Cryptography Reference
In-Depth Information
Figure 4.5 - Performance of the algebraic decoding of the (15,7) BCH code. 4-PSK
transmission on a Gaussian channel.
Using the Bayes' rule and assuming all the codewords equiprobable, the above
inequality can also be written:
c
)
,
=
c
∈
c
=
c
if
p
(
r
|
c
)
>p
(
r
|
∀
c
C
(
n, k
)
(4.26)
where
p
(
r
c
)
is the probability density function of observation
r
conditionally
to codeword
c
.
For a Gaussian channel, probability density function
p
(
r
|
|
c
)
is equal to:
⎛
E
s
c
j
)
2
⎞
c
)=
n
n−
1
1
√
2
πσ
b
1
2
σ
b
⎝
−
⎠
p
(
r
|
exp
(
r
j
−
j
=0
where
σ
b
is the variance of the noise.
Replacing the two probability density functions by their respective expres-
sions in inequality (4.26) and after some basic computation, we obtain:
n−
1
n−
1
r
j
c
j
,
=
c
∈
c
=
c
⇔
r
j
c
j
>
∀
c
C
(
n, k
)
j
=0
j
=0
The decoded codeword is the one that maximizes the scalar product
.We
could also show that the decoded codeword is the one that minimizes the square
of the Euclidean distance
r
r
,
c
−
√
E
s
c
2
.
