114 research outputs found
Spatially Coupled Turbo Codes: Principles and Finite Length Performance
In this paper, we give an overview of spatially coupled turbo codes (SC-TCs),
the spatial coupling of parallel and serially concatenated convolutional codes,
recently introduced by the authors. For presentation purposes, we focus on
spatially coupled serially concatenated codes (SC-SCCs). We review the main
principles of SC-TCs and discuss their exact density evolution (DE) analysis on
the binary erasure channel. We also consider the construction of a family of
rate-compatible SC-SCCs with simple 4-state component encoders. For all
considered code rates, threshold saturation of the belief propagation (BP) to
the maximum a posteriori threshold of the uncoupled ensemble is demonstrated,
and it is shown that the BP threshold approaches the Shannon limit as the
coupling memory increases. Finally we give some simulation results for finite
lengths.Comment: Invited paper, IEEE Int. Symp. Wireless Communications Systems
(ISWCS), Aug. 201
Threshold Analysis of Non-Binary Spatially-Coupled LDPC Codes with Windowed Decoding
In this paper we study the iterative decoding threshold performance of
non-binary spatially-coupled low-density parity-check (NB-SC-LDPC) code
ensembles for both the binary erasure channel (BEC) and the binary-input
additive white Gaussian noise channel (BIAWGNC), with particular emphasis on
windowed decoding (WD). We consider both (2,4)-regular and (3,6)-regular
NB-SC-LDPC code ensembles constructed using protographs and compute their
thresholds using protograph versions of NB density evolution and NB extrinsic
information transfer analysis. For these code ensembles, we show that WD of
NB-SC-LDPC codes, which provides a significant decrease in latency and
complexity compared to decoding across the entire parity-check matrix, results
in a negligible decrease in the near-capacity performance for a sufficiently
large window size W on both the BEC and the BIAWGNC. Also, we show that
NB-SC-LDPC code ensembles exhibit gains in the WD threshold compared to the
corresponding block code ensembles decoded across the entire parity-check
matrix, and that the gains increase as the finite field size q increases.
Moreover, from the viewpoint of decoding complexity, we see that (3,6)-regular
NB-SC-LDPC codes are particularly attractive due to the fact that they achieve
near-capacity thresholds even for small q and W.Comment: 6 pages, 8 figures; submitted to 2014 IEEE International Symposium on
Information Theor
Spatially Coupled LDPC Codes Constructed from Protographs
In this paper, we construct protograph-based spatially coupled low-density
parity-check (SC-LDPC) codes by coupling together a series of L disjoint, or
uncoupled, LDPC code Tanner graphs into a single coupled chain. By varying L,
we obtain a flexible family of code ensembles with varying rates and frame
lengths that can share the same encoding and decoding architecture for
arbitrary L. We demonstrate that the resulting codes combine the best features
of optimized irregular and regular codes in one design: capacity approaching
iterative belief propagation (BP) decoding thresholds and linear growth of
minimum distance with block length. In particular, we show that, for
sufficiently large L, the BP thresholds on both the binary erasure channel
(BEC) and the binary-input additive white Gaussian noise channel (AWGNC)
saturate to a particular value significantly better than the BP decoding
threshold and numerically indistinguishable from the optimal maximum
a-posteriori (MAP) decoding threshold of the uncoupled LDPC code. When all
variable nodes in the coupled chain have degree greater than two,
asymptotically the error probability converges at least doubly exponentially
with decoding iterations and we obtain sequences of asymptotically good LDPC
codes with fast convergence rates and BP thresholds close to the Shannon limit.
