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29
Random Sparse Linear Systems Observed Via Arbitrary Channels: A Decoupling Principle
"... Abstract—This paper studies the problem of estimating the vector input to a sparse linear transformation based on the observation of the output vector through a bank of arbitrary independent channels. The linear transformation is drawn randomly from an ensemble with mild regularity conditions. The c ..."
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Abstract—This paper studies the problem of estimating the vector input to a sparse linear transformation based on the observation of the output vector through a bank of arbitrary independent channels. The linear transformation is drawn randomly from an ensemble with mild regularity conditions. The central result is a decoupling principle in the largesystem limit. That is, the optimal estimation of each individual symbol in the input vector is asymptotically equivalent to estimating the same symbol through a scalar additive Gaussian channel, where the aggregate effect of the interfering symbols is tantamount to a degradation in the signaltonoise ratio. The degradation is determined from a recursive formula related to the score function of the conditional probability distribution of the noisy channel. A sufficient condition is provided for belief propagation (BP) to asymptotically produce the a posteriori probability distribution of each input symbol given the output. This paper extends the authors ’ previous decoupling result for Gaussian channels to arbitrary channels, which was based on an earlier work of Montanari and Tse. Moreover, a rigorous justification is provided for the generalization of some results obtained via statical physics methods. I.
On the duality between SlepianWolf coding and channel coding
 IEEE International Symposium on Information Theory
"... Abstract — A codebooklevel duality between SlepianWolf coding and channel coding is established. Specifically, it is shown that using linear codes over ZM (the ring of integers mod M), each SlepianWolf coding problem is equivalent to a channel coding problem for a semisymmetric additive channel ..."
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Abstract — A codebooklevel duality between SlepianWolf coding and channel coding is established. Specifically, it is shown that using linear codes over ZM (the ring of integers mod M), each SlepianWolf coding problem is equivalent to a channel coding problem for a semisymmetric additive channel under optimal decoding, belief propagation decoding, and minimum entropy decoding. Various notions of symmetric channels are discussed and their connections with semisymmetric additive channels are clarified. I.
Modern coding theory: the statistical mechanics and computer science point of view
, 2007
"... These are the notes for a set of lectures delivered by the two authors at the Les Houches Summer School on ‘Complex Systems’ in July 2006. They provide an introduction to the basic concepts in modern (probabilistic) coding theory, highlighting connections with statistical mechanics. We also stress c ..."
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These are the notes for a set of lectures delivered by the two authors at the Les Houches Summer School on ‘Complex Systems’ in July 2006. They provide an introduction to the basic concepts in modern (probabilistic) coding theory, highlighting connections with statistical mechanics. We also stress common concepts with other disciplines dealing with similar problems that can be generically referred to as ‘large graphical models’. While most of the lectures are devoted to the classical channel coding problem over simple memoryless channels, we present a discussion of more complex channel models. We conclude with an overview of the main open challenges in the field.
Characterization and Optimization of LDPC codes for the 2user Gaussian Multiple Access Channel
, 2007
"... In this paper we address the problem of designing good LDPC codes for the Gaussian multiple access channel (MAC). The framework we choose is to design multiuser LDPC codes with joint Belief Propagation decoding on the joint graph of the 2user case. Our main result compared to existing work is to ex ..."
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Cited by 7 (0 self)
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In this paper we address the problem of designing good LDPC codes for the Gaussian multiple access channel (MAC). The framework we choose is to design multiuser LDPC codes with joint Belief Propagation decoding on the joint graph of the 2user case. Our main result compared to existing work is to express analytically EXIT functions of the multiuser decoder with two different approximations of the Density Evolution. This allows us to propose a very simple linear programming optimization for the complicated problem of LDPC code design with joint multiuser decoding. The stability condition for our case is derived and used in the optimization constraints. The codes that we obtain for the 2user case are quite good for various rates, especially if we consider the very simple optimization procedure.
Finitedimensional bounds on Zm and binary LDPC codes with beliefpropagation decoders
 IEEE Trans. on Information Theory
, 2007
"... This paper focuses on finitedimensional upper and lower bounds on decodable thresholds of Zm and binary LDPC codes, assuming belief propagation decoding on memoryless channels. Two noise measures will be considered: the Bhattacharyya noise parameter and the soft bit value for a MAP decoder on the u ..."
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This paper focuses on finitedimensional upper and lower bounds on decodable thresholds of Zm and binary LDPC codes, assuming belief propagation decoding on memoryless channels. Two noise measures will be considered: the Bhattacharyya noise parameter and the soft bit value for a MAP decoder on the uncoded channel. For Zm LDPC codes, an iterative mdimensional bound is derived for maryinput/symmetricoutput channels, which gives a sufficient stability condition for Zm LDPC codes and will be complemented by a matched necessary stability condition introduced herein. Applications to the coded modulations and to codes with nonequiprobable distributed codewords will also be discussed. For binary codes, two new lower bounds are provided for symmetric channels, including a twodimensional iterative bound and a onedimensional noniterative bound, the latter of which is the best known bound that is tight for BSCs. By adapting the reverse channel perspective, a pair of upper and lower bounds on the decodable Bhattacharyya noise parameter is derived for nonsymmetric channels, which coincides with the existing bound for symmetric channels.
