Cryptography Reference
In-Depth Information
computation of
L
(
x
i,j
)
, we obtain:
α
i−
1
(
s
)exp
X
l,k
L
a
(
x
i,k
)
β
i
(
s
)
⎡
⎤
k
=1
m
−
|y
i
−z
i
|
2
σ
w
+
⎣
⎦
(
s
,s
)
/x
i,j
=1
α
i−
1
(
s
)exp
k
=1
X
l,k
L
a
(
x
i,k
)
β
i
(
s
)
L
(
x
i,j
)=ln
(11.21)
m
−
|y
i
−z
i
|
2
σ
w
+
(
s
,s
)
/x
i,j
=0
We can then factorize the
a priori
information term in relation to the bit
x
i,j
considered, both in numerator (
X
l,j
=1
) and in denominator (
X
l,j
=0
), which
gives:
L
(
x
i,j
)=
L
a
(
x
i,j
)
α
i−
1
(
s
)exp
X
l,k
L
a
(
x
i,k
)
β
i
(
s
)
⎡
⎤
+
k
=
j
−
|y
i
−z
i
|
2
σ
w
⎣
⎦
(
s
,s
)
/x
i,j
=1
α
i−
1
(
s
)exp
+
k
=
j
X
l,k
L
a
(
x
i,k
)
β
i
(
s
)
+ln
(11.22)
−
|y
i
−z
i
|
2
σ
w
(
s
,s
)
/x
i,j
=0
L
e
(
x
i,j
)
Finally, we see that the extrinsic information is obtained quite simply by sub-
tracting 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
)
(11.23)
This remark concludes the description of the MAP equalizer. As we have
presented it, this algorithm proves to be dicult to implement on a circuit due
to the presence of numerous multiplication operations. In order to simplify
the computations, we can then envisage transposing the whole algorithm into
the logarithmic domain (Log-MAP algorithm), the advantage being that the
multiplications are then converted into additions, which are simpler to do. If we
wish to further reduce the processing complexity, we can also use a simplified
(but sub-optimal) version, the Max-Log-MAP (or Sub-MAP) algorithm. These
two variants were presented in the context of turbo codes in Chapter 7. The
derivation is quite similar in the case of the MAP equalizer. Reference [11.5]
presents a comparison in performance between these different algorithms in a
MAP turbo equalization scenario. In particular, it turns out that the Max-Log-
MAP equalizer offers the best performance/complexity compromise when the
estimation of the channel is imperfect.
Example of performance
In order to illustrate the good performance offered by MAP turbo equalization,
we chose to simulate the following transmission scenario: a binary source gener-
ates messages of 16382 bits of information, which are then protected by a rate