Further, the gap to capacity decreases as the density of the graph increases,
opening up a new way to construct capacity achieving codes on memoryless
binary-input symmetric-output (MBS) channels with low-complexity BP decoding.Comment: Submitted to the IEEE Transactions on Information Theor
Spatially Coupled Turbo Codes
In this paper, we introduce the concept of spatially coupled turbo codes
(SC-TCs), as the turbo codes counterpart of spatially coupled low-density
parity-check codes. We describe spatial coupling for both Berrou et al. and
Benedetto et al. parallel and serially concatenated codes. For the binary
erasure channel, we derive the exact density evolution (DE) equations of SC-TCs
by using the method proposed by Kurkoski et al. to compute the decoding erasure
probability of convolutional encoders. Using DE, we then analyze the asymptotic
behavior of SC-TCs. We observe that the belief propagation (BP) threshold of
SC-TCs improves with respect to that of the uncoupled ensemble and approaches
its maximum a posteriori threshold. This phenomenon is especially significant
for serially concatenated codes, whose uncoupled ensemble suffers from a poor
BP threshold.Comment: in Proc. 8th International Symposium on Turbo Codes & Iterative
Information Processing 2014, Bremen, Germany, August 2014. To appear. (The
PCC ensemble is changed with respect to the one in the previous version of
the paper. However, it gives identical thresholds
Nonbinary Spatially-Coupled LDPC Codes on the Binary Erasure Channel
We analyze the asymptotic performance of nonbinary spatially-coupled
low-density parity-check (SC-LDPC) codes built on the general linear group,
when the transmission takes place over the binary erasure channel. We propose
an efficient method to derive an upper bound to the maximum a posteriori
probability (MAP) threshold for nonbinary LDPC codes, and observe that the MAP
performance of regular LDPC codes improves with the alphabet size. We then
consider nonbinary SC-LDPC codes. We show that the same threshold saturation
effect experienced by binary SC-LDPC codes occurs for the nonbinary codes,
hence we conjecture that the BP threshold for large termination length
approaches the MAP threshold of the underlying regular ensemble.Comment: Submitted to IEEE International Conference on Communications 201
Randomly Punctured LDPC Codes
In this paper, we present a random puncturing analysis of low-density parity-check (LDPC) code ensembles. We derive a simple analytic expression for the iterative belief propagation (BP) decoding threshold of a randomly punctured LDPC code ensemble on the binary erasure channel (BEC) and show that, with respect to the BP threshold, the strength and suitability of an LDPC code ensemble for random puncturing is completely determined by a single constant that depends only on the rate and the BP threshold of the mother code ensemble. We then provide an efficient way to accurately predict BP thresholds of randomly punctured LDPC code ensembles on the binary- input additive white Gaussian noise channel (BI-AWGNC), given only the BP threshold of the mother code ensemble on the BEC and the design rate, and we show how the prediction can be improved with knowledge of the BI-AWGNC threshold. We also perform an asymptotic minimum distance analysis of randomly punctured code ensembles and present simulation results that confirm the robust decoding performance promised by the asymptotic results. Protograph-based LDPC block code and spatially coupled LDPC code ensembles are used throughout as examples to demonstrate the results
Randomly Punctured Spatially Coupled LDPC Codes
In this paper, we study random puncturing of protograph-based spatially coupled low-density parity-check (SC- LDPC) code ensembles. We show that, with respect to iterative decoding threshold, the strength and suitability of an LDPC code ensemble for random puncturing over the binary erasure channel (BEC) is completely determined by a single constant that depends only on the rate and iterative decoding threshold of the mother code ensemble. We then use this analysis to show that randomly punctured SC-LDPC code ensembles display near capacity thresholds for a wide range of rates. We also perform an asymptotic minimum distance analysis and show that, like the SC-LDPC mother code ensemble, the punctured SC-LDPC code ensembles are also asymptotically good. Finally, we present some simulation results that confirm the excellent decoding performance promised by the asymptotic results
Optimized Bit Mappings for Spatially Coupled LDPC Codes over Parallel Binary Erasure Channels
In many practical communication systems, one binary encoder/decoder pair is
used to communicate over a set of parallel channels. Examples of this setup
include multi-carrier transmission, rate-compatible puncturing of turbo-like
codes, and bit-interleaved coded modulation (BICM). A bit mapper is commonly
employed to determine how the coded bits are allocated to the channels. In this
paper, we study spatially coupled low-density parity check codes over parallel
channels and optimize the bit mapper using BICM as the driving example. For
simplicity, the parallel bit channels that arise in BICM are replaced by
independent binary erasure channels (BECs). For two parallel BECs modeled
according to a 4-PAM constellation labeled by the binary reflected Gray code,
the optimization results show that the decoding threshold can be improved over
a uniform random bit mapper, or, alternatively, the spatial chain length of the
code can be reduced for a given gap to capacity. It is also shown that for
rate-loss free, circular (tail-biting) ensembles, a decoding wave effect can be
initiated using only an optimized bit mapper
Non-Binary LDPC Codes with Large Alphabet Size
We study LDPC codes for the channel with input and
output . The aim of this paper is to evaluate
decoding performance of -ary non-binary LDPC codes for large . We give
density evolution and decoding performance evaluation for regular non-binary
LDPC codes and spatially-coupled (SC) codes. We show the regular codes do not
achieve the capacity of the channel while SC codes do
Spatially-Coupled Nearly-Regular LDPC Code Ensembles for Rate-Flexible Code Design
Spatially coupled regular LDPC code ensembles have outstanding performance with belief propagation decoding and can perform close to the Shannon limit. In this paper we investigate the suitability of coupled regular LDPC code ensembles with respect to rate-flexibility. Regular ensembles with good performance and low complexity exist for a variety of specific code rates. On the other hand it can be observed that outside this set of favorable rational rates the complexity and performance become unreasonably high. We therefore propose ensembles with slight irregularity that allow us to smoothly cover the complete range of rational rates. Our simple construction allows a performance with negligible gap to the Shannon limit while maintaining complexity as low as for the best regular code ensembles. At the same time the construction guarantees that asymptotically the minimum distance grows linearly with the length of the coupled blocks
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