On Codes that Correct Asymmetric Errors with Graded Magnitude Distribution
"... Abstract—In multilevel flash memories, the dominant cell errors are asymmetric with limitedmagnitude. With such an error model in mind, Cassuto et al. recently developed bounds and constructions for codes correcting t asymmetric errors with magnitude no more than ℓ. However, a more refined model o ..."
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Cited by 6 (1 self)
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Abstract—In multilevel flash memories, the dominant cell errors are asymmetric with limitedmagnitude. With such an error model in mind, Cassuto et al. recently developed bounds and constructions for codes correcting t asymmetric errors with magnitude no more than ℓ. However, a more refined model of these memory devices reflects the fact that typically only a small number of errors have large magnitude while the remainder are of smaller magnitude. In this work, we study such an error model, in which at most t1 errors of maximum magnitude ℓ1 and at most t2 errors of maximum magnitude ℓ2, with ℓ1 < ℓ2, can occur. We adapt the analysis and code construction of Cassuto, et al. for the refined error model and assess the relative efficiency of the new codes. We then consider in more detail specific constructions for the case where t1 = t2 = 1, ℓ1 = 1, and ℓ2> 1. I.
On finitedimensional bounds for LDPClike codes with iterative decoding
 in Proc. Int’l Symp. Inform. Theory & its Applications
, 2004
"... This paper focuses on finitedimensional upper and lower bounds on decodable thresholds assuming iterative decoding. Two noise measures will be considered: the Bhattacharyya noise parameter, and the expected soft bit of the considered channel. An iterative mdimensional lower bound is derived for Zm ..."
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Cited by 6 (4 self)
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This paper focuses on finitedimensional upper and lower bounds on decodable thresholds assuming iterative decoding. Two noise measures will be considered: the Bhattacharyya noise parameter, and the expected soft bit of the considered channel. An iterative mdimensional lower bound is derived for Zm LDPC codes in symmetric channels, which gives a sufficient stability condition for Zm LDPC codes and will be complemented by a new necessary stability condition introduced herein. Two new lower bounds are provided for binaryinput/symmetricoutput channels, including a twodimensional iterative bound and a onedimensional noniterative bound, the latter of which is the best known bound that is tight for BSCs. A reverse channel perspective is then used to generalize existing bounds from binaryinput/symmetricoutput channels to binaryinput/nonsymmetricoutput channels. 1.
A decision feedback based scheme for SlepianWolf coding of sources with hidden Markov correlation
 IEEE Commun. Letters
, 2006
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Dirty Paper Coding using Signbit Shaping and LDPC Codes
"... Abstract—Dirty paper coding (DPC) refers to methods for presubtraction of known interference at the transmitter of a multiuser communication system. There are numerous applications for DPC, including coding for broadcast channels. Recently, latticebased coding techniques have provided several desi ..."
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Abstract—Dirty paper coding (DPC) refers to methods for presubtraction of known interference at the transmitter of a multiuser communication system. There are numerous applications for DPC, including coding for broadcast channels. Recently, latticebased coding techniques have provided several designs for DPC. In latticebased DPC, there are two codes a convolutional code that defines a lattice used for shaping and an error correction code used for channel coding. Several specific designs have been reported in the recent literature using convolutional and graphbased codes for capacityapproaching shaping and coding gains. In most of the reported designs, either the encoder works on a joint trellis of shaping and channel codes or the decoder requires iterations between the shaping and channel decoders. This results in high complexity of implementation. In this work, we present a latticebased DPC scheme that provides good shaping and coding gains with moderate complexity at both the encoder and the decoder. We use a convolutional code for signbit shaping, and a lowdensity parity check (LDPC) code for channel coding. The crucial idea is the introduction of a onecodeword delay and careful parsing of the bits at the transmitter, which enables an LDPC decoder to be run first at the receiver. This provides gains without the need for iterations between the shaping and channel decoders. Simulation results confirm that at high rates the proposed DPC method performs close to capacity with moderate complexity. As an application of the proposed DPC method, we show a design for superposition coding that provides rates better than timesharing over a Gaussian broadcast channel. I.
Approaching the SlepianWolf Limit with LDPC Coset Codes
"... We consider the SlepianWolf code design based on LDPC (lowdensity paritycheck) coset codes. We derive the density evolution formula for the SlepianWolf coding, equipped with a concentration theorem. An intimate connection between the SlepianWolf coding and channel coding is established. Specifi ..."
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We consider the SlepianWolf code design based on LDPC (lowdensity paritycheck) coset codes. We derive the density evolution formula for the SlepianWolf coding, equipped with a concentration theorem. An intimate connection between the SlepianWolf coding and channel coding is established. Specifically we show that, under density evolution, each SlepianWolf source coding problem is equivalent to a channel coding problem for a binaryinput symmetricoutput channel. With this connection, many classic results in channel coding can be easily translated into the SlepianWolf setting.