers,

networked control, networked control,

signal coding in networked control, signal coding in networked control,

channel capacity issues in networked control, channel capacity issues in networked control,

applications in audio compression and control over applications in audio compression and control over

communication channels.communication channels.

33

OutlineOutline

1.1. Recall QuantizationRecall Quantization

2.2. Predictive and Noise Shaping QuantizersPredictive and Noise Shaping Quantizers

3.3. Application to Audio CompressionApplication to Audio Compression

4.4. Networked ControlNetworked Control

5.5. Modelling Communication LinkModelling Communication Link

6.6. Predictive and Noise Shaping CodingPredictive and Noise Shaping Coding

7.7. Experimental ResultsExperimental Results

8.8. ConclusionsConclusions

44

Recall Basic Idea of SamplingRecall Basic Idea of Samplingand Quantizationand Quantization

QuantizationQuantization

SamplingSampling

t1 t3t2 t4

t0

123456

55

QuantizationQuantization(Actually we saw some aspects of this in relation to (Actually we saw some aspects of this in relation to

coefficient quantization in lecture 2.)coefficient quantization in lecture 2.)

Here:Here: Fix the sampling pattern (say uniform for simplicity) Fix the sampling pattern (say uniform for simplicity) and examine the quantization of the samples.and examine the quantization of the samples.

Approaches:Approaches:

1.1. NonlinearNonlinear – quantization is an inherently nonlinear – quantization is an inherently nonlinear process.process.

2.2. LinearLinear – approximate quantization errors as noise. – approximate quantization errors as noise.

To illustrate ideas we will follow route 2.To illustrate ideas we will follow route 2.((Generally gives design insights.Generally gives design insights.))

66

Signal to Noise Ratio Model for Signal to Noise Ratio Model for QuantizationQuantization

bb bit quantizer bit quantizer levelslevelsAssume quantization errors areAssume quantization errors arewhite noise uniformly distributedwhite noise uniformly distributed

We want small probability that signal amplitude exceedsWe want small probability that signal amplitude exceedsthe range of the quantizer. Assume variance of signal is , the range of the quantizer. Assume variance of signal is , then e.g. 4 s.d. rule says that .then e.g. 4 s.d. rule says that .

HenceHence

2 1bL é ù= -ë û

22

12q

Qs =

,2 2Q Qé ù-

ê úë û

2vs

[ ]22 2 216 16

; 2 1 6 /3 3

bq uk k L dB bits s

-- é ù= = = -ë û

4 / 2v LQs =

range

b = 3L = 7

Q

Uniform Quantizer

77

OutlineOutline

1.1. QuantizationQuantization








88

Predictive and Noise Shaping Quantizers Predictive and Noise Shaping Quantizers ((Quantization errors modeled as additive white noiseQuantization errors modeled as additive white noise))

L1

L2

L3

NN

U DEC +-

Quantizer

Utilizing the power of feedback!

1 3 3

2 21 1

L L LD C N

L L

1 2

2 21 1

L LE C N

L L

(We will return to this approximation later!)

1

21

LC

L

cf W

Note that the feedback loop is related to the delta operator (lecture 2) since we subtract what we already “know” before quantizing/approximating.

99

Focus on Frequency Weighted (Focus on Frequency Weighted (WW) ) Noise Power in Noise Power in DD

Use normalized transfer functions; G(0) = 1

2

2 3

2

Noise Power in 1N

WLWD dw

L

2 2

31

2 21 1c

WLLk dw dw

L L

2

3

21E

WLkP dw

L

cwhere is the input signal spectrum

1010

Heuristic Explanation of the Heuristic Explanation of the Optimal DesignOptimal Design

Spectrum of Spectrum of C C and characteristics of and characteristics of WW are known. are known.

We have 3 filters to design.We have 3 filters to design.

One degree of freedom removed by One degree of freedom removed by “Perfect Reconstruction” requirement i.e., “Perfect Reconstruction” requirement i.e.,

With remaining 2 degrees of freedom can (With remaining 2 degrees of freedom can (ii) shape ) shape EE to to have minimal variance (prediction) and (ii) shape have minimal variance (prediction) and (ii) shape component of due to component of due to NN to have minimal variance to have minimal variance (noise shaping).(noise shaping).

