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Let us consider the particular case of low pass filter attack when its frequency
response is
h ( k )= 0 f K 2 ≤ k ≤ N − K 2
1 otherwise
(26)
for some integer K h , with 0
1.
By substitution of (26) into (22) and (25) we get
µ =
≤ K h ≤ N −
K oh · N
η − K oh
K h
η − K h
N
=
(27)
with K oh = K h
and η = η w
η a
. If there is no filtering attack ( K h = N ), from (27),
N
N
η −
µ 0 =
(28)
1
which coincides, indeed, with the result presented in [2]. We can see, from (27)
and (28), that filtering attack results in a degradation of the WM system. It
requires at least 1 /K h times more WM elements N in order to achieve the
values of P m and P fa obtained in a WM system without filtering attack.
2.2 The WM Is a “Colored Noise” Sequence
and Additive Attack Noise Is a “White Noise” Sequence
Under the current conditions, the stochastic process W has an arbitrary (al-
though not uniform as in the case of a “white noise” sequence) power density
spectrum φ w . The relations (9), (10), (11) and (13) still hold. Nevertheless, the
variances of Λ 0 and Λ 1 does not coincide in general. However we may claim only
that Var ( Λ 1 )
Var ( Λ 0 ). The relations (20) and (21) determine lower bounds
for probabilities P m and P fa .
Let us consider the particular case in which
φ w ( k )= 0if K 2
≤ k ≤ N − K 2
α 2 N
K w
otherwise
with K w ≤ N −
1. Here, the WM can be obtained by passing the “white noise”
sequence given by (15) through a low-frequency filter with frequency response
= 0if K 2
h w ( k )
≤ k ≤ N − K 2
2
N
K w
otherwise
(this condition produces the same signal-to-noise ratio η w = σ
α 2 for any K w ).
In the case of a linear filtering attack with frequency response h satisfy-
ing (26) we obtain from (11), (13) and (16):
E ( Λ 0 )=0
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