Digital Signal Processing Reference
In-Depth Information
To analytically approximate the optimal power allocation problem, the following
optimization problem is considered:
⎧
⎨
min
L
=
1
G
2
−
L
=
1
H
2
G
2
2
σ
v
H
2
G
2
+
(5.27)
subject to
q
σ
w
≤
0
,
⎩
G
≥
=
0
,
1
,
2
,...,L
(
1
β
2
1
D
)
−
1
. Thus, the optimal power allocated to the sensor nodes can be
derived following the same procedure as in Sect.
5.3.1
. The corresponding optimal
solution to the problem (
5.27
) is given by:
where
q
=
+
⎧
⎨
H
k
N
1
j
2
H
k
σ
v
1
H
j
1
if
k
≤
N
1 and
L>
2
σ
v
q
,
σ
w
=
1
2
σ
v
q)
−
(N
1
−
G
k
=
(5.28)
if
k>N
1 and
L>
2
σ
v
q
,
0
⎩
if
L<
2
σ
v
q
infeasible
where
N
1 is unique and is defined such that
f(N
1
)<
1 and
f(N
1
+
1
)
≥
1for
2
σ
v
q)
H
k
j
=
1
(k
−
L
.
f(k)
1
≤
N
1
≤
=
.
1
H
j
In the optimal solution, the number of active sensors should be greater than 2
σ
v
q
in order to satisfy the required fusion error probability at the fusion center. This
solution suggests also that some sensors should remain inactive in order to minimize
the total power consumption.
Next, we propose a numerical approach to find the optimal power allocation
when local observation are arbitrary correlated. The solution proposed in this work
is based on the variation of the BBO algorithm.
5.4 Constrained BBO for Optimal Power Allocation
This section details the description of the basic Biogeography-Based Optimization
algorithm, adapted from [
21
], in a nutshell. A brief description of the constrained op-
timization problem and a review of several popular constraint-handling approaches
are presented, followed by a detailed description of the algorithms proposed in this
work.
5.4.1 Standard Unconstrained Biogeography-Based Optimization
(BBO)
The Biogeography-based optimization (BBO) algorithm, developed by Dan Simon
[
21
], was strongly influenced by the
equilibrium theory
of island biogeography [
15
].
The basic premise of this theory is that the rate of change in the number of species on
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