L1

L2

L3

N

U DEC +-

Quantizer

'DW

'D

1 3

2

11

L L

L=

+

1111

Perfect Reconstruction ConstraintPerfect Reconstruction Constraint

Minimizing variance of Minimizing variance of EE

Minimize variance of Minimize variance of WDWD due to due to NN

Solution:Solution:

1 3

2

11

L L

L=

+

1

12

2

12

3

1c

c

L W

L W

L

f

f

=

= -

=

11 2

21 c

L

Lf=

+

3

2

11

WL

L=

+(Noise shaping)

(Whitening Filter: Predictive coding)

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Predictive CoderPredictive Coder

Choose Choose W W = 1= 1

Optimal choices areOptimal choices are

12

3

12

2

1

1

1

c

c

L

L

L

This solution corresponds to This solution corresponds to

Minimum Variance Control Minimum Variance Control

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Noise Shaping QuantizerNoise Shaping Quantizer(Sigma-delta)(Sigma-delta)

Add extra constraint Add extra constraint LL33 = 1= 1

Optimal Choices: Optimal Choices:

ThenThen (Noise Shaping)(Noise Shaping)

(Achieved Performance)(Achieved Performance)

1

2 1

L W

L W

1D C N

W

WD W C N

1414

The Role of OversamplingThe Role of Oversampling

Say we choose Say we choose LL33 = 1 and = 1 and WW as ideal low pass filter as ideal low pass filter

Then apparently, all we need do is make an ideal Then apparently, all we need do is make an ideal

high pass filter to “push” “quantization noise” outside the high pass filter to “push” “quantization noise” outside the

band of interest.band of interest.

Does this make sense?Does this make sense?

2

1

1 L

W

1

1515

Insights from Feedback TheoryInsights from Feedback Theory

is a sensitivity function.is a sensitivity function.2

1

1 L

Thus making the sensitivity arbitrarily small in some Thus making the sensitivity arbitrarily small in some frequency range automatically means that it will be frequency range automatically means that it will be arbitrarily large somewhere else!arbitrarily large somewhere else!

2

1ln 0

1 L

We know from Bode integral thatWe know from Bode integral that

((Water Bed EffectWater Bed Effect))

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Indeed, this goes back to the early simplifying assumption thatIndeed, this goes back to the early simplifying assumption that

1

21

LE C

L

1 2

2 21 1

L LE C N

L L

In fact it should have beenIn fact it should have been

2 2

31

2 22

2

2

1 1

11

c

WLLk dw dw

L LWD

Lk dw

L

andand

Noise Power in Noise Power in

More Complex (but more realistic) optimization problem.More Complex (but more realistic) optimization problem.

It turns out to be convex!

1717

In summary – we can design an “optimal”In summary – we can design an “optimal”

quantizer which:quantizer which:

(i)(i) minimizes the impact of quantization noise on minimizes the impact of quantization noise on the output, andthe output, and

(ii)(ii) takes account of the fact that quantization takes account of the fact that quantization errors ultimately need themselves to be errors ultimately need themselves to be quantized due to the feed back structure.quantized due to the feed back structure.

1818

OutlineOutline









1919

Application : Audio CompressionApplication : Audio Compression

Elvis Presley

Original

N=0

N=1

N=2

Stop

44.1 kHzBits 3

2020

Other Insights From Control TheoryOther Insights From Control Theory

(i)(i) Bode integral Bode integral

Spectrum of Errors due to Quantization

2121

OutlineOutline









2222

Network Control SystemsNetwork Control SystemsIn a Networked Control System (NCS) controller and In a Networked Control System (NCS) controller and plant are connected via a plant are connected via a communication linkcommunication link..Therefore, signals transmitted:Therefore, signals transmitted:– Have to be quantizedHave to be quantized– May be delayedMay be delayed– May get lostMay get lost

The communication link constitutes a performance The communication link constitutes a performance bottleneck.bottleneck.When designing NCS’s the characteristics of the network When designing NCS’s the characteristics of the network should be accounted for to ensure acceptable should be accounted for to ensure acceptable performance levels.performance levels.When comparing to traditional control loops, in NCS’s When comparing to traditional control loops, in NCS’s there exist additional degrees of freedom to be designed.there exist additional degrees of freedom to be designed.It is useful to investigate:It is useful to investigate:– Architectural issuesArchitectural issues– Signal coding methodsSignal coding methods

2323

Networked Control ProblemNetworked Control Problem

(a)

(b)

2424

Useful analog to think about:Useful analog to think about:

2525

Nominal Nominal Control DesignControl Design

We will consider the situation where an LTI We will consider the situation where an LTI controller has already been designed for a SISO controller has already been designed for a SISO LTI plant model.LTI plant model.

