Abstract: Turbo Codes can approach the Shannon limit very closely with the help of its special iterative decoding algorithm. This paper establishes a nonlinear dynamic system to analyze the relationship between Turbo decoding output and the number of iterations. Here, the number of iterations is taken as the time axis, decoding output as the state variable, SNR and information bits N as system parameters. It is shown that with SNR increasing, the decoding algorithm undergoes three stages, namely the indecisive fix-point, singular region and unequivocal fix-point. Bifurcations occur during the transformation from the indecisive fix-point to the singular region. It is first proposed that fold, period doubling and Neimark-Sacker bifurcation all have the possibility to occur, depending on the value of N. In the singular region, phase trajectories may appear as period-two, period-three, quasiperiod and chaos. This paper first observed and confirmed the existence of period-three and chaos. Singular region deteriorates the performance of Turbo codes under low SNR. This paper proposes a time-delay feedback control method to stablize the fix-point. Simulation results show that this method achieves 0.1—0.3 dB improvement for Turbo codes under low SNR condition.

Keywords: 
• Turbo decoding algorithm /
• dynamic system /
• bifurcation /
• chaos /
• time-delay feedback control