Cryptography Reference
In-Depth Information
Replacing the probability densities by their respective expression we obtain:
exp
s jp ) 2
π j 1
N
p =1 ( r p
N
1
N 0
s j ( t )
if
πN 0
exp
s np ) 2
n 1
N
p =1 ( r p
N
1
N 0
πN 0
After simplification:
N
N
s j ( t )
if
r p s jp + C j >
r p s np + C n
n
= jn =1 , 2 ,
···
,M
(2.59)
p =1
p =1
p =1
N
E j
2
N 2
( s jp ) 2 .
where C j =
ln( π j )
with E j =
Noting that:
T
T
N
N
r ( t ) s j ( t ) dt =
r p ν p ( t )
s jm ν m ( t ) dt
p =1
m =1
0
0
and recalling that the functions ν p ( t ) are normed and orthogonal, we obtain:
T
N
r ( t ) s j ( t ) dt =
r p s jp
p =1
0
In the same way:
T
T
N
N
s j ( t ) dt =
s jp ν p ( t )
s jm ν m ( t ) dt
p =1
m =1
0
0
and finally:
T
N
s j ( t ) dt =
s jp
p =1
0
Taking into account the above, the MAP criterion can again be written in the
form:
0
T
0
T
s j ( t ) if
r ( t ) s j ( t ) dt + C j >
r ( t ) s n ( t ) dt + C n
(2.60)
n
= jn =1 , 2 ,
···
,M
0
T
E j
2
N 2
s j ( t ) dt .
where C j =
ln( π j )
with E j =
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