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IEEE Information Theory Workshop (ITW)

Finite-length scaling of irregular LDPC code ensembles

Ala Eddine Kaddour Amraoui, Rudiger L. Urbanke, Andrea Montanari A

Published Jan 1, 2005• 22 citations
Publisher

Abstract

We investigate the finite-length scaling methodology for irregular LDPC code ensembles when transmission takes place over the binary erasure channel (BEC). We first show how the necessary computations, namely the covariance evolution and the computation of the finite-length shift, can be accomplished in the irregular case. We then investigate how the obtained approximation can be used to predict the performance of irregular code ensembles and to optimize the degree distributions for finite-length codes.

DOI
10.1109/itw.2005.1531845

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We investigate the finite-length scaling methodology for irregular LDPC code ensembles when transmission takes place over the binary erasure channel (BEC). We first show how the necessary computations, namely the covariance evolution and the computation of the finite-length shift, can be accomplished in the irregular case. We then investigate how the obtained approximation can be used to predict the performance of irregular code ensembles and to optimize the degree distributions for finite-length codes.
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In this paper, we explore a novel approach to evaluate the inherent UEP (unequal error protection) properties of irregular LDPC (low-density parity-check) codes over BECs (binary erasure channels). Exploiting the finite-length scaling methodology, suggested by Amraoui et. al., we introduce a scaling approach to approximate the bit erasure rates of variable nodes with different degrees in the waterfall region of the peeling decoder. Comparing the bit erasure rates obtained from Monte Carlo simulation with the proposed scaling approximations, we demonstrate that the scaling approach provides a close approximation for a wide range of code lengths (between 1000 and 8000). In view of the complexity associated with the numerical evaluation of the scaling approximation, we also derive simpler upper and lower bounds.
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.
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.
This paper is devoted to the analysis of finite-length iterative performance of punctured LDPC ensembles in the waterfall region, assuming the transmission over the binary erasure channel (BEC). The analysis is carried out using the scaling approach proposed in. Two punctured ensembles are considered: (a) randomly punctured ensembles, in the sense that each bit of a codeword is punctured with some puncturing probability; (b) ensembles with a fixed puncturing fraction of bits of each degree. In both cases, parameters of the scaling approximation are completely determined in terms of the ensemble parameters such as left, right and puncturing degree distributions.