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.

[1]  Jung-Fu Cheng,et al.  Turbo Decoding as an Instance of Pearl's "Belief Propagation" Algorithm , 1998, IEEE J. Sel. Areas Commun..

[2]  H. Vincent Poor,et al.  Iterative (turbo) soft interference cancellation and decoding for coded CDMA , 1999, IEEE Trans. Commun..

[3]  Lihua Xie,et al.  The sector bound approach to quantized feedback control , 2005, IEEE Transactions on Automatic Control.

[4]  Minyue Fu,et al.  Ergodic Properties for Multirate Linear Systems , 2007, IEEE Transactions on Signal Processing.

[5]  John Baillieul,et al.  Feedback Designs in Information-Based Control , 2002 .

[6]  Norman C. Beaulieu,et al.  Estimating the distribution of a sum of independent lognormal random variables , 1995, IEEE Trans. Commun..

[7]  John Cocke,et al.  Optimal decoding of linear codes for minimizing symbol error rate (Corresp.) , 1974, IEEE Trans. Inf. Theory.

[8]  William E. Ryan.  A Turbo Code Tutorial , 1997 .

[9]  Stephan ten Brink,et al.  Convergence behavior of iteratively decoded parallel concatenated codes , 2001, IEEE Trans. Commun..

[10]  S. Sahai,et al.  The necessity and sufficiency of anytime capacity for control over a noisy communication link , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[11]  Alex J. Grant,et al.  Convergence of linear interference cancellation multiuser receivers , 2001, IEEE Trans. Commun..

[12]  Desmond P. Taylor,et al.  Convergence and errors in turbo-decoding , 2001, Proceedings. 2001 IEEE International Symposium on Information Theory (IEEE Cat. No.01CH37252).

[13]  Minyue Fu,et al.  Stochastic analysis of turbo decoding , 2005, IEEE Transactions on Information Theory.

[14]  Dariush Divsalar,et al.  Iterative turbo decoder analysis based on density evolution , 2001, IEEE J. Sel. Areas Commun..

[15]  Hesham El Gamal,et al.  Analyzing the turbo decoder using the Gaussian approximation , 2001, IEEE Trans. Inf. Theory.

[16]  Sang Joon Kim,et al.  A Mathematical Theory of Communication , 2006 .

[17]  Sekhar Tatikonda,et al.  Control over noisy channels , 2004, IEEE Transactions on Automatic Control.

[18]  Minyue Fu,et al.  QUANTIZED FEEDBACK CONTROL FOR SAMPLED-DATA SYSTEMS , 2005 .

[19]  Daniel Liberzon,et al.  Quantized feedback stabilization of linear systems , 2000, IEEE Trans. Autom. Control..

[20]  M. Fu.  Robust stabilization of linear uncertain systems via quantized feedback , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[21]  Richard E. Blahut,et al.  Generalized EXIT chart and BER analysis of finite-length turbo codes , 2003, GLOBECOM '03. IEEE Global Telecommunications Conference (IEEE Cat. No.03CH37489).

[22]  Nicola Elia,et al.  Stabilization of linear systems with limited information , 2001, IEEE Trans. Autom. Control..

[23]  Radford M. Neal,et al.  Near Shannon limit performance of low density parity check codes , 1996 .

[24]  Alex Grant,et al.  Iterative Implementations for Linear Multiuser Detectors , 1999 .

[25]  A. Glavieux,et al.  Near Shannon limit error-correcting coding and decoding: Turbo-codes. 1 , 1993, Proceedings of ICC '93 - IEEE International Conference on Communications.

[26]  Wing Shing Wong,et al.  Systems with finite communication bandwidth constraints. II. Stabilization with limited information feedback , 1999, IEEE Trans. Autom. Control..

[27]  Thomas J. Richardson,et al.  The geometry of turbo-decoding dynamics , 2000, IEEE Trans. Inf. Theory.

[28]  Rüdiger L. Urbanke,et al.  Design of capacity-approaching irregular low-density parity-check codes , 2001, IEEE Trans. Inf. Theory.

[29]  Lihua Xie,et al.  Performance Control of Linear Systems Using Quantized Feedback , 2003, 2003 4th International Conference on Control and Automation Proceedings.