The harmonic solution
• Source signal :
• x(t) = cos2πft
• Because of the h(t) of the free space channel:
• y(t) = cos2πf(t – R/c)/2πR
• 1/ 2πR can be replaced by ρ in order to take into
account additional attenuation
Carrier and modulation
• x(t) = a(t)cos2πft
• In the ideal channel the group delay and the
phase delay are equal
y(t) = a(t-R/c)cos2πf(t – R/c)/2πR
R is present in the phase delay and in the
modulation delay
Free Space Propagation
 λ 
 R0 
PR = PT GT GR 
 = P(R0)  
 4π R 
 R
2
2
L(R) = (4πR/λ)2 Isotropic attenuation
In area systems this law is hardly verified
If the exponent 2 is substuituted with α the experimental
verification is extended
Dominant Path Loss Component
Ro is the limit distance for using the formulas
Pr = Pt +Gt + Gr - L(R).
In dB normalizzato ad 1dBm
Example of link budget
Pt
dBm 30
Gt
Gr
dB
dB
2
2
P = Pt +Gt + Gr dBm 34
Pr
dBm - 100
L(d) = P – Pr
dB
134
TROPOSFERIC PROPAGATION
Spatial wave with modifications with respect to free
space
* Refraction and Atmospheric Absorption
* Ground and Terrain Effect
* Obstacles Diffraction
* Multipath phenomena
Multipath propagation often occurs only
in local area
SRB
If it is the case there is a
dominant path loss
component
Macrocellular enviroment
Mob.
Base station
antenna
Starting point
Spot x1
x
Mean received power P(x)
Right scale
Left scale
P(x1)
s(x1)
0
x1
Starting
(spot x1)
point
x
Distance from the base station, km
Received signal as a function of the
distance driven x, s(x)
Distance from the antenna is practically proportional to
the driven one
PATH LOSS COMPONENTS
1-Dominant component decreasing as a
function of the distance
2-Slow fluctuations (scale of obstacle
dimensions)
3-Fast fluctuations (scale of
wavelenght)
1- Average propagation conditions in the
considered area differs from free space
(exponent > 2) propagation
2-Obstruction phenomena are present in
various locations with different effect on
propagation conditions (shadowing)
3- The final received signal is the result of
different components reaching the
receiver following different propagation
paths (multipath)
Right scale
Left scale
P(x1)
s(x1)
0
Starting
point
x1
x
Μ(spot
eanx1received
power α=3.5
)
_____
Distance from the base station, km
Mean received power P(x) α=2
Received signal as a function of the
distance x, s(x)
1-Dominant component decreasing
as a function of the distance
 λ 
 R0 
PR = PT GT GR 
 = P( R 0)  
 4π R 
R
2
 λ 
 R0
PR = PT GT GR 
 = P(R0)  
4π R
R
2
R0
PR = P(R0)  
R
α
 R0 
PR = P( R 0)  
R
α
2
2
Mean received power P(x)
Right scale
Left scale
P(x1)
s(x1)
0
x1
Starting
(spot x1)
point
Slow
fluctuations
x
Distance from the base station, km
Received signal as a function of the
distance x, s(x)
Scale obstructions dimensions
Attenuation Components
2. Slow oscillations (1/2)
Slow
oscillations
:
Slow oscillations can be represented by a lognormal distribution
f L (l ) =
1
2π σl
⋅e
−
(l − µ )2
2σ 2
Attenuation Components
2. Slow oscillations (2/2)
In a mobile communication there are obstacles present wich can
produce diffraction loss
Lognormal shadowing
BS
MS
Mean received power P(x)
Right scale
Left scale
P(x1)
s(x1)
0
x1
Starting
(spot x1)
point
fast
fluctuations
x
Scale wavelength dimension
Distance from the base station, km
Received signal as a function of the
distance x, s(x)
3-Propagation characteristics in mobile
radio systems
Multipath phenomena
Fast signal variation with distance
Frequency variation
Fast variations in a small area
Signal strength (dB)
15
Measured data
5 o 11.200 MHz
x 836 MHz
-5
-15
-25
-35
x
o
x
o
xo
xo
x
xo
xo
x o
Rayleigh
Distribution
99.9999.9699.8 99 96 90 70 30
1 0.01
Cumulative distribution for 836 and 11200 MHz
Rayleigh distribution
for the absolute value of the
field E (variable r)
over some tens of wave-lengths
 r 
p (r ) = 2 exp − 2 
σ
2
σ


r
2
p(r) =
r
σ2
 r2 
exp − 2 
 2σ 
prob[r ≤ R ] = P(R )
R
= ∫ p ( R) dr
prob[r ≤ R ] = P (R )
R
=
∫ p( R) dr
0
− R 2 
= 1 - exp
2 
 2σ 
0
 R 
= 1 - exp − 2 
 2σ 
2
{ }
E r = 2σ
2
E{r} = σ
2
π
2
{ }
E r =2 σ
2
E{r} = σ
2
π
2
= 1 .2533
σ
= 1.2533σ
rM = median value = 1.1774 σ
p(r)
σ
123
1= median value 1.1774 σ
2= mean value 1.2533 σ
r
3=RMS 1.41 σ
The fast fading component can be computed
only in same simple cases.
The evaluation of its effects are to be done in
a statistic way: tipically as a margin.
In fact the impairments produced in the
receiver are function of the characteristics of
signal, modulation etc. The margin is defined
by the system designer.
In planning design this component is not
considered: the field strenght to be offered is
intended as the mean value over a distance
of same wavelengths.