Radiowave Propagation
Introduction
• The main textbook supporting these lectures is: R.E.
Collin, Antennas and Radiowave Propagation, New
York: McGraw-Hill, 1985
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
Radio Spectrum
Symbol Frequency range Wavelength,  Comments
ELF
< 300 Hz
> 1000 km
ULF
300 Hz – 3 kHz
Earth-ionosphere waveguide
1000 – 100 km propagation
VLF
3 kHz – 30 kHz
100 – 10 km
LF
30 – 300 kHz
10 – 1 km
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
UHF
300 MHz – 3 GHz 1 m – 100 mm
SHF
3 – 30 GHz
100 – 10 mm
Space waves, scattering by objects
similarly sized to, or bigger than, a freespace wavelength, increasingly affected
by tropospheric phenomena
EHF
30 – 300 GHz
10 – 1 mm
c  f  ;
Ground wave propagation
c  3 108 ms 1
Electromagnetic waves
• Spherical waves

 
1
– Intensity (time-average) Wm   S  2 E  H
2
– Conservation of energy; the inverse square law
Electromagnetic waves
• Conservation of energy; the inverse square law
– Energy cannot flow perpendicularly to, but flows along
“light rays”



 r2  A1 r12
 2  PA1   r1 A1   r2 A2  PA2
 
 r1  A2 r2
 

1
1
  r   2  Er  
r
r
 Ptransmitted in an angular sector of l steradians
 r  
lr2
 Ptransmitted
 r  
4 r 2
Free-space propagation
Tx
R
• Transmitted power Ptx
• EIPR (equivalent isotropically radiated power) Gtx Ptx
• Power density at receiver
• Received power

Gtx Ptx
S rx 
4 R 2
2
Gtx Ptx rx

rx
Prx 

A
;
A
e
e  Grx
2
4 R
4
• Friis power transmission formula
  
Prx

 GtxGrx 
Ptx
 4 R 
2
Rx
Free-space propagation (cont.)
• Taking logarithms gives
 4 R 
10 log 10 Prx  10 log 10 Ptx  10 log 10 Gtx  10 log 10 Grx  20 log 10 

  
Prx dBW   Ptx dBW   Gtx dBi   Grx dBi   L0 dB
where L0 is the free-space path loss, measured in decibels
 4 R 
L0  20 log 10 
 dB
  
L0 dB  32.4  20 log 10 f MHz  20 log 10 d km
• Maths reminder
log c b
c
log a b   c  log a b, log a b  
, log a b  c   log a b  log a c
log c a
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
L0 dB  32.4  20 log 10 2400  20 log 10 0.1  80.0


Prx dBW   10 log 10 10 2  10 log 10 2  10 log 10 2  80.0  94.0
• 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 N  k BTB
where k B  1.38 1023 JK 1 is Boltzman’s constant, B is the
receiver bandwidth and T is the room temperature in Kelvin
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
N dBW   10 log 10 k BTB   10 log 10 F 


N dBW   10 log 10 1.38 10 23  300  200 103  10 log 10 10
N  140.8 dBW  110.8 dBm
• Thus the signal to noise ratio (SNR) is given by
SNRdB  PdBW   N dBW    94.0  140.8
SNR  46.8 dB
Basic calculations (cont.)
Propagation over a flat earth
• The two ray model (homogeneous ground)
z
Tx
r1
ht

Rx
r2

P
hr
air, e0, m0
ground, er, m0, s
d
– 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)
x
Propagation over flat earth
• The path difference between the direct and ground-reflected
paths is r  r2  r1 and this corresponds to a phase difference
  k r2  r1 
• The total electric field at the receiver is given by



Er ,    E1 r ,    E 2 r ,  

exp  j t  r1 c 
Er ,    60 Prad 
 eˆ  gT  ,    eˆ  gT  ,  
r1
exp  j t  r2 c 
 60 Prad 
 eˆ  gT  ,    eˆ  gT  ,  . Γ
r2
• 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
Reflection of plane waves
• Reflection coefficient is a tensor
r
i
E  Γ.E
• The reflection coefficient can be resolved
into two canonical polarisations, TE and TM
and has both a magnitude and phase
   exp  j 
 TE    
 TM
cos  
cos  
e r  js
||
 
e r  js
e r  js e 0   sin 2 
e r  js e 0   sin 2 
e 0  cos   e r  js
e 0  cos   e r  js
e 0   sin 2 
e 0   sin 2 
Plane of
incidence
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,
Pseudo-Brewster angle
e r  15, s  102 Sm 1
15
Mobihoc '03 Radio Channel Modelling
Tutorial
Propagation over flat earth
• This expression can be simplified considerably for vertical and
horizontal polarisations for large ranges d >> ht, hr, ,
2kht hr
2
2 
2
2

