quantization and transmission in wireless multi-hop networks

QUANTIZATIONANDTRANSMISSIONINWIRELESSMULTI-HOPNETWORKS

Behzad M. Dogahe

Department of Electrical and Computer Engineering

Outline

•  Mo6va6on•  ProblemStatement•  BackgroundandProposedSolu6on•  Simula6onResults•  Remarks

Motivation

Applica6ons

Voice

StreamingMul6media

Gaming

WirelessAd-HocNetworksRemoteCompu6ng

Requirements(QoS)

DataRate

Reliability

End-to-EndDelay

Powerlevel

Reproduc6onFidelity

Classifica6on

Resources

Power

Bandwidth

•  Co-existèShareresourcesèConges6onControl

•  DigitalèQuan6za6on

Problem Statement

•  Goal:Transmitdatainwirelessmul6-hopnetworkwithQoSdependonrate,end-to-enddelay.Atthedecoder,notonlyreconstruc6ngquan6zedsignalbutalsoclassifythem.

QoS

Rate

End-to-endDelayReproduc6onFidelityClassifica6on

Resources

Power

Linkcapacity

Problem Statement Breakdown

MainGoal(rate,delay,power,quan6za6on,classifica6on)

Problem1(rate,delay,power)

Problem2(Quan6za6on,Classifica6on)

CombinedSolu6on

QoSinsharedresourceconstrainednetwork

1.TransportandPhysical

2.Presenta6onLink

Problem 1




CombinedSolu6on



2.Presenta6onLink

Problem 2




CombinedSolu6on



2.Presenta6onLink

Combining2Problems




CombinedSolu6on



2.Presenta6onLink

Problem 1

•  Problem1:BalancingPowerandRatetoAchieveBoundedAverageDelayinWirelessNetworks

QoS

Rate

End-to-endDelayReproduc6onFidelity

Classifica6on

Resources

Power

LinkCapacity

Problem 1

q  Application running on a wireless multi-hop network q  Link capacities are interference dependent. q  How can we balance power control and congestion control to improve the overall network performance (Data Rate and Delay Requirements of the Sessions)?

1x

2x 3x

4x

Background

q  Congestion Control: A mechanism that regulates the allowed rates so that the total traffic load on a link does not exceed the capacity. (Example: Congestion-Avoidance in TCP) q  Congestion avoidance in TCP approximates a distributed algorithm that implicitly solves network utility maximization (NUM) problems.

QoS

Ratev

End-to-endDelay

Reproduc6onFidelity

Classifica6on

Resources

Power

LinkCapacity

Developing Problem Statement

Network Utility Maximization q  Each source has a Utility Function

§  A Quality Measure §  Example: TCP-Vegas

q  Most Basic NUM: )log()( sss xxU α=

lcx

xU

lsLlss

ss

ssxs

∀≤∑

∑

∈

∀

,)(:

}:{)(max

Subject to:

sx : Data Rate of Session s

lc : Capacity of Link l


q  Capacities of the Wireless Links are dependent on the Transmission Power and Interference : Transmitter Power on Link l

: Noise on Link l Kisaconstantthatdependsonthemodula6onandtherequiredbiterror.Gpathgainandpathloss.

QoS

Rate

End-to-endDelay

Reproduc6onFidelity

Classifica6on

Resources

Powerv

LinkCapacityv

Parameters

QoS

Rate✔

End-to-endDelay

Reproduc6onFidelity

Classifica6on

Resources

Power✔

LinkCapacity✔

Background - Delay

End-to-EndDelay:

Propaga6onDelay TransmissionDelay QueuingDelay

Forapplica6onsproducingburstydata,queuingdelaycanbequitesignificant,

thereforeitisthedominantcomponentoftotaldelay.


The New Requirement: q  Bounded Average Queuing Delay

§  Each Link Modeled as an M/G/1 Queue §  General Packet Length Distribution

•  Mean 1/µ , and Variance 𝜎↑2 

∑∈

−+

−=

)(:

//)1()(

sLlssll

l xccTE µβµβ

2/)1( 22σµβ +=

ll dTE ≤)(ld : Maximum Allowed Delay on Link l

(.)E : Expected Value

Problem Statement

ll dTE ≤)(

DelayRequirementofSessionsMinimumRateRequirement

Non-convexè high-SIR&changeofvariable

Problem Statement (Lagrangian Dual)

Lagrangian lλ : Lagrange Multipliers

Lagrange Dual Function:

Dual Problem:

Problem Statement (source & link)

