A Comparative Study of Modified BBO Variants and Other Metaheuristics for Optimal Power Allocation in Wireless Sensor Networks - Advances in Heuristic Signal Processing and Applications - page 87

The covariance matrix v of the observation noise is not diagonal. Consequently,
it is difficult to evaluate − n in closed form in ( 5.9 ) for a general v . One answer
to this problem is the tridiagonal approximation of v for sufficiently small ρ ,this
corresponds to the case where only adjacent node observations are correlated.
Following a similar procedure as in [ 27 ], we present the upper bound on the
probability of error at the fusion center using the tridiagonal approximation matrix
as well as Bergstrom's inequality [ 1 ] and finally considering the case where the
correlation coefficients are sufficiently small.
According to Bergstrom's inequality, for any positive definite matrices P and Q :

( e T ( P Q ) − 1 e )( e T Q − 1 e ) +

e T P − 1 e

≥

Q ) − 1 e .

(5.21)

e T Q − 1 e e T ( P

− +

From Eqs. ( 5.4 ) and ( 5.9 ), m 2 e T A − n Ae m 2 e T (Σ v + σ w A − 2 ) − 1 e .Let P = =

σ w A − 2 ) and consider the matrix Q given by: (Σ v +

⎛
⎝

⎞
⎠

− ρ d ... − ρ d(L − 2 ) − ρ d(L − 1 ) 1

ρ d ρ d(L − 3 ) ρ d(L − 2 ) − 1 ... − −

σ v

Q = . (5.22)

. . . . . . . . .

. . . . . .

ρ d(L − 1 ) ρ d(L − 2 ) ρ d − − ... − 1

From ( 5.21 ) it can be shown that
e T v +

− 1

σ w A − 2 − 1 e

1 1

D

≥ − (5.23)

L = 1

H 2 G 2
2 σ v H 2 G 2 +

σ w

e T Q − 1 e . Therefore, from ( 5.9 ) and ( 5.23 ), the fusion error probability
can be bounded from above by:

where D =

m
2

−

2 .

1 1

D

P e ≤ Q − (5.24)

L = 1

H 2 G 2
2 σ v H 2 G 2 +

σ w

L/σ v , we get When ρ = 0, D =

1 1

D

L
σ v

lim
G

− = . (5.25)

L = 1

H 2 G 2
2 σ v H 2 G 2 +

→∞

σ w

For the correlated observations, the optimization problem ( 5.11 ) can be reformu-
lated to the following equivalent statement:
⎧
⎨

min L = 1 G 2
subject to β 2

e T AΣ − n Ae

(5.26)

− ≤ 0 ,

⎩

G ≥ 0 , = 1 , 2 ,...,L.

Since it is difficult to obtain an analytical closed form of the power allocation
problem, as it was explained in Sect. 5.3.1 , it is useful to have an analytical approx-
imation of the problem that minimizes the fusion error probability bound ( 5.24 ).