The focus of this talk is on analyzing the phase trajectories 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. In conjunction with extensive simulations, this parameterization is used to show that the entire SNR range can be subdivided into three regions with the waterfall region in the middle. These three regions have distinctive phase trajectories, and in most cases, the transient behavior of a phase trajectory can be used to accurately predict its asymptotic behavior. The existence and the properties of fixed points in these three SNR regions will also be discussed.

It is shown that the turbo decoding algorithm has two main types of fixed points. In a wide range of SNRs (corresponding to bit-error rates less than 1E-1), the decoding algorithm has `unequivocal' fixed points which correspond to mostly correct decisions on the information bits. Within this range, towards 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 is demonstrated that the indecisive fixed points bifurcate and disappear for SNRs in the waterfall region. We associate the qualitative transition of phase trajectories in the waterfall region to the bifurcation of indecisive fixed points. The bifurcation of these fixed points explains the quasi-periodic and periodic phase trajectory of turbo decoding as observed in simulations.

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