Cryptography Reference
In-Depth Information
7.6
Analysis tools
7.6.1 Theoretical performance
Figure 1.6 shows two essential parameters allowing the performance of an error
correcting code and its decoder to be evaluated:
•
the
asymptotic gain
measuring the behaviour of the coded system at low
error rates. This is mainly dictated by the MHD of the code (see Sec-
tion 1.5). A low value of the MHD leads to a great change in the slope
(
flattening
) in the error rate curve. When the asymptotic gain is reached,
the BER(
E
b
/N
0
) curve with coding becomes parallel to the curve without
coding.
•
the
convergence threshold
defined as the signal to noise ratio from which
the coded system becomes more ecient than the non-coded transmission
system;
In the case of turbo codes and the iterative process of their decoding, it is
not always easy to estimate the performance either of the asymptotic gain or of
the convergence. Methods for estimating or determining the minimum distance
proposed by Berrou
et al
. [7.18], Garello
et al.
[7.27] and Crozier
et al
. [7.22]
are presented in the rest of this chapter. The EXIT diagram method proposed
by ten Brink [7.46] to estimate the convergence threshold is also introduced.
7.6.2 Asymptotic behaviour
Determining the performance of error correcting codes with low error rates by
simulation requires high calculation power. It is, however, possible to estimate
this performance when the MHD
d
min
and the multiplicity are known (see Sec-
tion 1.5). Thus, the packet error rate with high signal to noise ratio
E
b
/N
0
is
given by the first term of the
union bound
(UB). The expression of the UB is de-
scribed by relation (3.21), and estimation of the PER, given by Equation (1.16),
is shown again here:
2
N
(
d
min
)
erfc
Rd
min
E
b
1
PER
≈
(7.57)
N
0
where
N
(
d
min
)
, the multiplicity, represents the number of codewords at the
minimum distance.
The minimum distance of a code is not, in the general case, simple to de-
termine except if the number of codewords is low enough for us to make an
exhaustive list of them, or if particular properties of the code enable us to es-
tablish an analytical expression of this value (for example, the minimum distance
