173 research outputs found
A Scaling Law to Predict the Finite-Length Performance of Spatially-Coupled LDPC Codes
Spatially-coupled LDPC codes are known to have excellent asymptotic
properties. Much less is known regarding their finite-length performance. We
propose a scaling law to predict the error probability of finite-length
spatially-coupled ensembles when transmission takes place over the binary
erasure channel. We discuss how the parameters of the scaling law are connected
to fundamental quantities appearing in the asymptotic analysis of these
ensembles and we verify that the predictions of the scaling law fit well to the
data derived from simulations over a wide range of parameters. The ultimate
goal of this line of research is to develop analytic tools for the design of
spatially-coupled LDPC codes under practical constraints
How to Find Good Finite-Length Codes: From Art Towards Science
We explain how to optimize finite-length LDPC codes for transmission over the
binary erasure channel. Our approach relies on an analytic approximation of the
erasure probability. This is in turn based on a finite-length scaling result to
model large scale erasures and a union bound involving minimal stopping sets to
take into account small error events. We show that the performances of
optimized ensembles as observed in simulations are well described by our
approximation. Although we only address the case of transmission over the
binary erasure channel, our method should be applicable to a more general
setting.Comment: 13 pages, 13 eps figures, enhanced version of an invited paperat the
4th International Symposium on Turbo Codes and Related Topics, Munich,
Germany, 200
Finite-Length Scaling of Spatially Coupled LDPC Codes Under Window Decoding Over the BEC
We analyze the finite-length performance of spatially coupled low-density
parity-check (SC-LDPC) codes under window decoding over the binary erasure
channel. In particular, we propose a refinement of the scaling law by Olmos and
Urbanke for the frame error rate (FER) of terminated SC-LDPC ensembles under
full belief propagation (BP) decoding. The refined scaling law models the
decoding process as two independent Ornstein-Uhlenbeck processes, in
correspondence to the two decoding waves that propagate toward the center of
the coupled chain for terminated SC-LDPC codes. We then extend the proposed
scaling law to predict the performance of (terminated) SC-LDPC code ensembles
under the more practical sliding window decoding. Finally, we extend this
framework to predict the bit error rate (BER) and block error rate (BLER) of
SC-LDPC code ensembles. The proposed scaling law yields very accurate
predictions of the FER, BLER, and BER for both full BP and window decoding.Comment: Published in IEEE Transactions on Communications (Early Access). This
paper was presented in part at the IEEE Information Theory Workshop (ITW),
Visby, Sweden, August 2019 (arXiv:1904.10410
Tree-Structure Expectation Propagation for LDPC Decoding over the BEC
We present the tree-structure expectation propagation (Tree-EP) algorithm to
decode low-density parity-check (LDPC) codes over discrete memoryless channels
(DMCs). EP generalizes belief propagation (BP) in two ways. First, it can be
used with any exponential family distribution over the cliques in the graph.
Second, it can impose additional constraints on the marginal distributions. We
use this second property to impose pair-wise marginal constraints over pairs of
variables connected to a check node of the LDPC code's Tanner graph. Thanks to
these additional constraints, the Tree-EP marginal estimates for each variable
in the graph are more accurate than those provided by BP. We also reformulate
the Tree-EP algorithm for the binary erasure channel (BEC) as a peeling-type
algorithm (TEP) and we show that the algorithm has the same computational
complexity as BP and it decodes a higher fraction of errors. We describe the
TEP decoding process by a set of differential equations that represents the
expected residual graph evolution as a function of the code parameters. The
solution of these equations is used to predict the TEP decoder performance in
both the asymptotic regime and the finite-length regime over the BEC. While the
asymptotic threshold of the TEP decoder is the same as the BP decoder for
regular and optimized codes, we propose a scaling law (SL) for finite-length
LDPC codes, which accurately approximates the TEP improved performance and
facilitates its optimization
Turbo EP-based Equalization: a Filter-Type Implementation
This manuscript has been submitted to Transactions on Communications on
September 7, 2017; revised on January 10, 2018 and March 27, 2018; and accepted
on April 25, 2018
We propose a novel filter-type equalizer to improve the solution of the
linear minimum-mean squared-error (LMMSE) turbo equalizer, with computational
complexity constrained to be quadratic in the filter length. When high-order
modulations and/or large memory channels are used the optimal BCJR equalizer is
unavailable, due to its computational complexity. In this scenario, the
filter-type LMMSE turbo equalization exhibits a good performance compared to
other approximations. In this paper, we show that this solution can be
significantly improved by using expectation propagation (EP) in the estimation
of the a posteriori probabilities. First, it yields a more accurate estimation
of the extrinsic distribution to be sent to the channel decoder. Second,
compared to other solutions based on EP the computational complexity of the
proposed solution is constrained to be quadratic in the length of the finite
impulse response (FIR). In addition, we review previous EP-based turbo
equalization implementations. Instead of considering default uniform priors we
exploit the outputs of the decoder. Some simulation results are included to
show that this new EP-based filter remarkably outperforms the turbo approach of
previous versions of the EP algorithm and also improves the LMMSE solution,
with and without turbo equalization
Finite-Length Scaling of SC-LDPC Codes With a Limited Number of Decoding Iterations
We propose four finite-length scaling laws to predict the frame error rate (FER) performance of spatially-coupled low-density parity-check codes under full belief propagation (BP) decoding with a limit on the number of decoding iterations and a scaling law for sliding window decoding, also with limited iterations. The laws for full BP decoding provide a choice between accuracy and computational complexity; a good balance between them is achieved by the law that models the number of decoded bits after a certain number of BP iterations by a time-integrated Ornstein-Uhlenbeck process. This framework is developed further to model sliding window decoding as a race between the integrated Ornstein-Uhlenbeck process and an absorbing barrier that corresponds to the left boundary of the sliding window. The proposed scaling laws yield accurate FER predictions
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