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Propagation Mechanisms 1
1. Propagation Mechanisms
Contents:
• The main propagation mechanisms• Point sources in free-space• Complex representation of waves• Polarization• Electric field pattern• Antenna characteristics• Free-space propagation loss• Uniform plane wave in free-space
Propagation Mechanisms 2
1. Propagation Mechanisms
Contents (cont’d):
• Uniform plane wave in a lossy medium• Reflection and transmission• Propagation over flat earth• Scattering from rough surfaces• Diffraction over/around obstacles• The uniform geometrical theory of diffraction
Propagation Mechanisms 3
Multipath propagation:
Diffraction
Reflections
Line of sight
path
Shadowing
Scattering
Uplink
Downlink
The main propagation mechanisms
Propagation Mechanisms 4
Spherical coordinate system:
x1
x2
x3
Ωar Ω( )
S
O
ah Ω( )
φ
θ
1
Tx
av Ω( )
T Ω( ) Definitions and remarks:
• : Sphere of radius 1centred at .
• : Direction
• Azimuthal and coelevationangle, resp.
• : Outward unit normal to at
• : Tangent plane to at
Transverse plane to
• :Unit vectors pointing in thedirection of increment ofand , resp.
SO
Ω S∈φ θ,
ar Ω( ) S Ω
T Ω( ) S Ωar Ω( )
ah Ω( ) av Ω( ), T Ω( )∈
φθ
Point sources in free space
Propagation Mechanisms 5
Vertically polarized isotropic point source:
d
GT
PT
d0
av Ω( )ar Ω( )
ah Ω( )
E v Ω d t, ,( )
H h Ω d t, ,( )
d
λ
Polarization
planeΩ
Far zone
region
Sphere of
equal phase
S
Near zone
region
Point sources in free space
Propagation Mechanisms 6
Vertically polarized isotropic point source (cont’d):
Field equations:
E v Ω d t, ,( ) av Ω( )E v d( ) 2πft kd ϕv+–( ) Electric field [V/m]cos=
H h Ω d t, ,( ) ah Ω( )H h d( ) 2πft kd ϕv+–( ) Magnetic field [A/m]cos=
Point sources in free space
Propagation Mechanisms 7
Characteristics of the radiated wave:• The surfaces of equal phase are spheres
-> Spherical wave
• The wave amplitude on the equal phase surfaces is constant-> Uniform wave
• The electric and magnetic fields are orthogonal and belong to .-> Transverse electromagnetic (TEM) wave
•
T Ω( )
Z0H h d( ) E v d( )=
av Ω( )E v d( )
ar Ω( )
ah Ω( )H h d( )
Point sources in free space
Propagation Mechanisms 8
Electrical characteristics of free-space:
• Permittivity
• Permeability
• Intrinsic impedance
Wave constants:
• Frequency [Hz]
• Propagation constant
• Phase velocity [m/s]
• Wavelength [m]
ε0 8.86 1012–
[F/m]⋅≈
μ0 4π 109–
[H/m]⋅≈
Z0
μ0
ε0
-----⎝ ⎠⎛ ⎞ 1 2⁄
≡ 120π Ω[ ]≈
f
k 2πf μ0ε0( )1 2⁄≡ 2πλ
------= m1–[ ]
c μ0ε0( ) 1 2⁄–≡ 3 108
m/s[ ]⋅≈
λ c f⁄≡
Point sources in free space
Propagation Mechanisms 9
Wave’s Poynting vector:
Radiated power:
: sphere of radius .
