Transcript Improved Spread Spectrum

Improved Spread Spectrum:
A New Modulation Technique
for Robust Watermarking
IEEE Trans. On Signal Processing, April 2003
Multimedia Security
Spread-spectrum based watermarking,
where b is the bit to be embedded
• u: the chip sequence (reference pattern), with zero
mean and whose elements are equal to +σu or -σu (1bit message coding)
• inner product: <x,u> ＝
1
N 1
N
i 0
 xi u i
• norm: <x,x> ＝ ||x||
• Embedding: s = x + bu
• Distortion in the embedded signal:
D = ||s - x|| = ||bu|| = ||u|| = σu 2
• Channel noise: y = s + n
3
Detection is performed by first computing
the (normalized) sufficient statistic r :
normalized correlation
<y,u>
<bu+x+n,u>
＝
r ＝
<u,u>
σu 2
<x,u> <n,u>
~
~
＝ b +
＝ b + x + n
+
||u||
||u||
and estimating the embedded bit by
^
b = sign( r )
4
We usually assume simple statistical
models for the original signal x and the
attack noise n:
both to be samples from uncorrelated white
Gaussian random process
xi ～ N(0,σx2)
ni ～ N(0,σn2)
5
Then, it is easy to show that the sufficient
statistic r is also Gaussian, i.e.,
r ～ N(mr,σr2)
where
mr = E( r ) = b
b {0,1}
2 +σ 2
σ
n
σr2 ＝ x
Nσu2
6
Let’s consider the case
when b = 1.
Then, an error occurs
when r < 0, and therefore,
the error probability p is
given by
for b = 1, mr = E( r ) = b = 1
p  Pr(bˆ  0 | b  1)
 mr 
1

 erfc 


2
r 2 
2

1

u N
 erfc 
 2  x2   n2
2


2

N

u

 1
 x2
1
 erfc 
2
2

2

1 n2

x

where erfc  is the












complement ary error function.
7
The same error probability is obtained
under the assumption that b = -1 but r^> 0
If we want an error
prob. Better than 10-3,
then we need
-3
mr/σr > 3

Nσu2 > 9(σx2+σn2)
( mr/σr )
8
In general, to achieve an error probability p,
we need
Nσu2 > 2 (erfc-1(p))2(σx2+σn2)
 One can trade the length of the chip
sequence N with the energy of the
sequence σu2 !!
9
New Approach via Improved-SS
Main idea:
by using the encoder knowledge about the
signal x (or more precisely, the projection
of x on the watermark), one can enhance
performance by modulating the energy of
the inserted watermark to compensate for
the signal interference.
10
We vary the amplitude of the inserted chip
sequence by a function μ(x,b):
~
s = x + μ(x,b)u
where, as before
x~ = <x,u> / ||u|| : signal interference
 SS is a special case of the ISS in which
~
the function μ is made independent of x.
11
Linear Approximation:
μ is a linear function of x
~
s = x + (αb-λx)u
The parameters α and λ control the
distortion level and the removal of the
carrier distortion on the detection statistics.
Traditional SS is obtained by setting α= 1
and λ= 0.
12
With the same channel noise model as
before, the receiver sufficient statistic is
r
＝
<y,u>
||u||
＝
αb + (1 - λ) x~+ n~
The closer we make λ to 1, the more the
~
influence of x is removed from r.
The detector is the same as in SS, i.e., the
detected bit is sign(r).
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The expected distortion of the new system is given by
E D   E


sx

2
2
 E b   ~
x u

 2  2 x 2 
 u2
   
2 
N

u

1
To make the average distortion of the new system to equal
that of traditional SS, we force E[D]=σu2, and therefore

N u   2 x
2
N u
2
2
14
To compute the error probability, all we need is
the mean and variance of the sufficient statistic r.
They are given by mr  b
r2
 n 2  (1   ) 2  x 2

2
N u
Therefore, the error probability p is
p  Prr  0 | b  1
 mr 
1

 erfc 

2
r 2 
2
2
2


N




1
u
x


 erfc
 2( 2  (1   ) 2  2 ) 
2
n
x


15
We can also write p as a function of the relative
power of the SS sequence Nσu2/σx2 and
the SNR σx2/σn2


 1
1
p  erfc 
2
 2



u2
2
N 2  

x
 n2
 x2


1
 1
  erfc 
2
2 
 2
 (1   ) 



1
 2
power  ss
1
 (1   ) 2
SNR







By proper selection of the parameter λ, the error
probability in the proposed method can be made
several orders of magnitude better than using
traditional SS.
16
Solid lines
represent a 10dB SIR and dash
lines represent a
7-d SIR
SIR: Signal-tointerference ratio
The three lines correspond to values to 5, 10,
and 20dB SNR (with higher values having
smaller error probability).
17
As can be inferred from the above figure, the error
probability varies with λ, with the optimum value
usually close to 1.
The expression for the optimum value for λ can be
p
0
computed from the error probability p by

and is given by
opt

2
2
N u
1    n
  1  2 
2   x
 x2





  n 2 N u 2
1 
  2   2
x
x

2
2

N u
 4
2


x






Note: for N large enough, λopt → 1 as SNR → 
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