MASSIMO FRANCESCHETTIUniversity of California at Berkeley

The wandering photon, a probabilistic model of

wave propagation

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

2D: exact solution as a series of Bessel polynomials3D: approximated solution

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

how many photons reach a given distance?each photon is a sleepy drunk,

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

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

Isotropic sourceuniform scattering obstacles

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

Transport theory numerical integration

plots in Ishimaru, 1978

Wandering Photon analytical results

r2 densityr2 flux sat

t

sW

0

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

Validation

Classic approachClassic approachwave propagation in random media

Random walksRandom walks

Model with lossesModel with losses

ExperimentsExperiments

comparison

relates

analytical solutionanalytical solution

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)

Data Collectionlocation

Collected data

Fitting the data

2

1

r 2

1

r

14.0

10.0

10.0

13.0

Power FluxPower Flux Power DensityPower Density

r

eP

r

r

(dB/m losses at large distances)

Simplified formula

based on the theoretical, wandering photon model

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

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

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

The wandering photon

can do more

We need to characterize the channel

•Power loss•Bandwidth•Correlations

BN

PBC

0

1log

Random walks with echoes

Channel

impulse response of a urban wireless channel

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