Chapter 3: Log-Normal Shadowing Models Motivation for dynamical channel models Log-Normal dynamical...

Chapter 3: Log-Normal Shadowing Models

Motivation for dynamical channel models

Log-Normal dynamical models

Short-term dynamical models

The Models are Used in the Shot-Noise Representation of Wireless Channels

Chapter 3: Motivation for Dynamical Channel Models

Short-term Fading

Varying environmentObstacles on/off

Area 2Area 1

Transmitter

Log-normalShadowing

Varying environmentObstacles on/off

Mobiles move

Chapter 3: S.D.E.’s in Modeling Log-Normal Shadowing

Dynamical spatial log-normal channel model Geometric Brownian motion model

Spatial correlation

Dynamical temporal channel model Mean-reverting log-normal model

Space-time mean-reverting log-normal model Mean-reverting log-normal model

Chapter 3: Log-Normal Shadowing Model

Transmittern,1

Receiver

k,1

or

d

n,3n,2

k,2

k,3

k,4 one subpath

LOS

path k

path n

d(t)

vmR(t)

n

Chapter 3: Static Log-Normal Model

20

( )[ ]

Power path-loss in dB's:

Gaussian random variable

( )[ ] ( )[ ] , (0, ),

Signal Attenuation Coefficient:

Log-Normal random variable

( ) , - ln10 / 20

d

kPL d dB

PL d dB PL d dB X X N d d

r d e k

d : time-delay equivalent to distance d=vc

speed of light

S(t,) and X (t,) : random processes modeled using S.D.E.’s with respect to t and

Chapter 3: Dynamical Log-Normal Model

0

0

0,

0,

( , )0

Power path-loss: Gaussian process

( ) ( , ) ,

Signal Attenuation coefficient: log-normal process

( ) ( , )

( , ) , 0, ,

t

t

kX t

PL d dB X t

r d S t

S t e t

Chapter 3: Dynamical Spatial Log-Normal Model

Transmitter

Receiver

S(t, 3)

S(t, 2)

S(t, 1)

S(t, 4)

S(t, 5)

• Time, t: fixed (snap shot at propagation environment)• {S(t,)}|t=fixed S.D.E. w.r.t.


2 1 1

1

For time : fixed

, , , ,, ,

, ,

% changes: independent random variables, decompose into

a systematic (drift): evolution of local mean and

a random part (diffusion): variability aro

m m

m

t

S t S t S t S t

S t S t

und mean

( , ) Geometric Brownian Motion (GBM) w.r.t.

Percentage changes are independent and

identically distributed, log-normal distribution

( , )( , )

( , )

t fixedS t

dS tt d

S t

0 0

0

( , ) , ( , ),

0, : standard B.M., independent of S( , ).

t dW S t

W N t

Need specific S.D.E.s for {X(t,), S(t,)} where

{X(t,)}|t=fixed => At every , B.M. with non-zero drift

{S(t,)}| t=fixed => At every , G.B.M.

: models loss characteristics of propagation environment


0

( , )

20 0

0

( , )

Choose a S.D.E. for ( , ) as follows:

10( , ) ( , ) ,

ln10

( , ) [ ];

0, : standard B.M., independent of X( , ).

kX tS t e

X t

dX t d t dW

X t N PL d dB

W N t

Properties of spatial log-normal model

S(t,) = ekX(t,) : Geometric Brownian Motion w.r.t.


0 0

0

0

0 0

( , ) ( , ) 10 log 10 log

( ) ,

for given ( ) and if ( , ) is Gaussian or fixed

( , ) : evolves like a B.M. with non-zero drift w.r.t.

( , ) 10 log 10 log

c c

c c

X t X t v v

t W W

t X t

X t

E X t PL d v v

20( , ) ( )Var X t t

Properties of spatial log-normal modelS(t,) = ekX(t,) => Using Ito’s differential rule

S(t,) = ekX(t,) : Geometric Brownian Motion w.r.t.


0

2

( , )0

( , )

,( , ), ,

( , ) 2 2

( , )

Solution by substitution: ( , );

kX t

X t

k tdS td k t dW

S t

S t e

S t e

Chapter 3: Spatial Log-Normal Model Simulations

• Time t: fixed• Snap-shot at propagation environment• {X(t,)}|t=fixed : increases logarithmically with d or • S(t,) = ekX(t,) : Log-Normal

Experimental Data (Pahlavan)

Chapter 3: Spatial Correlation of Log-Normal Model

Spatial correlation characteristics:Indicate what proportion of the environment remains

the same from one observation instant or location to the next, separated by the sampling interval.

Consider

Since the mobile is in motion it implies that the above correlation corresponds to the spatial correlation.

cov ( , ) ( , ) [ ( , )]

( , ) [ ( , )]

X t t t E X t E X t

X t t E X t t

Reported spatial correlation decreases exponentially with d

X2: covariance of power-loss process

d, t : distance, time between consecutive samples v: velocity of mobile Xc: density of propagation environment

Chapter 3: Experimental Correlation

2 2cov cov ( , ) cc v X td XX X Xt X t t t e e

Consider the following linear process


0

0

0

20

0

22 ( )( ) ( ) 2

( , ) ( , ) ( , ) ( , ) ,

( , ) 0; ( )


Consider the case: ( , ), ( , ) ( ), ( ) then

( )cov ( )

2 ( )

t

t t ttX

dX t t X t dt t dW t

X t N

W t N t t

t t

t e e e

0

0

0

2 ( )( )

22

2 ( )

1

( )Choose the variace of the initial condition: ( )

2 ( )

then cov ( )

