3.4 Basic Propagation Mechanisms & Transmission Impairments

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3.4 Basic Propagation Mechanisms & Transmission Impairments . (1) Reflection: propagating wave impinges on object with size >>  examples include ground, buildings, walls. (2) Diffraction: transmission path obstructed by objects with edges

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3.4 Basic Propagation Mechanisms & Transmission Impairments

• (1) Reflection: propagating wave impinges on object with size >> 
• examples include ground, buildings, walls
• (2) Diffraction: transmission path obstructed by objects with edges
• 2ndry waves are present throughout space (even behind object)
• gives rise to bending around obstacle (NLOS transmission path)
• (3) Scattering propagating wave impinges on object with size < 
• number of obstacles per unit volume is large (dense)
• examples include rough surfaces, foliage, street signs, lamp posts

at high frequencies  diffraction & reflections depend on

• geometry of objects
• EM wave’s, amplitude, phase, & polarization at point of intersection
• Models are used to predict received power or path loss (reciprocal)
• based on refraction, reflection, scattering
• Large Scale Models
• part of the wave is transmitted
• part of the wave is reflected
• (1) plane-wave incident on a perfect dielectric(non-conductor)
• part of energy is transmitted (refracted) into 2nd medium
• part of energy is transmitted (reflected) back into 1st medium
• assumes no loss of energy from absorption (not practically)
• (2) plane-wave incident on a perfect conductor
• all energy is reflected back into the medium
• assumes no loss of energy from absorption (not practically)

incident wave

reflected wave

refracted wave

boundary between dielectrics (reflecting surface)

• (3)  = Fersnel reflection coefficient relates Electric Field intensity
• of reflected & refracted waves to incident wave as a function of:
• material properties,
• polarization of wave
• angle of incidence
• signal frequency

E||

E

• (4) Polarization:EM waves are generally polarized
• instantaneous electric field components are inorthogonal directions
• in space represented as either:
• (i) sum of 2 spatially orthogonal components (e.g. vertical
• & horizontal)
• (ii) left-handed or right handed circularly polarized components
• reflected fieldsfrom a reflecting surface can be computed using
• superposition for any arbitrary polarization

plane of incidence

reflecting surface= boundary between dielectrics

i

r

t

• 3.5.1 Reflection from Dielectrics
• assume no loss of energy from absorption
• EM wave with E-fieldincident at i with boundary between 2 dielectric media
• some energy is reflected into 1st media at r
• remaining energy is refracted into 2nd media at t
• reflections vary with the polarization of the E-field

plane of incidence = plane containing incident, reflected, & refracted rays

Evi

plane of incidence

Ehi

i

r

boundary between dielectrics (reflecting surface)

t

• Two distinct cases are used to study arbitrary directions of polarization
• (1) Vertical Polarization: (Evi) E-field polarization is
• parallelto theplane of incidence
• normal component to reflecting surface
• (2) Horizontal Polarization: (Ehi) E-field polarization is
• perpendicular to the plane of incidence
• parallel component to reflecting surface

i

r

t

i

r

t

Ei

Er

Hr

Er

Ei

Hi

1,1, 1

Hi

Hr

1,1, 1

2,2, 2

2,2, 2

Et

Et

Vertical Polarization: E-field in the plane of incidence

Horizontal Polarization: E-field normal to plane of incidence

• Ei & Hi = Incident electric and magnetic fields
• Er& Hr = Reflected electric and magnetic fields
• Et = Transmitted (penetrating)electric field

dielectric constant for perfect dielectric (e.g. perfect reflector of lossless material) given by

• 0= 8.85 10-12 F/m
• (1) EM Parameters of Materials
•  = permittivity (dielectric constant): measure of a materials ability
• to resist current flow
•  = permeability: ratio of magnetic induction to magnetic field
• intensity
•  = conductance: ability of a material to conduct electricity,
• measured in Ω-1

highly conductive materials 

• r&  are generally insensitive to operating frequency

lossy dielectricmaterials will absorb power  permittivity described with complex dielectric constant

 = 0 r -j’

(3.17)

(3.18)

where ’ =

often permittivity of a material,  is related to relative permittivityr

 = 0 r

• 0 and r are generally constant
•  may be sensitive to operating frequency

(2) Reflections, Polarized Components & Fresnel Reflection

• Coefficients
• because of superposition – only 2 orthogonal polarizations need be
• considered to solve general reflection problem
• Maxwell’s Equationboundary conditions used to derive (3.19-3.23)
• Fresnel reflection coefficientsfor E-field polarization at reflecting
• surface boundary
• ||represents coefficient for || E-field polarization
• represents coefficient for  E-field polarization

