EE 447
Mobile and Wireless Communications
Fall 2006
Outdoor Propagation Models
Richard S. Wolff, Ph. D.
rwolff@montana.edu
406 994 7172
509 Cobleigh Hall
Fall 2006
Small scale and large scale fading
Fall 2006
Free space propagation
• Friis free space equation (fancy way of
saying that energy is conserved!)
Gt Gr 
Pr  Pt
2 2
(4 ) d L
2
Fall 2006
Free Space Path Loss
The total received power
Pr  Ar pr
Gr 
Antenna effective area
4Ar
2
  
Pr  
 Pt Gt Gr
 4r 
2
  
Pr (dB)  20log
  Pt  Gt  Gr
 4r 
Fall 2006
Free Space Path Loss
Free Space Path Loss
 4r 
Lp  

  
2
  
Lp (dB)  20log

 4r 
General Path Loss formula
d 
Lp (d )  Lp (d 0 )  10 log   S
 d0 
where Lp(do) is path loss at the reference distance d0
loss exponent γ is the slope of the average increase in path loss with
dB-distance,
Shadowing S denotes a zero-mean Gaussian random variable with
standard deviation σ.
Fall 2006
A few conditions and useful terms
• Applies to received power in far-field
(Fraunhofer region):
d  df 
2D

2
D = largest dimension of
transmitting antenna
•Received power P0 reference point d0
2
 d0 
Pr (d )  Pr (d 0 )
 , d  d0  d f
 d 
Fall 2006
Log notation frequently used
 Pr (d 0 ) 
 d0 
Pr (d )dBm  10 log

20
log
 

 0.001W 
d 
where Pr (d 0 ) is in units of watts
 Pr ( d 0 ) 
P at hloss in dB  10 log

 Pr ( d ) 
Fall 2006
Free space path loss: practical application
Convenient tool:
http://www.terabeam.com/support/calculations/free-spaceloss.php
Fall 2006
Typical large-scale path loss exponents
Fall 2006
Measured large-scale path loss
Fall 2006
Basic propagation mechanisms
• Reflection
– Dimensions of reflector are large compared to 
– Applies to surface of earth, buildings, etc.
• Diffraction
– Obstacle with sharp edges in path between T and R
(could be totally blocking the path)
– Depends on geometry of object, , phase, polarization,
etc.
• Scattering
– Objects small compared to  in path between T and R
– Caused by rough surfaces, foliage, etc.
• Absorption
– Attenuation by solid materials (walls, etc.)
– Rain
Fall 2006
Reflection from smooth surface
Fall 2006
Typical electromagnetic properties of
materials
Fall 2006
Reflection coefficients for parallel and
perpendicular polarized fields
Reflected wave will 100% polarized perpendicular to plane of
Incidence when qi is equal to the Brewster angle
Fall 2006
Classical 2-ray ground bounce model
Fall 2006
Path loss over reflecting surface


Es  1  re jq E  Ejq
Direct path
r  1
ht
hr
indirect path
q
r
is reflection coefficient
Is phase difference between
direct path and indirect path
 4 
q   ht hr
 r 
2
   PtGtGr
Pr  
j

q

 4r  Lo
2
2
 h h  PG G
Pr   t 2 r  t t r
Lo
 r 
 ht hr 
Pr (dB)  20log 2   Pt  Gt  Gr  Lo
 r 
Fall 2006
Propagation near the earth’s surface
W h en d 
ht hr
Pr ( d )  Pt Gt Gr
ht2 hr2
d4
Note fourth power dependence with distance!
Pathloss (dB)  40 log d 10 log Gt 10 log Gr  20 log hr  20 log ht
Fall 2006
Received signal power as a function of distance
Fall 2006
Effect of antenna height on received power
Fall 2006
Diffraction
• Allows radio waves to propagate over the
horizon
• Radio waves can propagate into shadowed
(obstructed) areas
• Governed by Huygen’s principle:
– all points on a wave front can be considered as
point sources to produce secondary wavelets
– Secondary wavelets combine (vector sum) to form
a new wave in the direction of propagation
Fall 2006
Huygen’s wavelet approach
Wavelets form
on knife edge,
transmit a new
wave into
shadowed zone
Fall 2006
Fresnel zones: locus of points of equal path
length (phase) relative to direct path
r n
n  d1 d 2
fo r d1 , d 2  r n
d1  d 2
ex cess p at h len gt h  n /2
Fall 2006
Examples of Fresnel diffraction
geometries
Figure 4.12 Illustration of Fresnel zones for different knife-edge diffraction scenarios.
Fall 2006
Fresnel zone clearance: practical application
Fresnel Zone
1. Typically, 20% Fresnel Zone blockage introduces little signal
loss to the link. Beyond 40% blockage, signal loss will
become significant
http://www.terabeam.com/support/calculations/fresnel-zone.php
Fall 2006
Effect of obstructions: treat with knifeedge diffraction
Fall 2006
Knife-edge diffraction loss
2(d1  d 2 )
 h
d1d 2
Fall 2006
Multiple knife-edge diffraction –
used to calculate propagation in rough terrain
Fall 2006
Propagation modeling for diffraction - RF
Propcalc
Fall 2006
Scattering
• Important when the dimensions of
obstructions or surface features are small
relative to 