•  Source Problem

•  Link Problem

•  Dual Problem

§  At each source

Problem Statement (link: delay & power)

•  Delay Distribution Problem

•  Power Allocation Problem

§  Update Equation

Solution Steps

Main Problem

Source Problem Link Problem

Dual Decomposition Lagrange Multipliers lλ

Solved at Each Source

Power Allocation

Delay Distribution

sx

lP

De-Centralized Delay Distribution

Dual Decomposition sν

ld

Derived Algorithm

1.  Initialize

2.  At Each Source:

3.  Each Transmitter Calculates ‘m’ locally and Transmits it to All other Transmitters by a Flooding Protocol

4.  Each Transmitter Updates its Power

Derived Algorithm

4.1. Initialize

4.2. Link Updates its Delay Share

4.3. Delay Price Update

5. Each Link Updates its Link Price:

Simulations

q  5 Nodes, 4 Sessions q  Delay Requirement: 10 ms q  Maximum Power: 0.12 W q  Utility Function:

Simulations

with Delay Requirement (Dashed Lines) vs. No Delay Requirement (Solid Lines)

DataRate

Simulations

Power

with Delay Requirement (Dashed Lines) vs. No Delay Requirement (Solid Lines)

Simulations

QueuingDelay

Observations

q  Sessions reduced data rates to achieve lower queuing delays based on the constraints.

q  Some transmitters consume more power to handle the delay requirements of the sessions.

q  Some transmitters reduce power to reduce the interference on other links.

Remarks

q  Presented problem of allocating resources of bandwidth and power in a multi-hop wireless network. q  Incorporated average queuing delay requirement of sessions in the NUM Problem.

q  Transferred the non-convex problem to a convex optimization by high-SIR and change of variable. Presented a distributed iterative algorithm.

q  This augmented NUM formulation allowed application to tradeoff rate, power and queuing delay according to its needs.

Problem 2




CombinedSolu6on



2.Presenta6onLink

Problem 2

q Problem2:Quan6za6onForClassifica6onAccuracyInHigh-RateQuan6zers

QoS

Rate

End-to-endDelayReproduc6onFidelity

Classifica6on

Resources

Power

LinkCapacity

Problem 2 (more motivation)

q Quan6za6onofsignalsrequiredformanyapplica6onsq Originalsignalquan6zedatencoder.Atdecoderareplicathat

shouldresembleoriginalsignalinsomesenseisrecoveredq Presentquan6zersmakeefforttoreducethedistor6onof

signalinthesenseofreproduc6onfidelityq Considerscenariosinwhichsignalsaregeneratedfrom

mul6pleclasses.Theencoderfocusesonthetaskofquan6za6onwithoutanyregardstotheclassofthesignal

q Thequan6zedsignalreachesthedecoderwherenotonlytherecoveryofthesignalshouldtakeplacebutalsoadecisionistobemadeontheclassofthesignalbasedonthequan6zedversionofthesignalonly

Goal

q Goal:Designofaquan6zerthatisop6mizedforthetaskofreproduc6onfidelityandclassifica6onatthedecoder

q Applica6onScenarios:§  Wanttohavegoodsoundfidelity(goodvoice/audio

quality)butalsowanttobeabletoperformspeakerrecogni6on

§  Sensornetworkwherethesensorshavelowcomplexity,simplequan6zers,butthedecoder/sensorsinknodedoesmoresophis6catedprocessing(sotherawsignalvalueisneeded,butwealsowanttobeabletoclassifythesensedsignal)

Design of a Quantizerq  GoalofQuan6za6onistominimize:

q  whereisDistor6onMeasure

q  ExamplesofDistor6onMeasure:§  MSE§  LogSpectralDistor6on

2ˆ)ˆ,( xxxxd −=

Quan6zerx )(ˆ xQx =

Design of a Quantizer – High-Rate theory

x

x̂

x

)(xp

x

)(xλ

x

x̂

x

)(xp

x

)(xλ

Inhigh-ratetheorypointdensityfunc6onrepresentsthedensityofcodebookpointsinanyregionforaquan6zer.Thedesignofaquan6zerisequivalenttodesignoftheop6malpointdensityfunc6on. )(xp :ProbabilityDensityFunc6on

Background

q  Followingthestepsin[GardnerandRao]pointdensityfunc6onwillbederivedas

(nisthedimensionofx)W.R.GardnerandB.D.Rao,“Theore6calanalysisofthehigh-ratevectorquan6za6onoflpcparameters,”SpeechandAudioProcessing,IEEETransac7onson,vol.3,no.5,pp.367–381,sep1995.