(i) E-field in plane of incidence (vertical polarization)

|| =

(3.19)

(ii) E-field not in plane of incidence (horizontal polarization)

=

(3.20)

• i = intrinsic impedance of the ith medium
• ratio of electric field to magnetic field for uniform plane wave in
• ith medium
• given by i=

Fersnel reflection coefficients given by

velocity of an EM wave given by

(3.21)

i = r

(3.22)

Er =  Ei

(3.23a)

Et = (1 +  )Ei

(3.23b)

boundary conditionsat surface of incidence obey Snell’s Law

•  is either || or  depending on polarization
• |  |  1  for a perfect conductor, wave is fully reflected
• |  |  0  for a lossy material, wave is fully refracted

||=

(3.24)

=

(3.25)

Simplification of reflection coefficients for vertical and horizontal

polarization assuming:

• radio wave propagating in free space (1st medium is free space)
• 1 = 2
• Elliptically Polarized Waves have both vertical & horizontal components
• waves can be depolarized (broken down) into vertical & horizontal
• E-field components
• superposition can be used to determine transmitted & reflected
• waves

spatial vertical axis

||

orthogonal components

of propagating wave

spatial horizontal axis

• (3) General Case ofreflection or transmission
• horizontal & vertical axes of spatial coordinates may not coincide
• with || & axes of propagating waves
• for wave propagating out of the page  define angle 
• measured counter clock-wise from horizontal axes

vertical & horizontal

polarized components

components perpendicular

& parallel to plane of incidence

EiH , EiV

EdH, EdV

(3.26)

relationship of vertical & horizontal field components at the dielectric boundary

EdH,EdVEiH , EiV= Time Varying Components of E-field

• EdH, EdV= depolarized field components along the horizontal &
• vertical axes
• EiH , EiV= horizontal & vertical polarized components of incident
• wave

- E-field components may be represented by phasors

R = transformation matrix that maps E-field components

,  = angle between two sets of axes

R =

DC= depolarization matrix

DC=

• for case of reflection:
• D= 
• D|| ||= ||
• for case of refraction (transmission):
• D= 1+ 
• D|| ||= 1+ ||

1.0

0.8

0.6

0.4

0.2

0.0

||||

r=12

r=4

vertical polarization

(E-field in plane of incidence)

0 10 20 30 40 50 60 70 80 90

Brewster Angle (B)

for r=12

angle of incidence (i)

Plot of Reflection Coefficients for Parallel Polarization for r= 12, 4

for i < B: a larger dielectric constant smaller || & smaller Er

for i > B: a larger dielectric constant larger || & larger Er

||

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

r=12

horizontal polarization

(E-field not in plane of incidence)

r=4

0 10 20 30 40 50 60 70 80 90

angle of incidence (i)

Plot of Reflection Coefficients for Perpendicular Polarization for r= 12, 4

for given i: larger dielectric constant larger  and larger Er

||=

=

• e.g. let medium 1 = free space & medium 2 = dielectric
• if i  0o (wave is parallel to ground)
• thenindependent of r, coefficients ||  1 and ||||  1
• thus, if incident wave grazes the earth
• ground may be modeled as a perfect reflector with || = 1
• regardless of polarization or ground dielectric properties
• horizontal polarization results in 180 phase shift

B satisfies

sin(B) =

(3.27)

• if 1st medium = free space & 2nd medium has relative permittivity r
• then (3.27) can be expressed as

sin(B) =

(3.28)

3.5.2 Brewster Angle = B

• Brewster angle only occurs for vertical (parallel) polarization
• angle at which no reflection occurs in medium of origin
• occurs when incident angle iis such that || = 0  i = B

Let r = 15 

= 0.25

sin(B) =

Let r = 4 

B= sin-1(0.25) = 14.5o

= 0.44

sin(B) =

B= sin-1(0.44) = 26.6o

e.g. 1st medium = free space

3.6 Ground Reflection – 2 Ray Model

• Free Space Propagation model is inaccuratefor most mobile RF
• channels
• 2 Ray Ground Reflection model considers both LOS path & ground
• reflected path
• based on geometric optics
• reasonably accurate for predicting large scale signal strength for
• distances of several km
• useful for
• - mobile RF systems which use tall towers (> 50m)
• - LOS microcell channels in urban environments
• Assume
• maximum LOS distances d  10km
• earth is flat

