Algorithms for the Iterative Estimation of Discrete-Valued Sparse Vectors

In Compressed Sensing, a real-valued sparse vector has to be estimated from an underdetermined system of linear equations. In many applications, however, the elements of the sparse vector are drawn from a finite set. For the estimation of these discrete-valued vectors, matched algorithms are required which take the additional knowledge of the discrete nature into account. In this paper, the estimation problem is treated from a communications engineering point of view. A powerful new algorithm incorporating techniques known from digital communications and information theory is derived. For comparison, Turbo Compressed Sensing is adapted to the discrete setup and a simplified and generalized notation is presented. The performance of the algorithms is covered by numerical simulations.

[1]  Johannes B. Huber.  Iterated Soft-Decision Interfer-ence Cancellation for CDMA , 1998 .

[2]  Xiaojun Yuan,et al.  Turbo Compressed Sensing with Partial DFT Sensing Matrix , 2014, IEEE Signal Processing Letters.

[3]  M. Reinhardt,et al.  Signal estimation for interference cancellation and decision feedback equalization , 1997, 1997 IEEE 47th Vehicular Technology Conference. Technology in Motion.

[4]  Robert F. H. Fischer,et al.  MMSE-based version of OMP for recovery of discrete-valued sparse signals , 2016 .

[5]  Steven Kay,et al.  Fundamentals Of Statistical Signal Processing , 2001 .

[6]  Andrea Montanari,et al.  Message passing algorithms for compressed sensing: I. motivation and construction , 2009, 2010 IEEE Information Theory Workshop on Information Theory (ITW 2010, Cairo).

[7]  Johannes B. Huber,et al.  On improved multiuser detection with iterated soft decision interference cancellation , 1999, 1999 IEEE Communications Theory Mini-Conference (Cat. No.99EX352).

[8]  T. Blumensath,et al.  Iterative Thresholding for Sparse Approximations , 2008 .

[9]  David L Donoho,et al.  Compressed sensing , 2006, IEEE Transactions on Information Theory.

[10]  Andrea Montanari,et al.  Message-passing algorithms for compressed sensing , 2009, Proceedings of the National Academy of Sciences.

[11]  Xiaojun Yuan,et al.  On the Performance of Turbo Signal Recovery with Partial DFT Sensing Matrices , 2015, IEEE Signal Processing Letters.

[12]  S. Kay.  Fundamentals of statistical signal processing: estimation theory , 1993 .

[13]  Robert F. H. Fischer,et al.  Peak-to-Average Power Ratio Reduction in OFDM via Sparse Signals: Transmitter-Side Tone Reservation vs. Receiver-Side Compressed Sensing , 2012 .

[14]  Georgios B. Giannakis,et al.  Exploiting Sparse User Activity in Multiuser Detection , 2011 .

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

[16]  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.

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

[18]  I. Daubechies,et al.  Accelerated Projected Gradient Method for Linear Inverse Problems with Sparsity Constraints , 2007, 0706.4297.

[19]  Xiaohu You,et al.  Generalized turbo signal recovery for nonlinear measurements and orthogonal sensing matrices , 2015, 2016 IEEE International Symposium on Information Theory (ISIT).

[20]  Y. C. Pati,et al.  Orthogonal matching pursuit: recursive function approximation with applications to wavelet decomposition , 1993, Proceedings of 27th Asilomar Conference on Signals, Systems and Computers.

[21]  Volkan Cevher,et al.  Model-Based Compressive Sensing , 2008, IEEE Transactions on Information Theory.

[22]  Olgica Milenkovic,et al.  Weighted Euclidean superimposed codes for integer compressed sensing , 2008, 2008 IEEE Information Theory Workshop.

[23]  Robert F. H. Fischer,et al.  Adapting Compressed Sensing Algorithms to Discrete Sparse Signals , 2014, WSA.

[24]  Robert F. H. Fischer,et al.  Soft-feedback OMP for the recovery of discrete-valued sparse signals , 2015, 2015 23rd European Signal Processing Conference (EUSIPCO).

[25]  Laurence A. Wolsey,et al.  Integer and Combinatorial Optimization , 1988 .

[26]  T. Ostertag,et al.  Optimized Sinus Wave generation with Compressed Sensing for Radar Applications , 2013 .

[27]  Robin Fay,et al.  Introducing the counter mode of operation to Compressed Sensing based encryption , 2016, Inf. Process. Lett..

[28]  Przemyslaw Dymarski,et al.  Sparse signal modeling in a scalable CELP coder , 2013, 21st European Signal Processing Conference (EUSIPCO 2013).

[29]  Defeng Huang,et al.  A Concise Representation for the Soft-in Soft-out LMMSE Detector , 2011, IEEE Communications Letters.

[30]  Andrea Montanari,et al.  The dynamics of message passing on dense graphs, with applications to compressed sensing , 2010, ISIT.

[31]  Olgica Milenkovic,et al.  Sparse weighted Euclidean superimposed coding for integer compressed sensing , 2008, CISS.