The turbo decoding algorithm and its phase trajectories

This paper analyzes phase trajectories and fixed points of the turbo decodi ng algorithm as a function of signal-to-noise ratio (SNR), By exploiting th e large length of turbo codes, the turbo decoding algorithm is treated as a single-parameter dynamical system, parameterized (approximately) by the SN R, This parameterization, along with extensive simulations at practical SNR s and asymptotic analysis as SNR goes to zero and infinity, is used to subd ivide the entire SNR range into three regions with the "waterfall region" i n the middle, The turbo decoding algorithm has distinctive phase trajectori es and convergence properties in these three SNR regions. This paper also i nvestigates existence and properties of fixed points in these SNR regions. The main fixed points of the turbo decoding algorithm are classified into t wo categories, In a wide range of SNRs (corresponding to bit-error rates le ss than 10(-1)), the decoding algorithm has an "unequivocal" fixed points w hich correspond to mostly correct decisions on the information bits, Within this range, toward the lower values of SNR, there is another fixed point w hich 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 th e waterfall region. This paper associates the qualitative transition of pha se trajectories in the waterfall region to the bifurcation of indecisive fi xed points. These bifurcations also explain empirically observed quasi-peri odic and periodic phase trajectories of the turbo decoding algorithm.