Interference from Interference from Large Wireless Networks Large Wireless Networks

under Correlated Shadowingunder Correlated Shadowing

PhD DefencePhD DefenceSCE Dept., Carleton UniversitySCE Dept., Carleton University

Friday, January 7Friday, January 7thth, 2011, 2011

Sebastian S. SzyszkowiczSebastian S. Szyszkowicz, M.A.Sc., M.A.Sc.Prof. Prof. Halim YanikomerogluHalim Yanikomeroglu

Place in Current Research Place in Current Research (Ch 1)(Ch 1)

Many Interferers Many Interferers ((asymptoticasymptotic))

Uniform infinite layoutUniform infinite layout Independent shadowingIndependent shadowing May not correspond to realityMay not correspond to reality AnalyticalAnalytical Long simulations: Long simulations: O O ((N N ))

Few interferers (Few interferers (complexitycomplexity)) Any layoutAny layout Correlated shadowingCorrelated shadowing More realisticMore realistic Numerical / AnalyticalNumerical / Analytical RRaappiid td too s siimmuullaatte : e : O O ((NN2~32~3))

Any number of InterferersAny number of Interferers Any layoutAny layout Correlated shadowingCorrelated shadowing More realisticMore realistic SimulationSimulation Lengthy simulationsLengthy simulations Doubtful correlation modelDoubtful correlation model

VeryVery

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Plan of Plan of ArgumentArgument

Choosing a shadowing correlation model

System, channel, and interference model

Basic simulation setup

Fast approximate simulation algorithm

Analytical approximation for cluster geometry

Ch 2: Ch 2: TVT Nov’10TVT Nov’10

Ch 3Ch 3

Ch 5.1Ch 5.1

Ch 4.1, 5.2: WCNC’08, TCom Ch 4.1, 5.2: WCNC’08, TCom Dec’09, J. in prep.Dec’09, J. in prep.

Ch 4.2, 4.3, 5.3: Ch 4.2, 4.3, 5.3: VTC’S10, TVT subm.VTC’S10, TVT subm.

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The Importance of The Importance of Channel ModelingChannel Modeling

Channel Model

Channel Model

SINR Pe, Pout

• The channel model must be ‘good enough’ for the application.

• A test: increase your channel model detail by one ‘level’ of complexity:

• If the results do not change much, probably the model is good enough.

• If they change a lot, increase your channel complexity, and restart.

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Physical Argument for Physical Argument for CorrelationCorrelation

Viterbi ’94, Saunders ’96, …Viterbi ’94, Saunders ’96, … Three independent propagation areas: Three independent propagation areas:

WW, , WW11, , WW22 correlation: correlation: Consistent with measurements: Consistent with measurements:

– Graziano ’78; Gudmunson ’91; Sorensen Graziano ’78; Gudmunson ’91; Sorensen ’98,’99; and several more, recently.’98,’99; and several more, recently.

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Intuitive Physical Intuitive Physical ConstraintsConstraints

hh decreases with distance and angle decreases with distance and angle hh≥0 [contradicted by some ≥0 [contradicted by some

measurements!]measurements!] hh small for angle approaching 180° small for angle approaching 180° Continuity (bounded dh/dr) Continuity (bounded dh/dr) Not dependent on only.Not dependent on only.

Choice of Shadowing Choice of Shadowing Correlation ModelCorrelation Model Variation of model proposed in [1]Variation of model proposed in [1]

We argue it is the best model among ~17 found We argue it is the best model among ~17 found in literature : in literature : physically plausiblephysically plausible and and +ive +ive semidefinitesemidefinite..

2 parameters: 2 parameters: flexibleflexible, can , can approximateapproximate other other models.models.

InvariantInvariant under rotation and scaling under rotation and scaling Correlation shape Correlation shape fast implementationfast implementation for for

shadowing fields. shadowing fields. 77

Total InterferenceTotal Interference

Pathloss

Pathloss

Shadowing

Shadowing

Correlation

RX

ISs

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Classic Simulation (Ch Classic Simulation (Ch 5.1)5.1) Matrix Factorisation (e.g., Matrix Factorisation (e.g.,

Cholesky Factorisation – Cholesky Factorisation – O O ((N N 33), ), less for sparse matrices less for sparse matrices O O ((N N ~2~2), .), .

Correlated Shadowing

iid Gaussian(0,1)99

Analytical Analytical ApproximationApproximation

Lognormal approximation for large Lognormal approximation for large interference clusterinterference cluster

Based on exchangeabilityBased on exchangeability Ch 4.1, 5.2Ch 4.1, 5.2

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Limit TheoremLimit Theorem

Sum of Sum of exchangeableexchangeable and and augmentableaugmentable joint lognormalsjoint lognormals

Converges to a lognormal Converges to a lognormal

N

1

1

1

2

K

LNeN

DN

i

Vi 1

1

1111

1

1

1

2

K

σ = 6 dB

ρ = 0.05

N = 1

210

100 1000 10000

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Application of Limit Application of Limit Thm to Interference Thm to Interference ProblemProblem

Individual interferences are Individual interferences are not exchangeable not exchangeable when IS positions are when IS positions are statistically fixed.statistically fixed.

