The turbo decoding algorithm and its phase trajectories

Abstract

This paper analyzes phase trajectories and fixed points of the turbo decoding algorithm as a function of 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 an "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 of 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.