We will refer to this design as the We will refer to this design as the nominal nominal designdesign and we will assume that it gives and we will assume that it gives satisfactory performance in a non-networked satisfactory performance in a non-networked setting.setting.

We will show how to minimize the impact of the We will show how to minimize the impact of the communication link on closed loop performance.communication link on closed loop performance.

2626

Design RelationsDesign Relations

The tracking error is given by:The tracking error is given by:

wherewhere

are the loop sensitivity functions.are the loop sensitivity functions.

e r y S z r S z d T z n

0 0

1,

1 1

G z C zT z S z

G z C z G z C z

C z G zReference

Controller Plant

Disturbance d

y

nNoise

Plant Output

r + ++

++

-

2727

Design RelationshipsDesign Relationships

In non-networked situation we have:In non-networked situation we have:

1 1

1 1 1

G z C ze r d n

G z C z G z C z G z C z

To achieve good reference following and disturbance To achieve good reference following and disturbance

attenuation, attenuation, CC((zz)) is typically chosen such that the open is typically chosen such that the open

loop gain, is large at frequencies where loop gain, is large at frequencies where

and are significant. and are significant.

To handle measurement noise and plant model To handle measurement noise and plant model

inaccuracies, the open loop gain should be reduced at inaccuracies, the open loop gain should be reduced at

appropriate frequencies.appropriate frequencies.

j jG e C e

jR e jD e

2828

OutlineOutline









2929

The Communication LinkThe Communication LinkThe novel ingredient in an NCS, when compared to a The novel ingredient in an NCS, when compared to a traditional control loop, is the communication link.traditional control loop, is the communication link.

It constitutes a significant bottleneck in the achievable It constitutes a significant bottleneck in the achievable performance.performance.

From a design perspective, this opens the possibility of From a design perspective, this opens the possibility of investigating:investigating:

Where do I place the processing power?

What information do I send?

NCS Architectures

Signal Coding

3030

Channel ModelChannel ModelWe will consider an additive Noise model:We will consider an additive Noise model:

23SNR .2 ; : number of bits/sampl

6e.

1bv

q

b

v w

q

Channel

zero-mean stationary white noise with variance q

The channel has a given signal-to-noise ratio, say SNR:The channel has a given signal-to-noise ratio, say SNR:

The above characterization encompasses, e.g.,The above characterization encompasses, e.g.,– AWGN channels AWGN channels – Bit-rate limited channels (networks), where transmitted signals Bit-rate limited channels (networks), where transmitted signals

are passed through an appropriately scaled memoryless are passed through an appropriately scaled memoryless quantizerquantizer..

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OutlineOutline









3232

Recall Predictive and Noise shaping quantizerRecall Predictive and Noise shaping quantizer

1 32

11

1L L

L

æ ö÷ç =÷ç ÷ç ÷+è ø

Use this idea in Source CodingUse this idea in Source Coding

LL11 & & LL22 become part of source become part of source codercoder

LL33 becomes part of the source becomes part of the source decoderdecoder

Make transparent to Make transparent to

nominal control loop nominal control loop

((i.e.,i.e., Perfect Reconstruction) Perfect Reconstruction)

Noise Shaping Quantizer

x wv x1L

vbQ

2L

3L

3333

IllustrationIllustrationChannel in the DownlinkChannel in the Downlink

0

22

1

11

1

SGC

SL

Constraint: 1 2 3 1L S L

Noise Shaping NCS Architecture

r wv w

2L

1LC 3L G

chan

nel

q

u y

- -cu

Communication Link

3434

AnalysisAnalysis

Hence, variance of output error due to quantization Hence, variance of output error due to quantization errors iserrors is

0 3 2 0e r y S GL S q S r

However, from However, from SNRSNR model model

(a)

(b)

22 20 3 2 2eq q S GL S

22 vq SNR

NowNow

1 1 13 0 3 0 1 2

2 22 1 2 1 13 0 3 0 1 2

1 12 2 2 1 3

2 2

with 1 and using

v r q

v L CS r L CS GL T q

L C

T S S T T

S L CS GL T

(c)

3535

From (a), (b), (c)From (a), (b), (c)

2 213 0 0 3 2 22 2

2

2 0 21

r

eq

L CS S GL S

SNR S S

3636

Expression is essentially as for the Expression is essentially as for the Predictive and Noise Shaping Quantizer Predictive and Noise Shaping Quantizer Design save that now the Weighting Design save that now the Weighting Function is determined by the Nominal Function is determined by the Nominal Loop Sensitivity.Loop Sensitivity.

