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InformationTheoretically Optimal Compressed Sensing via Spatial Coupling and Approximate Message Passing
, 2011
"... We study the compressed sensing reconstruction problem for a broad class of random, banddiagonal sensing matrices. This construction is inspired by the idea of spatial coupling in coding theory. As demonstrated heuristically and numerically by Krzakala et al. [KMS+ 11], message passing algorithms ca ..."
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Cited by 51 (5 self)
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We study the compressed sensing reconstruction problem for a broad class of random, banddiagonal sensing matrices. This construction is inspired by the idea of spatial coupling in coding theory. As demonstrated heuristically and numerically by Krzakala et al. [KMS+ 11], message passing algorithms can effectively solve the reconstruction problem for spatially coupled measurements with undersampling rates close to the fraction of nonzero coordinates. We use an approximate message passing (AMP) algorithm and analyze it through the state evolution method. We give a rigorous proof that this approach is successful as soon as the undersampling rate δ exceeds the (upper) Rényi information dimension of the signal, d(pX). More precisely, for a sequence of signals of diverging dimension n whose empirical distribution converges to pX, reconstruction is with high probability successful from d(pX) n + o(n) measurements taken according to a band diagonal matrix. For sparse signals, i.e. sequences of dimension n and k(n) nonzero entries, this implies reconstruction from k(n)+o(n) measurements. For ‘discrete ’ signals, i.e. signals whose coordinates take a fixed finite set of values, this implies reconstruction from o(n) measurements. The result
The effect of spatial coupling on compressive sensing
 in Communication, Control, and Computing (Allerton
"... Abstract — Recently, it was observed that spatiallycoupled LDPC code ensembles approach the Shannon capacity for a class of binaryinput memoryless symmetric (BMS) channels. The fundamental reason for this was attributed to a threshold saturation phenomena derived in [1]. In particular, it was show ..."
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Cited by 46 (9 self)
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Abstract — Recently, it was observed that spatiallycoupled LDPC code ensembles approach the Shannon capacity for a class of binaryinput memoryless symmetric (BMS) channels. The fundamental reason for this was attributed to a threshold saturation phenomena derived in [1]. In particular, it was shown that the belief propagation (BP) threshold of the spatially coupled codes is equal to the maximum a posteriori (MAP) decoding threshold of the underlying constituent codes. In this sense, the BP threshold is saturated to its maximum value. Moreover, it has been empirically observed that the same phenomena also occurs when transmitting over more general classes of BMS channels. In this paper, we show that the effect of spatial coupling is not restricted to the realm of channel coding. The effect of coupling also manifests itself in compressed sensing. Specifically, we show that spatiallycoupled measurement matrices have an improved sparsity to sampling threshold for reconstruction algorithms based on verification decoding. For BPbased reconstruction algorithms, this phenomenon is also tested empirically via simulation. At the block lengths accessible via simulation, the effect is quite small and it seems that spatial coupling is not providing the gains one might expect. Based on the threshold analysis, however, we believe this warrants further study. I.
Threshold Saturation on BMS Channels via Spatial Coupling
"... We consider spatially coupled code ensembles. A particular instance are convolutional LDPC ensembles. It was recently shown that, for transmission over the binary erasure channel, this coupling increases the belief propagation threshold of the ensemble to the maximum apriori threshold of the unde ..."
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Cited by 36 (7 self)
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We consider spatially coupled code ensembles. A particular instance are convolutional LDPC ensembles. It was recently shown that, for transmission over the binary erasure channel, this coupling increases the belief propagation threshold of the ensemble to the maximum apriori threshold of the underlying component ensemble. We report on empirical evidence which suggests that the same phenomenon also occurs when transmission takes place over a general binary memoryless symmetric channel. This is confirmed both by simulations as well as by computing EBP GEXIT curves and by comparing the empirical BP thresholds of coupled ensembles to the empirically determined MAP thresholds of the underlying regular ensembles. We further consider ways of reducing the rateloss incurred by such constructions.
