Wireless Communication
Channels: Large-Scale Pathloss
Diffraction
Diffraction
Diffraction allows radio signals to propagate behind
obstacles between a transmitter and a receiver
ht
hr
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Huygen’s Principle & Diffraction
All points on a wavefront
can be considered as
point sources for the
production of secondary
wavelets. These wavelets
combine to produce a
new wavefront in the
direction of propagation.
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Knife-Edge Diffraction Geometry
α
h
Tx
Rx
γ
β
d2
d1
hobs
ht
hr
2
  d 12  h 2  d 2 2  h 2  d 1  d 2  d 1
h 2  d1  d 2 
 
 h  d1 ,d 2
2  d1d 2 
2
h
 h 
1     d 2 1     d1  d 2
 d1 
 d2 
where 1  x  1 
x
for x
2
<<
1
Δ: Excess Path Length (Difference between Diffracted Path and Direct Path)
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Fresnel Zone Diffraction Parameter (ν)
Ф: Phase Difference between
Diffracted Path and Direct Path)
2 h 2



 2
 d1  d 2 


d
d
 1 2 
Assume tan   
tan   
2
    
h
h h  d1  d 2 


d1 d 2
d1d 2
Fresnel Zone Diffraction
Parameter (ν)

   2
2
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 h
2  d1  d 2 
 d1d 2
2d1d 2

  d1  d 2 
 ν2=2, 6, 10 … corresponds to destructive
interference between direct and diffracted paths
 ν2=4, 8, 12 … corresponds to constructive
interference between direct and diffracted paths
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Fresnel Zones
Fresnel Zones:
Successive regions
where secondary waves
have a path length from
the transmitter to receiver
which is nλ/2 greater than
the total path length of a
line-of-sight path
From “Wireless Communications: Principles and Practice” T.S. Rappaport
n rn2  d1  d 2 


 rn 
2
2 d1d 2
n d1d 2
 d1  d 2 
rn: Radius of the nth Fresnel Zone
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Diffraction Loss
Diffraction Loss occurs from the blockage of secondary waves such
that only a portion of the energy is diffracted around the obstacle
Tx
l1
l2
Rx
d
ht
hr
First Fresnel Zone Points  l1+l2-d =(λ/2)
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Diffraction Loss
Diffraction Loss occurs from the blockage of secondary waves such
that only a portion of the energy is diffracted around the obstacle
Tx
l2
l1
Rx
d
ht
hr
First Fresnel Zone Points  l1+l2-d =(λ/2)
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Diffraction Loss
Diffraction Loss occurs from the blockage of secondary waves such
that only a portion of the energy is diffracted around the obstacle
Tx
l1
l2 Rx
d
ht
hr
First Fresnel Zone Points  l1+l2-d =(λ/2)
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Diffraction Loss
Diffraction Loss occurs from the blockage of secondary waves such
that only a portion of the energy is diffracted around the obstacle
Tx
Rx
l1
d
ht
l2
hr
First Fresnel Zone Points  l1+l2-d =(λ/2)
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Diffraction Loss
Diffraction Loss occurs from the blockage of secondary waves such
that only a portion of the energy is diffracted around the obstacle
Tx
Rx
l1
d
l2
ht
hr
First Fresnel Zone Points  l1+l2-d =(λ/2)
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Diffraction Loss
Diffraction Loss occurs from the blockage of secondary waves such
that only a portion of the energy is diffracted around the obstacle
Tx
l2
l1
Rx
d
ht
hr
Second Fresnel Zone Points  l1+l2-d = λ
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Diffraction Loss
Diffraction Loss occurs from the blockage of secondary waves such
that only a portion of the energy is diffracted around the obstacle
Tx
l2
l1
Rx
d
ht
hr
Third Fresnel Zone Points  l1+l2-d = (3λ/2)
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Knife-Edge Diffraction Scenarios
Tx
Rx
h (-ve)
d1
ht
d2
hr
 h & ν are –ve
 Relative Low Diffraction Loss
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Knife-Edge Diffraction Scenarios
Tx
Rx
h =0
d1
ht
d2
hr
 h =0
 Diffraction Loss = 0.5
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Knife-Edge Diffraction Scenarios
Tx
Rx
h (+ve)
d1
d2
ht
hr
 h & ν are +ve
 Relatively High Diffraction Loss
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Knife-Edge Diffraction Model
The field strength at
point Rx located in the
shadowed region is a
vector sum of the fields
due to all of the
secondary Huygen’s
sources in the plane
above the knife-edge
Electric Field Strength, Ed, of a Knife-Edge Diffracted Wave is given By:
E0: Free-Space Field Strength in absence of Ground Reflection and Knife-Edge Diffraction
F(ν) is called the complex Fresnel Integral
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Diffraction Gain
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Diffraction Gain Approximation
𝐺𝑑 𝑑𝐵 = 0
𝜈 ≤ −1
𝐺𝑑 𝑑𝐵 = 20 log 0.5 − 0.62𝜈
−1≤𝜈 ≤0
𝐺𝑑 𝑑𝐵 = 20 log 0.5𝑒𝑥𝑝 −0.95𝜈
𝐺𝑑 𝑑𝐵 = 20 log 0.4 − 0.1184 − 0.38 − 0.1𝜈
𝐺𝑑 𝑑𝐵 = 20 log
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0.225
𝜈
0≤𝜈≤1
2
1 ≤ 𝜈 ≤ 2.4
𝜈 > 2.4
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Multiple Knife-Edge Diffraction
Tx
Rx
d
ht


hr
In the practical situations, especially in hilly terrain, the propagation
path may consist of more than one obstruction.
Optimistic solution (by Bullington): The series of obstacles are
replaced by a single equivalent obstacle so that the path loss can be
obtained using single knife-edge diffraction models.
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Scattering
Scattering
The actual received signal in a mobile radio environment
is often stronger than what is predicted by reflection and
diffraction
Reason:
When a radio wave impinges on a rough surface, the
reflected energy is spread in all directions due to
scattering
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Reflection Vs Scattering
 Reflection: Flat surfaces that have much larger dimension
than wavelength
 Scattering: When the medium consists of objects with
dimensions that are small compared to the wavelength
Testing Surface Roughness using Rayleigh Criterion
hc : Critical Height of Surface Protuberance
Θi : Angle of Incidence
λ : Wavelength
Smooth Surface  Minimum to maximum protuberance h is less than hc
Rough Surface  Minimum to maximum protuberance h is greater than hc
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Reflection Coefficient for Rough Surfaces
Γrough: Reflection Coefficient for Rough Surfaces
Γ
: Reflection Coefficient for Smooth Surfaces
ρS : Scattering Loss Factor
σh : Standard deviation of the surface height h about the mean surface height
I0(.) : Bessel Function of the first kind and zero order
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