Interference from Large Wireless Networks under Correlated Shadowing

1. 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

2. Place in Current Research (Ch 1) • Many Interferers (asymptotic) • Uniform infinite layout • Independent shadowing • May not correspond to reality • Analytical • Long simulations: O (N ) • Few interferers (complexity) • Any layout • Correlated shadowing • More realistic • Numerical / Analytical • Rapid to simulate : O (N2~3) Interference from Poisson Fields Sum of Lognormal RVs Our Work Very • Any number of Interferers • Any layout • Correlated shadowing • More realistic • Simulation • Lengthy simulations • Doubtful correlation model • Sousa ‘90,’92, ’08 • Ilow, Hatzinakos ’98 • IEEE JSAC Sept’09 • Win, Pinto, Shepp ’09 • [ di Renzo, Imbriglio, Graziosi, Santucci, TCom 2008– … ] • [ Tellambura, Senaratne • TCom May’10] Shadowing Fields • [ Wang, Tameh, Nix, • TVT Jan’08 ] • [ Catrein, Mathar, • IEEE ATNAC Dec’08 ] Optimised Better model

3. Plan of Argument Ch 4.1, 5.2: WCNC’08, TCom Dec’09, J. in prep. Ch 4.2, 4.3, 5.3: VTC’S10, TVT subm. Analytical approximation for cluster geometry Fast approximate simulation algorithm Basic simulation setup Ch 5.1 System, channel, and interference model Ch 3 Choosing a shadowing correlation model Ch 2: TVT Nov’10

4. Channel Model Channel Model 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. • If they change a lot, increase your channel complexity, and restart. SINR Pe, Pout

5. Physical Argument for Correlation • 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.

6. Intuitive Physical Constraints • 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.

7. Choice of Shadowing Correlation Model 6 dB • 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. 60° [1] Klingenbrunn, Mogensen, VTC F’99

8. Total Interference Shadowing Correlation ISs Pathloss Shadowing Pathloss RX

9. Classic Simulation (Ch 5.1) • Matrix Factorisation (e.g., Cholesky Factorisation – O (N 3), less for sparse matrices O (N ~2), . Correlated Shadowing iid Gaussian(0,1)

10. Analytical Approximation • Lognormal approximation for large interference cluster • Based on exchangeability • Ch 4.1, 5.2

11. Limit Theorem • Sum of exchangeable and augmentable joint lognormals • Converges to a lognormal

12. N = 1 σ = 6 dB ρ = 0.05 2 10 100 1000 10000

13. Application of Limit Thm to Interference Problem • 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

14. Using numerical integration For large N

16. Bad Approximation for non-Cluster Geometries • Not ~lognormal for high N

17. Fast Simulation for General Case • Ch 4.2, 4.3, 5.3

18. Shadowing Fields  iid Gaussian field • Separable triangular correlation: separable box filters. • Log-polar geometric transformation. • Similar approaches for other correlation models. • Place ISs ( ) on area and read shadowing value. • Cost: high constant + O (N )  2D FIR Filter

19. Study of Moments (Ch 4.2) • 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

20. Repetitive Simulations • 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

21. Time Performance ~ 1 day  16 seconds

22. Optimisations in Journal Version (in development) -----------------------------------------------one day----------------------------------------------- • Random Sample reuse: • reduce time by constant factor --------------------------------------------one hour --------------------------------------------- • Extrapolation for N > 500 • Cumulative gains • Mixed simualtion/numerical/analysis approach • Any correlation model => 16 seconds Break-even @ ~ 30 interferers N (# interferers)

23. Little Loss in Accuracy (~1dB)

24. Main Contributions • 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).

25. Future Work • Analysis and simulation can be extended for more complex problems (Ch 6.2): • 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 • …

26. Thank You!

27. Mathematical Constraint • Every correlation matrix must be positive semidefinite (psd) • Generating correlated shadowing • H= [hij] • Solve CCT=H (any solution) • S = Z*C • Solutions forC 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 psdH. • All 2x2 correlation matrices are psd • Not necessarily for N=3,… • We can identify models such that allH are psd, for all N. • We developed various tests related to the Fourier transforms of the model in different dimensions.

28. What model to choose? Best! b=0, a=1

29. Levels of Channel Detail Independent Shadowing Big Gap! [our work] Correlated Shadowing Realism ??? Complexity Ray-Tracing Small Gap [some recent papers] Real-World Measurements