Digital Signal Processing Reference
In-Depth Information
Let us assume that the fusion center is equipped with the knowledge of channel
fading coefficients.
3
Then, the probability of the fusion error (
5.9
) is simplified to:
m
2
.
L
H
2
G
2
σ
v
H
2
G
2
+
P
e
=
Q
(5.13)
σ
w
=
1
It is interesting to note
tha
t the probability of error at the fusion center has a
performance floor of
(
√
Lγ
2
)
when
G
2
tends to infinity (
Q
=
1
,
2
,...,L
), i.e.,
L
H
2
G
2
L
σ
v
lim
G
→∞
σ
v
H
2
G
2
+
σ
w
=
,
(5.14)
=
1
√
Lγ
0
2
.
lim
G
→∞
P
e
=
Q
Intuitively, (
5.14
) says that, for a fixed
L
, the performance attained is determined
mainly by the observation quality at local sensor nodes regardless of the quality of
the wireless channel.
Using the fusion error probability given in (
5.13
), when the local observations
are i.i.d., the first inequality in (
5.11
) can be expressed as
L
H
2
G
2
σ
v
H
2
G
2
+
β
≤
(5.15)
σ
w
=
1
where
2
m
Q
−
1
(ε).
β
=
(5.16)
Then, the optimization problem can be stated as:
⎧
⎨
min
L
=
1
G
2
subject to
β
2
−
L
=
1
H
2
G
2
(5.17)
σ
v
H
2
G
2
+
σ
w
≤
0
,
⎩
G
≥
0
,
=
1
,
2
,...,L.
The Lagrangian cost function is given by
L
L
L
H
2
G
2
σ
v
H
2
G
∗
2
+
G
k
+
λ
0
β
2
L
(G,λ
0
,μ
k
)
=
−
+
μ
k
(
−
G
k
)
(5.18)
σ
w
k
=
1
k
=
1
k
=
1
3
The assumption that the transmission is idealized, i.e., the information sent from local sensors
is assumed to be received intact at the fusion center may be reasonable for some applications,
but it may not be realistic for many WSNs where the transmitted information has to endure both
channel fading and noise/interference. Acquiring channel state information may be too costly for a
resource-constrained sensor network. It may also be impossible to accurately estimate the quality
of a fast-changing channel. Hence, it can be argued to be reasonable to assume that this information
is available at the fusion center.
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