Information Technology Reference
In-Depth Information
2.3 Both WM and Additive Attack Noise
Are “Colored Noise” Sequences
In this case, the sequence
ε
is no longer i. i. d. but it can be of zero mean, with
autocorrelation function
ϕ
ε
(
n
1
− n
2
)=
E
(
ε
(
n
1
)
ε
(
n
2
))
.
We get from (8) and (7):
E
(
Λ
0
)=0
,E
(
Λ
1
)=
N−
1
k
=0
|
h
(
k
)
2
φ
wo
(
k
)
|
The relation (16) does not hold in this case and it has to be changed as follows:
N−
k
=0
|
1
2
φ
w
(
k
)+
N−
1
h
(
k
)
Var (
Λ
0
)=
σ
2
|
∆
n
1
n
2
where
ε
n
1
=0
n
2
=
n
1
N−
1
∆
n
1
n
2
=
ϕ
ε
(
n
1
− n
2
)
ϕ
w
(
n
3
− n
4
)
h
(
n
1
− n
3
)
h
(
n
2
− n
4
)
n
3
,n
4
=0
It is easy to see that if there is no a filtering attack, e. g.
h
(
n
)=0if
n
= 0, and
WM is a white noise sequence, e. g.
ϕ
w
(
n
3
− n
4
)=0if
n
3
=
n
4
, then necessarily
∆
n
1
n
2
= 0 for
n
1
=
n
2
. But in a general case, we have
∆
n
1
n
2
>
0 and this entails
a degradation of the WM system.
2.4 Calculation of the Probabilities
P
m
and
P
fa
for the Case of Tile-Based WM
Let us select a
spreading form
of the WM sequence in which it takes constant
values on blocks of
m
consecutive elements
1)
a
[
m
]
,
W
(
n
)=
α
(
−
n
=0
,...,N −
1
where
α>
0,
a
=(
a
n
1
)
[
N
−
1
]
n
1
=0
is a
{
0
,
1
}
i. i. d. sequence,
x →
[
x
]isthe
integer
m
part
map and
m
1 is an integer. If an attacker uses a low-pass filter with
frequency response close to (26), then for any parameter
K
h
of that filter, it
is possible to select an appropriated parameter
m
such that the WM sequence
after the attack by that filter practically is not corrupted. In this setting it is
very reasonable to select an attack additive noise
≥
ε
with a similar “variability”:
ε
(
n
)=
ε
n
m
,
n
=0
,...,N −
1
ε
is a zero mean i. i. d. sequence with variance
σ
2
ε
where
.
Since this model corresponds to the case of no filtering attack we can directly
exploit the results of [2] by using the correlation WM-detector that proceeds by
comparing with a given threshold
λ
the value
