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Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W: www.eee.bham.ac.uk/ConstantinouCC/ E: [email protected]
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Page 1: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

Introduction to Radiowave Propagation

Dr Costas ConstantinouSchool of Electronic, Electrical & Computer Engineering

University of BirminghamW: www.eee.bham.ac.uk/ConstantinouCC/

E: [email protected]

Page 2: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

Introduction

• For an overview, see Chapters 1 – 4 of L.W. Barclay (Ed.), Propagation of Radiowaves, 2nd Ed., London: The IEE, 2003

• The main textbook supporting these lectures is: R.E. Collin, Antennas and Radiowave Propagation, New York: McGraw-Hill, 1985

Page 3: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

Introduction (cont.)

• Simple free-space propagation occurs only rarely• For most radio links we need to study the influence

of the presence of the earth, buildings, vegetation, the atmosphere, hydrometeors and the ionosphere

• In this lectures we will concentrate on simple terrestrial propagation models only

Page 4: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

Radio SpectrumSymbol Frequency range Wavelength, Comments

ELF < 300 Hz > 1000 km Earth-ionosphere waveguide propagationULF 300 Hz – 3 kHz 1000 – 100 km

VLF 3 kHz – 30 kHz 100 – 10 km

LF 30 – 300 kHz 10 – 1 km Ground wave propagation

MF 300 kHz – 3 MHz 1 km – 100 m

HF 3 – 30 MHz 100 – 10 m Ionospheric sky-wave propagation

VHF 30 – 300 MHz 10 – 1 m Space waves, scattering by objects similarly sized to, or bigger than, a free-space wavelength, increasingly affected by tropospheric phenomena

UHF 300 MHz – 3 GHz 1 m – 100 mm

SHF 3 – 30 GHz 100 – 10 mm

EHF 30 – 300 GHz 10 – 1 mm

8 1; 3 10 msc f c

Page 5: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

Electromagnetic waves

• Spherical waves– Intensity (time-average)– Conservation of energy; the inverse square law

HES

212Wm

Page 6: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

Electromagnetic waves

• Conservation of energy; the inverse square law– Energy cannot flow perpendicularly to, but flows along

“light rays”

2

dtransmitte

2

steradians ofsector angular an in dtransmitte

2

221122

21

2

1

1

2

4

11

21

r

P

rl

Prr

PAAPr

r

A

A

l

AA

r

r

rEr

rrr

r

Page 7: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

Free-space propagation

• Transmitted power• EIPR (equivalent isotropically radiated power)• Power density at receiver

• Received power

• Friis power transmission formula

txP

txtx PG

2txtx

rx 4 R

PG

S

4;

4

2

rxrxrx

2txtx

rx GAAR

PGP ee

2

rxtxtx

rx

4

RGG

P

P

Tx Rx

R

Page 8: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

Free-space propagation (cont.)• Taking logarithms gives

where is the free-space path loss, measured in decibels

• Maths reminder

R

GGPP4

log20log10log10log10log10 10rx10tx10tx10rx10

cbcb aaa logloglog ,loglog bcb ac

a

dBdBidBidBWdBW 0rxtxtxrx LGGPP

0L

dB4

log20 100

R

L

kmdfL 10MHz100 log20log204.32dB

,log

loglog

a

bb

c

ca

Page 9: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

Basic calculations

• Example: Two vertical dipoles, each with gain 2dBi, separated in free space by 100m, the transmitting one radiating a power of 10mW at 2.4GHz

• This corresponds to 0.4nW (or an electric field strength of 0.12mVm-1)

• The important quantity though is the signal to noise ratio at the receiver. In most instances antenna noise is dominated by electronic equipment thermal noise, given by where

is Boltzman’s constant, B is the receiver bandwidth and T is the room temperature in Kelvin

0.801.0log202400log204.32dB 10100 L

0.940.802log102log1010log10dBW 10102

10rx P

TBkN B123 JK1038.1 Bk

Page 10: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

Basic calculations (cont.)

• The noise power output by a receiver with a Noise Figure F = 10dB, and bandwidth B = 200kHz at room temperature (T = 300K) is calculated as follows

• Thus the signal to noise ratio (SNR) is given by

FTBkN B 1010 log10log10dBW

10log10102003001038.1log10dBW 10323

10 N

dBm 8.110dBW 8.140 N

8.1400.94dBWdBWdB NPSNR

dB 8.46SNR

Page 11: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

Basic calculations (cont.)

