stochastic modeling of complex environments for wireless signal propagation massimo franceschetti
TRANSCRIPT
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STOCHASTIC MODELING OF COMPLEX ENVIRONMENTS FOR STOCHASTIC MODELING OF COMPLEX ENVIRONMENTS FOR WIRELESS SIGNAL PROPAGATIONWIRELESS SIGNAL PROPAGATION
Massimo Franceschetti
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MOTIVATION
No simple solution for complex environments
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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 ?
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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
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x
y
x
y
MODEL 1. Percolation Theory
Do we measure a non-zero field inside the city ?
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(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?
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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
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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
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0n,rn
n
1mm0n xrr ,...2,1,0n
1mmm rrx ,...3,2,1m
Stochastic process0r
1x
2x
1nx
nx
...
rn
MODEL 1. Mathematical formulation
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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
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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
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0i
1i
ir 0Pr
MODEL 1. Mathematical formulation
10 qirPr
iatarrivesiatreflectsiatarrives |PrPr
ir 0Pr
1
11
i
je
jeipqp
1jtanjje
ppp
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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
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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
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suburbs suburbscity center
x
y
TX RX
MODEL 1. Application: macrocells
x
y
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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
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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:
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3%
410
MODEL 1. Validation
o45
Analytical solution
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BUILDING PROFILE:
INCREASING EXPONENTIAL
BUILDING PROFILE:
DECREASING EXPONENTIAL
MODEL 1. Validation
ERROR PLOTS
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[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
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MODEL 2. Random walks
DIFFUSIVE OBSTACLES12
d
A low transmitting antenna is immersed in an environment of small scatterers
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MODEL 2. Application: microcells
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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
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MODEL 2. Mathematical formulation
• Amount of clutter • Amount of absorption
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Channel
• Impulse waveform• Time spread• Time delay• Attenuation
MODEL 2. Power delay profile
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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
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MODEL 2. Joint probabilty computation
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Can solve also this analytically !
MODEL 2. Power delay profile computation
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Coherent response
Incoherent response
Exponential tail
MODEL 2. Results
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MODEL 2. Tail of the response
Exponential decay in time and distance
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1m0.95T ~ 1nsecR ~ 6 m
MODEL 2. Validation
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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
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CONCLUSION
Finding the quality of being intricate and compounded
Modeling complex propagation environments