Results 1  10
of
41
Threshold Saturation via Spatial Coupling: Why Convolutional LDPC Ensembles Perform so well over the BEC
, 2010
"... Convolutional LDPC ensembles, introduced by Felström and Zigangirov, have excellent thresholds and these thresholds are rapidly increasing functions of the average degree. Several variations on the basic theme have been proposed to date, all of which share the good performance characteristics of c ..."
Abstract

Cited by 137 (13 self)
 Add to MetaCart
Convolutional LDPC ensembles, introduced by Felström and Zigangirov, have excellent thresholds and these thresholds are rapidly increasing functions of the average degree. Several variations on the basic theme have been proposed to date, all of which share the good performance characteristics of convolutional LDPC ensembles. We describe the fundamental mechanism which explains why “convolutionallike” or “spatially coupled” codes perform so well. In essence, the spatial coupling of the individual code structure has the effect of increasing the beliefpropagation threshold of the new ensemble to its maximum possible value, namely the maximumaposteriori threshold of the underlying ensemble. For this reason we call this phenomenon “threshold saturation”. This gives an entirely new way of approaching capacity. One significant advantage of such a construction is that one can create capacityapproaching ensembles with an error correcting radius which is increasing in the blocklength. Our proof makes use of the area theorem of the beliefpropagation EXIT curve and the connection between the maximumaposteriori and beliefpropagation threshold recently pointed out by Méasson, Montanari, Richardson, and Urbanke. Although we prove the connection between the maximumaposteriori and the beliefpropagation threshold only for a very specific ensemble and only for the binary erasure channel, empirically a threshold saturation phenomenon occurs for a wide class of ensembles and channels. More generally, we conjecture that for a large range of graphical systems a similar saturation of the “dynamical ” threshold occurs once individual components are coupled sufficiently strongly. This might give rise to improved algorithms as well as to new techniques for analysis.
LDPC block and convolutional codes based on circulant matrices
 IEEE TRANS. INFORM. THEORY
, 2004
"... A class of algebraically structured quasicyclic (QC) lowdensity paritycheck (LDPC) codes and their convolutional counterparts is presented. The QC codes are described by sparse paritycheck matrices comprised of blocks of circulant matrices. The sparse paritycheck representation allows for prac ..."
Abstract

Cited by 93 (8 self)
 Add to MetaCart
(Show Context)
A class of algebraically structured quasicyclic (QC) lowdensity paritycheck (LDPC) codes and their convolutional counterparts is presented. The QC codes are described by sparse paritycheck matrices comprised of blocks of circulant matrices. The sparse paritycheck representation allows for practical graphbased iterative messagepassing decoding. Based on the algebraic structure, bounds on the girth and minimum distance of the codes are found, and several possible encoding techniques are described. The performance of the QC LDPC block codes compares favorably with that of randomly constructed LDPC codes for short to moderate block lengths. The performance of the LDPC convolutional codes is superior to that of the QC codes on which they are based; this performance is the limiting performance obtained by increasing the circulant size of the base QC code. Finally, a continuous decoding procedure for the LDPC convolutional codes is described.
Iterative Decoding Threshold Analysis for LDPC Convolutional Codes
 ACCEPTED FOR PUBLICATION IN IEEE TRANSACTIONS ON INFORMATION THEORY
"... An iterative decoding threshold analysis for terminated regular LDPC convolutional (LDPCC) codes is presented. Using density evolution techniques, the convergence behavior of an iterative belief propagation decoder is analyzed for the binary erasure channel and the AWGN channel with binary inputs. I ..."
Abstract

Cited by 63 (10 self)
 Add to MetaCart
An iterative decoding threshold analysis for terminated regular LDPC convolutional (LDPCC) codes is presented. Using density evolution techniques, the convergence behavior of an iterative belief propagation decoder is analyzed for the binary erasure channel and the AWGN channel with binary inputs. It is shown that for a terminated LDPCC code ensemble, the thresholds are better than for corresponding regular and irregular LDPC block codes.
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 ..."
Abstract

Cited by 51 (5 self)
 Add to MetaCart
(Show Context)
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 ..."
Abstract

Cited by 46 (9 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 36 (7 self)
 Add to MetaCart
(Show Context)
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.
Terminated LDPC convolutional codes with thresholds close to capacity
 in Proceedings of the IEEE International Symposium on Information Theory
, 2005
"... Abstract — An ensemble of LDPC convolutional codes with paritycheck matrices composed of permutation matrices is considered. The convergence of the iterative belief propagation based decoder for terminated convolutional codes in the ensemble is analyzed for binaryinput outputsymmetric memoryless ..."
Abstract

Cited by 30 (3 self)
 Add to MetaCart
(Show Context)
Abstract — An ensemble of LDPC convolutional codes with paritycheck matrices composed of permutation matrices is considered. The convergence of the iterative belief propagation based decoder for terminated convolutional codes in the ensemble is analyzed for binaryinput outputsymmetric memoryless channels using density evolution techniques. We observe that the structured irregularity in the Tanner graph of the codes leads to significantly better thresholds when compared to corresponding LDPC block codes. I.
A simple proof of threshold saturation for coupled scalar recursions
 in Proc. Intl. Symp. on Turbo Codes and Iter. Inform. Proc. (ISTC), 2012
"... ar ..."
Approaching Capacity with Asymptotically Regular LDPC Codes
"... Abstract—We present a family of protograph based LDPC codes that can be derived from permutation matrix based regular (J, K) LDPC convolutional codes by termination. In the terminated protograph, all variable nodes still have degree J but some check nodes at the start and end of the protograph have ..."
Abstract

Cited by 25 (1 self)
 Add to MetaCart
(Show Context)
Abstract—We present a family of protograph based LDPC codes that can be derived from permutation matrix based regular (J, K) LDPC convolutional codes by termination. In the terminated protograph, all variable nodes still have degree J but some check nodes at the start and end of the protograph have degrees smaller than K. Since the fraction of these stronger nodes vanishes as the termination length L increases, we call the codes asymptotically regular. The density evolution thresholds of these protographs are better than those of regular (J, K) block codes. Interestingly, this threshold improvement gets stronger with increasing node degrees (at a fixed rate) and it does not decay as L increases. Terminated convolutional protographs can also be derived from standard irregular protographs and may exhibit a significant threshold improvement. I.
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 ..."
Abstract

Cited by 24 (10 self)
 Add to MetaCart
(Show Context)
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.