Hence can readily determine optimal Hence can readily determine optimal values of values of LL11, , LL22 and and LL33 as before! as before!

3737

Special Case (Predictive Coding)Special Case (Predictive Coding)(PCM)(PCM)

Fix Fix LL22 = 0 = 0

ThenThen

0 01T S

2 213 0 0 3 22 2

2

0 2

r

eq

L CS S GL

SNR T

3838

Relationship to Channel Capacity Relationship to Channel Capacity ConstraintsConstraints

The theory shows that for stability when deploying an The theory shows that for stability when deploying an AWGN channel, one needs:AWGN channel, one needs:

On the other hand, the channel capacity of an AWGN On the other hand, the channel capacity of an AWGN channel is:channel is:

Therefore, if we redesign the controller, the smallest Therefore, if we redesign the controller, the smallest channel capacity consistent with stability is:channel capacity consistent with stability is:

2

0SNR T

1log 1

2C SNR

2

01

1log 1 log

2

pn

ii

C T p

where {where {ppii} are the unstable poles of the plant.} are the unstable poles of the plant.

3939

Optimal Results 1: Optimal Results 1: PCM Coder in DownlinkPCM Coder in Downlink

Optimal performance for the down-link architectureOptimal performance for the down-link architecture

The minimum loss function is given by:The minimum loss function is given by:

The optimal encoder satisfies:The optimal encoder satisfies:

where where kkDD is any positive (fixed) real number. is any positive (fixed) real number.

2

0 02

1 1

2opt j j jD DJ S e T e e d

SNR T

2

3 , ,j

opt jD j j

D

G eL e k

C e e

4040

Optimal Results 2: Optimal Results 2: Up-Link NCS ArchitectureUp-Link NCS Architecture

For alternative architecture where the For alternative architecture where the communication system is located in the up-link, communication system is located in the up-link, i.ei.e., between plant output and controller input.., between plant output and controller input.

chan

nel

Communication Link

C z G z

Controller Plant

d

y

n

r + ++

++

-

1UF z UF z

EncoderDecoder

4141

Optimal CodingOptimal CodingProceeding as before, we can characterize Proceeding as before, we can characterize optimal coders via:optimal coders via:

where is the power spectral density of where is the power spectral density of the signalthe signal

2

, ,j j

opt jU U j

U

G e C eF e k

e

d n G z C z r

2j

U e

2

0 02

1 1

2opt j j jU UJ S e T e e d

SNR T

The corresponding optimal loss function is:The corresponding optimal loss function is:

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Special CaseSpecial Case

Internal Model ControlInternal Model ControlChoose Choose CC such that such that

Random Walk disturbancesRandom Walk disturbances

Then Then

i.ei.e., no need for coding in this special case.., no need for coding in this special case.

( )1k

GCz

=-

( )2

1u

k

zW =

-

constantoptuF =

4343

Optimal Results 3:Optimal Results 3:Predictive and Noise Shaping Coder Predictive and Noise Shaping Coder

in Downlinkin DownlinkThe optimal noise shaping parameters are given byThe optimal noise shaping parameters are given by

where are generalized Blaschke products where are generalized Blaschke products

for and , respectively.for and , respectively.

2 01

2 0

1 03

1 0

2 3 1

ˆ

ˆ

ˆ ˆ ˆ 1

opt

z

ropt

r z

opt opt opt

z S z G zL z

z S z G z

z C z S z zL z

z C z S z z

L z L z L z

1 2,z z 0S z G z 0 rC z S z z

4444

The corresponding optimal loss function is given byThe corresponding optimal loss function is given by

2

1 2 0 0

1ˆopt r z

J z z T z S z zSNR

4545

Some ObservationsSome Observations

1.1. For PCM coding, if disturbances dominate For PCM coding, if disturbances dominate ((rr = 0), then up-link and down-link architectures = 0), then up-link and down-link architectures give same give same optimaloptimal performance. performance.

2.2. For PCM coding, if |For PCM coding, if |GCGC| = || = |DD| then optimal coder | then optimal coder for for up-linkup-link case is unity ( case is unity (i.ei.e., no need for ., no need for coding).coding).

3.3. If approximately constant as a function If approximately constant as a function of frequency, then (i.e., PCM optimal), of frequency, then (i.e., PCM optimal), otherwise ‘Predictive Noise Shaping Coding’ otherwise ‘Predictive Noise Shaping Coding’ necessary to achieve optimal performance.necessary to achieve optimal performance.