Page 12: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

Propagation over a flat earth• The two ray model (homogeneous ground)

– Valid in the VHF, band and above (i.e. f 30MHz where ground/surface wave effects are negligible)

– Valid for flat ground (i.e. r.m.s. roughness z < , typically f 30GHz)– Valid for short ranges where the earth’s curvature is negligible (i.e. d <

10–30 km, depending on atmospheric conditions)

z

ht

hr

d

r1

r2 air, 0, 0

ground, r, 0,

Tx

Rx

P

x

Page 13: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

Propagation over flat earth• The path difference between the direct and ground-reflected

paths is and this corresponds to a phase difference

• The total electric field at the receiver is given by

• The angles and are the elevation and azimuth angles of the direct and ground reflected paths measured from the boresight of the transmitting antenna radiation pattern

12 rrr 12 rrk

Γ.ee

eeE

,ˆ,ˆexp

60

,ˆ,ˆexp

60,

2

2rad

1

1rad

TT

TT

ggr

crtjP

ggr

crtjPr

,,, 21 rrr EEE

Page 14: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

Reflection of plane waves• Reflection coefficient is a tensor

• The reflection coefficient can be resolved into two canonical polarisations, TE and TM and has both a magnitude and phase

2

0

20TE

sincos

sincos

j

j

r

r

2

00

200||TM

sincos

sincos

jj

jj

rr

rr

ir EΓE

.

jexp

Plane of incidence

Page 15: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

15 Mobihoc '03 Radio Channel Modelling Tutorial

Reflection of plane waves• Typical reflection

coefficients for ground as a function of the grazing angle (complement of the angle of incidence). In this instance,

12 Sm10,15 r

Pseudo-Brewster angle

Page 16: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

Propagation over flat earth

• This expression can be simplified considerably for vertical and horizontal polarisations for large ranges d >> ht, hr, ,

d

hkhhhdhhdkrrk rt

rtrt

2222212

onpolarisati h.for ˆ

onpolarisati for v.cosˆ,ˆ,ˆ

txy

txzTT G

Ggg

e

eee

pol. h.for ˆ

pol. for v.cosˆ,ˆ,ˆ

TE

TM

txy

txzTT G

Ggg

e

eee.Γ

1hTEvTM

dhhdrrt

11122

1

dhhdr

rt

11122

2

Page 17: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

Propagation over flat earth

• There are two sets of ranges to consider, separated by a breakpoint

jEE hvhv exp1 ,0,

2sin4exp1 20

2

0 rxrxrx PjPP

d

hhPP rt

rxrx 2

sin4 20

22sin&

4

22

b

rt dhh

d

22

4sin&22

2

bdd

Page 18: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

Propagation over flat earth• Thus there are two simple propagation path loss laws

where l is a rapidly varying (fading) term over distances of the scale of a wavelength, and

This simplifies to

• The total path loss (free space loss + excess path loss) is independent of frequency and shows that height increases the received signal power (antenna height gain) and that the received power falls as d-4 not d-2

cddlLL for0.3dB 0

cddLL forlog20dB 100

d

hhdL rt

4

log204

log20dB 1010

rt hhdL 101010 log20log20log40dB

Page 19: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

Propagation over flat earthTypical ground (earth), withr = 15= 0.005Sm-1

ht = 20m andhr = 2m

deep fade1/d2 power law regime (d < dc)

1/d4 power law regime (d > dc)

Page 20: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

Propagation over flat earth

• When ht = 0 or hr = 0

• This implies that no communication is possible for ground based antennas – (not quite true in practice)

• Furthermore, for perfectly conducting ground and vertical polarisation at grazing incidence,

02

sin4 20

d

hhPP rt

rxrx

1vTM

d

hhPP rt

rxrx 2

cos4 20

Page 21: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

Propagation over flat earth

• Problem: A boat has an elevated antenna mounted on a mast at height ht above a highly conducting perfectly flat sea. If the radiation pattern of the antenna approximates that of a vertically polarised current element, i.e. , determine the in-situ radiation pattern of the antenna and in particular the radiation pattern nulls as a function of the elevation angle above the horizon.