• Rayleigh criterion: hc 
8 sin i
A surface is smooth if the peak to peak protuberances
Are less than hc
Fall 2006
Measured results: scattering from a stone
wall
Fall 2006
Absorption: attenuation caused by rain
Fall 2006
Log-normal (Gaussian) shadowing
• Loss along two different paths with same d
can vary greatly
• Measured signals with same d can deviate
from average given by path loss equation
• Measurements show that PL(d ) is random
and distributed log-normally (normal in
dB) about the mean, PL(d )
Fall 2006
Log-normal shadowing
d
PL(d )[dB]  PL(d )  X   PL(d 0 )  10n log   X 
 d0 
WhereX is a zero- mean Gaussian distributed
random variable(in dB)
with standarddeviation in dB
Pr (d )[dBm]  Pt [dBm]  PL(d )[dB]
(antennagains included in PL(d))
Fall 2006
Gaussian distribution
PL(d ) and Pr (d ) are random variableswith normaldistribution
arounda distant- dependentmean.
 y2
1
2 2
p( y) 
e
 2
Normalized Gaussian distribution, zero mean
Fall 2006
Q, erf and erfc functions
y0

1
 y 2 /( 2 2 )
Pr( y  y0 )  
e
dy
2
y0 
Fall 2006
Q, erf and erfc functions
Define z  y /  ,

Q( z ) 

z
1
 y2 / 2
e
dy
2
Note:
Q(-z) = 1-Q(z)
Q(0)= 1/2
If the distribution has a non-zero mean m,
z =(y-m)/
Fall 2006
Q, erf and erfc functions
Define theerrorfunct ion, erf ( z )
erf ( z ) 
z
2


e
 y2 / 2
dy
0
Define thecomplimentary errorfunction,erfc( z )
erfc( z ) 
2



e
 y2 / 2
dy
z
Note erfc(z) = 1-erf(z)
Fall 2006
Q, erf and erfc functions
Some useful relationships:
1
z  1
z
Q( z )  1  erf ( )  erfc( )
2
2  2
2
erfc( z )  2Q ( 2 z )
erf ( z )  1  2Q ( 2 2 z )
Fall 2006
Log-normal shadowing
Probability that the received signal level (in dB) will exceed a
level :
   Pr (d ) 

Pr[Pr (d )   ]  Q




Probability that the received signal level (in dB) will be less than
a level :
 Pr (d )  
Pr[Pr (d )   ]  Q






Fall 2006
Log-normal shadowing - example
Suppose at a distance d, the mean received power
level P r(d) is -70dBm and the standard deviation  is
10 dB.
Find the probability that the received signal level (in
dB) will exceed a level  60dBm:
Pr[P r(d)>60]=Q{(-60+70)/10}=Q(1)=1/2erfc(1/1.414)
Pr[P r(d)> -60]=1/2{1-erf(.707)}
Pr[P r(d)> -60]= .16
Fall 2006
Multiple received rays due to scattering
Ricean: Nirect and scattered rays combine at receiver
Rayleigh: No direct ray (only scattered rays reach receiver)
Fall 2006
Rayleigh distribution

f ( )  2 e
r
2

2
2
r
,  0
Fall 2006
Comparison of Rayleigh and Ricean distributions
a 2  A2
a ( 2 R2 ) aA
f ( a ) 
e
I0 ( 2 )
2
R

R
A represents the power in the direct signal
Fall 2006