Digital Signal Processing Reference
In-Depth Information
5.3 Optimal Power Allocation
Power allocation plays a key role in improving the system performance. In this sec-
tion, the optimal power allocation among the sensors in the distributed detection
system is considered. The objective is to minimize the total power spent by the
whole sensor network to achieve a desired detection performance. As it was dis-
cussed in the previous paragraphs, the distributed nature of observations coupled
with bandwidth and power constraints requires a means of combining local sensor
observations while keeping the fusion error as small as possible.
The main question that one seeks to answer is:
What is the optimal power al-
location, for a given threshold, at the fusion center?
The problem of finding the
optimal power allocation scheme can be posed as follows:
Find a set of sensor
gains (G
1
,...,G
,...,G
L
) which solves the following constrained optimization
problem
:
⎧
⎨
min
L
=
1
G
2
subject to
P
e
=
Q(
2
m
2
e
T
A
−
n
Ae
)
≤
ε,
G
≥
(5.11)
⎩
0
,
=
1
,
2
,...,L.
The objective is to minimize the total power while keeping the fusion error prob-
ability under a required threshold
ε
. We consider two situations: (i) where the local
observations are i.i.d., and (ii) where the observations of different nodes are cor-
related. For both cases, we determine the optimum power allocation schemes that
minimize the total power required to satisfy a certain performance level. In situation
(i), the optimal solution to the gain allocation is analytically derived. In situation
(ii), an approximate analytical solution to the power allocation problem that mini-
mizes the fusion error probability bound in (
5.24
) is derived for small correlations. It
is also shown that, under arbitrary correlated observations, the numerical approach
becomes suitable to find the optimal power allocation, since it gets difficult to solve
this problem analytically.
5.3.1 Independent Observations
In the special case when local observations and the receiver noise are both i.i.d.,
v
=
σ
v
I
and
w
=
σ
w
I
, where
I
is the
L
L
identity matrix, the mean-square
error (MSE) based on the received signal (
5.3
) is given by:
×
L
−
1
H
2
G
2
σ
v
H
2
G
2
+
MSE
=
.
(5.12)
σ
w
=
1
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