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 19 th 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: h density g absorption
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
One dimension x After a random length x with probability g stop with probability (1 -g )/2 continue in each direction
One dimension x
One dimension x
One dimension x
One dimension x
One dimension x
One dimension x
One dimension P(absorbed at x) ? x pdf of the length of the first step 1/h is the average step length g is the absorption probability
One dimension P(absorbed at x) = f (|x|, g, h) x pdf of the length of the first step 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
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 r P(absorbed at r) = f (r, g, h) 2 D: exact solution as a series of Bessel polynomials 3 D: approximated solution
Derivation (2 D) 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 (2 D) FT FT-1
Derivation (2 D) The integrals in the series I 1 are Bessel Polynomials!
Derivation (2 D) Closed form approximation:
Relating f (r, g, h) to the power received each photon is a sleepy drunk, how many photons reach a given distance?
Relating f (r, g, h) to the power received Flux model All photons absorbed past distance r, per unit area o Density model All photons entering a sphere at distance r, per unit area 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)
Validation Random walks Classic approach wave propagation in random media relates Model with losses analytic solution comparison Experiments
Propagation in random media small scattering objects 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.
Isotropic source uniform scattering obstacles
Transport theory numerical integration plots in Ishimaru, 1978 r 2 D(r) r 2 F(r) Wandering Photon analytical results
Transport theory numerical integration plots in Ishimaru, 1978 r 2 density r 2 flux Wandering Photon analytical results
Validation Random walks Classic approach wave propagation in random media relates Model with losses analytical solution comparison Experiments
Urban microcells Collected in Rome, Italy, by Antenna height: 6 m Power transmitted: 6. 3 W Frequency: 900 MHZ Measured average received power over 50 measurements Along a path of 40 wavelengths (Lee method)
Data Collection location
Collected data
Power Loss empirical formulas Cellular systems Hata (1980) Microcellular systems Double regression formulas Typical values:
Fitting the data Power Flux Power Density
Simplified formula (d. B/m losses at large distances) based on 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
Simplified formula (d. B/m losses at large distances) based on theoretical, wandering photon model Transport capacity of an ad hoc wireless network L. Xie and P. R. Kumar “A network information theory for wireless Communication”
The wandering photon can do more
We need to characterize the channel • Power loss • Bandwidth • Correlations
Random walks with echoes impulse response of a urban wireless channel Channel
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. In preparation Download from: WWW. . . edu/~massimo Or send email to: massimo@paradise. caltech. edu
Скачать презентацию The wandering photon a probabilistic model of wave
fc1d555e0c1314225a6a39b96113455b.ppt