Problem Statement

q Wehavetoselectadistor6onmeasurethatiswelldefinedforclassifica6onpurposes

q WechosethesymmetricKullback-Leiblerdivergencemeasure

betweenprobabilityofclassgiventhesignalbeforeandajerquan6za6on

Problem Statement & Solution

Weassumeagenera6vemodelforclassifier.Henceandareknownapriori.

Trade-offDistor6onMeasure:

Optimal Mismatched Distortion Measure

q Studyperformanceofaquan6zertrainedbyminimizingthedistor6ond1butmeasuredbydistor6ond2.

q Thera6onaleforsuchanalysisistodesignaquan6zerwithdistor6onmeasureofrelevanced2butforprac6calandimplementa6onreasonswechoosetoquan6zewithdistor6onmeasured1.

q Showedthattheop6maldiagonalmatrixD1wouldhavethediagonalelementsofD2.

Simulations

q Signalisfromtwoclasseswithknowncondi6onalPDFs

q Dashedlinesrepresentthedecisionboundaries

q  Pointdensityfunc6ondedicatescodebookpointstotheboundaries

Simulations

q  onlydedicatescodebookpointswherethesignalisconcentrated

q ByintroducingtradeoffbetweenMSEandclassifica6on,codebookpointsmovetotheclassifica6onboundaries

Simulations

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

KL Tradeoff(a=0.2)Tradeoff(a=0.8) MSE

10Bits

8Bits

6Bits

Classifica@onError(%)

q Thehigherthebitrateofquan6zerthebemerclassifica6onaccuracy

q AswemovefromMSEtoKL,theclassifica6onaccuracyimproves

Simulations

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0KL Tradeoff(a=0.2)Tradeoff(a=0.8) MSE

10Bits

8Bits

6Bits

Distor@on(dB)

q PureKLperformspoorlyasfarasthedistor6onofthesignal

q However,introducingtheslightesttradeoffwithMSEimprovesdistor6onsignificantly

Simulations

q  onlydedicatescodebookpointswherethesignalisconcentrated

q ByintroducingtradeoffbetweenMSEandclassifica6on,codebookpointsmovetotheclassifica6onboundaries

Simulations


q AswemovefromMSEtoKL,theclassifica6onaccuracyimproves


0

0.1

0.2

0.3

0.4

0.5

0.6

KL Tradeoff(a=0.2) Tradeoff(a=0.8) MSE

12Bits

10Bits

8Bits

Simulations

q PureKLperformspoorlyasfarasthedistor6onofthesignal

q However,introducingtheslightesttradeoffwithMSEimprovesdistor6onsignificantly

-70

-60

-50

-40

-30

-20

-10

0KL Tradeoff(a=0.2) Tradeoff(a=0.8) MSE12Bits

10Bits

8Bits

Distor@on(dB)

Simulations (2-D Signal)

Simulations (2-D Signal)


q KLperformsbemerthanKL-DiagonalwhichperformsbemerthanMSE.


0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

KL KL-Diag MSE

15Bits

14Bits

12Bits

Simulations

q KLperformspoorlyasfarasthedistor6onofthesignalfollowedbyKL-DiagandthenMSE.

-35

-30

-25

-20

-15

-10

-5

0KL KL-Diag MSE15Bits

14Bits

12Bits

Distor@on(dB)

Real Data Set

q DataSet§  IrisdatasetfromUCImachinelearningdepository.§  4amributes:SepalandPetallengthandwidth§  3classesofIrisplant,50instancesofeachclass

q Simula6onScheme§  50Instancesofeachclassesdividedinto30fortrainingand

20fortes6ng§  Wedothispar66oningrandomlyandrepeattheexperiment

106mes.

Real Data Sets (2 attributes)

q FiongGaussiansforeachclass

q Trainingsamplesincircle

Real Data Setsq Random

Codebooks§  MSE:Blue

Plus§  KL:RedCircle§  KL-Diag:

BlackCrossq Repeat

genera6ngrandomcodebooks1006mes

Simulations (Iris Data Set)


q KLperformsbemerthanKL-DiagonalandMSEinthemiddle


0

0.5

1

1.5

2

2.5

3

3.5

KL KL-Diag MSE

9Bits

8Bits

7Bits

Simulations (Iris Data Set)

q Thehigherthebitrateofquan6zerthebemerdistor6on

q MSEperformsbemer

-30

-25

-20

-15

-10

-5

0KL KL-Diag MSE9Bits

8Bits

7Bits

Distor@on(dB)

