Propagation in random media - Paradise

The wandering photon,
a probabilistic model of
wave propagation
MASSIMO FRANCESCHETTI
University of California at Berkeley
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
The true logic of this world is in the calculus of probabilities.
James Clerk Maxwell
Maxwell Equations
in complex environments
• No closed form solution
• Use approximated numerical solvers
We need to characterize the channel

P 
C  B log 1 

 N0 B 
•Power loss
•Bandwidth
•Correlations
Simplified theoretical model
solved analytically
Everything should be as simple as possible, but not simpler.
Albert Einstein
Simplified theoretical model
solved analytically
2 parameters:
h density
g absorption
The photon’s stream
The wandering photon
Walks straight for a random length
Stops with probability g
Turns in a random direction with probability (1-g)
One dimension
One dimension
x
After a random length x
with probability g stop
with probability (1-g )/2 continue in each direction
One dimension
x
One dimension
x
One dimension
x
One dimension
x
One dimension
x
One dimension
x
One dimension
P(absorbed at x)
?
x
q( x) 
he h x
2
pdf of the length of the first step
1/h is the average step length
g is the absorption probability
One dimension
P(absorbed at x) = f (|x|,g,h) 
gh
2
e
 gh x
x
q( x) 
he h x
2
pdf of the length of the first step
1/h is the average step length
g is the absorption probability
The sleepy drunk
in higher dimensions
The sleepy drunk
in higher dimensions
After a random length,
with probability g stop
with probability (1-g ) pick a random direction
The sleepy drunk
in higher dimensions
The sleepy drunk
in higher dimensions
The sleepy drunk
in higher dimensions
The sleepy drunk
in higher dimensions
The sleepy drunk
in higher dimensions
The sleepy drunk
in higher dimensions
The sleepy drunk
in higher dimensions
The sleepy drunk
in higher dimensions
The sleepy drunk
in higher dimensions
The sleepy drunk
in higher dimensions
r
P(absorbed at r) = f (r,g,h)
2D: exact solution as a series of Bessel polynomials
3D: approximated solution
Derivation (2D)
he hr
q(r ) 
2r
g (r )   g i
pdf of hitting an obstacle at r in the first step
pdf of being absorbed at r
i
g 0 (r )  gq (r )
Stop first step
g1 (r )  (1  g )q * g 0 (r )
Stop second step
Stop third step
g 2 (r )  (1  g )q * (1  g )q * g 0 (r )
...
g (r )  gq(r )  (1  g )q * g (r )
Derivation (2D)
g (r )  gq(r )  (1  g )q * g (r )
FT
G ( ) 
gh
h 2   2  (1  g )h
FT-1
gh
g (r ) 
[K 0 ( h 2   2 r )  I1 ]
2
  h (1  g )

J 0 (r )
d
2
2 n 1 / 2
(  h )
0
I1    2 n 
n
Derivation (2D)
The integrals in the series I1 are Bessel Polynomials!
hr
gh 2
e
g (r ) 
[(1  g ) K 0 ( h 2   2 r ) 
(1  hr  cnn (hr )]
2
hr
n
Derivation (2D)
Closed form approximation:
gh
g (r ) 
[(1  g )hrK 0 ( 1  (1  g ) 2hr )  e [1(1g )
2r
2
]hr
]
Relating f (r,g,h) to the power received
each photon is a sleepy drunk,
how many photons reach a given distance?
Relating f (r,g,h) to the power received
Flux model
1
4r 2
Density model
f (r , g ,h )
 f (r , g ,h )rdr sin dd
All photons absorbed past
distance r, per unit area
o
gh
All photons entering a sphere
at distance r, per unit area
o
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)
Validation
Random walks
Classic approach
wave propagation
in random media
relates
Model with losses
analytic solution
comparison
Experiments
Propagation in random media
small
scattering
objects
Transport theory
Ishimaru A., 1978.
Wave propagation and scattering in random Media. Academic press.
Chandrasekhar, S., 1960, Radiative Transfer. Dover.
Ulaby, F.T. and Elachi, C. (eds), 1990.
Radar Polarimetry for Geoscience Applications.
Artech House.
Isotropic source
uniform scattering obstacles
Transport theory
numerical integration
Wandering Photon
analytical results
plots in Ishimaru, 1978
D(r ) 
1 
hr (1  g )e 
2 
4r 
h r

2



rK1 ( r )
2ghr
 g (1  g )

 h r
(
h
r

1
)
e

K
(

r
)

0
1  (1  g ) 2




F (r )  41r 2 

2
gh
1
/
2

10

r


Erfc( r )



1  10 r
  1  (1  g ) 2h
r2 D(r)
r2 F(r)
 t   a   s W0 
s
t
Transport theory
numerical integration
plots in Ishimaru, 1978
r2 density
s
W






2
0
t
a
s
r flux
t
Wandering Photon
analytical results
Validation
Random walks
Classic approach
wave propagation
in random media
relates
Model with losses
analytical solution
comparison
Experiments
Urban microcells
Collected in Rome, Italy, by
Antenna height: 6m
Power transmitted: 6.3W
Frequency: 900MHZ
Measured average received power over 50 measurements
Along a path of 40 wavelengths (Lee method)
Data Collection
location
Collected data
Power Loss
empirical formulas
Cellular systems
Hata (1980)
Microcellular systems
Double regression formulas
Typical values:
 2
g  4  10
1
P 
R
 2
1
( R  Rb )

R
1
( R  Rb )
g
R
Fitting the data
Power Flux
Power Density
1
r2
h  0.09
g  0.17
1
r2
h  0.12
g  0.12
Simplified formula
 r
e
Pr  
r
(dB/m losses at large distances)
based on the theoretical, wandering photon model
Fitting the data
dashed blue line: wandering photon model
 std  3.75dB
red line: power law model, 4.7 exponent
 std  6.05dB
staircase green line: best monotone fit
 std  2.04dB
Simplified formula
 r
e
Pr  
r
(dB/m losses at large distances)
based on the theoretical, wandering photon model
Transport capacity of an ad hoc wireless network
L. Xie and P.R. Kumar “A network information theory for
wireless Communication”
The wandering photon
can do more
We need to characterize the channel

P 
C  B log 1 

 N0 B 
•Power loss
•Bandwidth
•Correlations
Random walks with echoes
impulse response of a urban wireless channel
Channel
Papers:
Microcellular systems, random walks and wave propagation.
M. Franceschetti J. Bruck and L. Shulman
Short version in Proceedings IEEE AP-S ’02.
A pulse sounding thought experiment
M. Franceschetti.
In preparation
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.edu/~massimo
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massimo@paradise.caltech.edu

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