0 0 rT S *2 0L

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OutlineOutline









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1. Simulated Example1. Simulated ExampleWe consider a continuous time plant given by We consider a continuous time plant given by , sampled ever using a , sampled ever using a zero order hold at its input. The corresponding discrete zero order hold at its input. The corresponding discrete time transfer function istime transfer function is

1

0 2 5 1G s 1T s

0.36254

0.8187G z

z

We will consider two different reference signals, We will consider two different reference signals, rr11 and and rr22

with PSD’s given bywith PSD’s given by

1

2

22

2

2

0.02

1

0.03

0.9 0.7

jr j

jr j j

ee

ee e

For the control of For the control of GG((zz)) we choose the PI controller we choose the PI controller

2.44 0.4871

1

zC z

z

4848

The Case of The Case of rr11

In this case, In this case,

is approximately is approximately constant for all . Then, constant for all . Then, the PCM based scheme the PCM based scheme should have a should have a performance which is performance which is close to that of the noise close to that of the noise shaping based scheme.shaping based scheme.

10 0

j j jrT e S e e

Tracking error sample variance as a function of the channel bit-rate (r = r1).

4949

The Case of The Case of rr22

In this case, is far from In this case, is far from being constant. Therefore, the noise shaping being constant. Therefore, the noise shaping coder system outperforms PCM.coder system outperforms PCM.

20 0

j j jrT e S e e

5050

Finally, you may wonder about the simplification Finally, you may wonder about the simplification made by approximating the channel quantization made by approximating the channel quantization errors (a nonlinear phenomenon) by a SNR errors (a nonlinear phenomenon) by a SNR constrained noise source.constrained noise source.

The following figure compares the theoretical The following figure compares the theoretical tracking error (using the ‘noise model’ tracking error (using the ‘noise model’ expressions) with the practical (empirical) errors.expressions) with the practical (empirical) errors.

5151

Comparison between theoretical variance of the racking error as given by Theorem 1 and empirical tracking error sample variation with r = r2 and optimal noise shaping coding.

5252

2. Laboratory Results2. Laboratory Results

5353

DetailsDetails

Communications:Communications:

IEEE 802.3 EthernetIEEE 802.3 Ethernet

TCP/IP protocolTCP/IP protocol

Process ACTProcess ACT

6 second sampling interval – word length 2 bits 6 second sampling interval – word length 2 bits

bits/second.bits/second.1

3

5454

Measured response when the channel is in the down-link: measured plant output (with respect to the operating point – dotted line) and plant input (solid line).

5555

Measured response when the channel is in the up-link: measured plant output (with respect to the operating point – dotted line) and plant input (solid line).

5656

Table: for the proposed loops. Table: for the proposed loops. e t dtnon idealnon ideal

down-linkdown-link

non idealnon ideal

up-linkup-link

without disturbancewithout disturbance 7.27.2 5.55.5

with disturbancewith disturbance 194194 162162

As predicted by the theory: In the absence of As predicted by the theory: In the absence of coder/decoder- better to put channel in up-link coder/decoder- better to put channel in up-link ((i.ei.e., controller immediately before plant).., controller immediately before plant).

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OutlineOutline









5858

ConclusionsConclusions

This lecture has focused on quantization.This lecture has focused on quantization.

Key Tool – Predictive and Noise Shaping Quantizers – widely used Key Tool – Predictive and Noise Shaping Quantizers – widely used in Signal Processing and Telecommunication, and very recently in in Signal Processing and Telecommunication, and very recently in control and other areas e.g. Power Electronics (State of the Art).control and other areas e.g. Power Electronics (State of the Art).

Applications to Audio Compression and Networked ControlApplications to Audio Compression and Networked Control..

Recent work includes extension to multivariable systems and co-Recent work includes extension to multivariable systems and co-design of controller and coder/decoder pairs.design of controller and coder/decoder pairs.

All results in this lecture can be given alternative interpretation via All results in this lecture can be given alternative interpretation via Information Theory (Mutual Information, Source Coding, Channel Information Theory (Mutual Information, Source Coding, Channel Coding).Coding).

