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arXiv:2003.04421 · IEEE Transactions on Communications

Finite-Length Scaling of Spatially Coupled LDPC Codes Under Window Decoding Over the BEC

Roman Sokolovskii, Alexandre Graell i Amat, Fredrik Brännström

Published Jul 21, 2020• 10 citations
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Abstract

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.

DOI
10.1109/tcomm.2020.3010958

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IEEE International Symposium on Information Theory (ISIT)•Jul 1, 2012•2 citations
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.
We propose four finite-length scaling laws to predict the frame error rate (FER) performance in the waterfall region of spatially-coupled low-density parity-check code ensembles 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 for the semi-structured code ensembles proposed by Olmos and Urbanke.
In this paper, we compare the finite-length performance of non-binary spatially coupled low-density parity-check (NB SC-LDPC) codes constructed from protographs to non-binary LDPC block codes (NB LDPC-BCs). A sliding window decoding architecture with a stopping rule based on a soft bit-error-rate (BER) estimate for the NB SC-LDPC codes is considered. It is demonstrated that NB SC-LDPC codes with sliding window decoding outperform NB LDPC-BCs with no increase in decoding complexity when the decoding latency of the SC-LDPC codes equals the block length of the LDPC-BCs. We also investigate the relationship between the protograph lifting factor, the decoding window size, and the decoding performance of NB SC-LDPC codes when the decoding latency is fixed. Simulation results for several (3,6)-regular NB code examples confirm that NB SC-LDPC codes can significantly outperform both binary LDPC-BCs and binary SC-LDPC codes with the same decoding latency.
The finite-length performance of multi-edge spatially coupled low-density parity-check (SC-LDPC) codes over the binary erasure channel (BEC) is analyzed. Existing scaling laws are extended to arbitrary protograph base matrices that include puncturing patterns and multiple edges between nodes. A regular protograph-based SC-LDPC construction based on the (4; 8)-regular LDPC block code works well in the waterfall region compared to more involved rate-1/2 structures proposed to improve the threshold to minimum distance trade-off. Scaling laws are also used for code design and to estimate the block length of a given SC-LDPC code ensemble to match the performance of some other code. Estimates on the performance degradation are developed if the chain length varies.
Spatially-coupled low-density parity-check (SC-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 code 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 SC-LDPC codes under practical constraints.
The equivalence of peeling decoding (PD) and Belief Propagation (BP) for low-density parity-check (LDPC) codes over the binary erasure channel is analyzed. Modifying the scheduling for PD, it is shown that exactly the same variable nodes (VNs) are resolved in every iteration than with BP. The decrease of erased VNs during the decoding process is analyzed instead of resolvable equations: This quantity can also be derived with density evolution, resulting in a drastic decrease in complexity. Finally, a scaling law using this quantity is established for spatially coupled LDPC codes.
In this work, we analyze the finite-length performance of low-density parity check (LDPC) ensembles decoded over the binary erasure channel (BEC) using the tree-expectation propagation (TEP) algorithm. In a previous paper, we showed that the TEP improves the BP performance for decoding regular and irregular short LDPC codes, but the perspective was mainly empirical. In this work, given the degree-distribution of an LDPC ensemble, we explain and predict the range of code lengths for which the TEP improves the BP solution. In addition, for LDPC ensembles that present a single critical point, we propose a scaling law to accurately predict the performance in the waterfall region. These results are of critical importance to design practical LDPC codes for the TEP decoder.