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Λ
1
=
N−
1
((
Wh
)(
n
)+
ε
(
n
))(
Wh
)(
n
)
.
(7)
n
=0
and if the WM is not embedded into CM, the value of
Λ
coincides with
Λ
0
=
N−
1
ε
(
n
)(
Wh
)(
n
)
.
(8)
n
=0
The main characteristic of our approach is the assumption that the WM-sequence
W
is chosen randomly. Hence, the
Central Limit Theorem
can be applied in order
to obtain good approximations to both
Λ
0
and
Λ
1
as Gaussian variables. Thus,
the following relations are obtained for probabilities
P
m
and
P
fa
:
λ
−
E
(
Λ
1
)
Var (
Λ
1
)
P
m
=1
− Q
(9)
λ
−
E
(
Λ
0
)
Var (
Λ
0
)
P
fa
=
Q
(10)
where
E
is the mathematical expectation of the corresponding random variable
and for all real
x
,
Q
(
x
)=
+
∞
x
e
−
t
2
dt
. Thus, in order to complete this set
of formulas let us calculate
E
(
Λ
0
),
E
(
Λ
1
), Var (
Λ
0
) and Var (
Λ
1
).
For simplicity and without any loss of generality, let us assume
E
(
ε
(
n
))=0,
for all
n
=0
,...,N −
1
√
2
π
1. Then,
E
(
Λ
0
) = 0
(11)
and, from the definition given by eq. (7)
N−
1
((
Wh
)(
n
))
2
E
(
Λ
1
)=
E
n
=0
N−
1
W
(
n
1
)
W
(
n
2
)
h
(
n − n
1
)
h
(
n − n
2
)
=
N−
1
N−
1
E
n
=0
n
1
=0
n
2
=0
=
N−
1
N−
1
N−
1
ϕ
w
(
n
1
,n
2
)
h
(
n − n
1
)
h
(
n − n
2
)
(12)
n
=0
n
1
=0
n
2
=0
where
ϕ
w
(
n
1
,n
2
)=
E
(
W
(
n
1
)
W
(
n
2
)) is the
autocorrelation function
of the WM
sequence
W
. We assume that
W
is a wide-sense discrete time zero mean stochas-
tic process. Thus, we may write the autocorrelation function as
ϕ
w
(
n
1
,n
2
)=
ϕ
w
(
n
1
− n
2
). From eq. (12) we may express:
E
(
Λ
1
)=
N−
1
k
=0
|
h
(
k
)
|
2
φ
w
(
k
)
(13)
