Digital Signal Processing Reference
In-Depth Information
T
consists of three unknowns. Results will be given for moving least
squares approximations of order
n
=[
a
0
a
1
a
2
]
a
0
,
1 and 2.
The filtered RSSI at the node
N
i
is then given by:
ˆ
=
a
i
RSSI
i
= ˆ
p
(
x
i
).
(4.2)
Locality is obtained by weighting the neighboring nodes' contributions using the
distance-based weight function
w
ij
with local support
h
:
(
1
−
r
ij
)
3
if
r
ij
≤
1,
w
ij
=
(4.3)
0
otherwise,
where
r
ij
=
x
i
−
x
j
)/ h
.
So, although in practice it is only needed to fit a bi-variate polynomial at each
nodal position, the resulting
(
ˆ
RSSI is defined everywhere and will be
C
2
-continuous
thanks to the above defined smoothly vanishing weight function. This weight func-
tion decays fast enough to establish a true local fit [57]. Note that each smart node
computes its smoothed
ˆ
RSSI
i
estimate independently and only requires local RSSI
and GPS localization information to do so, which can be obtained by a single com-
munication step with its neighboring nodes if
h
is equal or smaller than the com-
munication range of the secondary network users. In the remainder, we will assume
that
h
is the communication range. This step hence requires
N
communications in
the network.
It is easy to see that for order
n
=
0, the minimizer of Eq.
4.1
equals the weighted
average measured RSSI:
j
w
ij
RSSI
j
j
w
ij
ˆ
RSSI
i
=
,
(4.4)
and thus the moving least squares approximation coincides with the well-known
moving average approximation method.
Intuitively, when the main goal of the smoothing operation is to average out spa-
tial variations, the moving average will result in adequate results. Alternatively, in
case larger neighborhoods are used and more measurement points are collected, it
might make sense to fit a local trend (e.g., linear or quadratic). Such a trend will
reduce the smoothing effect on the one hand, but will improve accuracy on the other
hand since there are more degrees of freedom for the parameters. This is illustrated
in Fig.
4.4
, left, using a simulated scenario (see also Sect.
4.3.5
) where there is
a known contour with a systematically increased noisiness of the RSSI measure-
ments. As can be seen on the figure, when less smoothing is required (less noise
power), a quadratic MLS approximates the propagation surface best. However, with
increasing noise power more smoothing is required and linear and constant MLS
start to outperform.
Choosing the order of approximation as function of the neighborhood size is a
delicate trade-off. The choice of local support
h
as used in Eq.
4.3
influences the
approximation quality. As shown in Fig.
4.4
, right, an optimal kernel width can be
found for all approximation orders and the optimal width typically increases with
approximation order.
