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The turbo decoding algorithm and its phase trajectories
Written by:
Dakshi Agrawal
and
Alexander Vardy.
Citation:
IEEE Transaction on Information Theory, 47:699-722, February 2001.
Copyright © (2001) by IEEE. Permission to make digital or
hard copies of part or all of this work for personal or classroom use
is granted without fee provided that copies are not made or distributed
for profit. To copy otherwise, to republish, to post on servers, or to
redistribute to lists, requires prior specific permission and/or a fee.
Abstract:
This paper analyzes phase trajectories and fixed points of the turbo
decoding algorithm as a function of the signal-to-noise ratio (SNR). By
exploiting the large length of turbo codes, the turbo decoding algorithm
is treated as a single-parameter dynamical system, parameterized
(approximately) by the SNR. This parameterization, along with extensive
simulations at practical SNRs and asymptotic analysis as SNR goes to
zero and infinity, is used to subdivide the entire SNR range into three
regions with the waterfall region in the middle. The turbo decoding
algorithm has distinctive phase trajectories and convergence properties
in these three SNR regions. This paper also investigates existence and
properties of fixed points in these SNR regions. The main fixed points
of the turbo decoding algorithm are classified into two categories. In
a wide range of SNRs (corresponding to bit-error rates less than 10-1 ),
the decoding algorithm has unequivocal fixed points which correspond to
mostly correct decisions on the information bits. Within this range,
toward the lower values of SNR, there is another fixed point which
corresponds to many erroneous decision on the information bits. Fixed
points of this type are referred to as indecisive fixed points. It turns
out that the indecisive fixed points bifurcate and disappear for SNRs in
the waterfall region. This paper associates the qualitative transition
o phase trajectories in the waterfall region to the bifurcation of
indecisive fixed points. These bifurcations also explain empirically
observed quasi-periodic and periodic phase trajectories of the turbo
decoding algorithm.
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