Simulations (Iris Data Set – 4 attributes)

q KLperformsbemer

thanKL-DiagonalandMSE


0

2

4

6

8

10

12

14

KL KL-Diag MSE

9Bits

8Bits

7Bits

Simulations (Iris Data Set – 4 attributes)

•  MSEperformsbemer

Distor@on(dB)

-12

-10

-8

-6

-4

-2

0

2

KL KL-Diag MSE

9Bits

8Bits

7Bits

Remarksq  Proposedaquan6zerforthepurposeofobtainingamore

accurateclassifica6onandreconstruc6onatthedecoderq  Employedhigh-ratetheoryforquan6zerdesign.q  Op6malpointdensityfunc6onwasderivedq  Theperformanceofthismethodwasexaminedandobserved

tobesuperiorinthetaskofclassifica6onofsignalsatthedecoder

q  Thetradeoffbetweenthereproduc6onfidelityandclassifica6onaccuracywasstudiedaswell

Interrela6onshipsofthe2solu6ons




CombinedSolu6on



2.Presenta6onLink

Interrela6onshipsofthe2solu6ons




DateRate

Quan6zer

Bits/vector

Simulation

q Databeingtransmimedisthefouramributesoftheirisplant.

DateRate

Bits/vector

Simulation

q Convergenceof

bits/vectorrate.

DateRate

Bits/vector

Simulation

q  Averageresultofclassifica6onerror.

q  Rate&Classifica6on:S1>S4>S3>S2

q  Classifica6on:KL>KL-Diag>MSE

q  WithDelay:Slightlyworseclassifica6on.

0

2

4

6

8

10

12

14

16

18

KL KL-Diag MSE

Classifica@

onError(%

)

Quan@za@onMethod

Session1

Session1(delay)

Session2

Session3

Session4

Session4(delay)

Simulation

q  Averageresultofdistor6onmeasure.

q  Rate&Distor6on:S1>S4>S3>S2

q  Distor6on:MSE>KL-Diag>KL

q  Withdelay:Slightlyworsedistor6on

-25

-20

-15

-10

-5

0

5

KL KL-Diag MSE

Distor@o

n(dB)

Quan@za@onMethod

Session1

Session1(delay)

Session2

Session3

Session4

Session4(delay)

Simulationq Short-term

behavior:Movingaverageofclassifica6onerrorfor4sessions.

q Classifica6on:KL>KL-Diag>MSE

Simulationq  Short-term

behavior:Movingaverageofdistor6onfor4sessions.

q  Distor6on:MSE>KL-Diag>KL

Contributionsq  Westudiedtheproblemofresourcealloca6onandquan6za6ontoachieveQoS

inasharedresourceconstrainednetwork.

q  Brokedownproblemintotwoproblemstobesolvedatdifferentnetworklayers.§  Firsttoachievedatarateanddelayrequirementswithpowercontrolofthe

transmimers.•  AugmentedNUMproblem•  Changeofvariableandhigh-SIRapproxima6ontotransformtoaconvex

problem.•  Presentedadistributedsolu6on.

§  Secondtodesignaquan6zertoachievesignalreconstruc6onandbemerclassifica6onatthedecoder.•  Introducedanewdistor6onmeasure,KLdivergence.•  Designedaquan6zerbasedonthenewdistor6onmeasurewithhigh-

ratetheory.

Contributions§  Secondtodesignaquan6zertoachievesignalreconstruc6onandbemer

classifica6onatthedecoder.•  Introducedanewdistor6onmeasure,KLdivergence.•  Designedaquan6zerbasedonthenewdistor6onmeasurewithhigh-

ratetheory.•  Derivedop6malmismatcheddistor6onmeasure

q  Demonstratedthelinkageandinterrela6onshipsofthetwosolu6ons.

Proposed Future Workq  Designofquan6zersthatperformwellwhenpairedwithmorenovelclassifiers

likeSupportVectorMachine.

q  DesignofajointNUMandquan6za6onmethodthroughalargeraugmenta6onofthebasicNUMproblem.•  AddQoSobtainedthroughclassifica6onandre-construc6ontotheobjec6ve

func6onorconstraints.•  Includethedistor6onmeasureofthequan6zerintheNUM.•  Highlynon-convexandwouldrequirechangeofvariablesandother

methodsbeyondwhatwepresentedinthisdisserta6ontoconverttheproblemintoaconvexop6miza6onproblem.

q  StudyoftheconvergenceofNUMframeworkwhenmessagesarequan6zed.