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A Final ObservationA Final ObservationNote thatNote that

Multivariable sampling (Multivariable sampling (lecture 1lecture 1) )

Delta operator (Delta operator (lecture 2lecture 2),),

Asymptotic sampling zero dynamics (Asymptotic sampling zero dynamics (lecture 2lecture 2),),

Predictive/Noise Shaping Quantizers (Predictive/Noise Shaping Quantizers (lecture 3lecture 3),),

Networked Control (Networked Control (lecture 3lecture 3) )

are all examples of a common principle- are all examples of a common principle-

““Don’t waste limited resources describing (storing, transmitting, Don’t waste limited resources describing (storing, transmitting, calculating….) things that are either (i) already known or (ii) calculating….) things that are either (i) already known or (ii) predetermined by a-priori knowledge regarding the signal or system.”predetermined by a-priori knowledge regarding the signal or system.”

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ReferencesReferences Quantization

R.M. Gray and D.L. Neuhoff, ‘Quantization’, IEEE Transactions on InformationTheory, Vol.44, No.6, pp.2325-2383, 1998. M. Fu and L. Xie, ‘The sector bound approach to quantized feedback control,’ IEEE Transactions on Automatic Control, Vol.50, No.11, pp.1698-1711, 2005. A. Gersho and R.M. Gray, Vector Quantization and Signal Compression, Boston, MA:Kluwer Academic, 1992.

Predictive and Noise Shaping Quantizers S.R. Norsworthy, R. Schreier and G.C. Temes, Eds, Delta-Sigma Data Converters: Theory, Design and Simulation. Piscataway, NJ: IEEE Press, 1997. S.K. Tewksbury and R.W. Hallock, ‘Oversampled, linear predictive and noise-shaping coders of order N>1,’ IEEE Transactions on Circuits and Systems, Vol.25, No.7, pp.436-447, 1978.

Audio Compression G.C. Goodwin, D.E. Quevedo and D. McGrath, ‘Moving-horizon optimal quantizer for audio signals,’ Journal Audio Engineering Society, Vol.51, No.3, pp.138-149, 2003. D.E. Quevedo and G.C. Goodwin, ‘Multistep optimal analog-to-digital conversion’, IEEE Transactions on Circuits and Systems I, Vol.52, No.4, pp.503-515, 2005. N. Gilchrist and C. Grewin, Eds., Collected Papers on Digital Audio Bit-rate Reduction. New York: Audio Engineering Society, 1996.

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ReferencesReferences Networked Control

D. Hristu-Varsakelis and W. Levine (Eds), Handbook of Networked and Embedded Systems. Boston:Birkhäuser 2005. ‘Special Issue on networked control systems’, IEEE Transactions on Automatic Control, Vol.49, No.9, 2004. H. Ishii and B.A. Francis, Limited Data Rate in Control Systems with Networks, Springer, 2002. N. Elia and S. Mitter, ‘Stabilization of linear systems with limited information’, IEEE Transactions on Automatic Control, Vol.46, No.9, pp.1384-1400, 2001. W.S. Wong and R.W. Brockett, ‘Systems with finite communication bandwidth constraints –II: Stabilization with limited information feedback,’ IEEE Transactions on Automatic Control, Vol.44, No.5, pp.1049-1053, 1999. G. Nair and R. Evans, ‘Stabilizability of stochastic linear systems with finite feedback data rates,’ SIAM Journal on Control and Optimization, Vol.43, No.2, pp.413-436, 2004. S. Tatikonda and S. Mitter, ‘Control under communication constraints,’ IEEE Transactions on Automatic Control, Vol.49, No.7, pp.1056-1068, 2004.

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ReferencesReferences

Modelling Communication Links J.H. Braslavsky, R.H. Middleton and J.S. Freudenberg, ‘Feedback stabilization over signal-to-noise ratio constrained channels,’ in Proceedings of the 2004 American Control Conference, Boston, USA, July 2004. D. Tse and P. Viswanath, Fundamentals of Wireless Communication, Cambridge University Press, 2005.

Predictive and Noise Shaping Coding G.C. Goodwin, D.E. Quevedo and E.I. Silva, “Architectures and coder design for networked control systems,” to appear Automatica, 2007. E.I. Silva, G.C. Goodwin, D.E. Quevedo and M.S. Derpich, ‘Optimal noise shaping for networked control systems’, European Control Conference, Kos, Greece 2-5 July 2007.

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Lecture 3Lecture 3Quantization in Signals and Quantization in Signals and

SystemsSystems

byby

Graham C. GoodwinGraham C. Goodwin

University of NewcastleUniversity of NewcastleAustraliaAustralia

Presented at the “Zaborszky Distinguished Lecture Series”December 3rd, 4th and 5th, 2007

Documents

Lecture 3 Quantization in Signals and Systems