807 research outputs found
Threshold Saturation for Spatially Coupled LDPC and LDGM Codes on BMS Channels
Spatially-coupled low-density parity-check (LDPC) codes, which were first introduced as LDPC convolutional codes, have been shown to exhibit excellent performance under low-complexity belief-propagation decoding. This phenomenon is now termed threshold saturation via spatial coupling. Spatially-coupled codes have been successfully applied in numerous areas. In particular, it was proven that spatially-coupled regular LDPC codes universally achieve capacity over the class of binary memoryless symmetric (BMS) channels under belief-propagation decoding. Recently, potential functions have been used to simplify threshold saturation proofs for scalar and vector recursions. In this paper, potential functions are used to prove threshold saturation for irregular LDPC and low-density generator-matrix codes on BMS channels, extending the simplified proof technique to BMS channels. The corresponding potential functions are closely related to the average Bethe free entropy of the ensembles in the large-system limit. These functions also appear in statistical physics when the replica method is used to analyze optimal decoding
Spatially-Coupled MacKay-Neal Codes and Hsu-Anastasopoulos Codes
Kudekar et al. recently proved that for transmission over the binary erasure
channel (BEC), spatial coupling of LDPC codes increases the BP threshold of the
coupled ensemble to the MAP threshold of the underlying LDPC codes. One major
drawback of the capacity-achieving spatially-coupled LDPC codes is that one
needs to increase the column and row weight of parity-check matrices of the
underlying LDPC codes.
It is proved, that Hsu-Anastasopoulos (HA) codes and MacKay-Neal (MN) codes
achieve the capacity of memoryless binary-input symmetric-output channels under
MAP decoding with bounded column and row weight of the parity-check matrices.
The HA codes and the MN codes are dual codes each other.
The aim of this paper is to present an empirical evidence that
spatially-coupled MN (resp. HA) codes with bounded column and row weight
achieve the capacity of the BEC. To this end, we introduce a spatial coupling
scheme of MN (resp. HA) codes. By density evolution analysis, we will show that
the resulting spatially-coupled MN (resp. HA) codes have the BP threshold close
to the Shannon limit.Comment: Corrected typos in degree distributions \nu and \mu of MN and HA
code
A Simple Proof of Maxwell Saturation for Coupled Scalar Recursions
Low-density parity-check (LDPC) convolutional codes (or spatially-coupled
codes) were recently shown to approach capacity on the binary erasure channel
(BEC) and binary-input memoryless symmetric channels. The mechanism behind this
spectacular performance is now called threshold saturation via spatial
coupling. This new phenomenon is characterized by the belief-propagation
threshold of the spatially-coupled ensemble increasing to an intrinsic noise
threshold defined by the uncoupled system. In this paper, we present a simple
proof of threshold saturation that applies to a wide class of coupled scalar
recursions. Our approach is based on constructing potential functions for both
the coupled and uncoupled recursions. Our results actually show that the fixed
point of the coupled recursion is essentially determined by the minimum of the
uncoupled potential function and we refer to this phenomenon as Maxwell
saturation. A variety of examples are considered including the
density-evolution equations for: irregular LDPC codes on the BEC, irregular
low-density generator matrix codes on the BEC, a class of generalized LDPC
codes with BCH component codes, the joint iterative decoding of LDPC codes on
intersymbol-interference channels with erasure noise, and the compressed
sensing of random vectors with i.i.d. components.Comment: This article is an extended journal version of arXiv:1204.5703 and
has now been accepted to the IEEE Transactions on Information Theory. This
version adds additional explanation for some details and also corrects a
number of small typo
Spatially Coupled LDPC Codes for Decode-and-Forward in Erasure Relay Channel
We consider spatially-coupled protograph-based LDPC codes for the three
terminal erasure relay channel. It is observed that BP threshold value, the
maximal erasure probability of the channel for which decoding error probability
converges to zero, of spatially-coupled codes, in particular spatially-coupled
MacKay-Neal code, is close to the theoretical limit for the relay channel.
Empirical results suggest that spatially-coupled protograph-based LDPC codes
have great potential to achieve theoretical limit of a general relay channel.Comment: 7 pages, extended version of ISIT201
Efficient Termination of Spatially-Coupled Codes
Spatially-coupled low-density parity-check codes attract much attention due
to their capacity-achieving performance and a memory-efficient sliding-window
decoding algorithm. On the other hand, the encoder needs to solve large linear
equations to terminate the encoding process. In this paper, we propose modified
spatially-coupled codes. The modified (\dl,\dr,L) codes have less rate-loss,
i.e., higher coding rate, and have the same threshold as (\dl,\dr,L) codes
and are efficiently terminable by using an accumulator
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