The turbo principle applied to equalization and detection - Codes and Turbo Codes - page 378

target bit error rate of 10 − 5 , the iterative process provides a gain of the order of
6.2 dB compared with the performance of the conventional receiver performing
the equalization and decoding disjointly, given by the curve at the 1 st iteration.
This performance is very similar to that presented in reference [11.7].
These results give rise to a certain number of remarks, since the example
considered here presents the characteristic behaviour of turbo systems. In par-
ticular, we see that the gain provided by the iterative process only appears
beyond a certain signal to noise ratio (convergence threshold, equal to 3 dB
here). Beyond this threshold, we observe a rapid convergence of the turbo
equalizer towards the asymptotic performance of the system, given by the error
probability after decoding on a non-selective AWGN channel. To improve the
global performance of the system, we can envisage using a more powerful error
correcting code. Experience shows that we then come up against the necessity
of finding a compromise in choosing the code, between rapid convergence of the
iterative process and good asymptotic performance of the system (at high signal
to noise ratios). The greater the correction capacity of the code, the higher the
convergence threshold. On this topic, we point out that today there exist semi-
analytical tools such as EXIT ( EXtrinsic Information Transfer ) charts [11.49],
enabling the value of the convergence threshold to be predicted precisely, as
well as the error rate after decoding for a given transmission scenario, under
the hypothesis of ideal interleaving (infinite size). A second solution involves
introducing a feedback effect in front of the equivalent discrete-time channel,
by inserting an adequate precoding scheme at transmission. Cascading the pre-
encoder with the channel produces a new channel model, recursive in nature,
leading to a performance gain that is greater, the larger the dimension of the
interleaver considered. This phenomenon is known as "interleaving gain" in the
literature dedicated to serial turbo codes. Subject to carefully choosing the pre-
encoder, we can then exceed the performance of classical non-recursive turbo
equalization schemes as has been shown in [11.35] and [11.39].

Complexity of the MAP turbo equalizer and alternative solutions

The complexity of the MAP turbo equalizer is mainly dictated by the complexity
of the MAP equalizer. Now, the latter increases proportionally with the number
of branches to examine at each instant in the trellis. Considering a modulation
with M states and a discrete channel with L coecients, the total number of
branches per section of the trellis rises to M

M L− 1 = M L . We therefore see
that the processing cost associated with the MAP equalizer increases exponen-
tially with the number of states of the modulation and the length of the impulse
response of the channel. As an illustration, EDGE ( Enhanced Data Rate for
GSM Evolution ) introduces the use of 8-PSK modulation on channels with 6
coecients maximum, that is, slightly more than 262000 branches to examine
at each instant! MAP turbo equalization is therefore an attractive solution for
modulations with a low number of states (typically BPSK and QPSK) on chan-

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