• Answer:

cose

tan

2coscosˆ th

f e

,2,1,0,4

12

n

h

n

t

Page 22: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

Path clearance on LOS paths

• Assume that in the worst case scenario we get the strongest possible scattering from the sub-path obstacle: specular reflection at grazing incidence

ht

d

r0

r1

Tx

Rx

P

hc

r01r02

hr

h

r11 r22

d1 d2

Page 23: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

Path clearance on LOS paths• The electrical path difference between the direct and

scattered rays from the top of the obstacle is,

• Since typically

02

220201

2201

0201121101

rhrrhrk

rrrrkrrkk

cc

chrr 0201,

21

2

21

2

0201

2

0202

2

020101

2

01

2

11

2

11

2

22

dd

dkh

dd

kh

rr

kh

rr

hrr

r

hrkk

c

cc

cc

Page 24: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

Path clearance on LOS paths• Additionally, comparing similar parallelograms gives,

• Under the assumptions made, the direct and scattered waves have similar magnitudes and differ in phase by due to the grazing incidence reflection

• If the electrical path difference is ≤ this corresponds to a first Fresnel zone path clearance

• Problem: Verify that the breakpoint distance in the two ray model corresponds to the point at which the first Fresnel zone touches the ground

cos21

h

d

dhdhh tr

c

d

ddhc

21

Page 25: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

Site shielding

• We consider the two-dimensional problem of site shielding by an obstacle in the line-of-sight path for simplicity (rigorous diffraction theory is beyond the scope of these introductory lectures)

• We invoke the Huygens-Fresnel principle to describe wave propagation:– Every point on a primary wavefront serves as the source of spherical

secondary wavelets such that the primary wavefront at some later time is the envelope of these wavelets. Moreover, the wavelets advance with a speed andfrequency equal to that of the primary wave at each point in space. Huygens's principle was slightly modified by Fresnel to explain why no back wave was formed, and Kirchhoff demonstrated that the principle could be derived from the wave equation

Page 26: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

Site shielding

Page 27: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

T

R

P

d1

d2

d1

d1

r = d2 + P

O

observation plane

perfectly absorbing knife-edge

du

u0 (u0 > 0 path obstraction) (u0 < 0 path clearance)

u

Site shielding

Page 28: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

Site sheilding

• The Kirchhoff integral describing the summing of secondary wavefronts in the Huygens-Fresnel principle yields the field at the receiver

where k1 describes the transmitter power, polarisation and radiation pattern, f(r) describes the amplitude spreading factor for the secondary waves (2D cylindrical wave f(r) = r1/2, 3D spherical wave f(r) = r) and u1 is a large positive value of u to describe a distant upper bound on the wavefront

1

0

1

expu

u

jkrE R k du

f r

Page 29: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

Site shielding

• Stationary phase arguments (since the exponent is oscillatory, especially for high frequencies) show that only the fields in the vicinity of the point O contribute significantly to the field at R

• If point O is obstructed by the knife-edge, then only the fields in the vicinity of the tip of the knife-edge contribute significantly to the field at R

• Using the cosine rule on the triangle TPR, gives

2 2 22

2 2 2

2 1 2 1 1 2 11

2 cos

2 cos

r PR TP TR TP TR

ud d d d d d d

d

Page 30: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

Site shielding

• If we assume that d1, d2 >> , u (stationary phase and far-field approximations), then u/d1, << 1 and 2 <<

• Thus, using stationary phase arguments, we may only keep the fast varying exponential term inside the Kirchhoff integral and evaluate the slowly varying f(r) term at the stationary phase point O, to give,

1

22 2 2 2 22 2 1 2 1 2 1 2 2

1

2 1 2

1 2

2 2 2 2 12

2

ud d d d d d d d d

d

d du

d d

1

0

1 2

2

expexp

u

u

k jkdE R jk u du

f d

Page 31: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

Site shielding

• Since , we make the substitution

which simplifies the integral to the form,

where we have used the stationary phase argument to make the upper limit

• Using the definition of the complex Fresnel integral,

21 2

1 2

d dk u u

d d

21 2

21 2 2

2&

2

d d du k u k du

d d k

0

1 2 2

2 2

expexp 2

k jkdE R j d

k f d

2

0

exp 2x

F x j d

Page 32: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

Site shielding

• To determine k3 we let – and use F(–)= – F() and the fact that in this case we have free-space propagation (i.e. E(R) = E0(R)) , to get,

1 23

2 2

3 0

3 0

exp

1

2

k jkdk

k f d

E R k F F

jE R k F

0 3

0 03

1

11 2

E R k j

E R E Rk j

j

Page 33: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

Site shielding

• Therefore,

where,

• The path-gain factor, F, is given by,

• Useful engineering approximations:

0

0 21 exp 22

E RE R j j d

1 20 0

1 2

2 d du

d d

0

2

0

1exp 2

2

E RF j d

E R

10 10 0 0

210 0 0 0

210 0 0 0

20log 13 20log 2.4

20log 6.02 9.11 1.27 0 2.4

20log 6.02 9.0 1.65 0.8 0

F

F v

F v

Page 34: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

Site shielding

Page 35: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

Multipath propagation

• Mobile radio channels are predominantly in the VHF and UHF bands– VHF band (30 MHz f 300 MHz, or 1 m 10 m)– UHF band (300 MHz f 3 GHz, or 10 cm 1 m)

• In an outdoor environment electromagnetic signals can travel from the transmitter to the receiver along many paths– Reflection– Diffraction– Transmission– Scattering

Page 36: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

Multipath propagation

• Narrowband signal (continuous wave – CW) envelope

Area mean or path loss (deterministic or empirical)

Local mean, or shadowing, or slow fading (deterministic or statistical)

Fast or multipath fading (statistical)

Page 37: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

Multipath propagation• The total signal consists of

many components– Each component

corresponds to a signal which has a variable amplitude and phase

– The power received varies rapidly as the component phasors add with rapidly changing phases

Averaging the phase angles results in the local mean signal over areas of the order of 102

Averaging the length (i.e. power) over many locations/obstructions results in the area mean

The signals at the receiver can be expressed in terms of delay, and depend on polarisation, angle of arrival, Doppler shift, etc.

Page 38: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

Area mean models

• We will only cover the Hata-Okumura model, which derives from extensive measurements made by Okumura in 1968 in and around Tokyo between 200 MHz and 2 GHz

• The measurements were approximated in a set of simple median path loss formulae by Hata

• The model has been standardised by the ITU as recommendation ITU-R P.529-2

Page 39: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

Area mean models

• The model applies to three clutter and terrain categories– Urban area: built-up city or large town with large buildings

and houses with two or more storeys, or larger villages with closely built houses and tall, thickly grown trees

– Suburban area: village or highway scattered with trees and houses, some obstacles being near the mobile, but not very congested

– Open area: open space, no tall trees or buildings in path, plot of land cleared for 300 – 400 m ahead, e.g. farmland, rice fields, open fields

Page 40: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

Area mean models

where

cities small tomediumfor 8.0log56.17.0log1.1

MHz300 cities, largefor 1.154.1log29.8

MHz300 cities, largefor 97.475.11log2.3

94.40log33.18log78.4

4.528log2

log55.69.44

log82.13log16.2655.69

2

2

2

2

cmc

cm

cm

cc

c

b

bc

fhfE

fhE

fhE

ffD

fC

hB

hfA

DRBAL

CRBAL

ERBAL

logdB :areasopen

logdB :areassuburban

logdB :areasurban

Page 41: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

Area mean models

• The Hata-Okumura model is only valid for:– Carrier frequencies: 150 MHz fc 1500 MHz

– Base station/transmitter heights: 30 m hb 200 m

– Mobile station/receiver heights: 1 m hm 10 m

– Communication range: R > 1 km

– A large city is defined as having an average building height in excess of 15 m

Page 42: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

Local mean model• The departure of the local mean power from the area mean

prediction, or equivalently the deviation of the area mean model is described by a log-normal distribution

• In the same manner that the theorem of large numbers states that the probability density function of the sum of many random processes obeys a normal distribution, the product of a large number of random processes obeys a log-normal distribution

• Here the product characterises the many cascaded interactions of electromagnetic waves in reaching the receiver

• The theoretical basis for this model is questionable over short-ranges, but it is the best available that fits observations

Page 43: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

Local mean model• Working in logarithmic units (decibels, dB), the total path loss is

given by

where X is a random variable obeying a lognormal distribution with standard deviation (again measured in dB)

• If x is measured in linear units (e.g. Volts)

where mx is the mean value of the signal given by the area mean model

XdLdPL

2dB

2

dB

2exp2

1 XXp

2dBdB 2

lnlnexp

2

1

xmx

xxp

Page 44: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

Local mean model

• Cumulative probability density function

• This can be used to calculate the probability that the signal-to-noise ratio will never be lower than a desired threshold value. This is called an outage calculation

• Typical values of dB = 10 dB are encountered in urban outdoor environments, with a de-correlation distance between 20 – 80 m with a median value of 40 m

2erfc

2

11

2exp2

1cdf 2

dB2

dB

Threshold

dLL

dXXLPL

T

dLLT

Page 45: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

Fast fading models• Constructive and destructive

interference– In spatial domain– In frequency domain– In time domain (scatterers, tx and rx in

relative motion)• Azimuth dependent Doppler shifts

– Each multipath component travels corresponds to a different path length.