( ) / : inversely proportional to density of the environment

t t

t

ttX

c

t e

v X

Since the mobile is in motion, covariance with respect to t spatial covariance

Identification of parameters {(), ()}

Use experimental correlation data identify (),From variance of initial condition and () identify (),

Note: variance of initial condition of power loss process increase with distance.

equivalent to:() increases or () decreases (denser environment)


Chapter 3: Dynamical Temporal Log-Normal Models

Sn(tm ,)

• {S(t,)}|=fixed S.D.E. w.r.t. t

Sn(tm-1,)

Transmitter

Receiver

• T-R separation distance d or fixed

Chapter 3: Dynamical Temporal Log-Normal Model

T-R separation distance or fixed:

( , ) Should vary around the mean predicted by

the spacial log normal model

log-normally distributed

( , ) power

fixed

fixed

d

S t

X t

loss: normally distributed

Variations are due to changes in the propagation

environment between the transmitter and the receiver

i.e. cars may obstruct the line of sight between the

transmitter and the receiver

Need specific S.D.E.s for {X(t,), S(t,)} where

{X(t,)}|=fixed => At every instant of time t, is Gaussian

{S(t,)}|=fixed => At every instant of time t, is Log-Normal

{(t,), (t,), (t,)}: model propagation environment


0

( , )

20

0

( , )

Choose: ( , ) mean-reverting linear S.D.E.

( , ) ( , ) ( , ) ( , ) ( , ) ,

( , ) [ ];


kX t

d t

S t e

X t

dX t t t X t dt t dW t

X t N PL d dB

W t N t t

Properties of mean-reverting process


( , ) ( , ) ( , ) ( , ) ( , ) ,

( , ) : models time-varying avg. power loss at distance

from transmitter: ( )[ ]( )

( , ) : speed of adjustment towards mean,

d


t

d PL d dB t

t

controls density of environment

Inversely proportional to deensity of prop. env.

( , ) : instantaneous variance or variability of processt

Properties of mean-reverting process


0 0

0

0

0 0

0

, , , ,

0

0 0 0

, , , ,

0

2 ,

( , )

, , , ( )

, , , , ( , )

At every instant of time ( , ) is Gaussian

( , ) , ,

( , )

tt t u t

t

t

t

tt t u t

t

t

X t e X e

u u du u dW u

t t u du X X t

X t

E X t e X e u u du

Var X t e

0 0

00

, 2 , , 2 2,tt u t

tte u du

S(t,) = ekX(t,) => Using Ito’s differential rule


02 ( , )

0

( , )

1( , ) ( , ) ( , ) ( , ) ln ( , )

1 ( , ) ( , ) , ( , )

2

Solution by substitution: ( , );

kX t

X t

dS t S t k t t S tk

k t dt k t dW t S t e

S t e

Illustration of mean reverting model

(t,) high: not-dense environment(t,) low: dense environment

Chapter 3: Temporal Log-Normal Model Simulations

low

high

Chapter 3: Dynamical Temporal-Spatial Log-Normal Model

vmT (t)

d

d(t)x

y

vmR (t)

(0,0)

n

Transmitter

Receiver

(t)

Propagation environment varies (t) Transmitter-Receiver relative motion d(t)

Chapter 3: Temporal-Spatial Log-Normal Model Sim.

0

0 0

20

Define average power path-loss ( , )

10 log ( )

( ): temporal variations of the environment

( , ) ( , ) ( , ) ( , ) ( , ) ,

( , ) [ ];

( , ) : Normally distri

d

d t

t PL d t dB

PL d dB d t d t

t


X t N PL d dB

X t

0

( , )

2 ( )

buted

( , ) : Log-normally distributed

( , ) ( , ) ( , )

cov ( )

kX t

ttX

S t e

X t X t E X t

t e

Chapter 3: Temporal-Spatial Log-Normal Model Sim.

n (t)

vm (t)

d

Transmitter

d(t)

Receiver

d2

d3

d1

() : inversely proportional to the density of the propagation environment


M. Gudamson. Correlation model for shadow fading in mobile radio systems. Electronics Letters, 27(23):2145-2146, 1991.D. Giancristofaro. Correlation model for shadow fading in mobile radio channels. Electronics Letters, 32(11):956-958, 1996.F. Graziosi, R. Tafazolli. Correlation model for shadow fading in land-mobile satellite systems. Electronics Letters, 33(15):1287-1288, 1997.A.J. Coulson, G. Williamson, R.G. Vaughan. A statistical basis for log-normal shadowing effects in multipath fading channels. IEEE Transactions in Communications, 46(4):494-502, 1998.R.S. Kennedy. Fading Dispersive Communication Channels. Wiley Interscience, 1969.S.R. Seshardi. Fundamentals of Transmission Lines and Electromagnetic Fields. Addison-Wesley, 1971.L. Arnold. Stochastic Differential Applications: Theory and Applications. Wiley Interscience, New York 1971.D. Parsons. The mobile radio propagation channel. John Wiley & Sons, New York, 1992.

Chapter 3: References

C.D. Charalambous, N. Menemenlis. Stochastic models for long-term multipath fading channels. Proceedings of 38th IEEE Conference on Decision and Control, 5:4947-4952, December 1999.C.D. Charalambous, N. Menemenlis. General non-stationary models for short-term and long-term fading channels. EUROCOMM 2000, pp 142-149, April 2000.C.D. Charalambous, N. Menemenlis. Dynamical spatial log-normal shadowing models for mobile communications. Proceedings of XXVIIth URSI General Assembly, Maastricht, August 2002.

Chapter 3: References

Chapter 3: Log-Normal Shadowing Models Motivation for dynamical channel models Log-Normal dynamical...

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