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E
(
Λ
1
)=
α
2
N
K
h
if
K
h
≤ K
w
K
w
α
2
N
otherwise
Var (
Λ
0
)=
σ
2
ε
α
2
N
K
h
if
K
h
≤ K
w
K
w
σ
2
ε
α
2
N
otherwise
Now, we can use the formulas (20) and (21) to calculate
P
m
and
P
fa
by taking
Var (
Λ
0
)
=
σ
ε
N
K
h
K
w
E
(
Λ
1
)
if
K
h
≤ K
w
µ
=
(29)
√
N
α
σ
ε
otherwise
In the current case, using the Fourier transform, eq. (5) gives:
K
h
≤ K
w
⇒ η
a
=
1
N
−
1
N−
1
σ
2
ε
σ
2
C
K
h
η
w
K
w
φ
co
(
k
)+
+
(30)
k
=
K
h
+1
K
h
≥ K
w
⇒ η
a
=
1
N
−
1
N−
1
1
η
w
σ
2
ε
σ
2
C
φ
co
(
k
)+
+
(31)
k
=
K
h
+1
We see from (29), (30) and (31) that the case
K
h
>K
w
is useless to the attacker
since it results in a decreasing of both
P
m
and
P
fa
when compared with the case
of no filtering attack (
K
h
=
N
). Considering just the case
K
h
≤ K
w
, from (30):
=
φ
co
(
k
)
−
2
N−
1
α
σ
ε
K
h
K
w
η
w
N
η −
−
(32)
k
=
K
h
+1
If the attack filter parameter is chosen in such a way to provide no noticeable
distortion of CM we can neglect the last summand in the denominator of (32).
Substituting the corresponding value into (29) we get for
K
h
≤ K
w
:
µ
=
NK
h
1
(33)
−
K
h
K
w
K
w
The designer of the WM system is able to select the parameter
K
w
such that
any use of the attack filter given by (26), for a parameter
K
h
≤ K
w
would result
in an intolerable distortion of CM. If we let
K
h
=
K
w
then from (33):
N
η −
µ
=
(34)
1
The value of
µ
in (34) coincides with
µ
0
as given by (28) which corresponds to
the case of no filtering attack. We conclude that the use of a “colored noise”
WM sequence discourages filtering attack. This is consistent with the simulation
results shown in [1]. Unfortunately it does not mean that any combination of
filtering and additive noise attack can be cancelled by an appropriated choice of
the colored noise WM sequence. We consider this problem in the next subsection.
