Information Technology Reference
In-Depth Information
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