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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

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Models are used to predict received power or path loss (reciprocal)based on refraction, reflection, scattering

• Large Scale Models

• Fading Models

at high frequencies diffraction & reflections depend on

• geometry of objects

• EM wave’s, amplitude, phase, & polarization at point of intersection

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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)

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(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

boundary between dielectrics (reflecting surface)

reflected wave

refracted wave

incident wave

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(4) Polarization: EM waves are generally polarized

• instantaneous electric field components are in orthogonal

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 fields from a reflecting surface can be computed using superposition for any arbitrary polarizationE||

E

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3.5.1 Reflection from Dielectrics • assume no loss of energy from absorption

EM wave with E-field incident 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

reflecting surface= boundary between dielectrics

irt

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

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Two distinct cases are used to study arbitrary directions of polarization

(1) Vertical Polarization: (Evi) E-field polarization is

• parallel to the plane 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

plane of incidence

irt

EviEhi

boundary between dielectrics (reflecting surface)

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• Ei & Hi = Incident electric and magnetic fields

• Er & Hr = Reflected electric and magnetic fields

• Et = Transmitted (penetrating) electric field

Hi Hr

Ei Er

i r

t

1,1, 1

2,2, 2

Et

Vertical Polarization: E-field in the plane of incidence

HiHrEi Er

i r

t

1,1, 1

2,2, 2

Et

Horizontal Polarization: E-field normal to plane of incidence

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(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

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

0 = 8.85 10-12 F/m

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

= 0 r

lossy dielectric materials will absorb power permittivity described with complex dielectric constant

(3.18) where ’ =f

2

(3.17) = 0 r -j’

highly conductive materials

r & are generally insensitive to operating frequencyr

f

0

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

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Material r /r0 f (Hz)Poor Ground 4 0.001 2.82 107 108

Typical Ground 15 0.005 3.77 107 108

Good Ground 25 0.02 9.04 107 108

Sea Water 81 5 6.97 109 108

Fresh Water 81 0.001 1.39 106 108

Brick 4.44 0.001 2.54 107 4109

Limestone 7.51 0.028 4.21 108 4109

Glass, Corning 707 4 0.00000018 5.08 103 106

Glass, Corning 707 4 0.000027 7.62 105 108

Glass, Corning 707 4 0.005 1.41 108 1010

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• because of superposition – only 2 orthogonal polarizations need be considered to solve general reflection problem

Maxwell’s Equation boundary conditions used to derive (3.19-3.23)

Fresnel reflection coefficients for E-field polarization at reflecting surface boundary

• || represents coefficient for || E-field polarization

• represents coefficient for E-field polarization

(2) Reflections, Polarized Components & Fresnel Reflection Coefficients

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Fersnel reflection coefficients given by

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

|| =it

it

i

r

E

E

sinsin

sinsin

12

12

(3.19)

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

=ti

ti

i

r

E

E

sinsin

sinsin

12

12

(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 = ii

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velocity of an EM wave given by 1

boundary conditions at surface of incidence obey Snell’s Law

)90sin()90sin( 222111 (3.21)

i = r (3.22)

Er = Ei (3.23a)

Et = (1 + )Ei (3.23b)

is either || or depending on polarization

• | | 1 for a perfect conductor, wave is fully reflected• | | 0 for a lossy material, wave is fully refracted

)90sin(sin90

2

11it

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• radio wave propagating in free space (1st medium is free space) • 1 = 2

|| =irir

irir

2

2

cossin

cossin

(3.24)

=iri

iri

2

2

cossin

cossin

(3.25)

Simplification of reflection coefficients for vertical and horizontal polarization assuming:

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

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(3) General Case of reflection 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

spatial horizontal axis

spatial vertical axis

||orthogonal components

of propagating wave

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vertical & horizontal polarized components

components perpendicular & parallel to plane of incidence

EiH , Ei

V EdH , Ed

V

• EdH , Ed

V = depolarized field components along the horizontal & vertical axes

• EiH , Ei

V = horizontal & vertical polarized components of incident wave

relationship of vertical & horizontal field components at the dielectric boundary

EdH, Ed

V EiH , Ei

V = Time Varying Components of E-field

iv

iH

CT

dv

dH

E

ERDR

E

E(3.26)

- E-field components may be represented by phasors

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for case of reflection:

• D =

• D|| || = ||

for case of refraction (transmission): • D = 1+

• D|| || = 1+ ||

R =

cossin

sincos, = angle between two sets of axes

DC =

|| ||0

0

D

D

R = transformation matrix that maps E-field components

DC = depolarization matrix

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1.00.80.60.40.20.0

0 10 20 30 40 50 60 70 80 90

||||

r=12

r=4

angle of incidence (i)Brewster Angle (B)

for r=12

vertical polarization (E-field in plane of incidence)

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

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

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

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r=12

r=4

||1.00.90.80.70.60.50.40.3

0 10 20 30 40 50 60 70 80 90angle of incidence (i)

horizontal polarization (E-field not in plane of

incidence)

for given i: larger dielectric constant larger and larger Er

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

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e.g. let medium 1 = free space & medium 2 = dielectric

