Dynamic Modeling and Analysis of Iterative Decoding for Turbo Codes

Turbo codes and low density parity check codes are two classes of most powerful error correcting codes. What makes these codes so powerful is the use of the so-called iterative decoding or turbo decoding. Roughly speaking, an iterative decoding process is an iterative learning process for a complex system where the objective is to provide a good suboptimal estimate of a desired signal. Iterative decoding is used when the true optimal estimation is impossible due to prohibitive computational complexities. Despite that iterative decoding algorithms are known to be very successful, there is no satisfactory understanding of their “magical” power. In fact, the behavior of iterative decoding is a big mystery in the coding theory. The aim of this presentation is to show how to model and analyze an iterative decoding process using a system-theory based approach. More specifically, we can view the iterative decoding process as a feedback system. With this view, we propose a stochastic framework for dynamic modeling and analysis of iterative decoding. By using appropriate statistical parameters to describe the signals in an iterative decoding process, we show that the process can be adequately approximated by a two-input, two-output nonlinear dynamic model. We have discovered that a typical decoding process is much more intricate than previously known, involving two regions of attractions, several fixed points, and a stable equilibrium manifold at which all decoding trajectories converge. This new modeling approach is useful in gaining new knowledge on iterative decoding and devising better decoding algorithms.