Coupled graphical models and their thresholds
 in Proc. of the IEEE Inform. Theory Workshop
, 2010
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Improvement of bpbased cdma multiuser detection by spatial coupling
 2011, coRR
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A simple proof of threshold saturation for coupled scalar recursions
 in Proc. Intl. Symp. on Turbo Codes and Iter. Inform. Proc. (ISTC), 2012
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Asymptotically regular LDPC codes with linear distance growth and thresholds close to capacity
 in Proc. Inform. Theory and App. Workshop
, 2010
"... Abstract—Families of asymptotically regular LDPC block code ensembles can be formed by terminating (J, K)regular protographbased LDPC convolutional codes. By varying the termination length, we obtain a large selection of LDPC block code ensembles with varying code rates and substantially better it ..."
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Cited by 24 (10 self)
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Abstract—Families of asymptotically regular LDPC block code ensembles can be formed by terminating (J, K)regular protographbased LDPC convolutional codes. By varying the termination length, we obtain a large selection of LDPC block code ensembles with varying code rates and substantially better iterative decoding thresholds than those of (J, K)regular LDPC block code ensembles, despite the fact that the terminated ensembles are almost regular. Also, by means of an asymptotic weight enumerator analysis, we show that minimum distance grows linearly with block length for all of the ensembles in these families, i.e., the ensembles are asymptotically good. We find that, as the termination length increases, families of “asymptotically regular ” codes with capacity approaching iterative decoding thresholds and declining minimum distance growth rates are obtained, allowing a code designer to tradeoff between distance growth rate and threshold. Further, we show that the thresholds andthedistancegrowthratescanbeimprovedbycarefullychoosing the component protographs used in the code construction. I.
On the thresholds of generalized LDPC convolutional codes based on protographs
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Windowed decoding of protographbased LDPC convolutional codes over erasure channels
 IEEE TRANS. ON INFORMATION THEORY
, 2012
"... We consider a windowed decoding scheme for LDPC convolutional codes that is based on the beliefpropagation (BP) algorithm. We discuss the advantages of this decoding scheme and identify certain characteristics of LDPC convolutional code ensembles that exhibit good performance with the windowed deco ..."
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Cited by 20 (3 self)
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We consider a windowed decoding scheme for LDPC convolutional codes that is based on the beliefpropagation (BP) algorithm. We discuss the advantages of this decoding scheme and identify certain characteristics of LDPC convolutional code ensembles that exhibit good performance with the windowed decoder. We will consider the performance of these ensembles and codes over erasure channels with and without memory. We show that the structure of LDPC convolutional code ensembles is suitable to obtain performance close to the theoretical limits over the memoryless erasure channel, both for the BP decoder and windowed decoding. However, the same structure imposes limitations on the performance over erasure channels with memory.
Universality for the noisy SlepianWolf problem via spatial coupling
 in Proc. IEEE Int. Symp. Inform. Theory
, 2011
"... Abstract—We consider a noisy SlepianWolf problem where two correlated sources are separately encoded and transmitted over two independent binary memoryless symmetric channels. Each channel capacity is assumed to be characterized by a single parameter which is not known at the transmitter. The recei ..."
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Cited by 14 (6 self)
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Abstract—We consider a noisy SlepianWolf problem where two correlated sources are separately encoded and transmitted over two independent binary memoryless symmetric channels. Each channel capacity is assumed to be characterized by a single parameter which is not known at the transmitter. The receiver has knowledge of both the source correlation and the channel parameters. We call a system universal if it retains nearcapacity performance without channel knowledge at the transmitter. Kudekar et al. recently showed that terminated lowdensity paritycheck (LDPC) convolutional codes (a.k.a. spatiallycoupled LDPC ensembles) can have beliefpropagation thresholds that approach their maximum aposteriori thresholds. This was proven for binary erasure channels and shown empirically for binary memoryless symmetric channels. They also conjectured that the principle of spatial coupling is very general and the phenomenon of threshold saturation applies to a very broad class of graphical models. In this work, we derive an area theorem for the joint decoder and empirically show that threshold saturation occurs for this problem. As a result, we demonstrate nearuniversal performance for this problem using the proposed spatiallycoupled coding system. A similar result is also discussed briefly for the 2user multipleaccess channel. Index Terms—LDPC codes, spatial coupling, EXIT functions, density evolution, correlated sources, nonsystematic encoders,