1. MASSIMO FRANCESCHETTI University of California at Berkeley Stochastic rays: the cluttered environment

2. The true logic of this world is in the calculus of probabilities. James Clerk Maxwell From a long view of the history of mankind — seen from, say ten thousand years from now — there can be little doubt that the most significant event of the 19th century will be judged as Maxwell’s discovery of the laws of electrodynamics. The American Civil War will pale into provincial insignificance in comparison with this important scientific event of the same decade. Richard Feynman

3. Maxwell Equations No closed form solution Use approximated numerical solvers in complex environments

4. We need to characterize the channel Power loss Bandwidth Correlations

5. solved analytically Simplified theoretical model Everything should be as simple as possible, but not simpler.

6. solved analytically Simplified theoretical model 2 parameters:  density  absorption

7. The photon’s stream

8. The wandering photon Walks straight for a random length Stops with probability  Turns in a random direction with probability (1-  )

9. One dimension

10. After a random length x with probability  stop with probability (1-  )/2  continue in each direction x

11. One dimension x

12. x

13. x

14. x

15. x

16. x

17. x P(absorbed at x) ? pdf of the length of the first step  is the average step length  is the absorption probability

18. One dimension pdf of the length of the first step  is the average step length  is the absorption probability x = f (|x|,  ) P(absorbed at x)

19. The sleepy drunk in higher dimensions

20. The sleepy drunk in higher dimensions After a random length, with probability  stop with probability (1-  ) pick a random direction

21. The sleepy drunk in higher dimensions

22. The sleepy drunk in higher dimensions

23. The sleepy drunk in higher dimensions

24. The sleepy drunk in higher dimensions

25. The sleepy drunk in higher dimensions

26. The sleepy drunk in higher dimensions

27. The sleepy drunk in higher dimensions

28. The sleepy drunk in higher dimensions

29. The sleepy drunk in higher dimensions

30. The sleepy drunk in higher dimensions r P(absorbed at r) = f (r,  )

31. Derivation (2D) Stop first step Stop second step Stop third step pdf of hitting an obstacle at r in the first step pdf of being absorbed at r

32. Derivation (2D) FT -1 FT

33. Derivation (2D) The integrals in the series I 1 are Bessel Polynomials !

34. Derivation (2D) Closed form approximation:

35. Relating f (r,  ) to the power received Flux model Density model All photons absorbed past distance r, per unit area All photons entering a sphere at distance r, per unit area o o

36. It is a simplified model At each step a photon may turn in a random direction (i.e. power is scattered uniformly at each obstacle)

37. Validation Classic approach wave propagation in random media Random walks Model with losses Experiments comparison relates analytic solution

38. Transport theory numerical integration plots in Ishimaru, 1978 Wandering Photon analytical results r 2 density r 2 flux

39.      absorbing scattering no obstacles absorbing scattering no obstacles 3-D 2-D FluxDensity

41. Fitting the data Power Flux Power Density

42. Fitting the data dashed blue line: wandering photon model red line: power law model, 4.7 exponent staircase green line: best monotone fit

43. The wandering photon can do more

44. Random walks with echoes Channel impulse response of a urban wireless channel

45. Impulse response R is total path length in n steps r is the final position after n steps o r |r 1 | |r 2 | |r 3 | |r 4 |

46. Results Varying absorption Varying pulse width

47. Results Varying transmitter to receiver distance time delay and time spread evaluation

48. .edu/~massimo WWW.. Download from: Or send email to: massimof@EECS.berkeley.edu Papers: A random walk model of wave propagation M. Franceschetti J. Bruck and L. Shulman IEEE Transactions on Antennas and Propagation to appear in 2004 Stochastic rays pulse propagation M. Franceschetti Submitted to IEEE Trans. Ant. Prop.