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æ
4
π
d
ö
æ
4
π
df
ö
æ
ö
4
π
L
(dB)
=
20 log
=
20 log
=
20 log
-
20 log
f
-
20 log
d
ç
÷
ç
÷
ç ÷
0
10
10
10
10
10
è
ø
è
ø
è ø
λ
c
c
Once again, it is conventional for radio propagation engineers to use MHz and
km instead of Hz and m in describing frequencies and distances respectively, so the
convention is to label the relative quantities with a subscript denoting the use of
non-SI units (rather than adopt units such as a dBMHz). Doing so involves changing
the conventional units of measuring the speed of light in vacuum from
c
=´
3
10 ms
8
-
1
to
c
=
, which renders the above expression in its commonly used
form in engineering,
0.3 km·MHz
L
(dB)
º =+
L
(dB)
32.4 20 log
f
+
20 log
d
0
10 MHz
10
km
where
is the radio wave frequency in MegaHertz and
k
d
is the antenna separa-
f
MHz
tion in kilometers.
A second example of a simple, deterministic path-loss example occurs when the
two radio nodes are situated over a flat ground. For sufficiently long antenna separa-
tions, geometrical optics approximations apply and the method of images can be used
together with the fact that at near-grazing incidence the ground reflection coefficient
is asymptotically approaching the value −1, to give
L
(dB)
=
40 log
d
-
20 log
h
-
20 log
h
valid for
d hh
λ
>
4
/
10
10
tx
10
rx
tx rx
where
{tx,rx
h
is the transmitter/receiver antenna height over ground measured in the
same units as the antenna separation
d
and
l
is the radio carrier wavelength also
measured in the same units.
For more general radio propagation environments where multipath propagation can
occur and obstructions can give rise to spatially extended geographical areas of reduced
average signal strength (these correspond to geometrical optics shadow regions), we
resort to nondeterministic expressions for the point-to-point path-loss equation. Such
expressions have been empirically derived and verified through numerous measurement
campaigns for land mobile radio systems and in their majority cannot be extrapolated
to low antenna heights applicable to mobile ad hoc networking research [
4, 8
].
Such path-loss models typically take the form,
L A dhh x y
σ
(dB)
=+
{
20 log
γ
- - + +
αβ
} { } { }
10
tx
rx
where the parameters
A
,
a
,
b
, and the path-loss exponent
g
are empirically found to
be frequency-dependent and change depending on the environment (e.g., urban,
rural, open), whether there exists a line-of-sight (LOS) between the transmitter and
receiver and also may only be valid for specific ranges of the separation
d
. The
determination of these parameters is made through least-squares fitting of the above
expression on data obtained from extensive measurement campaigns in “typical”
propagation environments. Such an expression is only valid for the class of environ-
ments in which the measurements were performed and in the case of [
4
] which we
shall use in this work for low antenna heights as terms depending on the logarithm
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