791 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
Lossy Source Coding via Spatially Coupled LDGM Ensembles
We study a new encoding scheme for lossy source compression based on
spatially coupled low-density generator-matrix codes. We develop a
belief-propagation guided-decimation algorithm, and show that this algorithm
allows to approach the optimal distortion of spatially coupled ensembles.
Moreover, using the survey propagation formalism, we also observe that the
optimal distortions of the spatially coupled and individual code ensembles are
the same. Since regular low-density generator-matrix codes are known to achieve
the Shannon rate-distortion bound under optimal encoding as the degrees grow,
our results suggest that spatial coupling can be used to reach the
rate-distortion bound, under a {\it low complexity} belief-propagation
guided-decimation algorithm.
This problem is analogous to the MAX-XORSAT problem in computer science.Comment: Submitted to ISIT 201
Approaching the Rate-Distortion Limit with Spatial Coupling, Belief propagation and Decimation
We investigate an encoding scheme for lossy compression of a binary symmetric
source based on simple spatially coupled Low-Density Generator-Matrix codes.
The degree of the check nodes is regular and the one of code-bits is Poisson
distributed with an average depending on the compression rate. The performance
of a low complexity Belief Propagation Guided Decimation algorithm is
excellent. The algorithmic rate-distortion curve approaches the optimal curve
of the ensemble as the width of the coupling window grows. Moreover, as the
check degree grows both curves approach the ultimate Shannon rate-distortion
limit. The Belief Propagation Guided Decimation encoder is based on the
posterior measure of a binary symmetric test-channel. This measure can be
interpreted as a random Gibbs measure at a "temperature" directly related to
the "noise level of the test-channel". We investigate the links between the
algorithmic performance of the Belief Propagation Guided Decimation encoder and
the phase diagram of this Gibbs measure. The phase diagram is investigated
thanks to the cavity method of spin glass theory which predicts a number of
phase transition thresholds. In particular the dynamical and condensation
"phase transition temperatures" (equivalently test-channel noise thresholds)
are computed. We observe that: (i) the dynamical temperature of the spatially
coupled construction saturates towards the condensation temperature; (ii) for
large degrees the condensation temperature approaches the temperature (i.e.
noise level) related to the information theoretic Shannon test-channel noise
parameter of rate-distortion theory. This provides heuristic insight into the
excellent performance of the Belief Propagation Guided Decimation algorithm.
The paper contains an introduction to the cavity method
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
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