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MASSIMO FRANCESCHETTI University of California at Berkeley

Stochastic rays propagation. MASSIMO FRANCESCHETTI University of California at Berkeley. Maxwell Equations. in complex environments. No closed form solution Use approximated numerical solvers. We need to characterize the channel. Power loss Bandwidth Correlations.

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MASSIMO FRANCESCHETTI University of California at Berkeley

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  1. Stochastic rays propagation MASSIMO FRANCESCHETTI University of California at Berkeley

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

  3. We need to characterize the channel • Power loss • Bandwidth • Correlations

  4. The true logic of this world is in the calculus of probabilities. James Clerk Maxwell

  5. Simplified theoretical model solved analytically 2 parameters: hdensity gabsorption

  6. The photon’s stream

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

  8. The wandering photon

  9. The wandering photon After a random length, with probability g stop with probability (1-g ) pick a random direction

  10. The wandering photon

  11. The wandering photon

  12. The wandering photon

  13. The wandering photon

  14. The wandering photon

  15. The wandering photon

  16. The wandering photon

  17. The wandering photon

  18. The wandering photon

  19. The wandering photon r P(absorbed at r) = g(r,g,h)

  20. Derivation pdf of hitting an obstacle at r in the first step pdf of being absorbed at r Stop first step Stop second step Stop third step

  21. All photons entering a sphere at distance r, per unit area All photons absorbed past distance r, per unit area o o Relatingg(r,g,h)to the received power Density model Flux model

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

  23. Fitting the data Power Flux Power Density

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

  25. The wandering photon can do more

  26. Random walks with echoes impulse response of a urbanwireless channel Channel

  27. Impulse response |r3| R is total path length in n steps r |r2| r is the final position after n steps o |r1| |r0|

  28. Results Varying absorption Varying pulse width

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

  30. WWW. . .edu/~massimo 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. Download from: Or send email to: massimof@EECS.berkeley.edu

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