Interference from
Large Wireless Networks
under Correlated Shadowing
PhD Defence
SCE Dept., Carleton University
Friday, January 7th, 2011
Sebastian S. Szyszkowicz, M.A.Sc.
Prof. Halim Yanikomeroglu
Place in Current Research (Ch 1)
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Very 
Many Interferers (asymptotic)
Uniform infinite layout
Independent shadowing
May not correspond to reality
Analytical
Long simulations: O (N )
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Few interferers (complexity)
Any layout
Correlated shadowing
More realistic
Numerical / Analytical
Rapid to simulate : O (N2~3)
Any number of Interferers
Any layout
Correlated shadowing
More realistic
Simulation
Lengthy simulations
Doubtful correlation model
2
Plan of Argument
Ch 4.1, 5.2: WCNC’08, TCom
Dec’09, J. in prep.
Analytical approximation
for cluster geometry
Ch 5.1
Ch 3
Ch 4.2, 4.3, 5.3: VTC’S10,
TVT subm.
Fast approximate
simulation algorithm
Basic simulation setup
System, channel, and interference model
Ch 2:
TVT Nov’10
Choosing a shadowing correlation model
3
The Importance of
Channel Modeling
• 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.
SINR  Pe, Pout
Channel
Model
• If they change a lot, increase
your channel complexity, and
restart.
4
Physical Argument for
Correlation
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Viterbi ’94, Saunders ’96, …
Three independent propagation areas: W,
W1, W2  correlation:
Consistent with measurements:
– Graziano ’78; Gudmunson ’91; Sorensen ’98,’99;
and several more, recently.
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Intuitive Physical Constraints
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h decreases with distance and angle
h≥0 [contradicted by some measurements!]
h small for angle approaching 180°
Continuity (bounded dh/dr)
Not dependent on
only.
Choice of Shadowing
Correlation Model
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Variation of model proposed in [1]
We argue it is the best model among ~17 found in
literature : physically plausible and +ive semidefinite.
2 parameters: flexible, can approximate other models.
Invariant under rotation and scaling
Correlation shape  fast implementation for
shadowing fields.
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Total Interference
ISs
RX
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Classic Simulation (Ch 5.1)
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Matrix Factorisation (e.g., Cholesky
Factorisation – O (N 3), less for sparse
matrices O (N ~2), .
Correlated
Shadowing
iid Gaussian(0,1)
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Analytical Approximation
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Lognormal approximation for large
interference cluster
Based on exchangeability
Ch 4.1, 5.2
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Limit Theorem
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Sum of exchangeable and augmentable joint
lognormals
 1    
 1   

2

K 
N
     


   1 
Converges to a lognormal
1
N
N
D
Vi
e
  LN
i 1
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1

K   2



  
1   
 

  1

N=1
σ = 6 dB
ρ = 0.05
2
10
100
1000
10000
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Application of Limit Thm
to Interference Problem
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Individual interferences are not exchangeable
when IS positions are statistically fixed.
They are exchangeable when positions are iid
random
They are also augmentable
They are approximately lognormal (but not
jointly, because the conditional correlation
matrix is random)
Very similar to limit theorem
Good approximation for “cluster” geometries
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Using numerical integration
For large N
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Bad Approximation for
non-Cluster Geometries
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Not ~lognormal for high N
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Fast Simulation for
General Case
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Ch 4.2, 4.3, 5.3
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Shadowing Fields
 iid Gaussian field
 2D FIR Filter
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Separable triangular correlation: separable box filters.
Log-polar geometric transformation.
Similar approaches for other correlation models.
Place ISs ( ) on area and read shadowing value.
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Cost: high constant + O (N )
Study of Moments (Ch 4.2)
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First and second moments of total interference I
found through integrals in 2 and 4 dimensions
VAR (I ) = O (N 2): very different from
independent shadowing: O (N )!
I is a sum of exchangeable RVs  I /N
converges in distribution to something.
Intuition: the shape of the cdf of I should
stabilise after some N (~500)
Approach: simulate for moderate N, then
extrapolate for high N using moment-matching
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Repetitive Simulations
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Random sample reuse: both matrix factorisation
and shadowing fields generate channels (corr.
shadowing) and IS positions separately. 
generate less of each and mix-and–match them.
CPU parallelism: multi-core/multi CPU
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Time Performance
~ 1 day  16 seconds
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Optimisations in Journal Version (in development)
-----------------------------------------------one day-----------------------------------------------
•
Random Sample reuse:
reduce time by constant factor
--------------------------------------------one hour ---------------------------------------------
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Extrapolation for N > 500
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Cumulative gains
Mixed simualtion/numerical/analysis
approach
Any correlation model
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=> 16 seconds
Break-even @ ~ 30 interferers
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N (# interferers)
Little Loss in Accuracy (~1dB)
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Main Contributions
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Shadowing correlation is essential in large
interference problems (future systems).
Study of correlation models according to math.
and physical plausibility  best model.
A large interference cluster can be approximated
by a single lognormal interferer.
Large interference problems can be
reformulated for fast simulation (16s) with good
accuracy (1dB).
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Future Work
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Analysis and simulation can be extended for more
complex problems (Ch 6.2):
–
–
–
–
–
–
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Random N
Correlated IS positions
Fading
Variable TX power
Directional RX antenna
Correlation in time and frequency
The approach can be fine-tuned for many specific
emerging contexts:
–
–
–
–
–
Aggressive spectrum reuse and sharing
Wireless sensor networks
Femto-cells in cellular networks
Dynamic spectrum access / cognitive radio
…
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Thank You!
Mathematical Constraint
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Every correlation matrix must be positive semidefinite (psd)
Generating correlated shadowing
– H = [hij]
– Solve CC T=H (any solution)
– S = Z*C
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Solutions for C may not exist!
How to make sure that a solution always exists?
– Project H onto psd matrix space [UP Valencia 2006-07]
– Our approach: make sure h () always gives psd H.
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All 2x2 correlation matrices are psd
Not necessarily for N=3,…
We can identify models such that all H are psd, for all N.
– We developed various tests related to the Fourier transforms of
the model in different dimensions.
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What model to choose?
Best!
b=0, a=1
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Levels of Channel Detail
Independent
Shadowing
Correlated
Shadowing
Big Gap!
[our work]
Realism
???
Complexity
Ray-Tracing
Real-World
Measurements
Small Gap [some
recent papers]
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