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In-Depth Information
where
h
is the
frequency response of attack filter
:
h
(
k
)=
N−
1
n
=0
h
(
n
)
e
−
2
πk
N
i
,
0
≤ k ≤ N −
1, and
φ
w
is the
power density spectrum
of the process
W
,
φ
w
(
k
)=
N−
1
ϕ
w
(
n
)
e
−
2
πk
N
i
,
k
=0
,...,N −
1
.
(14)
n
=0
In order to calculate the formulas for Var (
Λ
0
) and Var (
Λ
1
) we should consider
particular cases of the stochastic processes
W
and
ε
.
2.1 The WM Is a Binary “White Noise” Sequence
and Additive Attack Noise Is a “White Noise” Sequence
In this case,
W
(
n
)=
απ
(
n
)
,
n
=0
,...,N −
1
,
(15)
where
α>
0 is some real valued number,
π
is a
±
1 valued i. i. d. sequence and
is a zero mean i. i. d. sequence with variance
σ
2
ε
besides
ε
.
Under these conditions, eqs. (8), (12) and (13) give
Var (
Λ
0
)=Var
N−
1
ε
(
n
)(
Wh
)(
n
)
=
N−
1
Var (
ε
(
n
)(
Wh
)(
n
))
n
=0
n
=0
=
N−
1
N−
1
Var (
ε
(
n
))
E
(
Wh
)
2
(
n
)
=
σ
2
ε
h
(
k
)
2
=0
|
|
φ
w
(
k
)
(16)
n
=0
k
For the white noise binary sequence
W
given by eq. (15) we have
φ
w
(
k
)=
α
2
,
k
=0
,...,N −
1
.
(17)
Substitution of eq. (17) into eq. (13) and (16) produces
α
2
N−
1
k
=0
|
h
(
k
)
2
E
(
Λ
1
)=
|
α
2
N−
1
k
=0
|
h
(
k
)
Var (
Λ
0
)=
σ
2
2
.
|
(18)
ε
Since
ε
and
W
are mutually independent:
Var (
Λ
1
)=Var(
Λ
0
)+Var
N−
1
(
Wh
)
2
(
n
)
(19)
n
=0
where Var (
Λ
0
) is given by eq. (18). It is easy to see from eq. (19) that for
any “short” pulse response of the attack filter, the second term in the right side
of (19) is close to zero. However, in a general case, this term can be much greater
than zero. Since it is very hard to prove its representation in a closed form, we
