Cryptography Reference
In-Depth Information
L
(
x
i,j
)=ln
X
l
/X
l,j
=1
Pr(
x
i
=
X
l
|z
i
)
X
l
/X
l,j
=0
Pr(
x
i
=
X
l
|z
i
)
(11.49)
=ln
X
l
/X
l,j
=1
P
(
z
i
|x
i
=
X
l
)
P
a
(
X
l
)
X
l
/X
l,j
=0
P
(
z
n
|x
i
=
X
l
)
P
a
(
X
l
)
The second equality results from applying Bayes' relation. It shows the
a priori
probability
P
a
(
X
l
)=Pr(
x
i
=
X
l
)
of having transmitted a given symbol
X
l
of
the modulation alphabet. This probability is calculated from the
a priori
infor-
mation available at the input of the equalizer (relations (11.25) and (11.26)). By
exploiting the above hypotheses, the likelihood of observation
z
i
conditionally to
the hypothesis of having transmitted the symbol
X
l
at instant
i
can be written:
exp
(11.50)
2
1
πσ
ν
−
|
z
i
−
g
Δ
X
l
|
P
(
z
i
|
x
i
=
X
l
)=
σ
ν
After simplification, the
a posteriori
LLR calculated by the demapping operation
becomes:
exp
X
l,k
L
a
(
x
i,k
)
⎛
⎞
k
=1
m
−
|z
i
−g
Δ
X
l
|
2
σ
ν
+
⎝
⎠
X
l
/X
l,j
=1
L
(
x
i,j
)=ln
exp
X
l,k
L
a
(
x
i,k
)
(11.51)
k
=1
m
−
|z
i
−g
Δ
X
l
|
2
σ
ν
+
X
l
/X
l,j
=0
Like in the case of the BCJR-MAP equalizer, we can factorize in the numerator
and denominator the
a priori
information term in relation to the considered bit,
in order to obtain the extrinsic information that is then provided to the decoder:
exp
X
l,k
L
a
(
x
i,k
)
⎛
⎞
⎠
+
k
=
j
−
|z
i
−g
Δ
X
l
|
2
σ
ν
⎝
X
l
/X
l,j
=1
L
(
x
i,j
)=
L
a
(
x
i,j
)+ln
exp
X
l,k
L
a
(
x
i,k
)
+
k
=
j
−
|z
i
−g
Δ
X
l
|
2
σ
ν
X
j
/X
j,i
=0
L
e
(
x
i,j
)
(11.52)
Finally, the extrinsic information is obtained quite simply by subtracting the
a
priori
information from the
a posteriori
LLR calculated by the equalizer:
L
e
(
x
i,j
)=
L
(
x
i,j
)
−
L
a
(
x
i,j
)
(11.53)
In the particular case of BPSK modulation, the SISO demapping equations are
simplified to give the following expression of the extrinsic LLR:
4
L
(
x
i
)=
g
Δ
Re
{
z
i
}
(11.54)
1
−
