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STOCHASTIC MODELING OF COMPLEX ENVIRONMENTS FOR STOCHASTIC MODELING OF COMPLEX ENVIRONMENTS FOR WIRELESS SIGNAL PROPAGATIONWIRELESS SIGNAL PROPAGATION
Massimo Franceschetti
MOTIVATION
No simple solution for complex environments
Stochastic approximation of the environmentFew parameters
Simple analytical solutions
The true logic of this world is in the calculus of probabilities.
James Clerk Maxwell
WHY RANDOM MEDIA ?
WAVE APPROACH
RAY APPROACH
[1] G. Franceschetti, S. Marano, and F. Palmieri, “Propagation without wave equation, toward an urban area model,” IEEE Trans. Antennas and Propagation, vol 47, no 9, pp. 1393-1404, Sept 1999
[2] M. Franceschetti, J. Bruck, and L. Schulman, “A random walk model of wave propagation,” IEEE Trans. Antennas and Propagation, vol 52, no 5, pp. 1304-1317, May 2004
STOCHASTIC MODELS
x
y
x
y
MODEL 1. Percolation Theory
Do we measure a non-zero field inside the city ?
(yes if p > pc 0.5972 in 2D ) G.Grimmet, Percolation. New York: Springer-Verlag, 1989
SUBCRITICAL PHASE SUPERCRITICAL PHASE
propagation not allowed propagation allowed
4.0p 6.0p
MODEL 1. Percolation Theory
is propagation possible?
Reflection (Snell law)
1d2
2d 0f
R A Y A P P R O A C H(E,H)
Diffraction
Scattering
and bsorptionRefraction
MODEL 1. Propagation Mechanism
Pr{cell (j, i) is occupied} = f(j) = qj =1–pj
.pj=p=0.7 pj = pτj p = 0.6 τ = 0.2
MODEL 1. Extension to inhomogeneous grid
j=1j=2
j=n
0n,rn
n
1mm0n xrr ,...2,1,0n
1mmm rrx ,...3,2,1m
Stochastic process0r
1x
2x
1nx
nx
...
rn
MODEL 1. Mathematical formulation
0:minˆ nn rorkrnN
krPr10
krPrkr
N
NN
irirkrkrklevelreachi
NN 00 Pr|PrPrPr
1
2 3 …
N-1
N
0
1
2 3
…
N-1
N
0
MODEL 1. Mathematical formulation
00rkrPr 0N 0r0
kr0 1krkrPr 0N
kr0 0 kr0krPr 0N 0r0 kr0 kr0 0
MODEL 1. Mathematical formulation
irkrN 0|Pr
Assume: xm indep. RV’s
MARTINGALE THEORY
ki1
ki0k/i
0i0
irkrPr 0N
0i
1i
ir 0Pr
MODEL 1. Mathematical formulation
10 qirPr
iatarrivesiatreflectsiatarrives |PrPr
ir 0Pr
1
11
i
je
jeipqp
1jtanjje
ppp
pej+ = pj tanθ · pj+1
qej+ = 1 - pej
+ = 1 - pj tanθ · pj+1
irPrirkrPrkrPr 0i
0NN
ki1
ki0k/i
0i0
irkrPr 0N
1ipqp
0iqirPr 1i
1j jeie1
1
0
ki
1i
1j jeie1
1k
1i
1i
1j jeie1N pqppqpk
ikrPr
MODEL 1. Mathematical formulation
1k
1j je1
1k
1i
1i
1j jeie1
N pppqik
pkrPr
MODEL 1. Mathematical formulation
1k
1j je1
1k
1i
1i
1j jeie1
N pppqik
pkrPr
General formula for any obstacle density profile qj =1-pj
not only the uniform grid
j=1j=2
j=n
suburbs suburbscity center
x
y
TX RX
MODEL 1. Application: macrocells
x
y
Exponential profile
0 x 0 x
MODEL 1. Application
Ljq
Ljqq
Lj
Lj
j
)1(
)1(
L
1k
1j je1
1k
1i
1i
1j jeie1
N pppqik
pkrPr
max 32K
max 1000H
MODEL 1. Ray tracing validation
1cell
kr
krkr
Nraytracing
NFormulaAnalyticalNraytracing
k
Prmax
PrPr
max
1max
1 K
iiK
max
1
2
max
1 K
iiK
ERROR ANALYSIS:
3%
410
MODEL 1. Validation
o45
Analytical solution
BUILDING PROFILE:
INCREASING EXPONENTIAL
BUILDING PROFILE:
DECREASING EXPONENTIAL
MODEL 1. Validation
ERROR PLOTS
[1] G. Franceschetti, S. Marano, and F. Palmieri, “Propagation without wave equation, toward an urban area model,” IEEE Trans. Antennas and Propagation, vol 47, no 9, pp. 1393-1404, Sept 1999
[2] S. Marano, F. Palmieri, G. Francescehetti, “Statistical characterization of wave propagation in a random lattice,” J. Optic Soc. Amer. , vol 16, no 10, pp. 2459-2464, 1999.
PERCOLATION MODEL REFERENCES
[3] S. Marano, M. Franceschetti, “Ray propagation in a random lattice, a maximum entropy, anomalous diffusion process,” IEEE Trans. Antennas and Propagation, second revision due, 2004.
[4] M. Conci, A. Martini, M. Franceschetti, and A. Massa. “Wave propagation in non-uniform random lattices,” Preprint, 2004.
Homogeneous lattice pj=p
Source inside lattice
Source inside lattice
Inhomogeneous lattice profiles
MODEL 2. Random walks
DIFFUSIVE OBSTACLES12
d
A low transmitting antenna is immersed in an environment of small scatterers
MODEL 2. Application: microcells
Emitted power envelope: density of photons spreading isotropically in the environment
MODEL 2. Mathematical formulation
Pdf of a photon hitting an obstacle at r
Each photon walks straight for a random lengthStops with probability
Turns in a random direction with probability
RANDOM WALK FORMULATION
MODEL 2. Mathematical formulation
• Amount of clutter • Amount of absorption
Channel
• Impulse waveform• Time spread• Time delay• Attenuation
MODEL 2. Power delay profile
dRRrpn ),(
0
),(n
trh
c
Rtf
n
R is total path length in n steps
r is the final position after n stepso
r
|r0||r1|
|r2|
|r3|
3210 rrrrR
c is the speed of light
MODEL 2. Power delay profile
MODEL 2. Joint probabilty computation
Can solve also this analytically !
MODEL 2. Power delay profile computation
Coherent response
Incoherent response
Exponential tail
MODEL 2. Results
MODEL 2. Tail of the response
Exponential decay in time and distance
1m0.95T ~ 1nsecR ~ 6 m
MODEL 2. Validation
RANDOM WALK REFERENCES
[1] M. Franceschetti, J. Bruck, and L. Schulman, “A random walk model of wave propagation,” IEEE Trans. Antennas and Propagation, vol 52, no 5, pp. 1304-1317, May 2004.
[2] M. Franceschetti, “Stochastic rays pulse propagation,” IEEE Trans. Antennas and Propagation, to appear, October 2004.
Path Loss
Impulse power delay profile
CONCLUSION
Finding the quality of being intricate and compounded
Modeling complex propagation environments