av Ω( )E v d( )
ah Ω( )H h d( )xxxxxx
⎧ ⎨ ⎩
Sr d( )
Sr Ω d,( ) av Ω( )E v d( ) ah Ω( )H h d( )× ar Ω( )E v d( )2
Z0
------------------= =
ar Ω( )Sr d( )
PT1
2--- Sr Ω d,( ) Sdd⟨ | ⟩
Sd
∫1
2--- Sr d( )d2 Ωd
S∫ 2πd
2Sr d( ) constant= = = =
Lossless medium (✥) Sd d
Point sources in free space
Propagation Mechanisms 10
Vertically polarized isotropic point source (cont’d):
It follows from (✥) that:
andxxxx
⎧ ⎨ ⎩
E v
E v d( )Z0PT
2π-------------
1
d---⋅ 60PT
1
d---⋅ E v
1
d---⋅=≈= H h d( )
E v
Z0
-------1
d---⋅=
E v Ω d t, ,( ) av Ω( )E v
d------- 2πft kd ϕv+–( )cos=
H h Ω d t, ,( ) ah Ω( )E v
Z0d--------- 2πft kd ϕv+–( )cos=
Point sources in free space
Propagation Mechanisms 11
Horizontally polarized isotropic point source:
GT
PT
d
d0
av Ω( )ar Ω( )
ah Ω( )
d
λ
E h Ω d t, ,( )
H v Ω d t, ,( )
Polarization
plane
Far zone
region
Ω
S
Near zone
region
Point sources in free space
Propagation Mechanisms 12
Horizontally polarized isotropic point source (cont’d):
Isotropic point source:
Usually, the wave radiated by a source is the superposition of a vertically
and of a horizontally polarized spherical wave:
Henceforth, we only consider the electric field of the waves.
E h Ω d t, ,( ) ah Ω( )E h
d------- 2πft kd ϕh+–( )cos=
H v Ω d t, ,( ) av Ω( )E h
Z0d--------- 2πft kd ϕh+–( )cos–=
E Ω d t, ,( ) E h Ω d t, ,( ) E v Ω d t, ,( )+=
Point sources in free space
Propagation Mechanisms 13
Complex representation of spherical waves:
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
E h d t,( )
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
E v d t,( )
xxxxxxx
⎧ ⎪ ⎨ ⎪ ⎩
xxxxxxx
⎫ ⎪ ⎬ ⎪ ⎭
xxxxxxxxxxxxxxx
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
Eh d( )
xxxxxxxxxxxxxxxx⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭
Ev d( )
E h Ω d t, ,( ) ah Ω( )ReE h
d------- jϕh( )exp jkd–( )exp j2πft( )exp⋅
⎩ ⎭⎨ ⎬⎧ ⎫
=
[Complex] electric fields
E v Ω d t, ,( ) av Ω( )ReE v
d------- jϕv( )exp jkd–( )exp j2πft( )exp⋅
⎩ ⎭⎨ ⎬⎧ ⎫
=
Time-dependent
part
Complex representation of waves
Propagation Mechanisms 14
Concise notation for spherical waves:
E h
d------- jϕh( )exp jkd–( )exp
E v
d------- jϕv( )exp jkd–( )exp
E h jϕh( )exp⋅
E v jϕv( )exp⋅
1
d--- jkd–( )exp⋅ ⋅==
E d( )Eh d( )
Ev d( )≡
E
xxxxxxxxxxx
⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭
and are complex 2-dim. vectorsE d( ) E
E d( ) E1
d--- jkd–( )exp⋅ ⋅=
Complex representation of waves
Propagation Mechanisms 15
Polarization of the electric field vector:
E v d t,( )
t
E h d t,( ) E Ω d t, ,( )
T Ω( )ar Ω( )
ah Ω( )
av Ω( )
Polarization
Propagation Mechanisms 16
Polarization of the electric field vector:• : linearly polarized wave
• and : circularly polarized wave
ϕh ϕv= T Ω( )
Trajectory of in
as a function ofwith fixed.
E d t,( )T Ω( ) t
d
E h
E v
E Ω d t, ,( ) ar Ω( ) E h d t,( )
E v d t,( )
ϕh ϕv– π 2⁄( ) mod π= E h E v=
CW RH ϕh, ϕv– 3π 2⁄ mod 2π( )=
T Ω( )E Ω d t, ,( )
ar Ω( )E h
E v
CCW LH ϕ, h ϕv– π 2⁄ mod 2π( )=E h d t,( )
E v d t,( )
Polarization
Propagation Mechanisms 17
Polarization of the electric field vector:
• Otherwise the wave is said to be elliptically polarized
T Ω( )
E h
E v
E Ω d t, ,( )
ar Ω( )
E v d t,( )
E h d t,( )
Polarization
Propagation Mechanisms 18
Anisotropic sources:Usually, the source does not radiate isotropically:
depend on , i.e.
