1. Interference from Large Wireless Networks under Correlated Shadowing PhD Defence SCE Dept., Carleton University Friday, January 7 th, 2011 Sebastian S. Szyszkowicz, M.A.Sc. Prof. Halim Yanikomeroglu

2. Place in Current Research (Ch 1) Many Interferers (asymptotic) Many Interferers (asymptotic) Uniform infinite layout Uniform infinite layout Independent shadowing Independent shadowing May not correspond to reality May not correspond to reality Analytical Analytical Long simulations: O (N ) Long simulations: O (N ) Few interferers (complexity) Few interferers (complexity) Any layout Any layout Correlated shadowing Correlated shadowing More realistic More realistic Numerical / Analytical Numerical / Analytical Rapid to simulate : O (N 2~3 ) Rapid to simulate : O (N 2~3 ) Any number of Interferers Any number of Interferers Any layout Any layout Correlated shadowing Correlated shadowing More realistic More realistic Simulation Simulation Lengthy simulations Lengthy simulations Doubtful correlation model Doubtful correlation model Very 2

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

4. The Importance of Channel Modeling Channel Model SINR  P e, P out 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. 4

5. Physical Argument for Correlation Viterbi ’94, Saunders ’96, … Viterbi ’94, Saunders ’96, … Three independent propagation areas: W, W 1, W 2  correlation: Three independent propagation areas: W, W 1, W 2  correlation: Consistent with measurements: Consistent with measurements: –Graziano ’78; Gudmunson ’91; Sorensen ’98,’99; and several more, recently. 5

6. Intuitive Physical Constraints h decreases with distance and angle h decreases with distance and angle h≥0 [contradicted by some measurements!] h≥0 [contradicted by some measurements!] h small for angle approaching 180° h small for angle approaching 180° Continuity (bounded dh/dr) Continuity (bounded dh/dr) Not dependent on only. Not dependent on only. 6

7. Choice of Shadowing Correlation Model Variation of model proposed in [1] Variation of model proposed in [1] We argue it is the best model among ~17 found in literature : physically plausible and +ive semidefinite. We argue it is the best model among ~17 found in literature : physically plausible and +ive semidefinite. 2 parameters: flexible, can approximate other models. 2 parameters: flexible, can approximate other models. Invariant under rotation and scaling Invariant under rotation and scaling Correlation shape  fast implementation for shadowing fields. Correlation shape  fast implementation for shadowing fields. 7

8. Total Interference Pathloss Shadowing Correlation RX ISs 8

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

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

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

12. σ = 6 dB ρ = 0.05 N = 1 2 10 100 100010000 12

13. Application of Limit Thm to Interference Problem Individual interferences are not exchangeable when IS positions are statistically fixed. Individual interferences are not exchangeable when IS positions are statistically fixed. They are exchangeable when positions are iid random They are exchangeable when positions are iid random They are also augmentable They are also augmentable They are approximately lognormal (but not jointly, because the conditional correlation matrix is random) They are approximately lognormal (but not jointly, because the conditional correlation matrix is random) Very similar to limit theorem Very similar to limit theorem Good approximation for “cluster” geometries Good approximation for “cluster” geometries 13

14. Using numerical integration For large N 14

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

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

18. Shadowing 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 + O (N ) Cost: high constant + O (N )  iid Gaussian field  2D FIR Filter 18

19. Study of Moments (Ch 4.2) First and second moments of total interference I found through integrals in 2 and 4 dimensions 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 )! 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. 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) 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 Approach: simulate for moderate N, then extrapolate for high N using moment-matching 19

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. 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 CPU parallelism: multi-core/multi CPU 20

21. Time Performance ~ 1 day  16 seconds 21

22. 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----------------------------------------------- 22 Break-even @ ~ 30 interferers

23. Little Loss in Accuracy (~1dB) 23

24. Main Contributions Shadowing correlation is essential in large interference problems (future systems). Shadowing correlation is essential in large interference problems (future systems). Study of correlation models according to math. and physical plausibility  best model. Study of correlation models according to math. and physical plausibility  best model. A large interference cluster can be approximated by a single lognormal interferer. 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). Large interference problems can be reformulated for fast simulation (16s) with good accuracy (1dB). 24

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

26. Thank You!

27. Mathematical Constraint Every correlation matrix must be positive semidefinite (psd) Every correlation matrix must be positive semidefinite (psd) Generating correlated shadowing Generating correlated shadowing –H = [h ij ] –Solve CC T =H (any solution) –S = Z*C Solutions for C may not exist! Solutions for C may not exist! How to make sure that a solution always exists? 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. All 2x2 correlation matrices are psd All 2x2 correlation matrices are psd Not necessarily for N=3,… Not necessarily for N=3,… We can identify models such that all H are psd, for all N. 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. 27

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

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