Incorporating prior information into semidefinite relaxation of quadratic optimization problems

This paper focuses on equalization as a representative of the large class of communications receiver problems that can all be cast as the optimization of a quadratic objective function subject to quadratic constraints (QCQP problems). It is well known that approximate solutions to QCQP problems can be obtained efficiently through a relaxation technique which results in a linear program subject to a semidefinite constraint. This paper extends the semidefinite relaxation (SDR) approach to incorporate prior probabilities on the bits. With this extension, the SDR approach may used in the setting of turbo equalization and in other forms of iterative receiver processing. In this paper, the soft-in soft-out SDR-based equalizer is compared to an optimal maximum a posteriori probability based equalizer implemented via the BCJR algorithm.

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

[2]  Georgios B. Giannakis,et al.  OFDM or single-carrier block transmissions? , 2004, IEEE Transactions on Communications.

[3]  Joachim Hagenauer,et al.  A Viterbi algorithm with soft-decision outputs and its applications , 1989, IEEE Global Telecommunications Conference, 1989, and Exhibition. 'Communications Technology for the 1990s and Beyond.

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

[5]  Giuseppe Caire,et al.  On maximum-likelihood detection and the search for the closest lattice point , 2003, IEEE Trans. Inf. Theory.

[6]  Stephen P. Boyd,et al.  Graph Implementations for Nonsmooth Convex Programs , 2008, Recent Advances in Learning and Control.

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

[8]  Emanuele Viterbo,et al.  A universal lattice code decoder for fading channels , 1999, IEEE Trans. Inf. Theory.

[9]  C. Helmberg,et al.  Solving quadratic (0,1)-problems by semidefinite programs and cutting planes , 1998 .

[10]  Zhi-Quan Luo,et al.  Semidefinite Relaxation of Quadratic Optimization Problems , 2010, IEEE Signal Processing Magazine.

[11]  Alexander Vardy,et al.  Closest point search in lattices , 2002, IEEE Trans. Inf. Theory.

[12]  Björn E. Ottersten,et al.  On the complexity of sphere decoding in digital communications , 2005, IEEE Transactions on Signal Processing.

[13]  Andrew C. Singer,et al.  Turbo Equalization: An Overview , 2011, IEEE Transactions on Information Theory.

[14]  Zhi-Quan Luo,et al.  Efficient Implementation of Quasi- Maximum-Likelihood Detection Based on Semidefinite Relaxation , 2009, IEEE Transactions on Signal Processing.

[15]  Franz Rendl,et al.  Nonpolyhedral Relaxations of Graph-Bisection Problems , 1995, SIAM J. Optim..

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

[17]  Claude Berrou,et al.  A low complexity soft-output Viterbi decoder architecture , 1993, Proceedings of ICC '93 - IEEE International Conference on Communications.

[18]  Sergio Verdú,et al.  Computational complexity of optimum multiuser detection , 1989, Algorithmica.

[19]  Andrew C. Singer,et al.  Minimum mean squared error equalization using a priori information , 2002, IEEE Trans. Signal Process..

[20]  Alain Glavieux,et al.  Turbo equalization over a frequency selective channel , 1997 .

[21]  Michael E. Pohst,et al.  On the computation of lattice vectors of minimal length, successive minima and reduced bases with applications , 1981, SIGS.

[22]  Amir K. Khandani,et al.  A near maximum likelihood decoding algorithm for MIMO systems based on semi-definite programming , 2005, ISIT.

[23]  John M. Cioffi,et al.  Turbo decision aided equalization for magnetic recording channels , 1999, Seamless Interconnection for Universal Services. Global Telecommunications Conference. GLOBECOM'99. (Cat. No.99CH37042).

[24]  Dan Raphaeli.  Linear Equalizers for Turbo Equalization A New Optimization Criterion for Determining the Equalizer Taps , 2000 .

[25]  David P. Williamson,et al.  Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming , 1995, JACM.