•
•
[Normalized] electric field pattern of a source:
E h v ϕh v,⇒ Ω
E h v E h v Ω( )→ ϕh v, ϕh v Ω( )→
E h v ϕh v( )exp⋅ Eh v Eh v Ω( )→=
f Ω( )f h Ω( )
f v Ω( )≡
f h v Ω( )Eh v Ω( )
E h v
--------------------≡
E h v maxΩ Eh v Ω( ){ }≡⎩⎪⎪⎨⎪⎪⎧
with
f h v Ω( )
1
Electric field pattern
Propagation Mechanisms 19
Electric field pattern of a vertical -dipole antenna:λ 2⁄
f v φ θ,( )
x1
x2
x3
φ
θ
π2--- θ( )coscos
θ( )sin-----------------------------------=
f h Ω( ) 0=
f v Ω( ) f v φ θ,( )=
Electric field pattern
Propagation Mechanisms 20
Spherical wave radiated by an anisotropic source:
xxxxxxx
⎧ ⎪ ⎨ ⎪ ⎩
E Ω( )
E Ω d,( )f h Ω( )E h
f v Ω( )E v
1
d--- jkd–( )exp⋅ ⋅=
Electric field pattern
Propagation Mechanisms 21
Far-zone region:
Electric field pattern:As already discussed.
[Normalized] power pattern of a linear [polarized] wave:
d d0 2D
2
λ------≡≥
: maximal dimension ofthe antenna in meter.
D λ≥
p Ω( ) S Ω d,( )maxΩS Ω d,( )---------------------------------≡
f Ω( ) 2E 2
maxΩ f Ω( ) 2E 2{ }------------------------------------------------= f Ω( ) 2
=
f Ω( ) f h v Ω( )≡
E Ω( ) E h v Ω( )≡
p Ω( ) ph v Ω( )≡
Antenna characteristics
Propagation Mechanisms 22
Gain of a [lossless] linear antenna:
(same input power)Gmax. radiated power/m
2by the antenna
radiated power/m2by an isotropic antenna
------------------------------------------------------------------------------------------------------≡
maxΩ S d Ω,( ){ }
Siso
d( )----------------------------------------=
S d Ω,( )maxΩ S d Ω,( ){ }
Siso
d( )
Antenna characteristics
Propagation Mechanisms 23
Gain of a [lossless] linear antenna (cont’d):
GmaxΩ S d Ω,( ){ }
Siso
d( )----------------------------------------=
Siso
d( )PT
2πd2
------------= S d Ω,( ) f Ω( ) 2E 2
Z0d2
----------------------------=
G 2πmaxΩ f Ω( ) 2E 2{ }
Z0PT------------------------------------------------⋅ 2π E 2
Z0PT-------------⋅= =
Antenna characteristics
Propagation Mechanisms 24
Effective area:
Received power:
[W]
Aλ2
4π------G= m
2[ ] AG----
λ2
4π------=
PR1
2---SA=
is the direction of maximumpower radiation, i.e.Ω
p Ω( ) 1= G p Ω( ),
PR
A
S S ar Ω'( )=
ar Ω( )
Ω Ω'–=
Antenna characteristics
Propagation Mechanisms 25
Isotropic antenna:
• Gain:
• Effective area:
Linear antenna:
• Power pattern:
• Effective area:
• Gain:
Spherical wave radiated by an linear antenna:
Giso
1=
Aiso λ2
4π( )⁄=
p Ω( ) f Ω( ) 2=
A λ2G( ) 4π( )⁄=
G 2π E 2Z0PT( )⁄⋅=
E Ω d,( ) 60GPT f Ω( ) 1
d--- jkd–( )exp⋅ ⋅=
Antenna characteristics
Propagation Mechanisms 26
Free-space transmission formula:
Free-space transmission loss [isotropic antennas]:
PR
PT-------
λ4πd----------⎝ ⎠⎛ ⎞ 2
GT GR=
TxRx
LFS 10 PT PR⁄( )log≡ 32.4 20 d [km]( )log 20 f [MHz]( ) [dB]log+ +=
LF
S–
dB
[]
d( )log f( )log
Slope: -20 dB/decade
Free-space propagation loss
Propagation Mechanisms 27
Approximation of a spherical wave by a plane wave:
d
E v Ω0 d t, ,( )
H h Ω0 d t, ,( )
d
λ
r0
av Ω0( )
ar Ω0( )
ah Ω0( )
r
r0 r+
Vertical polarization
O'
New reference point
Uniform plane waves in free-space
Propagation Mechanisms 28
Approximation of a spherical wave by a plane wave (cont’d):
Electric field at :
Approximations for :
1. , where
2. , where is the direction toward , i.e. .
is the wave’s propagation vector.
