1. STOCHASTIC MODELING OF COMPLEX ENVIRONMENTS FOR WIRELESS SIGNAL PROPAGATION Massimo Franceschetti

2. MOTIVATION No simple solution for complex environments

3. Stochastic approximation of the environment Few parameters Simple analytical solutions The true logic of this world is in the calculus of probabilities. James Clerk Maxwell WHY RANDOM MEDIA ?

4. WAVE APPROACH RAY APPROACH [1] G. Franceschetti, S. Marano, and F. Palmieri, “Propagation without wave equation, toward an urban area model,” IEEE Trans. Antennas and Propagation, vol 47, no 9, pp. 1393-1404, Sept 1999 [2] M. Franceschetti, J. Bruck, and L. Schulman, “A random walk model of wave propagation,” IEEE Trans. Antennas and Propagation, vol 52, no 5, pp. 1304-1317, May 2004 STOCHASTIC MODELS

5. x y x y MODEL 1. Percolation Theory Do we measure a non-zero field inside the city ?

6. (yes if p > p c  0.5972 in 2D ) G.Grimmet, Percolation. New York: Springer-Verlag, 1989 SUBCRITICAL PHASE SUPERCRITICAL PHASE propagation not allowedpropagation allowed 4.0p  6.0p  MODEL 1. Percolation Theory is propagation possible?

7.  Reflection ( Snell law ) R A Y A P P R O A C H (E,H)(E,H)  Diffraction  Scattering  and bsorption Refraction MODEL 1. Propagation Mechanism

8. 1 2 3 … N-1 N 0 1 2 3 … N 0

9. Pr{cell (j, i) is occupied} = f(j) = q j =1–p j. p j =p=0.7p j = p τj p = 0.6τ = 0.2 MODEL 1. Extension to inhomogeneous grid j=1 j=2 j=n

10. Stochastic process rnrn MODEL 1. Mathematical formulation

11. 1 2 3 … N-1 N 0 1 2 3 … N 0 MODEL 1. Mathematical formulation

12.    Assume: x m indep. RV’s MARTINGALE THEORY

13.   MODEL 1. Mathematical formulation

14. p ej + = p j tanθ · p j+1 q ej + = 1 - p ej + = 1 - p j tanθ · p j+1 MODEL 1. Mathematical formulation

15. General formula for any obstacle density profile q j =1-p j not only the uniform grid j=1 j=2 j=n

16. suburbs city center x y TX RX MODEL 1. Application: macrocells x y

17. Exponential profile 0x0x MODEL 1. Application L

18. MODEL 1. Ray tracing validation ERROR ANALYSIS:

19. MODEL 1. Validation Analytical solution

20. BUILDING PROFILE: INCREASING EXPONENTIAL BUILDING PROFILE: DECREASING EXPONENTIAL MODEL 1. Validation ERROR PLOTS

21. [1] G. Franceschetti, S. Marano, and F. Palmieri, “Propagation without wave equation, toward an urban area model,” IEEE Trans. Antennas and Propagation, vol 47, no 9, pp. 1393-1404, Sept 1999 [2] S. Marano, F. Palmieri, G. Francescehetti, “Statistical characterization of wave propagation in a random lattice,” J. Optic Soc. Amer., vol 16, no 10, pp. 2459-2464, 1999. PERCOLATION MODEL REFERENCES [3] S. Marano, M. Franceschetti, “Ray propagation in a random lattice, a maximum entropy, anomalous diffusion process,” IEEE Trans. Antennas and Propagation, second revision due, 2004. [4] M. Conci, A. Martini, M. Franceschetti, and A. Massa. “Wave propagation in non-uniform random lattices,” Preprint, 2004. Homogeneous lattice pj=p Source inside lattice Inhomogeneous lattice profiles

22. MODEL 2. Random walks DIFFUSIVE OBSTACLES A low transmitting antenna is immersed in an environment of small scatterers

23. MODEL 2. Application: microcells

24. Emitted power envelope: density of photons spreading isotropically in the environment MODEL 2. Mathematical formulation Pdf of a photon hitting an obstacle at r Each photon walks straight for a random length Stops with probability  Turns in a random direction with probability  RANDOM WALK FORMULATION

25. MODEL 2. Mathematical formulation Amount of clutter Amount of absorption

26. Channel Impulse waveform Time spread Time delay Attenuation MODEL 2. Power delay profile

27. R is total path length in n steps r is the final position after n steps o r |r 0 | |r 1 | |r 2 | |r 3 | c is the speed of light MODEL 2. Power delay profile

28. MODEL 2. Joint probabilty computation

29. Can solve also this analytically ! MODEL 2. Power delay profile computation

30. Coherent response Incoherent response Exponential tail MODEL 2. Results

31. MODEL 2. Tail of the response Exponential decay in time and distance

32.  1m  0.95 T ~ 1nsec R ~ 6 m MODEL 2. Validation

33. RANDOM WALK REFERENCES [1] M. Franceschetti, J. Bruck, and L. Schulman, “A random walk model of wave propagation,” IEEE Trans. Antennas and Propagation, vol 52, no 5, pp. 1304-1317, May 2004. [2] M. Franceschetti, “Stochastic rays pulse propagation,” IEEE Trans. Antennas and Propagation, to appear, October 2004. Path Loss Impulse power delay profile

34. CONCLUSION Finding the quality of being intricate and compounded Modeling complex propagation environments