Cryptography Reference
In-Depth Information
deduced from (7.58). The parameters obtained by the error impulse method
are:
•
d
min
=13
and
n
(
d
min
) = 752
for
R
=2
/
3
d
min
=8
and
n
(
d
min
) = 940
for
R
=4
/
5
For packet error rates of
10
−
7
, less than 0.2 dB separates the measured and
estimated curves.
•
Modified error impulse method
The approach of Garello
et al.
[7.27] is similar to the error impulse method pre-
sented above. It involves placing an impulse in row
i
in the "all zero" codeword.
This time, the amplitude of the impulse is high enough for the decoder not to
converge towards the "all zero" codeword but towards another sequence that
contains a 1 in position
i
. In addition, Gaussian noise is added to the input
sequence of the decoder, which tends to help the latter converge towards the
concurrent word having the lowest weight. This is what often happens when the
level of noise is well adjusted. In all cases, the weight of the codeword provided
by the decoder is an upper limit of the minimum distance of all the codewords
containing a 1 in row
i
. The minimum distance and the multiplicity are esti-
mated by sweeping all the positions. This algorithm works very well for small
and average distances.
Double error impulse method
The method proposed by Crozier
et al.
[7.22] is an improvement of the previous
method, at the expense of higher computation time. It involves placing a first
high level impulse at row
i
and a second at row
j
to the right of
i
and such that
j
i<r
. The upper limit of
r
is
2
D
where
D
is an upper bound of the distance
to be evaluated. Then, decoding is applied similar to that described above but
with a stronger probability of obtaining a codeword at the minimum distance.
The calculation time is increased by a ratio
r
.
−
7.6.3 Convergence
A SISO decoder can be seen as a processor that transforms one of its input
values, the LLR of the extrinsic information used as
a priori
information, into
an output extrinsic LLR. In iterative decoding, the characteristics of the extrinsic
information provided by decoder 1 depend on the extrinsic information provided
by decoder 2 and vice-versa. The degree of dependency between the input and
output extrinsic information can be measured by the mutual information (MI).
The idea implemented by ten Brink [7.46] is to follow the exchange of ex-
trinsic information through the SISO decoders working in parallel on a diagram,
