1. 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

2. Chapter 3: Motivation for Dynamical Channel Models
Short-term
Fading
Varying environment
Obstacles on/off
Area 2
Area 1
Transmitter
Log-normal
Shadowing
Mobiles move

3. Chapter 3: Motivation for Dynamical Channel Models
Complex low-pass representation of impulse response:

4. 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

5. Chapter 3: Log-Normal Shadowing Model
Transmitter
tn,1
Receiver
tk,1
t or d
tn,3
tn,2
tk,2
tk,3
tk,4
one subpath
LOS
path k
path n
d(t)
vmR(t) qn

6. Chapter 3: Static Log-Normal Model

7. Chapter 3: Dynamical Log-Normal Model
t d : time-delay equivalent to distance d=vct
speed of light
S(t,t) and X (t,t) : random processes
modeled using S.D.E.’s
with respect to t and t

8. Chapter 3: Dynamical Spatial Log-Normal Model*
S(t, t5)
Receiver *
S(t, t2) *
S(t, t4)
S(t, t3) *
S(t, t1) *
Time, t: fixed (snap shot at propagation
environment)
{S(t,t)}|t=fixed S.D.E. w.r.t. t
Transmitter

9. Chapter 3: Dynamical Spatial Log-Normal Model

10. Chapter 3: Dynamical Spatial Log-Normal Model
Need specific S.D.E.s for {X(t,t), S(t,t)} where
{X(t,t)}|t=fixed => At every t, B.M. with non-zero drift
{S(t,t)}| t =fixed => At every t, G.B.M.
a : models loss characteristics of propagation environment

11. Chapter 3: Dynamical Spatial Log-Normal Model
Properties of spatial log-normal model
S(t,t) = ekX(t,t) : Geometric Brownian Motion w.r.t. t

12. Chapter 3: Dynamical Spatial Log-Normal Model
Properties of spatial log-normal model
S(t,t) = ekX(t,t) => Using Ito’s differential rule
S(t,t) = ekX(t,t) : Geometric Brownian Motion w.r.t. t

13. Chapter 3: Spatial Log-Normal Model Simulations
Experimental Data (Pahlavan)
Time t : fixed
Snap-shot at propagation environment
{X(t,t)}|t=fixed : increases logarithmically with d or t
S(t,t) = ekX(t,t) : Log-Normal

14. 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.

15. Chapter 3: Experimental Correlation
Reported spatial correlation decreases exponentially with d
sX2: covariance of power-loss process
Dd, Dt : distance, time between consecutive samples
v: velocity of mobile
Xc: density of propagation environment

16. Chapter 3: Spatial Correlation of Log-Normal Model
Consider the following linear process

17. Chapter 3: Spatial Correlation of Log-Normal Model
Since the mobile is in motion, covariance with respect to t  spatial covariance
Identification of parameters {b(t), d(t)}
Use experimental correlation data  identify b(t),
From variance of initial condition and b(t)  identify d(t),
Note: variance of initial condition of power loss process increase with distance.
equivalent to:
d(t) increases or b(t) decreases (denser environment)

18. Chapter 3: Dynamical Temporal Log-Normal Models
Transmitter
Receiver
T-R separation distance d or t fixed
Sn(tm-1,t) *
Sn(tm ,t) *
{S(t,t)}|t=fixed S.D.E. w.r.t. t

19. Chapter 3: Dynamical Temporal Log-Normal Model

20. Chapter 3: Dynamical Temporal Log-Normal Model
Need specific S.D.E.s for {X(t,t), S(t,t)} where
{X(t,t)}|t=fixed => At every instant of time t, is Gaussian
{S(t,t)}|t=fixed => At every instant of time t, is Log-Normal
{b(t,t), g (t,t), d (t,t)}: model propagation environment

21. Chapter 3: Dynamical Temporal Log-Normal Model
Properties of mean-reverting process

22. Chapter 3: Dynamical Temporal Log-Normal Model
Properties of mean-reverting process

23. Chapter 3: Dynamical Temporal Log-Normal Model
S(t,t) = ekX(t,t) => Using Ito’s differential rule

24. Chapter 3: Temporal Log-Normal Model Simulations
Illustration of mean reverting model
b(t,t) high: not-dense environment
b(t,t) low: dense environment
low
high

25. Chapter 3: Dynamical Temporal-Spatial Log-Normal Model
vmT (t) d
d(t) x y
vmR (t)
(0,0) qn
Transmitter
Receiver
x(t)
Propagation environment varies x(t)
Transmitter-Receiver relative motion d(t)

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

27. Chapter 3: Temporal-Spatial Log-Normal Model Sim.
qn (t)
vm (t) d
Transmitter
d(t)
Receiver d2 d3 d1

28. Chapter 3: Spatial Correlation of Log-Normal Model
b(t) : inversely proportional to the density of the propagation environment

29. Chapter 3: References
M. Gudamson. Correlation model for shadow fading in mobile radio systems. Electronics Letters, 27(23): , 1991.
D. Giancristofaro. Correlation model for shadow fading in mobile radio channels. Electronics Letters, 32(11): , 1996.
F. Graziosi, R. Tafazolli. Correlation model for shadow fading in land-mobile satellite systems. Electronics Letters, 33(15): , 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): , 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.

30. 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: , December 1999.
C.D. Charalambous, N. Menemenlis. General non-stationary models for short-term and long-term fading channels. EUROCOMM 2000, pp , 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.