3.
r
E r( ) E Ω( ) 1
r0 r+------------------ jk r0 r+–( )exp⋅ ⋅≡
r0 r»
1
r0 r+------------------
1
r0
---------≈ 1
d0
-----= d0 r0≡
k r0 r+ kd0 k Ω0( ) r⟨ | ⟩+≈ Ω0 O ar Ω0( ) 1
d0
-----r0=
k Ω0( ) k ar Ω0( )≡
E Ω( ) E Ω0( )≈
Uniform plane waves in free-space
Propagation Mechanisms 29
Approximation of a spherical wave by a plane wave:
Equation of a uniform plane wave propagating in free-space:
(✮) are the solutions of the Maxwell equations in source-free free-space.
1. 2.+( ) E r( ) E Ω0( ) 1
d0
----- jkd0–( )exp j k Ω0( ) r⟨ | ⟩–( )exp⋅ ⋅ ⋅≈⇒
: Electric field atE r0
xxxxxxxxxxxxxxxxx
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
E r( ) E j k r⟨ | ⟩–( )exp⋅=
H r( ) EZ0
------ j k r⟨ | ⟩–( )exp⋅=(✮)k k Ω0( ) 2π
λ------ ar Ω0( )= =
Uniform plane waves in free-space
Propagation Mechanisms 30
Planes of equal phase:
O
ar Ω0( )
r
ar Ω0( ) x⟨ | ⟩ λ
k r⟨ | ⟩ cons ttan= ar Ω0( ) r⟨ | ⟩ r α( )cos=⇔ cons ttan=
k k Ω0( ) 2πλ
------ ar Ω0( )= =
α
Planes of equal phase
Uniform plane waves in free-space
Propagation Mechanisms 31
Characteristics of a lossy material:
• Permeability [H/m]
• Permittivity [F/m]
• Conductivity [S/m]
Equivalent characterization of a lossy material:• Permeability [H/m]
• Effective permittivity [F/m](with )
Comments:•We shall retain the symbol for the effective permittivity.
•Henceforth, we only consider non-magnetic material:
μεκ
μ
εeff ε jκ
2πf---------–=
ε κ, 0≥
εμ μ0≈
Uniform plane waves in a lossy medium
Propagation Mechanisms 32
Secondary constants of the medium:
• Intrinsic impedance [ ]
• Propagation constantwhere
(✮) are still the solutions of the Maxwell equations in an source-free lossymedium:
Zμε---⎝ ⎠⎛ ⎞ 1 2⁄
≡ Ω
k 2πf με( )1 2⁄k' jk''–= = m
1–[ ]k' k'', 0≥( )
xxxxxxxxxxxxxxx
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩Attenuation in the direc-
tion of propagation is now
complex
Z
E r( ) E k'' ar Ω0( ) r⟨ | ⟩–( )exp jk' ar Ω0( ) r⟨ | ⟩–( )exp⋅ ⋅=
H r( ) E1
Z--- k'' ar Ω0( ) r⟨ | ⟩–( )exp jk' ar Ω0( ) r⟨ | ⟩–( )exp⋅ ⋅ ⋅=
Uniform plane waves in a lossy medium
Propagation Mechanisms 33
Perpendicular (transverse magnetic) polarization:
E t v,
H t h,
kt
E r v,
H r h, kr
E i v,
H i h, ki
θr
θt
βθi
ε1 μ0,
ε2 μ0,
Reflection and transmission
Propagation Mechanisms 34
Angle of reflection, angle of transmission:
k1 θi( )sin k1 θr( )sin k2 θt( )sin= =
θi θr=
θt
ε1
ε2
----- β( )cos⎝ ⎠⎜ ⎟⎛ ⎞
asin=⎩⎪⎨⎪⎧
⇒
Snell’s law:
Reflection and transmission
Propagation Mechanisms 35
Perpendicular (transverse magnetic) polarization (cont’d):•Reflection coefficient:
•Transmission coefficient:
Rv
Er v,Ei v,----------≡
ε2
ε1
----- β( )sinε2
ε1
----- cos2 β( )––
ε2
ε1
----- β( )sinε2
ε1
----- cos2 β( )–+
-----------------------------------------------------------------=
T v
Et v,Ei v,---------≡
2ε2
ε1
----- β( )sin
ε2
ε1
----- β( )sinε2
ε1
----- cos2 β( )–+
-----------------------------------------------------------------=
Reflection and transmission
Propagation Mechanisms 36
Parallel (transverse electric) polarization:
E i h,
H i v,
ki
E r h,
H r v,kr
E t h,H t v,
kt
θr
θt
βθi
