lecture7 - improved spread spectrum

7/28/2019 Lecture7 - Improved Spread Spectrum

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Improved Spread Spectrum:

A New Modulation Techniquefor Robust Watermarking

IEEE Trans. On Signal Processing, April 2003

Multimedia Security


2/18

Spread-spectrum based watermarking,

where b is the bit to be embedded


3/18

3

u: the chip sequence (reference pattern), with zeromean and whose elements are equal to +uor -u (1-bit message coding)

inner product:

norm: ||x||

Embedding: s =x +bu

Distortion in the embedded signal:

D = ||s - x|| = ||bu|| = ||u|| =u2

Channel noise: y =s +n

=

1

0

1 N

iiuixN


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Detection is performed by first computing

the (normalized) sufficient statistic r:normalized correlation

u 2

||u|| ||u||and estimating the embedded bit by

b = sign( r )

r

b + + b + +x n~ ~

^


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We usually assume simple statistical

models for the original signal x and theattack noise n:

both to be samples from uncorrelated white

Gaussian random process

xi N(0,x2)

ni N(0,n2)


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Then, it is easy to show that the sufficientstatistic ris also Gaussian, i.e.,

r N(mr,r2)

where

mr= E( r ) = b b {0,1}

x2 +n

2

Nu2

r2


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Lets consider the casewhen b = 1.

Then, an error occurs

when r< 0, and therefore,the error probability p isgiven by

for b = 1, mr= E( r) =b = 1

( )

( )

function.errorarycomplement

theiserfcwhere

12

1

2

1

22

1

22

1

)1|0Pr(

2

2

2

2

22

2

+

=

+=

=

=


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8

The same error probability is obtained

under the assumption that b = -1 but r> 0^

( mr/r )

If we want an errorprob. Better than 10-3,

then we need

mr/

r> 3

N

u

2 > 9(x

2+n

2)

-3


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9

In general, to achieve an error probability p,

we need

Nu2 > 2 (erfc-1(p))2(x

2+n2)

One can trade the length of the chip

sequence N with the energy of thesequenceu

2 !!


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10

Main idea:by using the encoder knowledge about thesignal x (or more precisely, the projection

ofx on the watermark), one can enhanceperformance by modulating the energy of

the inserted watermark to compensate forthe signal interference.

New Approach via Improved-SS


11/18

11

We vary the amplitude of the inserted chip

sequence by a function(x,b):

s =x +(x,b)uwhere, as before

x = / ||u|| : signal interference

SS is a special case of the ISS in whichthe function is made independent ofx.

~

~

~


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12

Linear Approximation:

is a linear function of x

s

=x

+ (b-x

)u

The parameters and control thedistortion level and the removal of the

carrier distortion on the detection statistics.Traditional SS is obtained by setting= 1

and= 0.

~


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13

With the same channel noise model as

before, the receiver sufficient statistic is||u||

The closer we make to 1, the more theinfluence ofx is removed from r.

The detector is the same as in SS, i.e., thedetected bit is sign(r).

r b + (1 -) x +n~ ~

~


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14

The expected distortion of the new system is given by

To make the average distortion of the new system to

equal that of traditional SS, we force E[D]=

u

2

, andtherefore

[ ]

[ ]2

2

222

22

1

~

u

u

x

u

N

xbE

xsEDE

44 344 21

=

+=

=

=

2uN

222

xuN

=


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15

To compute the error probability, all we need isthe mean and variance of the sufficient statistic r.

They are given by

Therefore, the error probability p is

2

2222 )1(

u

xn

r

r

N

bm

+=

=

{ }

+

=

=

=


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16

We can also write p as a function of the relativepower of the SS sequence Nu

2/x2and

the SNR x2/n2

By proper selection of the parameter, the errorprobability in the proposed method can be made

several orders of magnitude better than usingtraditional SS.

+

=

+

=2

2

2

2

2

2

2

2

)1(1

1

21

21

)1(21

21

SNR

sspowererfc

N

erfcp

x

n

x

u


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17

The three lines correspond to values to 5, 10,and 20dB SNR (with higher values having

smaller error probability).

Solid linesrepresent a 10-

dB SIR and dashlines represent a7-d SIR

SIR: Signal-to-interference ratio


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18

As can be inferred from the above figure, the errorprobability varies with, with the optimum value

usually close to 1.The expression for the optimum value for canbe computed from the error probability p byand is given by

Note: for N large enough,opt 1 as SNR

0=

p

++

++=

2

22

2

2

2

2

2

2

2

2

4112

1

x

u

x

u

x

n

x

u

x

n

opt

NNN

lecture7 - improved spread spectrum

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