Cryptography Reference
In-Depth Information
L
i
(
j
)
can then be written, by reference to (7.22) and (7.23), as follows:
L
i
(
j
)=
1
2
(
A
i
(
j
)
−
B
i
)
(7.36)
Expressions (7.34) and (7.35) can be simplified by applying the so-called Max-
Log approximation:
ln(exp(
a
)+exp(
b
))
≈
max(
a, b
)
(7.37)
For
A
i
(
j
)
we get:
M
i
+1
(
s
)+
M
i
(
s
)+
M
i
(
s
,s
)
A
i
(
j
)
≈
min
(
s
,s
)
/
d
i
(
s
,s
)
(7.38)
≡
j
and for
B
i
:
M
i
+1
(
s
)+
M
i
(
s
)+
M
i
(
s
,s
)
=min
l
=0
···
2
m
B
i
≈
min
(
s
,s
)
−
1
A
i
(
l
)
(7.39)
and finally we get:
A
i
(
j
)
1
A
i
(
l
)
L
i
(
j
)=
1
2
−
min
l
=0
(7.40)
···
2
m
−
Note that these values are always positive or equal to zero.
Introduce the values
L
a
proportional to the logarithms of the
a priori
prob-
abilities Pr
a
:
σ
2
2
L
i
(
j
)=
ln
Pr
a
(
d
i
≡
−
j
)
(7.41)
Branch metrics
M
i
(
s
,s
)
can be written, according to (7.24) and (7.33):
M
i
(
s
,s
)=2
L
i
(
d
(
s
,s
))
σ
2
ln
p
(
v
i
|
−
u
i
)
(7.42)
If the statistic of the
a priori
transmission of the
m
-tuples
d
i
is uniform,
term
2
L
i
(
d
(
s
,s
))
can be omitted from the above relation since it is the same
value that is used in all the branch metrics. In the case of a transmission over
a Gaussian channel with binary inputs, we have according to (7.26):
m
+
m
M
i
(
s
,s
)=2
L
i
(
d
(
s
,s
))
−
v
i,l
·
u
i,l
(7.43)
l
=1
The forward and backward metrics are then calculated from the following
recurrence relations:
⎛
⎞
m
+
m
⎝
M
i−
1
(
s
)
⎠
(7.44)
M
i
(
s
)=
u
i−
1
,l
+2
L
i−
1
(
d
(
s
,s
))
min
s
=0
−
v
i−
1
,l
·
···
2
ν
−
1
l
=1