ε1 μ0,
ε2 μ0,
Reflection and transmission
Propagation Mechanisms 37
Parallel (transverse electric) polarization (cont’d):
•Reflection coefficient:
•Transmission coefficient:
Rh
Er h,Ei h,----------≡
β( )sinε2
ε1
----- cos2 β( )––
β( )sinε2
ε1
----- cos2 β( )–+
-----------------------------------------------------------=
T h
Et h,Ei h,----------≡ 2 β( )sin
β( )sinε2
ε1
----- cos2 β( )–+
-----------------------------------------------------------=
Reflection and transmission
Propagation Mechanisms 38
Comments:
1. total reflection; total transmission
2.
3.
4. (ideal conductor)
5. : (Brewster angle)
Rh v 1= T h v 1=
Rh v 1= T h v 1=
β 0→ Rv Rh, 1–→⇒
β π 2⁄= Rh Rv– 1 ε2 ε1⁄–( ) 1 ε2 ε1⁄+( )⁄= =⇒
σ2 ∞→ Rh 1–→ Rv 1→,⇒
σ1 σ2 0= = βB 1 1 ε2 ε1⁄+⁄( )asin≡ Rv 0=⇒
Reflection and transmission
Propagation Mechanisms 39
Behaviour of as a function of the grazing angle :Rh v β
Source: Grosskopf
Rh v( )arg
ββ
Rh v
Reflection and transmission
Propagation Mechanisms 40
Two-path model:
d
dd
Rx
Tx'
dr
Tx
hT
hR
Direct path
Reflected path
ε0 μ0,
ε1 μ0,
hT
dr
hT hR+
⎩⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎧
hT hR–
⎩⎪⎪⎨⎪⎪⎧
Propagation over flat earth
Propagation Mechanisms 41
Resulting field at the receiver location:Single polarization
Assumptions/approximations:
•
•
ER E dd( ) jkdd–( )exp RE dr( ) jkdr–( )exp+=
1
dd-----
1
dr-----
1
d---≈ ≈ E dd( ) E dr( ) E d( )≈ ≈⇒ E
d---=
hT hR– hT hR+, d« dr dd– 2hT hR
d------------≈⇒
Propagation over flat earth
Propagation Mechanisms 42
Approximated field at the receiver location:
Special case: Total reflection
ER d( ) EFS d( ) 1 R j4πhT hR
λd------------–⎝ ⎠
⎛ ⎞exp+≈
Field from free-space
propagation Tx-Rx
R 1–=( )
ER d( )EFS d( )------------------ 2 2π
hT hR
λd------------⎝ ⎠
⎛ ⎞sin≈
Propagation over flat earth
Propagation Mechanisms 43
Example [calculated]:
101
102
103
104
−120
−100
−80
−60
−40
−20
0
Free-space
propagation
Breakpoint
dBP
ER
d()
ER
1()
---------------
[dB
]
Distance [m]d
dBP
4hT hR
λ----------------≡
f c 900 MHz=
hR 1.6 m=
hT 9 m=
R 1–=
20– d( )log∼
40– d( )log ∼
Two ray model
Propagation over flat earth
Propagation Mechanisms 44
Example [measured]:
Source: COST 231 TD(95)78, J. Wiart
Measured signal strength
Two-path model
Free-space propagation
Propagation over flat earth
Propagation Mechanisms 45
Rayleigh and Fraunhofer criterions:
Conditions for a surface to be “flat”:
• Rayleigh criterion:
• Fraunhofer criterion:
Δhβ β
β β
Δhλ
------- β( )sin1
8---<4πΔh
λ------- β( )sin
π2---< ⇔
Δhλ
------- β( )sin1
32------<4πΔh
λ------- β( )sin
π8---< ⇔
Scattering from rough surfaces
Propagation Mechanisms 46
Rayleigh and Fraunhofer criterions (cont’d):
ki
β
kr
ΔhA
B
β
λ
C
B'
2Δh
λ
Path length difference :
Resulting phase difference:
AB ACB'– 2Δh β( )sin
2π2Δh β( )sin
λ--------------------------
2Δh β( )sin
Scattering from rough surfaces
Propagation Mechanisms 47
Rayleigh and Fraunhofer criteria (cont’d):
Comments:
• Investigations have shown that in the microwave range (~ 1GHz), the
Fraunhofer criterion is more appropriate than the one by Rayleigh.
