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will calculate it further by simulation. Combining eqs. (11), (13), (16) and (19)
for “short” pulse response, we obtain from (9) and (10):
− Q
(
λ
− µ
)
P
m
=1
(20)
P
fa
=
Q
(
λ
)
(21)
where
λ
Var (
Λ
0
)
=
λ
λ
=
N−
1
k
=0
h
(
k
)
ασ
ε
|
|
2
N−
k
=0
|
1
E
(
Λ
1
)
Var (
Λ
0
)
=
α
σ
ε
h
(
k
)
µ
=
|
2
.
(22)
Using Fourier transforms and considering
C
as a wide-sense stochastic process,
we can express (5) as follows:
N−
1
k
=0
|
N−
1
k
=0
|
σ
2
ε
σ
2
C
1
η
a
1
N
1
η
w
N
h
(
k
)
h
(
k
)
2
2
+
=
|
φ
co
(
k
)+
|
(23)
where
N−
1
1
σ
2
C
ϕ
C
(
n
)
e
−
2
πk
N
i
,
φ
co
(
k
)=
k
=0
,...,N −
1
,
n
=0
is the autocorrelation function of
C
,
σ
2
C
ϕ
C
= Var (
C
), and
h
0
(
k
)=
N−
1
− δ
(
n
))
e
−
2
πk
N
i
,
(
h
(
n
)
k
=0
,...,N −
1
,
n
=0
Using (23) we may express the reciprocal of factor in last member of (22) as
N−
1
k
=0
|
N−
1
k
=0
|
σ
α
η
η
a
1
N
η
N
h
(
k
)
h
(
k
)
=
−
|
2
−
|
2
φ
co
(
k
)
.
(24)
From now on we can compute the probabilities
P
m
and
P
fa
using (20)-(22)
and (24) for any frequency response of the attack filter
h
, power density spectrum
φ
C
of CM, number
N
of WM elements and distortion constraints
η
w
and
η
a
.
Since it is not easy to know the power density spectrum (
φ
co
(
k
))
N−
1
k
=0
of CM,
we may assume that the attack filter has to be chosen in such a way to provide
no noticeable distortions of CM. Then we can make zero the last term in the
right of (24) to get
N−
k
=0
|
1
σ
α
η
η
a
1
N
h
(
k
)
=
−
|
2
.
(25)
