Cryptography Reference
In-Depth Information
the corresponding transmitted data. However, unlike the classical linear
MMSE equalizer, the reliability information coming from the decoder is
here explicitly taken into account when calculating the coecients.
3. The equalizer is finally followed by a soft-input soft-output demapping
module whose role is to convert the equalized symbols into extrinsic LLRs
on the (interleaved) coded bits.
We now examine in greater detail the implementation of each of these three
functions.
SISO mapping
This operation involves calculating the soft estimate
x
i
, defined as the math-
ematical expectation of symbol
x
i
transmitted at instant
i
:
•
M
x
i
=
E
a
{
x
i
}
=
X
l
×
P
a
(
X
l
)
(11.24)
l
=1
The sum here concerns all of the discrete symbols in the constellation. The
term
P
a
(
X
l
)
denotes the
a priori
probability
Pr(
x
i
=
X
l
)
of symbol
X
l
being
transmitted at instant
i
. We have put index
a
at the level of the expectation
term to highlight the fact that these probabilities are deduced from the
a priori
information at the input of the equalizer. Indeed, provided the
m
bits making
up symbol
x
i
are statistically independent, it is possible to write:
m
P
a
(
X
l
)=
P
a
(
X
l,j
)
(11.25)
j
=1
where binary element
X
l,j
takes the value 0 or 1 according to the symbol
X
l
considered and the mapping rule. On the other hand, starting from the general
definition (11.4) of the LLR, we can show that the
a priori
probability and the
a priori
LLR are linked by the following relation:
1+(2
X
l,j
−
1) tanh
L
a
(
x
i,j
)
2
with
X
l,j
∈{
P
a
(
X
l,j
)=
1
2
0
,
1
}
(11.26)
In the particular case of a BPSK modulation, the above computations are
greatly simplified. We then obtain the following expression for the soft esti-
mate
x
i
:
x
i
=tanh
L
a
(
x
i
)
2
(11.27)
In the classical situation where we make the hypothesis of equiprobability
on the transmitted symbols, we have
L
a
(
x
i,j
)=0
and
x
i
=0
. On the other
hand, in the ideal case of perfect
a priori
information,
L
a
(
x
i,j
)
→±∞
and