• A surface will be effectively smooth if either or is small.
• The Rayleigh and Fraunhofer criteria can be applied to irregular surfaces
as well. In this case is replaced in the inequalities by the standard
deviation of the terrain (see next slide).
Δhλ
------- β
Δhς
Scattering from rough surfaces
Propagation Mechanisms 48
Characterization of random surfaces:
Correlation function of the terrain height:
(Usual) Gaussian assumption:
• : Correlation lengthcharacterizes the horizontal scale roughness
• : Standard deviation of the terrain height. This number
characterizes the vertical scale roughness.
E h d( )h d Δd+( )[ ] E h d( )[ ] 0=( )
E h d( )h d Δd+( )[ ] ς2 Δd2
l2---------–⎝ ⎠
⎛ ⎞exp=
l E h d( )h d l+( )[ ]( ) ς2⁄ e1–
=( )
ς Var h d( )[ ]≡
Scattering from rough surfaces
Propagation Mechanisms 49
Characterization of random surfaces (cont’d):
Meaning of the terrain parameters andς l
d d
ς2h d( ) h d( )
ς1 ς2<
ς1
d d
l2h d( ) h d( )
l1 l2<
l1
Scattering from rough surfaces
Propagation Mechanisms 50
Slightly rough surfaces, modified Fresnel reflection coefficients:
A surface is slightly rough if
(Fraunhofer criterion)
In this case the scattering mechanism can be approximated by a reflection
with the modified Fresnel reflection coefficients:
At the Fraunhofer limit (the above inequality sign is replaced by equality),
ςλ--- β( )sin
1
32------<
Rh vmod
Rh v 8π2 ςλ---⎝ ⎠⎛ ⎞ 2
β( )sin2
–⎝ ⎠⎛ ⎞exp=
Rh vmod
Rh v⁄ 0 926, 0.67dB–= =
Scattering from rough surfaces
Propagation Mechanisms 51
Coherent and diffuse scattering:
Computation methods:
• Coherent scattering: Kirchhoff method• Diffuse scattering: Small perturbation method
ki
Coherent scattering
Diffuse scattering
Scattering from rough surfaces
Propagation Mechanisms 52
Second Green’s theorem:
Illustration (one-dim. case):
Es r( ) j–
4πr---------e
j– ksr
⎝ ⎠⎛ ⎞ ks n z( ) E× z( )
Z0
ks------ks n z( ) H× z( )( )×– e
j ks z⟨ | ⟩dS
S∫∫×=
ki
n z( )
S
Dz
z
rks
E s v,
E s h,
E i v,
E i h,
dS
Scattering from rough surfaces
Propagation Mechanisms 53
Second Green’s theorem (cont’d):
• : normal unit vector to at
• , resultant electric and magnetic field at
Kirchhoff scalar approximation:Basic assumption:
• The surface boundary is sufficiently smooth
-> In a local region it may be looked upon as an inclined plane.
• Specular reflection occurs at these planes.
-> The resulting field at the surface is the sum of the incident and reflected fields.
