The role of entropy in wave propagation Stefano Marano Universita’ di Salerno Massimo...

The role of entropy in wave propagation

Stefano MaranoUniversita’ di Salerno

Massimo FranceschettiUniversity of California at Berkeley

Francesco PalmieriSeconda Universita’ di Napoli

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.

Maxwell Equations

• No closed form solution• Use approximated numerical solvers

in complex environments

Alternative approachStochastic characterization of the environment

Few parametersSimple 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

Alternative approachStochastic characterization of the environment

Few parametersSimple analytical solutions

Two recent modelsTwo 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

v

v v

v

vx

x

Percolation model

Penetration inside the medium Pk(q)

k penetration levelq density of occupied sitesincidence angle

Source inside the medium, Pn(m,k), generic ray reaches site (m,k) at the nth step

Percolation model

Random walk model

Pn(r), generic ray reaches coordinate r at the nth step

r

Walk straight for a random lengththen turn in a random direction

average step length is is a measure of the density of the clutter

r

eq

r

2)(

r

Random walk model[the wandering photon]

r

eq

r

2)(

r

)(2

2 2

)( 12/

12/2

rKr

nrP n

n

n

)(*...*)(*)(*)()( rrrr qqqqrPn nqFTFT )]}([{1 r

Where, after n steps?

Evaluation for large n

n

r

en

2

)(22

2

22

4)(

1

2

nrVar

nr

2

/ln

4

)/(1

11ln1

/ ,)1(2

~) (

2

2

2

4/12

rr

z

zz

rzz

e

zK

2~ )(

e

)(2

2 2

)( 12/

12/2

rKr

nrP n

n

n

2

22

22

2

)(2

2

1

2~)( n

r

n

n

r

n een

P

r

is the max entropy distribution satisfying the constraint

22222

),()( nn rryxfyx

Having fixed the density of the clutter, we have the “most random” distribution

in our model2

2 2

nrn

2

22

22

2

)(2

2

1

2~)( n

r

n

n

r

n een

P

r

is the max entropy distribution satisfying the constraint

22222

),()( nn rryxfyx

Having fixed the “time evolution” in the origin, we have the “most random” spatial behavior

in our model2

2 2

nrn

22

2

2 )0( r

nrPn

Percolation model

Pn(m,k) |)||(|~ kmf

||||

2

)0(1

)0(1 )0(),(

km

n

nnn P

PPkmP

is the max entropy distribution satisfying the constraint

nkm

n dDMkmPkm . .),(|)||(|,

Having fixed the “time evolution” in the origin, we have the “most random” spatial behavior

in our model

42

42)0(

2

2

nn

nnn

dd

ddP

nrP

n

n 2

1)0(

2

2

Fix one of the three and obtain the most random propagation spatial behavior

Time evolutionin origin

Euclidianmetric constraint

Environment parameter

Comparing the two models1. Random walk model

)(42

42)0(

2

2

qfdd

ddP n

nn

nnn

Comparing the two models

Time evolutionin origin

Manhattanmetric constraint

Environment parameter

2. Percolation model

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 propagationone location is enough

For papers, send me email:

massimof@EECS.berkeley.edu