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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
