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IEEE International Symposium on Information Theory (ISIT)

Finite-length performance of multi-edge protograph-based spatially coupled LDPC codes

Markus Stinner, Pablo M. Olmos

Published Jun 1, 2015• 2 citations
Publisher

Abstract

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.

DOI
10.1109/isit.2015.7282583

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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 block codes (SC-LDPC-BCs) are a class of LDPC codes in which certain ensembles achieve capacity under iterative belief propagation decoding. Prior work has considered these codes mostly from a protograph perspective. Following this viewpoint, our work presents new protographs for the design of protograph-based SC-LDPC-BCs. A reciprocal channel approximation (RCA) analysis of the terminated protographs yields the asymptotic performance limits of the ensembles in terms of their iterative decoding thresholds. Our protographs possess the properties of linear minimum distance growth rate and excellent thresholds. We consider the following channels in our work: binary-input additive white Gaussian noise channel (BI-AWGNC), binary erasure channel (BEC), and binary symmetric channel (BSC). Our work focuses on obtaining good designs, with two classes of ensembles being asymptotically regular, for all the three channels while minimizing rate loss.
The peeling decoding for spatially coupled low-density parity-check (SC-LDPC) codes is analyzed for a binary erasure channel. An analytical calculation of the mean evolution of degree-one check nodes of protograph-based SC-LDPC codes is given and an estimate for the covariance evolution of degree-one check nodes is proposed in the stable decoding phase where the decoding wave propagates along the chain of coupled codes. Both results are verified numerically. Protograph-based SC-LDPC codes turn out to have a more robust behavior than unstructured random SC-LDPC codes. Using the analytically calculated parameters, the finite-length scaling laws for these constructions are given and verified by numerical simulations.
The peeling decoding for spatially coupled low-density parity-check (SC-LDPC) codes is analyzed for a binary erasure channel. An analytical calculation of the mean evolution of degree-one check nodes of protograph-based SC-LDPC codes is given and an estimate for the covariance evolution of degree-one check nodes is proposed in the stable decoding phase where the decoding wave propagates along the chain of coupled codes. Both results are verified numerically. Protograph-based SC-LDPC codes turn out to have a more robust behavior than unstructured random SC-LDPC codes. Using the analytically calculated parameters, the finite- length scaling laws for these constructions are given and verified by numerical simulations.
The peeling decoding for spatially coupled low-density parity-check (SC-LDPC) codes is analyzed for a binary erasure channel. An analytical calculation of the mean evolution of degree-one check nodes of protograph-based SC-LDPC codes is given and an estimate for the covariance evolution of degree-one check nodes is proposed in the stable decoding phase where the decoding wave propagates along the chain of coupled codes. Both results are verified numerically. Protograph-based SC-LDPC codes turn out to have a more robust behavior than unstructured random SC-LDPC codes. Using the analytically calculated parameters, the finite- length scaling laws for these constructions are given and verified by numerical simulations.
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.
The finite length performance of codes on graphs constructed by connecting spatially coupled low-density parity-check (SC-LDPC) code chains is analyzed. Successive (peeling) decoding is considered for the binary erasure channel (BEC). The evolution of the undecoded portion of the bipartite graph remaining after each iteration is analyzed as a dynamical system. It is shown that, in addition to superior iterative decoding thresholds, connected chain ensembles have better performance than single chain ensembles of the same rate and length.