MASSIMO FRANCESCHETTI University of California at Berkeley The wandering photon, a probabilistic...
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MASSIMO FRANCESCHETTIUniversity of California at Berkeley
The wandering photon, a probabilistic model of
wave propagation
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The true logic of this world is in the calculus of probabilities.James Clerk Maxwell
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
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Maxwell Equations
• No closed form solution• Use approximated numerical solvers
in complex environments
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We need to characterize the channel
•Power loss•Bandwidth•Correlations
BN
PBC
0
1log
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solved analytically
Simplified theoretical model
Everything should be as simple as possible, but not simpler.
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solved analytically
Simplified theoretical model
2 parameters: density absorption
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The photon’s stream
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The wandering photon
Walks straight for a random lengthStops with probability
Turns in a random direction with probability (1-)
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One dimension
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One dimension
After a random length xwith probability stop
with probability (1-)/2continue in each direction
x
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One dimension
x
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One dimension
x
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One dimension
x
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One dimension
x
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One dimension
x
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One dimension
x
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One dimension
x
P(absorbed at x) ?
2)(
xexq
pdf of the length of the first stepis the average step lengthis the absorption probability
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One dimension
2)(
xexq
pdf of the length of the first stepis the average step lengthis the absorption probability
x
= f (|x|,) xe
2P(absorbed at x)
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The sleepy drunkin higher dimensions
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The sleepy drunkin higher dimensions
After a random length, with probability stop
with probability (1-) pick a random direction
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The sleepy drunkin higher dimensions
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The sleepy drunkin higher dimensions
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The sleepy drunkin higher dimensions
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The sleepy drunkin higher dimensions
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The sleepy drunkin higher dimensions
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The sleepy drunkin higher dimensions
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The sleepy drunkin higher dimensions
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The sleepy drunkin higher dimensions
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The sleepy drunkin higher dimensions
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The sleepy drunkin higher dimensions
2D: exact solution as a series of Bessel polynomials3D: approximated solution
r
P(absorbed at r) = f (r,)
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Derivation (2D)
...
)(*)1(*)1()(
)(*)1()(
)()(
02
01
0
rgqqrg
rgqrg
rqrg
Stop first step
Stop second step
Stop third step
r
erq
r
2)(
pdf of hitting an obstacle at r in the first step
i
igrg )( pdf of being absorbed at r
)(*)1()()( rgqrqrg
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Derivation (2D)
)(*)1()()( rgqrqrg
)1()(
22 G
])([2
)( 122
0 IrKrg
FT-1
FT
nn
n drJ
I0
2/12202
1 )(
)(
)1(
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Derivation (2D)
The integrals in the series I1 are Bessel Polynomials!
])(1()()1[(2
)( 220
2
nnn
r
rcrr
erKrg
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Derivation (2D)
Closed form approximation:
]))1(1()1[(2
)( ])1(1[20
2 rerrKr
rg
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Relating f (r,) to the power received
how many photons reach a given distance?each photon is a sleepy drunk,
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Relating f (r,) to the power received
Flux model Density model
ddrdrrfr
sin),,(4
12
All photons absorbed pastdistance r, per unit area
),,(rf
All photons entering a sphere at distance r, per unit area
o
o
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It is a simplified model
At each step a photon may turnin a random direction (i.e. power is scattered uniformly at each obstacle)
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Validation
Classic approachClassic approachwave propagation in random media
Random walksRandom walks
Model with lossesModel with losses
ExperimentsExperiments
comparison
relates
analytic solutionanalytic solution
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Propagation in random media
Ulaby, F.T. and Elachi, C. (eds), 1990. Radar Polarimetry for Geoscience Applications. Artech House.
Chandrasekhar, S., 1960, Radiative Transfer. Dover.
Ishimaru A., 1978. Wave propagation and scattering in random Media. Academic press.
Transport theory
small scatteringobjects
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Isotropic sourceuniform scattering obstacles
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Transport theory numerical integration
plots in Ishimaru, 1978
Wandering Photon analytical results
r2 D(r)r2 F(r) sat
t
sW
0
r
r
rer
erF
2
2
2)1(12
])1(1[
41
]1)1(1)[1()(
rr eerr
rD
])1(1[)1(1
2
22
)1(4
1)(
2)1(1
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Transport theory numerical integration
plots in Ishimaru, 1978
Wandering Photon analytical results
r2 densityr2 flux sat
t
sW
0
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absorbing
scattering
no obstacles
absorbing
scattering
no obstacles
r
e r
2
r
e r
2
r2
1
r2
1
r2
1
24 r
e r
24 r
e r
24
1
r r
1~
24
1
r 24
1
r
3-D
2-D
Flux Density
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Validation
Classic approachClassic approachwave propagation in random media
Random walksRandom walks
Model with lossesModel with losses
ExperimentsExperiments
comparison
relates
analytical solutionanalytical solution
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Urban microcells
Antenna height: 6mPower transmitted: 6.3WFrequency: 900MHZ
Collected in Rome, Italy, by
Measured average received power over 50 measurementsAlong a path of 40 wavelengths (Lee method)
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Data Collectionlocation
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Collected data
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Fitting the data
2
1
r 2
1
r
14.0
10.0
10.0
13.0
Power FluxPower Flux Power DensityPower Density
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r
eP
r
r
(dB/m losses at large distances)
Simplified formula
based on the theoretical, wandering photon model
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Power Lossempirical formulas
RP
1 2 Hata (1980)
Cellular systems
)(1
bRRR
)(1
bRRR
104
2
Typical values:Typical values:
Double regression formulas
Microcellular systems
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Fitting the data
dB
dB
dB
std
std
std
04.2
05.6
75.3
dashed blue line: wandering photon model
red line: power law model, 4.7 exponent
staircase green line: best monotone fit
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r
eP
r
r
(dB/m losses at large distances)
Simplified formula
based on the theoretical, wandering photon model
L. Xie and P.R. Kumar “A network information theory for wireless Communication”
Transport capacity of an ad hoc wireless network
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The wandering photon
can do more
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We need to characterize the channel
•Power loss•Bandwidth•Correlations
BN
PBC
0
1log
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Random walks with echoes
Channel
impulse response of a urban wireless channel
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Impulse response
dRRrpn ),(
1
),(n
trh
ct
r
n
c
Rtf
R is total path length in n steps
r is the final position after n stepso
r
|r1||r2|
|r3|
|r4|
4321 rrrrR
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Papers:Microcellular systems, random walks and wave propagation.M. Franceschetti J. Bruck and L. ShulmanShort version in Proceedings IEEE AP-S ’02.
A pulse sounding thought experimentM. Franceschetti, David Tse In preparation