massimo franceschetti university of california at berkeley stochastic rays: the cluttered...
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MASSIMO FRANCESCHETTIUniversity of California at Berkeley
Stochastic rays: the cluttered environment
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
Maxwell Equations
• No closed form solution• Use approximated numerical solvers
in complex environments
We need to characterize the channel
•Power loss•Bandwidth•Correlations
BN
PBC
0
1log
solved analytically
Simplified theoretical model
Everything should be as simple as possible, but not simpler.
solved analytically
Simplified theoretical model
2 parameters: density absorption
The photon’s stream
The wandering photon
Walks straight for a random lengthStops with probability
Turns in a random direction with probability (1-)
One dimension
One dimension
After a random length xwith probability stop
with probability (1-)/2continue in each direction
x
One dimension
x
One dimension
x
One dimension
x
One dimension
x
One dimension
x
One dimension
x
One dimension
x
P(absorbed at x) ?
2)(
xexq
pdf of the length of the first stepis the average step lengthis the absorption probability
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)
The sleepy drunkin higher dimensions
The sleepy drunkin higher dimensions
After a random length, with probability stop
with probability (1-) pick a random direction
The sleepy drunkin higher dimensions
The sleepy drunkin higher dimensions
The sleepy drunkin higher dimensions
The sleepy drunkin higher dimensions
The sleepy drunkin higher dimensions
The sleepy drunkin higher dimensions
The sleepy drunkin higher dimensions
The sleepy drunkin higher dimensions
The sleepy drunkin higher dimensions
The sleepy drunkin higher dimensions
r
P(absorbed at r) = f (r,)
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
Derivation (2D)
)(*)1()()( rgqrqrg
)1()(
22 G
])([2
)( 122
0 IrKrg
FT-1
FT
nn
n drJ
I0
2/12202
1 )(
)(
)1(
Derivation (2D)
The integrals in the series I1 are Bessel Polynomials!
])(1()()1[(2
)( 220
2
nnn
r
rcrr
erKrg
Derivation (2D)
Closed form approximation:
]))1(1()1[(2
)( ])1(1[20
2 rerrKr
rg
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
It is a simplified model
At each step a photon may turnin a random direction (i.e. power is scattered uniformly at each obstacle)
Validation
Classic approachClassic approachwave propagation in random media
Random walksRandom walks
Model with lossesModel with losses
ExperimentsExperiments
comparison
relates
analytic solutionanalytic solution
Transport theory numerical integration
plots in Ishimaru, 1978
Wandering Photon analytical results
r2 densityr2 flux sat
t
sW
0
Fitting the data
2
1
r 2
1
r
14.0
10.0
10.0
13.0
Power FluxPower Flux Power DensityPower Density
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
The wandering photon
can do more
Random walks with echoes
Channel
impulse response of a urban wireless channel
Impulse response
dRRrpn ),(
1
),(n
trh
c
Rtf
n
R is total path length in n steps
r is the final position after n stepso
r
|r1||r2|
|r3|
|r4|
4321 rrrrR
Results
Varying absorption Varying pulse width
Results
Varying transmitter to receiver distance
time delay and time spread evaluation
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Papers:A random walk model of wave propagationM. Franceschetti J. Bruck and L. ShulmanIEEE Transactions on Antennas and Propagation to appear in 2004
Stochastic rays pulse propagationM. FranceschettiSubmitted to IEEE Trans. Ant. Prop.