stochastic modeling of complex environments for wireless signal propagation massimo franceschetti

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


00rkrPr 0N 0r0

kr0 1krkrPr 0N

kr0 0 kr0krPr 0N 0r0 kr0 kr0 0


irkrN 0|Pr

Assume: xm indep. RV’s

MARTINGALE THEORY

ki1

ki0k/i

0i0

irkrPr 0N

0i

1i

ir 0Pr


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


1k

1j je1

1k

1i

1i

1j jeie1

N pppqik

pkrPr


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


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


• 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

stochastic modeling of complex environments for wireless signal propagation massimo franceschetti

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