Digital Signal Processing Reference
In-Depth Information
where
λ
0
≥
0 and
μ
k
≥
0for
k
=
0
,...,L
are the Lagrange multipliers associated
with the inequality constraints.
Given that both the objective function and the constraints are convex, the Karush-
Kuhn-Tucker (KKT) conditions are valid [
13
]. The optimal solution is derived as:
⎧
⎨
H
k
K
1
j
1
H
j
(K
1
−
β
2
σ
v
)
−
H
k
σ
v
1
if
k
σ
w
=
1
K
1
and
L>β
2
σ
v
,
≤
G
k
=
(5.19)
if
k>K
1
and
L>β
2
σ
v
,
0
⎩
if
L<β
2
σ
v
infeasible
where
K
1
is found such that
f(K
1
)<
1 and
f(K
1
+
1
)
≥
1for1
≤
K
1
≤
L
,
f(k)
=
β
2
σ
v
)
H
k
j
=
1
(k
−
. The proof of the uniqueness of such a
K
1
and the global optimality of
1
H
j
the solution (
5.19
) for the optimization problem (
5.17
)aregivenin[
27
].
Statistically, we model the fading coefficients
H
(
1
,...,L
) as unit mean
Rayleigh random variables and, without loss of generality, they are assumed to be
ranked in the descending order such that
H
1
≥
=
H
L
.
The solution given in (
5.19
) is feasible only if
L>β
2
σ
v
, i.e.,
γ
0
>
L
(
Q
−
1
(P
e
))
2
,
this implies that the probability of error
P
e
is lower-bounded by
H
2
≥···≥
(
√
Lγ
2
)
, which is
Q
consistent with (
5.14
).
5.3.2 Correlated Observations
While the popular assumption that the observations at the sensors are independent
is convenient for analysis, it does not necessarily hold for arbitrary sensor systems.
In practice, it is likely that the sensor observations are spatially correlated leading
to a nondiagonal covariance matrix
v
.
We consider here that the sensor nodes are equally spaced, along a straight line,
at a distance
d
and correlation between observations at node
i
and
j
is proportional
to
ρ
d
|
i
−
j
|
, where 0
<
1.
4
The observation noise covariance matrix
v
can
be written as a symmetric Hermitian Toeplitz matrix, referred to as Kac-Murdock-
Szegö matrix [
11
]:
|
ρ
|≤
⎛
⎝
⎞
⎠
ρ
d
... ρ
d(L
−
2
)
ρ
d(L
−
1
)
1
ρ
d
... ρ
d(L
−
3
)
ρ
d(L
−
2
)
1
σ
v
v
=
.
(5.20)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
ρ
d(L
−
1
)
ρ
d(L
−
2
)
... ρ
d
1
4
Correlation degree
ρ
=
1 means that two observations are perfectly correlated. Correlation degree
0
<ρ<
1 indicates that two observations are partially correlated (i.e., spatial correlation), while
ρ
=
0 implies that two observations are independent of each other.
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