ELOS

Ei

Er = Eg

d

E(d,t) =

(3.33)

i

0

(1) Determine Total Received E-field (in V/m) ETOT

ETOT is combination of ELOS & Eg

• ELOS= E-field of LOS component
• Eg= E-field of ground reflected component
• θi = θr

LetE0= free space E-field (V/m) at distance d0

• Propagating Free Space E-field at distance d > d0 is given by
• E-field’s envelope at distance d from transmitter given by
• |E(d,t)| = E0 d0/d

Eg(d”,t) =

(3.35)

d’

ELOS

Ei

ht

Eg

d”

h r

i

0

d

E-field for LOS and reflected wave relative to E0 given by:

ELOS(d’,t) =

(3.34)

and ETOT = ELOS + Eg

• assumes LOS & reflected waves arrive at the receiver with
• - d’ = distance of LOS wave
• - d” = distance of reflected wave

i = 0

Eg =  Ei

Et = (1+) Ei

(3.36)

(3.37a)

(3.37b)

ELOS

d’

Ei

Eg

d”

Et

i

0

From laws of reflection in dielectrics (section 3.5.1)

 = reflection coefficient for ground

resultant E-field is vector sum of ELOS and Eg

• total E-field Envelope is given by |ETOT| = |ELOS + Eg|

(3.38)

• total electric field given by

ETOT(d,t) =

(3.39)

• Assume
• i. perfect horizontal E-field Polarization
• ii. perfect ground reflection
• iii. small i (grazing incidence)  ≈ -1 & Et ≈ 0
• reflected wave & incident wave haveequal magnitude
• reflected wave is 180oout of phase with incident wave
• transmitted wave≈ 0

=

(3-40)

ELOS

h

d’

Eg

Ei

ht

h r

d”

0

i

ht +hr

if d >> hr + ht Taylor series approximations yields (from 3-40)

d

(3-41)



(2) Compute Phase Difference & Delay Between Two Components

• path difference  = d” – d’ determined from method of images

=

(3-42)

0 π 2π

d =

(3-43)

Δ

if Δ = /n   = 2π/n

• time delay

(3.44)

once  is known  we can compute

• phase difference
• As d becomes large   = d”-d’ becomes small
• amplitudes of ELOS & Eg are nearly identical & differ only in phase

reflected path arrives at receiver at

t = d”/c

=

=

(3) Evaluate E-field when reflected path arrives at receiver

ETOT(d,t)|t=d”/c =

(3.45)

|ETOT(d)|=

ETOT



=

(3.46)

=

(3.47)

(3.48)

=

(4) Determine exact E-field for 2-ray ground model at distance d

Use phasor diagram to find resultant E-field from combined direct & ground reflected rays:

-50

-60

-70

-80

-90

-100

-110

-120

-130

-140

fc = 3GHz

fc = 7GHz

fc = 11GHz

Propagation Lossht = hr = 1, Gt = Gr = 0dB

101 102 103 104 m

• As d increases ETOT(d) decreases in oscillatory manner
• local maxima 6dB > free space value
• local minima ≈ - dB (cancellation)
• once d is large enough θΔ< π & ETOT(d) falls off asymtotically
• with increasing d

this implies

(3.50)

V/m

ETOT(d) 

(3.51)

|ETOT(d)|

For phase difference,  < 0.6 radians (34o)sin(0.5 )  

(3.49)

d >

if d satisfies 3.50  total E-fieldcan be approximated as:

kis a constant related to E0 ht,hr, and 

e.g. at 900MHz  if  < 0.03m  total E-field decays with d2

Pr(d) =

(3.52a)

Pr(d) =

(3.52b)

• receive power & path loss become independent of frequency

if d >>

Received Power at d is related to square of E-field by 3.2, 3.15, & 3.51

PL =

PL(dB) = 40log d - (10logGt+ 10logGr + 20log ht + 20 log hr)

(3.53)

Path Loss for 2-ray model with antenna gains is expressed as:

• 3.50 must hold
• for short Tx-Rx distances use (3.39) to computetotal E field
• evaluate (3.42) for =  (180o) d = 4hthr/is where the ground
• appears in 1stFresnel Zone between Tx & Rx
• - 1st Fresnel distance zone is useful parameter in microcell path
• loss models