Cryptography Reference
In-Depth Information
The noise can also be represented in the form of a series of normed and orthog-
onal functions but of infinite length (Karhunen Loeve expansion). When the
noise is white, we show that the normed and orthogonal functions can be chosen
arbitrarily. We are therefore going to take the same orthonormed functions as
those used to represent the s j ( t ) signals, but after extension to infinity of this
base of functions:
b ( t )=
N
b p ν p ( t )+ b ( t )
b p ν p ( t )=
p =1
p =1
where b p is a scalar equal to the projection of b ( t ) on the function ν p ( t ) .
T
b p =
b ( t ) ν p ( t ) dt
0
The quantities b p are random non-correlated Gaussian variables, with zero mean
and variance σ 2 = N 0 / 2 .
T
T
b ( t ) b ( t )
ν p ( t ) ν n ( t ) dtdt
{
b p b n }
=
{
}
E
E
0
0
b ( t ) b ( t )
N 2
t ) and thus:
The noise being white, E
{
}
=
δ ( t
T
N 0
2
N 0
2
E
{
b p b n }
=
ν p ( t ) ν n ( t ) dt =
δ n,p
(2.56)
0
where δ n,p is the Kronecker symbol, equal to 1 if n = p and to 0 if n
= p .
Using the representations of the s j ( t ) signals and of the b ( t ) noise by their
respective series, we can write:
N
N
r p ν p ( t )+ b ( t )
r ( t )=
( s jp + b p ) ν p ( t )+
b p ν p ( t )=
p =1
p = N +1
p =1
Conditionally to the emission of the signal s j ( t ) , the quantities r p are random
Gaussian variables, with mean and variance N 0 / 2 . They are non-correlated to
the noise b ( t ) . Indeed, we have:
= E ( s jp + b p )
b n ν n ( t )
r p b ( t )
E
{
}
p =1 , 2 ,
···
,N
n = N +1
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