– Plot of power carried by each component against delay is called the power delay profile (PDP )of the channel.

– 2nd central moment of PDP is called the delay spread

P

Im

Re

Page 46: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

Fast fading models

• The relation of the radio system channel bandwidth Bch to the delay spread is very important– Narrowband channel (flat fading, negligible inter-symbol interference

(ISI), diversity antennas useful)– Wideband channel (frequency selective fading, need equalisation

(RAKE receiver) or spread spectrum techniques (W-CDMA, OFDM, etc.) to avoid/limit ISI)

• Fast fading refers to very rapid variations in signal strength (20 to in excess of 50 dB in magnitude) typically in an analogue narrowband channel– Dominant LOS component Rician fading– NLOS components of similar magnitude Rayleigh fading

1chB

1chB

Page 47: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

Fast fading models

• Working in logarithmic units (decibels, dB), the total path loss is given by

where Y is random variable which describes the fast fading and it obeys the distribution

for Rayleigh fading, where the mean value of Y is

YXdLdPL 10log20

80.012 Y

0,0

0,2

exp2

2

2

Y

YYY

Yp

Page 48: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

Fast fading models• For Rician fading

where ys is the amplitude of the dominant (LOS) component with power . The ratio is called the Rician K-factor. The mean value of Y is

The Rician K-factor can vary considerably across small areas in indoor environments

0,0

0,I2

exp202

22

2

Y

YYyyYY

Ypss

22sy 22

Rice 2syK

2exp2I2I12 10 KKKKKY

Page 49: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

Fading models

• Similar but much more complicated outage calculations– E.g. Rayleigh and log-normal distributions combine to give a Suzuki

distribution

• The spatial distribution of fades is such that the “length” of a fade depends on the number of dB below the local mean signal we are concerned with

Fade depth (dB) Average fade length ()

0 0.479

-10 0.108

-20 0.033

-30 0.010

Page 50: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

Tropospheric propagation

• Over long-distances, more than a few tens of km, and heights of up to 10 km above the earth’s surface, clear air effects in the troposphere become non-negligible

• The dielectric constant of the air at the earth’s surface of (approx.) 1.0003 falls to 1.0000 at great heights where the density of the air tends to zero

• A consequence of Snell’s law of refraction is that radiowaves follow curved, rather than straight-line trajectories

Page 51: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

Tropospheric propagation

• The variation of the ray curvature with refractive index is derived:AA: wavefront at time tBB: wavefront at time t + dt

AB and AB: rays normal to the wavefronts

: radius of curvature of AB

A

A

B

B

O

d

d dh

n + dn

n

c dtA B d v dt

nc dt

AB d d v dv dtn dn

d c c

dt n n dn d

Page 52: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

Tropospheric propagation

Retaining only terms which are correct to first order in small quantities,

But this is the curvature, C, of the ray AB, by definition. Furthermore,

For rays propagating along the earth’s surface is very small and we may take cos = 1. Moreover, n–1 1.

n n nd dn dnd

1 1

dn nd

dn

n d

cosdh d 1 1

cosdn

Cn dh

Page 53: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

Tropospheric propagation

• If n = constant, dn/dh = 0 C = 0 and the ray has zero curvature, i.e. the ray path is a straight line

• A ray propagating horizontally above the earth must have a curvature C = (earth’s radius)–1 = a–1 in order to remain parallel with the earth’s surface. But its actual curvature is given by C and not C.

• The difference between the two curvatures gives the curvature of an equivalent earth for which dn/dh = 0 and which has an effective radius ae,

dnC

dh

1 1 1

e

dn

a a dh ka

Page 54: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

Tropospheric propagation

• k is known as the k-factor for the earth• Typically, dn/dh –0.03910–6 m–1 1/(25,600 km)

• Therefore,

• The k-factor of the earth is k = 4/3

• The effective radius of the earth is ae = 4a/3

• These values are used in the standard earth model which explains why the radio horizon is bigger than the radio horizon

1 1 1 1

6,400 km 25,600 km 6,400 kmea k

Page 55: Introduction to Radiowave Propagation Dr Costas Constantinou School of Electronic, Electrical & Computer Engineering University of Birmingham W:

Tropospheric propagation

• Problem: Find the radio horizon of an elevated antenna at a height ht above the earth

• Answer: 2 e tR a h


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