The Wandering Photon: A Probabilistic Model of Wave Propagation

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 • 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: hdensity gabsorption

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

x One dimension After a random length x with probability g stop with probability (1-g )/2continue in each direction

x One dimension

x pdf of the length of the first step One dimension P(absorbed at x) ? 1/h is the average step length g is the absorption probability

x pdf of the length of the first step One dimension P(absorbed at x) = f (|x|,g,h) 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 r P(absorbed at r) = f (r,g,h) 2D: exact solution as a series of Bessel polynomials 3D: approximated solution

Derivation (2D) pdf of hitting an obstacle at r in the first step pdf of being absorbed at r Stop first step Stop second step Stop third step

Derivation (2D) FT-1 FT

Derivation (2D) The integrals in the series I1are Bessel Polynomials!

Derivation (2D) Closed form approximation:

Relatingf (r,g,h)to the power received each photon is a sleepy drunk, how many photons reach a given distance?

All photons entering a sphere at distance r, per unit area All photons absorbed past distance r, per unit area o o Relatingf (r,g,h)to the power received Density model Flux model

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)

Classic approach wave propagation in random media relates comparison Validation Random walks Model with losses analytic solution Experiments

Propagation in random media 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. small scattering objects

Isotropic source uniform scattering obstacles

r2D(r) r2F(r) Transport theory numerical integration plots in Ishimaru, 1978 Wandering Photon analytical results

r2 density r2 flux Transport theory numerical integration plots in Ishimaru, 1978 Wandering Photon analytical results

Classic approach wave propagation in random media relates comparison Validation Random walks Model with losses analytical solution 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 Cellular systems Hata (1980) Microcellular systems Double regression formulas Typical values: empirical formulas

Fitting the data Power Flux Power Density

Simplified formula (dB/m losses at large distances) based on the theoretical, wandering photon model

Fitting the data dashed blue line: wandering photon model red line: power law model, 4.7 exponent staircase green line: best monotone fit

The Wandering Photon: A Probabilistic Model of Wave Propagation

The Wandering Photon: A Probabilistic Model of Wave Propagation

Presentation Transcript

Massimo Franceschetti Kaushik Chakraborty University of California at San Diego

Michael Chamberlain University of California, Berkeley

MASSIMO FRANCESCHETTI University of California at Berkeley

University of California at Berkeley – Physics Department

Philip Marcus – University of California at Berkeley

University of California at Berkeley – Physics Department

University of California Berkeley

University of California Berkeley

MASSIMO FRANCESCHETTI University of California at Berkeley

University of California at Berkeley – Physics Department

Dan Werthimer University of California, Berkeley

MASSIMO FRANCESCHETTI University of California at San Diego

MASSIMO FRANCESCHETTI University of California at Berkeley

MASSIMO FRANCESCHETTI University of California at Berkeley

University of California Berkeley

University of California, Berkeley

Tom Henzinger University of California, Berkeley

University of California at Berkeley – Physics Department

Jaein Jeong University of California, Berkeley