  k r2  r1   k  d  ht  hr   d  ht  hr   


d
1
1
1
1
1
1




2
2
2
2
r1
d
r2
d
d  ht  hr 
d  ht  hr 
eˆ z Gtx   cos for v. polarisati on
eˆ  gT  ,    eˆ  gT  ,    
for h. polarisati on
 eˆ y Gtx  
eˆ z TM  Gtx   cos for v. pol.
Γ .eˆ  gT  ,    eˆ  gT  ,    
for h. pol.
 eˆ y TE  Gtx  
TM  v   TE  h   1
Propagation over flat earth
Ev ,h  E0 1  v ,h exp  j 
Prx  Prx0 1  exp  j   4 Prx0 sin 2  2
2
 2ht hr 
Prx  4 Prx0 sin 

 d 
• There are two sets of ranges to consider, separated by a
breakpoint
4ht hr
 
   

 d
 d b & sin 

2
2

2
 2 
 
2   

 d  d b & 4sin 
 2
2
2
 2 
2
Propagation over flat earth
• Thus there are two simple propagation path loss laws
LdB  L0  3.0  l for d  d c
where l is a rapidly varying (fading) term over distances of the
scale of a wavelength, and
LdB  L0  20 log 10   for d  d c
This simplifies to
 4ht hr 
 4d 
LdB  20 log 10 

  20 log 10 
  
 d 
LdB  40 log 10 d  20 log 10 ht  20 log 10 hr
• 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
Propagation over flat earth
Typical ground
(earth), with
er = 15
s= 0.005Sm-1
ht = 20m and
hr = 2m
1/d4 power law regime (d > dc)
1/d2 power law regime (d < dc)
deep fade
Propagation over flat earth
2  2ht hr

P

4
P
sin
• When ht = 0 or hr = 0

rx
rx 0
 d

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, TM  v   1
 2ht hr 
Prx  4 Prx0 cos 

 d 
2
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. ê cos , 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.
2ht


• Answer: f    eˆ  cos  cos
tan  
 


2n  1 
, n  0,1,2,
4 ht
Path clearance on LOS paths

Tx
r01
r02
r0
hc
r1
r11
Rx
r22
ht
hr
h
P
d2
d1
d
• Assume that in the worst case scenario we get the strongest
possible scattering from the sub-path obstacle: specular
reflection at grazing incidence
Path clearance on LOS paths
• The electrical path difference between the direct and
scattered rays from the top of the obstacle is,
k  k r1  r0   k r11  r12  r01  r02 
k
 r
2
01
r
 hc2  r01 
2
02
 hc2  r02

• Since typically r01, r02  hc

 

hc2
hc2
k  k  r01 
 r01    r02 
 r02 
2r01
2r02
 


khc2  1
1  khc2  1 1 
   
  

2  r01 r02 
2  d1 d 2 
khc2 d

2d1d 2
Path clearance on LOS paths
• Additionally, comparing similar parallelograms gives,
h d hd

hc   r 1 t 2  h  cos 
d


• 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
hc 
d1d 2
d
• Problem: Verify that the breakpoint distance in the two ray
model corresponds to the point at which the first Fresnel zone
touches the ground
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
Site shielding
Site shielding
P
d
u
d
T
a
u
O
1
R
d
u0 (u0 > 02  path
obstraction)
(u0 < 0  path
clearance)
d
d
r = d2 + 
1
1
P
perfectly
absorbin
g knifeedge
observati
on plane
Site sheilding
• The Kirchhoff integral describing the summing of secondary
wavefronts in the Huygens-Fresnel principle yields the field at
the receiver
u
exp   jkr 
E  R   k1 
du
f r 
u
1
0
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
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
r 2   PR   TP   TR   2 TP TR  cos a
2
 d2   
2
2
2
  d1  d 2    d1 
2
2
u 
 2  d1  d 2  d1  cos  
 d1 
Site shielding
• If we assume that d1, d2 >> , u (stationary phase and far-field
approximations), then u/d1, a << 1 and  2 << 
2


u
2
2
2
2
2
d 2  2d 2  
2d1  d 2  2d1d 2  2  d  d1d 2  1  2 
 2d1 
1

d1  d 2
u
2d1d 2
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,
k1 exp   jkd 2  u
E  R
exp  jk  u  du

f  d2 
u
1
0
Site shielding
• Since k  u 
 d1  d 2 2
u , we make the substitution
 d1d 2
2  d1  d 2 
 u
 d1d 2
 2
d
k 2 u  k 
& du 
2
k2
which simplifies the integral to the form,
k1 exp   jkd 2  
2
E  R
exp

j

2 d


k2 f  d 2   0
where we have used the stationary phase argument to make
the upper limit 
• Using the definition of the complex Fresnel integral,
x
F  x   exp  j 2 2 d
0
Site shielding
k3
k1 exp   jkd 2 
k2 f  d 2 
E  R
k3  F     F  0  
1  j

k3 
 F  0  
 2

• 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,
E  R
E0  R 
k3 1  j 
E0  R  E0  R 
k3 

1  j 
1 j
2
Site shielding
• Therefore, E  R 
where,  0  u0

E0  R 
1  j   exp  j 2 2 d
2
0
2  d1  d 2 
 d1d 2
• The path-gain factor, F, is given by,
F
E  R
1

