MASSIMO FRANCESCHETTI University 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

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 length Stops with probability  Turns in a random direction with probability (1-  )

One dimension

After a random length x with probability  stop with probability (1-  )/2  continue in each direction x

One dimension x

x

x

x

x

x

x P(absorbed at x) ? pdf of the length of the first step  is the average step length  is the absorption probability

One dimension pdf of the length of the first step  is the average step length  is the absorption probability x = f (|x|,  ) P(absorbed at x)

The sleepy drunk in higher dimensions

The sleepy drunk in higher dimensions After a random length, with probability  stop with probability (1-  ) 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 2D: exact solution as a series of Bessel polynomials 3D: approximated solution r P(absorbed at r) = f (r,  )

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

Derivation (2D) FT -1 FT

Derivation (2D) The integrals in the series I 1 are Bessel Polynomials !

Derivation (2D) Closed form approximation:

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 All photons absorbed past distance r, per unit area All photons entering a sphere at distance r, per unit area o 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)

It is a simplified model in microcellular systems this may be a better assumption than optical reflection (see Tarng & Ju, IEEE Trans. Electromag. Comp. 1999)

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

Propagation in random media Ulaby, F.T. and Elachi, C. (eds), Radar Polarimetry for Geoscience Applications. Artech House. Chandrasekhar, S., 1960, Radiative Transfer. Dover. Ishimaru A., Wave propagation and scattering in random Media. Academic press. Transport theory small scattering objects

Isotropic source uniform scattering obstacles

Transport theory numerical integration plots in Ishimaru, 1978 Wandering Photon analytical results r 2 D(r) r 2 F(r)

Transport theory numerical integration plots in Ishimaru, 1978 Wandering Photon analytical results r 2 density r 2 flux

     absorbing scattering no obstacles absorbing scattering no obstacles 3-D 2-D FluxDensity

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

Urban microcells Antenna height: 6m Power transmitted: 6.3W Frequency: 900MHZ Collected in Rome, Italy, by Measured average received power over 50 measurements Along a path of 40 wavelengths (Lee method)

Data Collection location

Collected data

Fitting the data Power Flux Power Density

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

Power Loss empirical formulas Hata (1980) Cellular systems Typical values: Double regression formulas Microcellular systems

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

(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

Random walks with echoes Channel impulse response of a urban wireless channel

Impulse response R is total path length in n steps r is the final position after n steps o r |r 1 | |r 2 | |r 3 | |r 4 |

.edu/~massimo Download from: Or send to: 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, David Tse In preparation