• Surface with small slope
-> Reduction from a vector formulation to a scalar formulation.
n z( ) S z
E z( ) H z( ) z
Scattering from rough surfaces
Propagation Mechanisms 54
Kirchhoff scalar approximation for slightly rough surfaces:“Definition” of a slightly rough (random Gaussian) surface:
• correlation length > wavelength
• average radius of curvature > wavelength
• RMS terrain slope < 0.25
• dimension of the surface >> wavelength
kl 6>
l2
ςλ------ 2.76> l
2
2.76ς-------------
ςl--
2
8-------> ς
l-- 2
Dx Dy, λ»
Scattering from rough surfaces
Propagation Mechanisms 55
Coherent scattering matrix:Surface-anchored coordinate system:
ki
E i v,
E i h,
ks
E s v,E s h,
θiθs
n
x
y
z
φs
φi
Dx
Dy
Elementary surface
Scattered waveImpinging wave
Scattering from rough surfaces
Propagation Mechanisms 56
Coherent scattering matrix (cont’d):
where is given by
with
Es Sc Ei=
Sc
Sc VRh θi( ) θi( )cos φs φi–( )cos– Rv θi( ) φs φi–( )sin–
Rh θi( ) θi( )cos θs( )cos φs φi–( )cos– Rv θi( ) θs( )cos φs φi–( )cos⋅=
•
•
V jk
2π------DxDy
k Dxζx( )sin
k Dxζx-----------------------------
k Dyζy( )sin
k Dyζy-----------------------------e
kςζz( )2 2⁄–=
ζx θs( ) φs( )cossin θi( ) φi( )cossin–( ) 2⁄=
ζy θs( ) φs( )sinsin θi( ) φi( )sinsin–( ) 2⁄=
ζz φs( )cos φi( )cos+=
Scattering from rough surfaces
Propagation Mechanisms 57
Huygens’ principle:
Tx Rxσ ∞=
ER
d
Radiating element
Surface A
d'
Ad
A'd
E 0
d1 d2
Δh
Diffraction over/around obstacles
Propagation Mechanisms 58
Kirchhoff’s mathematical formulation of Huygens’ principle:
Kirchhoff’s formula for diffraction simplifies in the situation considered
above to:
: amplitude of the field generated by the transmitter on the top of theobstacle.
Solving the integral above under some further geometrical simplifications
yields
ER E 01
d--- jkd–( )exp Ad
A∫≈
E 0
ER
EFS---------
jπ 4⁄( )exp
2---------------------------
j– π 4⁄( )exp
2------------------------------- C∗ w( )–⋅=
Diffraction over/around obstacles
Propagation Mechanisms 59
Kirchhoff’s mathematical formulation of Huygens’ principle (cont’d):• : Free-space electric field at Rx if the obstacle were suppressed.
• : Fresnel coefficient:
• : Cornu spiral:
Approximation for large positive values of :
EFS
w
w Δh2
λ---
1
d1
-----1
d2
-----+⎝ ⎠⎛ ⎞≡
C w( )
C w( ) jπ2---z
2
⎝ ⎠⎛ ⎞exp zd
0
w
∫≡
w
ER
EFS---------
1
2πw---------------≈ w 0.56≥
Diffraction over/around obstacles
Propagation Mechanisms 60
Diffraction attenuation:
Behaviour of the Cornu-spiral Behaviour of as a function ofER
EFS--------- w
Source: Grosskopf
w
w
5– 5
Line of sight No line of sight
jπ 4⁄{ }exp
2-----------------------------
Re
Im
Diffraction over/around obstacles
Propagation Mechanisms 61
Geometrical-optics representation of a wave:
Investigated special cases:
• Spherical wave:
• Plane wave:
Geometrical optics:
Eikonal equation:
E r( ) E1
r------- j– k r( )exp=
E r( ) E j– k r⟨ | ⟩( )exp=
E r( ) E0 r( ) j– kΦ r( )( )exp=
Amplitude term Eikonal
grad Φ r( )( ) 1=
The uniform geometrical theory of diffraction
Propagation Mechanisms 62
Geometrical-optics representation of a wave (cont’d):Astigmatic ray tube:
Wave equation:
d 0= d nλ=
rv
rh
dA0
dAd
Surfaces of constant phase(wave fronts)
Axial ray
direction ofpropagation
≡
Principal radii ofcurvature dA0
Caustics
Reference point
E d( ) E 0( )rhrv
rh d+( ) rv d+( )-------------------------------------- jkd–( )exp=
The uniform geometrical theory of diffraction
Propagation Mechanisms 63
Geometrical-optics representation of a wave (cont’d):Examples:• Spherical wave:
• Plane wave:
rh rv r= = E d( ) E 0( ) rr d+( )
---------------- jkd–( )exp=⇒
rh rv, ∞= E d( ) E 0( ) jkd–( )exp=⇒
The uniform geometrical theory of diffraction
Propagation Mechanisms 64
Key idea of the UTD:Extension of the classical geometrical optics to incorporate rays diffractedby [curved] edges.
