Cryptography Reference
In-Depth Information
•
good average performance for this code whose decoding complexity re-
mains very reasonable (around 18,000 gates per iteration plus the mem-
ory);
•
a certain coherence concerning the variation of performance with block
size (in agreement with the curves of Figures 3.6, 3.9, 3.10). The same
coherence could also be observed for the variation of performance with
coding rate;
•
quasi-optimality of decoding with low error rates. The theoretical asymp-
totic curve for 188 bytes has been calculated from the sole knowledge of
the minimum distance of the code (that is, 13 with a relative multiplicity
of
0
.
5
) and not from the total spectrum of the distances. In spite of this,
the difference between the asymptotic curve and the curve obtained by
simulation is only
0
.
2
dB for a PER of
10
−
7
.
16-state double-binary turbo code
The extension of the previous scheme to 16-state elementary encoders allows the
minimum distances to be greatly increased. We can, for example, choose:
⎡
⎤
⎡
⎤
⎡
⎤
0011
1000
0100
0010
11
01
00
01
1
1
1
0
⎣
⎦
⎣
⎦
⎣
⎦
G
=
,
C
=
,
R
=
For the rate 2/3 turbo code, again with blocks of 188 bytes, the minimum
distance obtained is equal to 18 (relative multiplicity of
0
.
75
) instead of 13 for
the 8-state code. Figure 7.22(b) shows the gain obtained for low error rates:
around 1 dB for a PER of
10
−
7
and 1.4 dB asymptotically, considering the
respective minimum distances. We can note that the convergence threshold is
almost the same for 8-state and 16-state decoders, the curves being practically
identical for a PER greater than
10
−
4
. The theoretical limit (TL), for
R
=2
/
3
and for a blocksize of 188 bytes, is 1.7 dB. The performance of the decoder in
this case is: TL + 0.9 dB for a PER of
10
−
4
and TL + 1.3 dB for a PER of
10
−
7
. These intervals are typical of what we obtain in most rate and blocksize
configurations.
Replacing 4-PSK modulation by 8-PSK modulation, in the so-called prag-
matic approach, gives the results shown in Figure 7.22(b), for blocks of 188
and 376 bytes. Here again, good performance of the double-binary code can
be observed, with losses compared to the theoretical limits (that are around
3.5 and 3.3 dB, respectively) close to those obtained with 4-PSK modulation.
Associating turbo codes with different modulations is described in Chapter 10.
For a particular system, the choice between an 8-state or 16-state turbo code
depends, apart from the complexity desired for the decoder, on the target error