E0  R 
2

2
exp

j

2 d
 
0
• Useful engineering approximations:
20 log10 F
13  20 log10  0
 0  2.4
20 log10 F
6.02  9.11 0  1.27v02
0   0  2.4
20 log10 F
6.02  9.0 0  1.65v02
 0.8   0  0
Site shielding
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
Multipath propagation
• Narrowband signal
(continuous wave
– CW) envelope
Area mean or path
loss (deterministic or
empirical)
Fast or multipath
fading (statistical)
Local mean, or shadowing, or slow
fading (deterministic or statistical)
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  102
 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.
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
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
Area mean models
urban areas :
LdB  A  B log R  E
suburban areas : LdB  A  B log R  C
open areas :
LdB  A  B log R  D
where
A  69.55  26.16 log f c  13.82 log hb
B  44.9  6.55 log hb
C  2log  f c 28  5.4
2
D  4.78log f c   18.33 log f c  40.94
2
E  3.2log 11.75hm   4.97 for large cities, f c  300MHz
2
E  8.29log 1.54hm   1.1
2
for large cities, f c  300MHz
E  1.1log f c  0.7 hm  1.56 log f c  0.8 for medium to small cities
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
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
Local mean model
• Working in logarithmic units (decibels, dB), the total path loss
is given by
PLd   Ld   X s
where Xs is a random variable obeying a lognormal
distribution with standard deviation s (again measured in dB)
1
2
p X s  
exp  X s2 2s dB
s dB 2
• If x is measured in linear units (e.g. Volts)
 ln x  ln mx 
1
p x  
exp 

2
2s dB 
s dB x 2



where mx is the mean value of the signal given by the area
mean model
Local mean model
• Cumulative probability density function
LT  L  d 
cdf PL  LThreshold   

1
s dB 2


2
exp  X 2 2s dB
dX
1
 L  Ld  
 1  erfc  T

2
2 

• This can be used to calculate the probability that the signal-tonoise ratio will never be lower than a desired threshold value.
This is called an outage calculation
• Typical values of sdB = 10 dB are encountered in urban
outdoor environments, with a de-correlation distance
between 20 – 80 m with a median value of 40 m
Fast fading models
Im
• Constructive and destructive
interference
– In spatial domain
– In frequency domain
– In time domain (scatterers, tx and rx in P
relative motion)
Re
• 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 
t
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
1
(ISI), diversity antennas useful) Bch  
– Wideband channel (frequency selective fading, need equalisation
(RAKE receiver) or spread spectrum techniques (W-CDMA, OFDM,
1
etc.) to avoid/limit ISI) Bch  
• 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
Fast fading models
• Working in logarithmic units (decibels, dB), the total path loss
is given by
PLd   Ld   X s  20 log 10 Y
where Y is random variable which describes the fast fading
and it obeys the distribution
Y
 Y2 
 2 exp   2 , Y  0
pY    
 2 
 0,
Y 0

for Rayleigh fading, where the mean value of Y is
Y    2  1    0.80
Fast fading models
• For Rician fading
Y
 Y 2  ys2   Yy s 
 I 0  2 , Y  0
 2 exp  
2
pY    
2    

 0,
Y 0

where ys is the amplitude of the dominant (LOS) component
with power ys2 2. The ratio K Rice  y s2 2  2 is called the Rician
K-factor. The mean value of Y is
Y    21  K  I0 K 2  K I1 K 2exp  K 2
The Rician K-factor can vary considerably across small areas in
indoor environments
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
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 nonnegligible
• 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
Tropospheric propagation
• The variation of the ray
curvature with refractive index is
derived:
B
B
A
d
AA: wavefront at time t
BB: wavefront at time t + dt
AB and AB: rays normal to the
wavefronts
 : radius of curvature of AB
c  dt
AB    d  v  dt 
n
c  dt
AB     d    d   v  dv   dt 
n  dn
d
c
c



dt n   n  dn    d  
n + dn

dh
A
n

d
O
Tropospheric propagation
n  n  nd    dn  dnd 
Retaining only terms which are correct to first order in small
quantities,
 dn  nd 
1
1 dn


n d
But this is the curvature, C, of the ray AB, by definition.
Furthermore, dh  d  cos 
1
1 dn
C

cos 

n dh
For rays propagating along the earth’s surface  is very small
and we may take cos = 1. Moreover, n–1  1.
Tropospheric propagation
C
dn

dh
• 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,
1 1 dn
1
 
ae a dh ka
Tropospheric propagation
• k is known as the k-factor for the earth
• Typically, dn/dh  –0.03910–6 m–1  1/(25,600 km)
• Therefore, 1
1
1
1



ae 6, 400 km 25,600 km k  6, 400 km 
• 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
Tropospheric propagation
• Problem: Find the radio horizon of an elevated antenna at a
height ht above the earth
• Answer: R  2ae ht