Transcript 1-erf(z)
EE 447
Mobile and Wireless Communications
Fall 2006
Outdoor Propagation Models
Richard S. Wolff, Ph. D.
[email protected]
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
4Ar
2
Pr
Pt Gt Gr
4r
2
Pr (dB) 20log
Pt Gt Gr
4r
Fall 2006
Free Space Path Loss
Free Space Path Loss
4r
Lp
2
Lp (dB) 20log
4r
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
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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 jq E Ejq
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
4r 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
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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
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Absorption: attenuation caused by rain
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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 )
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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))
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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
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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)
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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