Cryptography Reference
In-Depth Information
In this case, other schemes, falling in between "systematic" and "redundant"
schemes, can be defined.
The reception scheme corresponding to the transmitter of Figure 10.9 is
described in Figure 10.10. A standard turbo decoder is used, which requires
calculating a weighted estimation of each of the bits contained in the symbols
at the output of the demodulator before carrying out the decoding.
The weighted estimation of each bit
s
i
is obtained by calculating the log
likelihood ratio (LLR) defined by:
ln
Pr(
s
i
=1
=
σ
2
2
ln
Pr(
s
i
=1
s
i
=Λ(
s
i
)=
σ
2
2
|
r
)
|
I,Q
)
(10.2)
Pr(
s
i
=0
|
r
)
Pr(
s
i
=0
|
I,Q
)
0
,
1 otherwise) and its absolute value represents the weight, that is to say, the
reliability, associated with the decision. In the case of a transmission over a
Gaussian channel,
s
i
can be written:
The sign of the LLR provides the binary decision on
s
i
(0 if
Λ(
s
i
)
≤
exp
⎛
⎝
⎞
⎠
d
r,s
2
σ
2
−
s
i
=
σ
2
2
s
∈
S
i,
1
ln
exp
(10.3)
d
r,s
2
σ
2
−
s∈S
i,
0
where
S
i,
1
and
S
i,
0
represent the sets of points
s
of the constellation such that
the
i
th
bit
s
i
is equal to 1 or 0, and
d
r,s
is the Euclidean distance between the
received symbol
r
and the constellation point considered
s
.
In practice, the Max-Log approximation is commonly used to simplify the
calculation of the LLRs:
ln(exp(
a
)+exp(
b
))
≈
max(
a, b
)
(10.4)
and the weighted estimations are calculated as:
min
s∈S
i,
0
(
d
r,s
)
.
s
i
=
1
4
(
d
r,s
)
−
min
s∈S
i,
1
(10.5)
We note that it is not necessary to know the noise variance on the channel, when
this simplification is used.
The other blocks of the receiver perform the inverse operations of the blocks
of Figure 10.9. The depuncturing operation corresponds to the insertion of an
LLR equal to 0, this is to say, a zero reliability decision, at the input of the
decoder for all the non-transmitted bits.
The pragmatic turbo coded modulation approach enables performance very
close to that of TTCMs to be obtained. Figures 10.12 and 10.13 show the
performance of two pragmatic turbo coded modulations using the double-binary
