Analysis of the iterative decoding of LDPC and product codes using the Gaussian approximation

We propose a novel density evolution approach to analyze the iterative decoding algorithms of low-density parity-check (LDPC) codes and product codes, based on Gaussian densities. Namely, for these classes of codes we derive a one-dimensional (1D) map whose iterates directly represent the error probability both for the additive white Gaussian noise (AWGN) and the Rayleigh-fading channel. These simple models allow a qualitative analysis of the nonlinear dynamics of the decoding algorithm. As an application, we compute the decoding thresholds and show that they are consistent with the simulation results available in the literature.

[1]  Sae-Young Chung,et al.  Analysis of sum-product decoding of low-density parity-check codes using a Gaussian approximation , 2001, IEEE Trans. Inf. Theory.

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

[3]  M. Fossorier,et al.  Error performance analysis for reliability-based decoding algorithms , 2000, 2000 IEEE International Symposium on Information Theory (Cat. No.00CH37060).

[4]  Rüdiger L. Urbanke,et al.  The capacity of low-density parity-check codes under message-passing decoding , 2001, IEEE Trans. Inf. Theory.

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

[6]  David Chase,et al.  Class of algorithms for decoding block codes with channel measurement information , 1972, IEEE Trans. Inf. Theory.

[7]  S. Brink.  Convergence of iterative decoding , 1999 .

[8]  L. Bazzi,et al.  Exact thresholds and optimal codes for the binary symmetric channel and Gallager's decoding algorithm A , 2000, 2000 IEEE International Symposium on Information Theory (Cat. No.00CH37060).

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

[10]  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.

[11]  X. Jin.  Factor graphs and the Sum-Product Algorithm , 2002 .

[12]  Benjamin Van Roy,et al.  An analysis of belief propagation on the turbo decoding graph with Gaussian densities , 2001, IEEE Trans. Inf. Theory.

[13]  G. M. Maggio,et al.  An approximate analytical model of the message passing decoder of LDPC codes , 2002, Proceedings IEEE International Symposium on Information Theory,.

[14]  Steven H. Strogatz,et al.  Nonlinear Dynamics and Chaos , 2024 .

[15]  Niclas Wiberg,et al.  Codes and Decoding on General Graphs , 1996 .

[16]  P. A. Wintz,et al.  Error Free Coding , 1973 .

[17]  M. Aminshokrollahi.  New sequences of linear time erasure codes approaching the channel capacity , 1999 .

[18]  Peter Elias,et al.  Error-free Coding , 1954, Trans. IRE Prof. Group Inf. Theory.

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

[20]  Paul H. Siegel,et al.  Performance analysis and code optimization of low density parity-check codes on Rayleigh fading channels , 2001, IEEE J. Sel. Areas Commun..

[21]  Ramesh Pyndiah,et al.  Near optimum decoding of product codes , 1994, 1994 IEEE GLOBECOM. Communications: The Global Bridge.

[22]  Ramesh Pyndiah,et al.  Near-optimum decoding of product codes: block turbo codes , 1998, IEEE Trans. Commun..

[23]  Shu Lin,et al.  Error performance analysis for reliability-based decoding algorithms , 2002, IEEE Trans. Inf. Theory.

[24]  Robert G. Gallager,et al.  Low-density parity-check codes , 1962, IRE Trans. Inf. Theory.

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