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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
- 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
- Fading Models

3.5 Reflection: EM wave in 1st medium impinges on 2nd medium

- 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)

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

- (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

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

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

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

- 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 permittivityr

= 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

||

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

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+ ||

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

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

= 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

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

(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

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

=

(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:

-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

(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

(3.52a)

Pr(d) =

(3.52b)

- received power falls off at 40dB/decade
- 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(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

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