Interior Point Decoding for Linear Vector Channels Based on Convex Optimization

In the present paper, a novel decoding algorithm for low-density parity-check (LDPC) codes based on convex optimization is presented. The decoding algorithm, which is referred to hereinafter as interior point decoding, is designed for linear vector channels. The linear vector channels include several practically important channels, such as inter-symbol interference channels and partial response (PR) channels. It is shown that the maximum likelihood decoding (MLD) rule for a linear vector channel can be relaxed to a convex optimization problem, which is called a relaxed MLD problem. The proposed decoding algorithm is based on a numerical optimization technique known as the interior point method with barrier functions. Approximate variations of an interior point method based on the gradient descent and Newton methods are used to solve the relaxed MLD problem. Compared with a conventional joint message-passing decoder, from computer simulations, it is observed that the proposed decoding algorithm achieves better BER performance on PR channels with less decoding complexity in several cases. Furthermore, an extension of the proposed algorithm for high-order modulation formats, such as PAM and QAM, is presented.

[1]  Rudiger Urbanke,et al.  Approaching the AWGN channel capacity without active shaping , 1997, Proceedings of IEEE International Symposium on Information Theory.

[2]  A. Kolen.  Combinatorial optimization algorithm and complexity: Prentice-Hall, Englewood Cliffs, 1982, 496 pages, $49.50 , 1983 .

[3]  Shlomo Shamai,et al.  Superposition coding for side-information channels , 2006, IEEE Transactions on Information Theory.

[4]  Li Ping,et al.  Modified LMMSE turbo equalization , 2004, IEEE Communications Letters.

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

[6]  Paul H. Siegel,et al.  Joint message-passing decoding of LDPC Codes and partial-response channels , 2002, IEEE Trans. Inf. Theory.

[7]  Paul H. Siegel,et al.  Equalization on Graphs: Linear Programming and Message Passing , 2007, 2007 IEEE International Symposium on Information Theory.

[8]  Paul H. Siegel,et al.  Correction to "Joint Message-Passing Decoding of LDPC Codes and Partial-Response Channels" , 2003 .

[9]  Wayne E. Stark,et al.  Unified design of iterative receivers using factor graphs , 2001, IEEE Trans. Inf. Theory.

[10]  Li Ping,et al.  Coded modulation using superimposed binary codes , 2004, IEEE Transactions on Information Theory.

[11]  Jon Feldman,et al.  A New Linear Programming Approach to Decoding Linear Block Codes , 2008, IEEE Transactions on Information Theory.

[12]  Alain Glavieux,et al.  Iterative correction of intersymbol interference: Turbo-equalization , 1995, Eur. Trans. Telecommun..

[13]  Jinghu Chen,et al.  Density evolution for two improved BP-Based decoding algorithms of LDPC codes , 2002, IEEE Communications Letters.

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

[15]  F. A. Seiler,et al.  Numerical Recipes in C: The Art of Scientific Computing , 1989 .

[16]  Jon Feldman,et al.  Decoding error-correcting codes via linear programming , 2003 .

[17]  William H. Press,et al.  The Art of Scientific Computing Second Edition , 1998 .

[18]  Paul H. Siegel,et al.  Adaptive Methods for Linear Programming Decoding , 2008, IEEE Transactions on Information Theory.

[19]  Alexander Schrijver,et al.  Combinatorial optimization. Polyhedra and efficiency. , 2003 .

[20]  Alexandros G. Dimakis,et al.  Guessing Facets: Polytope Structure and Improved LP Decoder , 2009, IEEE Transactions on Information Theory.

[21]  John D. Villasenor,et al.  Turbo decoding of Gilbert-Elliot channels , 2002, IEEE Trans. Commun..

[22]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[23]  Tadashi Wadayama,et al.  A Coded Modulation Scheme Based on Low Density Parity Check Codes , 2001 .

[24]  Tadashi Wadayama.  An LP decoding algorithm based on primal path-following interior point method , 2009, 2009 IEEE International Symposium on Information Theory.

[25]  J. Bard,et al.  Interior-Point Method , 2014 .

[26]  Ralf Koetter,et al.  Towards Low-Complexity Linear-Programming Decoding , 2006, ArXiv.

[27]  Paul H. Siegel,et al.  Adaptive Linear Programming Decoding , 2006, 2006 IEEE International Symposium on Information Theory.

[28]  Yinyu Ye,et al.  Interior point algorithms: theory and analysis , 1997 .

[29]  K. Schittkowski,et al.  NONLINEAR PROGRAMMING , 2022 .

[30]  Pascal O. Vontobel,et al.  Interior-point algorithms for linear-programming decoding , 2008, 2008 Information Theory and Applications Workshop.

[31]  Hideki Imai,et al.  A new multilevel coding method using error-correcting codes , 1977, IEEE Trans. Inf. Theory.

[32]  Paul H. Siegel,et al.  Efficient implementation of linear programming decoding , 2008, 2008 46th Annual Allerton Conference on Communication, Control, and Computing.

[33]  R. Koetter,et al.  Turbo equalization , 2004, IEEE Signal Processing Magazine.

[34]  Martin J. Wainwright,et al.  Using linear programming to Decode Binary linear codes , 2005, IEEE Transactions on Information Theory.

[35]  Tadashi Wadayama.  Interior point decoding for linear vector channels based on convex optimization , 2010, IEEE Trans. Inf. Theory.

[36]  David Burshtein,et al.  Iterative Approximate Linear Programming Decoding of LDPC Codes With Linear Complexity , 2008, IEEE Transactions on Information Theory.

[37]  P. Vontobel,et al.  Graph-covers and iterative decoding of nite length codes , 2003 .

[38]  David J. C. MacKay,et al.  Encyclopedia of Sparse Graph Codes , 1999 .