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of the antenna heights are more commonplace when at least one of these heights
above the ground is significant. Unlike cellular radio systems, sensor networks have
lowlying antennas and only few empirical propagation models apply to them.
Finally, the last two terms, in curly brackets,
x
σ
and
y
are zero mean random
variables taken from a statistical distribution that describes shadow fading and
multipath fading, respectively. The first term in the curly brackets is commonly
referred to as the area mean path loss, the second term is referred to as the local
mean, shadowing or slow fading, and the third term is referred to as the fast or
multipath fading. The local mean,
x
σ
, arises from the fact that a statistically
significant number of measurement locations occurs when large (on the linear scale
of a wavelength
l
) obstacles exist between the transmitter and the receiver, severely
limiting the amount of radiation intensity at the receiver. At each and every location,
the variable
x
σ
takes definite values that are dependent on the specific buildings and
ground undulations of the area surrounding both the transmitter and receiver, but in
the absence of accurate topography information, it is treated as a stochastic variable.
This stochastic variable
x
σ
is empirically, typically found to obey a lognormal
distribution,
{
}
1
2
2
2
dB
dB
px
(
)
=
exp

x
σ
σ
2
σ
σπ
2
dB
where
d
σ
is the shadowing standard deviation measured in decibels. The reasons
for which
x
σ
obeys a normal (Gaussian) distribution with logarithmic units are not
wellunderstood, although plausibility arguments loosely based on the law of large
numbers have been advocated at times.
The multipath fading which arises from the constructive and destructive inter
ference of a large number of waves scattered by different objects in the propagation
environment,
y
, is found to obey various statistical distributions, e.g., Rayleigh,
Rician, Nakagami, Weighbull that are strongly environmentdependent (e.g., if
there is a dominant scattered wave component, if there is diffuse scattering) and its
length scale of variation is of the order of the wavelength
l
.
For the purposes of our discussion, it is important to point out that, when the
receiver is moving along a trajectory,
x
σ
and
y
vary on significantly different spatial
scales. The local mean variation is empirically found to be spatially correlated. Taking
many measured instances of
x
σ
as a function of distance along the trajectory allows us
to compute the ensemble averaged autocorrelation function
2
+D
,
which is empirically found to be approximated well by an exponential function
exp{
xdxd d x
σ
() (
)/
σ
σ
. The
e

correlation distance
d
is related to the 50% correlation distance
1
D
d
δ
/ }
d
δ=
. As is expected, this is strongly environmentdependent
(e.g., depends on ground undulations, density, and type of urban/suburban buildings)
and normally lies in the range
[
6
] by
/ln 2
d
50%
50%
d
<<
in urban environments [
48
].
On the other hand, the multipath fading is known to have a spatial correlation
distance of the order of
/λ
[
7
], which need not concern us at all, as most radio
systems tend to employ fade mitigation diversity techniques that smear out the
multipath fading pattern in the immediate vicinity of the receiver.
20 m
80 m
50%
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