They are They are exchangeableexchangeable when positions are when positions are iid iid randomrandom

They are also They are also augmentableaugmentable They are They are approximately lognormal approximately lognormal (but (but not not

jointlyjointly, because the conditional correlation , because the conditional correlation matrix is random)matrix is random)

Very similar to limit theoremVery similar to limit theorem Good approximation for “cluster” geometriesGood approximation for “cluster” geometries

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Using numerical integration

For large N

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Bad Approximation for Bad Approximation for non-Cluster non-Cluster GeometriesGeometries Not ~lognormal for high Not ~lognormal for high NN

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Fast Simulation for Fast Simulation for General CaseGeneral Case

Ch 4.2, 4.3, 5.3Ch 4.2, 4.3, 5.3

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Shadowing FieldsShadowing Fields

Separable triangular correlation: separable box filters.Separable triangular correlation: separable box filters. Log-polar geometric transformation.Log-polar geometric transformation. Similar approaches for other correlation models.Similar approaches for other correlation models. Place ISs ( ) on area and read shadowing value.Place ISs ( ) on area and read shadowing value. Cost: high constant + Cost: high constant + O O ((N N ) )

iid Gaussian field

2D FIR Filter

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Study of Moments (Ch Study of Moments (Ch 4.2)4.2)

First and second moments of total interference First and second moments of total interference I I found through integrals in 2 and 4 found through integrals in 2 and 4 dimensions dimensions

VAR (VAR (I I ) = ) = O O ((N N 22): very different from ): very different from independent shadowing: independent shadowing: O O ((N N )!)!

II is a sum of exchangeable RVs is a sum of exchangeable RVs I I //N N converges in distribution to something.converges in distribution to something.

Intuition: the shape of the cdf of Intuition: the shape of the cdf of I I should should stabilise after some stabilise after some N N (~500)(~500)

Approach: simulate for moderate Approach: simulate for moderate NN, then , then extrapolate for high extrapolate for high N N using moment-matchingusing moment-matching1919

Repetitive SimulationsRepetitive Simulations

Random sample reuse: both matrix Random sample reuse: both matrix factorisation and shadowing fields factorisation and shadowing fields generate generate channelschannels (corr. shadowing) and (corr. shadowing) and IS IS positionspositions separately. separately. generate less of generate less of each and mix-and–match them.each and mix-and–match them.

CPU parallelism: multi-core/multi CPU CPU parallelism: multi-core/multi CPU

2020

Time PerformanceTime Performance~ 1 day ~ 1 day 16 seconds 16 seconds

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Optimisations in Journal Version (in development)

N (# interferers)

• Random Sample reuse: reduce time by constant factor

• Extrapolation for N > 500

=> 16 seconds

- Cumulative gains- Mixed simualtion/numerical/analysis

approach- Any correlation model

--------------------------------------------one hour ---------------------------------------------

-----------------------------------------------one day-----------------------------------------------

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Break-even @ ~ 30 interferers

Little Loss in Accuracy Little Loss in Accuracy (~1dB)(~1dB)

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Main ContributionsMain Contributions

Shadowing correlation is essential in large Shadowing correlation is essential in large interference problems (future systems).interference problems (future systems).

Study of correlation models according to Study of correlation models according to math. and physical plausibility math. and physical plausibility best best model.model.

A large interference cluster can be A large interference cluster can be approximated by a single lognormal approximated by a single lognormal interferer.interferer.

Large interference problems can be Large interference problems can be reformulated for fast simulation (16s) with reformulated for fast simulation (16s) with good accuracy (1dB).good accuracy (1dB).

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Future Work Future Work

AnalysisAnalysis and and simulationsimulation can be extended for more can be extended for more complex problems (Ch 6.2):complex problems (Ch 6.2):– Random Random NN– Correlated IS positionsCorrelated IS positions– FadingFading– Variable TX powerVariable TX power– Directional RX antennaDirectional RX antenna– Correlation in time and frequencyCorrelation in time and frequency

The approach can be fine-tuned for many specific The approach can be fine-tuned for many specific emerging contexts:emerging contexts:– Aggressive spectrum reuse and sharingAggressive spectrum reuse and sharing– Wireless sensor networksWireless sensor networks– Femto-cells in cellular networksFemto-cells in cellular networks– Dynamic spectrum access / cognitive radioDynamic spectrum access / cognitive radio– ……

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Thank You!Thank You!

Mathematical Mathematical ConstraintConstraint Every correlation matrix must be positive semidefinite Every correlation matrix must be positive semidefinite

(psd)(psd) Generating correlated shadowing Generating correlated shadowing

– HH = [= [hhijij] ] – Solve Solve CCCC TT==H H (any solution)(any solution)– S = Z*S = Z*CC

Solutions forSolutions for CC may not exist! may not exist! How to make sure that a solution always exists?How to make sure that a solution always exists?

– Project Project HH onto psd matrix space [UP Valencia 2006-07] onto psd matrix space [UP Valencia 2006-07]– Our approach: make sure Our approach: make sure h h () always gives psd () always gives psd HH..

All 2x2 correlation matrices are psdAll 2x2 correlation matrices are psd Not necessarily for N=3,…Not necessarily for N=3,… We can identify models such that We can identify models such that allall HH are psd, for all are psd, for all

N.N.– We developed various tests related to the Fourier We developed various tests related to the Fourier

transforms of the model in different dimensions.transforms of the model in different dimensions. 2727

What model to What model to choose?choose?

Best! b=0, a=1

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Levels of Levels of Channel Channel DetailDetail

Independent Shadowing

Correlated Shadowing

Real-World Measurement

s

Ray-Tracing

Realism

Complexity

Big Gap! [our work]

Small Gap [some recent papers]

???

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