1. The role of entropy in wave propagation Stefano Marano Universita’ di Salerno Massimo Franceschetti University of California at Berkeley Francesco Palmieri Seconda Universita’ di Napoli

2. Why wave propagation? The capacity of a wireless network depends on the physics of propagation. We need to develop analytical models of propagation to compute the fundamental limits of wireless communication.

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

4. Alternative approach Stochastic characterization of the environment Few parameters Simple analytical solutions using these stochastic models, Shannon’s entropy is useful to understand the nature of propagation The true logic of this world is in the calculus of probabilities. James Clerk Maxwell

5. Alternative approach Stochastic characterization of the environment Few parameters Simple analytical solutions Two recent models “Wave Propagation Without Wave Equation” G. Franceschetti, S. Marano, F. Palmieri, IEEE Trans. Ant. Prop. 1999 “A Random Walk Model of Wave Propagation” M. Franceschetti, J. Bruck, L. Schulman, IEEE Trans. Ant. Prop., to appear. B. Hughes. Random walks and random environments, Vol.1, Oxford University Press, 1995 D. Stauffer, A. Aharony. Introduction to percolation theory, Taylor and Francis, London, 1994

6. v vv v vx x  Percolation model Penetration inside the medium P k (  q) k penetration level q density of occupied sites  incidence angle

7. Source inside the medium, P n (m,k), generic ray reaches site (m,k) at the n th step Percolation model

8. Random walk model P n (r), generic ray reaches coordinate r at the n th step r

9. Walk straight for a random length then turn in a random direction average step length is  is a measure of the density of the clutter r e q r    2 )(   r Random walk model [the wandering photon]

10. r e q r    2 )(   r )( 2 2 2 )( 12/ 12/ 2 rK r n rP n n n                    )(*...*)(*)(*)()(rrrrqqqqrP n  n qFT )]}([{ 1 r   Where, after n steps?

11. Evaluation for large n n r e n 2 )( 2 2 2     2 2 4 )( 1 2     n rVar n r    2 / ln 4 )/( 1 11 1 /, )1(2 ~) ( 2 2 2 4/12 r r z z z rz z e z K           2~ )( e   )( 2 2 2 )( 12/ 12/ 2 rK r n rP n n n                   

12. is the max entropy distribution satisfying the constraint Having fixed the density of the clutter, we have the “most random” distribution in our model 2 2 2  n r n 

13. is the max entropy distribution satisfying the constraint Having fixed the “time evolution” in the origin, we have the “most random” spatial behavior in our model 2 2 2  n r n  2 2 2 2 )0(r n rP n   

14. Percolation model P n (m,k) |)||(|~kmf 

15. is the max entropy distribution satisfying the constraint Having fixed the “time evolution” in the origin, we have the “most random” spatial behavior in our model 42 42 )0( 2 2    nn nn n dd dd P

17. Fix one of the three and obtain the most random propagation spatial behavior Time evolution in origin Euclidian metric constraint Environment parameter Comparing the two models 1. Random walk model

18. Comparing the two models Time evolution in origin Manhattan metric constraint Environment parameter 2. Percolation model

19. Conclusion Propagation modeled as a stochastic process Most random evolution given the model parameter given the metric constraint given the time evolution environment characteristic given the type of propagation one location is enough

20. For papers, send me email: massimof@EECS.berkeley.edu