Keller’s law of edge diffraction:
ki
kd
aw
βd
ki
βi π 2⁄=
Cone of diffractedrays
kdaw
ki aw⟨ | ⟩ kd aw⟨ | ⟩= βi⇔ βd=
βi
The uniform geometrical theory of diffraction
Propagation Mechanisms 65
Shadow boundaries:Reflection
shadowboundary
Incidentshadow
boundary
Et Ed=
Et Ei Ed+=Et Ei Er Ed+ +=
ki
ki
ki
krkd kd
kdθi
θiSpecial case:
Plane waveincidence
Incident wave Reflected wave Diffracted wave
The uniform geometrical theory of diffraction
Propagation Mechanisms 66
Edge-anchored coordinate system:
β
β
ϑi
ϑd
E i v,
E i h,
E d h,E d v,–
kd
ki dzP0
P
Qaw
P i h,
ai v,zQ
2m
–(
)π
P d h,
The uniform geometrical theory of diffraction
Propagation Mechanisms 67
Geometrical-optics representation of the incident wave:
• : Principal radius of curvature at of the wavefront in the plane
• : Principal radius of curvature at of the wavefront in the plane
spanned by and .
Special case: Plane wave
Ei z( ) Ei P0( )ri h, ri v,
ri h, z+( ) ri v, z+( )-------------------------------------------- jkz–( )exp=
ri h, P0 P i h,
ri v, P0
ki ai v,
ri h, ri v,, ∞→
Ei z( ) Ei P0( ) jkz–( )exp=
The uniform geometrical theory of diffraction
Propagation Mechanisms 68
Geometrical-optics representation of the diffracted wave:
where
Special case: Plane wave
Ed d( ) DEi Q( )rd
rd d+( )d----------------------- jkd–( )exp=
• : diffraction matrix
• : depends in particular on the principal radius of curvature of the edge at .
For a straight edge (curv. radius= ):
DDh β ϑi ϑd m, , ,( )– 0
0 Dv β ϑi ϑd m, , ,( )–≡
rd Q∞ rd ri h, zQ+=
Ed d( ) DEi Q( ) 1
d------- jkd–( )exp=
The uniform geometrical theory of diffraction
Propagation Mechanisms 69
Dyadic diffraction coefficients [diffraction on an edge]:
Dh v β ϑi ϑd m, , ,( ) jπ 4⁄–( )exp–
2m 2πk β( )sin-------------------------------------- ⋅=
π ϑd ϑi–( )+
2m--------------------------------⎝ ⎠⎛ ⎞cot F kDv
+ ϑd ϑi–( )[ ]π ϑd ϑi–( )–
2m--------------------------------⎝ ⎠⎛ ⎞cot F kDv
- ϑd ϑi–( )[ ]+⋅
π ϑd ϑi+( )+
2m---------------------------------⎝ ⎠⎛ ⎞cot F kDv
+ ϑd ϑi+( )[ ]π ϑd ϑi+( )–
2m--------------------------------⎝ ⎠⎛ ⎞cot F kDv
- ϑd ϑi+( )[ ]+⎩ ⎭⎨ ⎬⎧ ⎫
+−
F u( ) 2 j u ju( )exp1
2------- jπ 4⁄–( )exp C u( )–=
v+|-
u( ) 22mπN
+|-u–
2-------------------------------⎝ ⎠⎛ ⎞cos
2
=
N+|-
minu integer π± u– 2πm( )⁄( )arg=
Dd zQ
d zQ+---------------- β( )sin
2=
The uniform geometrical theory of diffraction
Propagation Mechanisms 70
Example:
ϑd °[ ]
E t h,
E d h,
E t v,
E d v,
•
•
•
β π 2⁄=
m 16 9⁄=
ϑi 55°=
•
•
f 3 GHz=
d 1 m=
Am
pli
tude
rel.
to
Ei
h,[d
B]
ϑd °[ ]A
mpli
tude
rel.
to
Ei
v,[d
B]
The uniform geometrical theory of diffraction