• if i 0o (wave is parallel to ground)

• then independent of r, coefficients || 1 and |||| 1

|| = 1cos

cos

cossin

cossin2

2

02

2

ir

ir

irir

irir

i

= 1cos

cos

cossin

cossin2

2

02

2

ir

ir

iri

iri

i

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

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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 i is such that || = 0 i = B

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

1

12

r

r

sin(B) = (3.28)

sin(B) = 21

1

(3.27

)B satisfies

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e.g. 1st medium = free space

Let r = 4

116

14

sin(B) = = 0.44

B = sin-1(0.44) = 26.6o

Let r = 15

115

1152

sin(B) = = 0.25

B = sin-1(0.25) = 14.5o

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3.6 Ground Reflection – 2 Ray Model

Free Space Propagation model is inaccurate for 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

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Let E0 = free space E-field (V/m) at distance d0

• Propagating Free Space E-field at distance d > d0 is given by

E(d,t) =

c

dtw

d

dEccos00 (3.33)

• E-field’s envelope at distance d from transmitter given by

|E(d,t)| = E0 d0/d

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

ELOS

Ei

E r = E g

i 0

d

ETOT is combination of ELOS & Eg

• ELOS = E-field of LOS component

• Eg = E-field of ground reflected component

• θi = θr

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d’

d”

ELOS

Ei

E gi 0

d

ht h r

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

and ETOT = ELOS + Eg

ELOS(d’,t) =

c

dtw

d

dEc

'cos

'00 (3.34)

Eg(d”,t) =

c

dtw

d

dEΓ c

"cos

"00 (3.35)

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

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From laws of reflection in dielectrics (section 3.5.1)

i = 0 (3.36)

Eg = Ei (3.37a)

Et = (1+) Ei (3.37b)

= reflection coefficient for ground

E g

d’

d”

ELOS

Ei

i 0

Et

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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

c

dtw

d

dEc

'cos

'00

c

dtw

d

dEc

"cos

")1( 00 (3.39)ETOT(d,t) =

Assume i. perfect horizontal E-field Polarization

ii. perfect ground reflection

iii. small i (grazing incidence) ≈ -1 & Et ≈ 0

• reflected wave & incident wave have equal magnitude

• reflected wave is 180o out of phase with incident wave

• transmitted wave ≈ 0

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

2222dhhdhh rtrt = (3-40)

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

d

hh rt2 (3-41)

(2) Compute Phase Difference & Delay Between Two Components

ELOS

d

d’

d”i 0

ht

h r

h

ht +

h r

Ei Eg

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once is known we can compute

• phase difference =c

wc

2

(3-42)

• time delay d =cfc

2

(3-43)

As d becomes large = d”-d’ becomes small

• amplitudes of ELOS & Eg are nearly identical & differ only in phase

"'000000

d

dE

d

dE

d

dE (3.44)

if Δ = /n = 2π/n 0 π 2π

Δ

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

0cos"

)1('"

cos'

0000

d

dE

c

ddw

d

dEc

(3.45)ETOT(d,t)|t=d”/c =

t = d”/c reflected path arrives at receiver at

1cos00

cw

d

dEc

1cos00 d

dE=

100 d

dE=

32

(3.46)

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200 sin1cos

d

dE=

2

2

0022

00 1 sind

dEcos

d

dE|ETOT(d)|=

=

=

2sin2 00

d

dE

cos2200

d

dE(3.47)

(3.48)

ETOT

"00

d

dE

'd

dE 00

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

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

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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

-50

-60-70

-80

-90

-100

-110-120

-130

-140101 102 103 104 m

fc = 3GHzfc = 7GHzfc = 11GHz

Propagation Loss ht = hr = 1, Gt = Gr = 0dB

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if d satisfies 3.50 total E-field can be approximated as:

k is a constant related to E0 ht,hr, and

radd

hh rt 3.022

2

1

2

(3.49)

d > (3.50)

rtrt hhhh 20

3

20 this implies

For phase difference, < 0.6 radians (34o) sin(0.5 )

22 00

d

dE|ETOT(d)|

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

200 22

d

k

d

hh

d

dE rt

(3.51)ETOT(d)

V/m

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

Pr(d) = (3.52b)

4120

)(

120

)( 2220 rR

eGdE

AdE

Pr(d) = 4

22

d

hhGGP rt

rtt (3.52a)

• received power falls off at 40dB/decade

• receive power & path loss become independent of frequency

rthhif d >>

36

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

• for short Tx-Rx distances use (3.39) to compute total E field

• evaluate (3.42) for = (180o) d = 4hthr/ is where the ground

appears in 1st Fresnel Zone between Tx & Rx

- 1st Fresnel distance zone is useful parameter in microcell path loss models

PL(dB) = 40log d - (10logGt + 10logGr + 20log ht + 20 log hr ) (3.53)

PL = 1

4

22

d

hhGG

P

P rtrt

r

t

• 3.50 must hold