On the design of LDPC code ensembles for BIAWGN channels (original) (raw)

A More Accurate One-Dimensional Analysis and Design of Irregular LDPC Codes

IEEE Transactions on Communications, 2004

We introduce a new one-dimensional (1-D) analysis of low-density parity-check (LDPC) codes on additive white Gaussian noise channels which is significantly more accurate than similar 1-D methods. Our method assumes a Gaussian distribution in message-passing decoding only for messages from variable nodes to check nodes. Compared to existing work, which makes a Gaussian assumption both for messages from check nodes and from variable nodes, our method offers a significantly more accurate estimate of convergence behavior and threshold of convergence. Similar to previous work, the problem of designing irregular LDPC codes reduces to a linear programming problem. However, our method allows irregular code design in a wider range of rates without any limit on the maximum variable-node degree. We use our method to design irregular LDPC codes with rates greater than 1 4 that perform within a few hundredths of a decibel from the Shannon limit. The designed codes perform almost as well as codes designed by density evolution.

Design of irregular LDPC codes with optimized performance-complexity tradeoff

IEEE Transactions on Communications, 2000

The optimal performance-complexity tradeoff for error-correcting codes at rates strictly below the Shannon limit is a central question in coding theory. This paper proposes a numerical approach for the minimization of decoding complexity for long-block-length irregular low-density parity-check (LDPC) codes. The proposed design methodology is applicable to any binary-input memoryless symmetric channel and any iterative message-passing decoding algorithm with a parallelupdate schedule. A key feature of the proposed optimization method is a new complexity measure that incorporates both the number of operations required to carry out a single decoding iteration and the number of iterations required for convergence. This paper shows that the proposed complexity measure can be accurately estimated from a density-evolution and extrinsicinformation transfer chart analysis of the code. A sufficient condition is presented for convexity of the complexity measure in the variable edge-degree distribution; when it is not satisfied, numerical experiments nevertheless suggest that the local minimum is unique. The results presented herein show that when the decoding complexity is constrained, the complexity-optimized codes significantly outperform threshold-optimized codes at long block lengths, within the ensemble of irregular codes.

Density Evolution for Nonbinary LDPC Codes Under Gaussian Approximation

IEEE Transactions on Information Theory, 2009

This paper extends the work on density evolution for binary low-density parity-check (LDPC) codes with Gaussian approximation to LDPC codes over GF (q). We first generalize the definition of channel symmetry for nonbinary inputs to include q-ary phase-shift keying (PSK) modulated channels for prime q and binary-modulated channels for q that is a power of 2. For the well-defined q-ary-input symmetric-output channel, we prove that under the Gaussian assumption, the density distribution for messages undergoing decoding is fully characterized by (q 0 1) quantities. Assuming uniform edge weights, we further show that the density of messages computed by the check node decoder (CND) is fully defined by a single number. We then present the approximate density evolution for regular and irregular LDPC codes, and show that the (q 0 1)-dimensional integration involved can be simplified using a dimensionality reduction algorithm for the important case of q = 2 p. Through application of approximate density evolution and linear programming, we optimize the degree distribution of LDPC codes over GF(3) and GF(4). The optimized irregular LDPC codes demonstrate performance close to the Shannon capacity for long codewords. We also design GF(q) codes for high-order modulation by using the idea of a channel adapter. We find that codes designed in this fashion outperform those optimized specifically for the binary additive white Gaussian noise (AWGN) channel for a short codewords and a spectral efficiency of 2 bits per channel use (b/cu). Index Terms-density evolution, Gaussian approximation, lowdensity parity-check (LDPC) codes. I. INTRODUCTION T HIS paper is motivated by the impressive performance of irregular low-density parity-check (LDPC) codes over a wide class of channels [1]-[5]. Irregular LDPC codes are commonly designed using a numerical technique called density evolution. Developed by Richardson and Urbanke [1], the method of density evolution is one of the most powerful tools known for analyzing the asymptotic performance of an LDPC Manuscript

Deterministic Design of Low-Density Parity-Check Codes for Binary Erasure Channels

IEEE GLOBECOM 2007-2007 IEEE Global Telecommunications Conference, 2007

We propose a deterministic method to design irregular Low-Density Parity-Check (LDPC) codes for binary erasure channels (BEC). Compared to the existing methods, which are based on the application of asymptomatic analysis tools such as density evolution or Extrinsic Information Transfer (EXIT) charts in an optimization process, the proposed method is much simpler and faster. Through a number of examples, we demonstrate that the codes designed by the proposed method perform very closely to the best codes designed by optimization. An important property of the proposed designs is the flexibility to select the number of constituent variable node degrees P. The proposed designs include existing deterministic designs as a special case with P = N-1, where N is the maximum variable node degree. Compared to the existing deterministic designs, for a given rate and a given δ > 0, the designed ensembles can have a threshold in δ-neighborhood of the capacity upper bound with smaller values of P and N. They can also achieve the capacity of the BEC as N, and correspondingly P and the maximum check node degree tend to infinity. Index Terms-channel coding, low-density parity-check (LDPC) codes, binary erasure channel (BEC), deterministic design. I. INTRDOUCTION Low-Density Parity-Check (LDPC) codes have received much attention in the past decade due to their attractive performance/complexity tradeoff on a variety of communication channels. In particular, on the Binary Erasure Channel (BEC), they achieve the channel capacity asymptotically [1-4]. In [1],[5],[6] a complete mathematical analysis for the performance of LDPC codes over the BEC, both asymptotically and for finite block lengths, has been developed. For other types of channels such as the Binary Symmetric Channel (BSC) and the Binary Input Additive White Gaussian Noise (BIAWGN) channel, only asymptotic analysis is available [7]. For irregular LDPC codes, the problem of finding ensemble

Design of irregular LDPC codes for BIAWGN channels with SNR mismatch

IEEE Transactions on Communications, 2000

Belief propagation (BP) algorithm for decoding lowdensity parity-check (LDPC) codes over a binary input additive white Gaussian noise (BIAWGN) channel requires the knowledge of the signal-to-noise ratio (SNR) at the receiver to achieve its ultimate performance. An erroneous estimation or the absence of a perfect knowledge of the SNR at the decoder is referred to as "SNR mismatch". SNR mismatch can significantly degrade the performance of LDPC codes decoded by the BP algorithm. In this paper, using extrinsic information transfer (EXIT) charts, we design irregular LDPC codes that perform better (have a lower SNR threshold) in the presence of mismatch compared to the conventionally designed irregular LDPC codes that are optimized for zero mismatch. Considering that min-sum (MS) algorithm is the limit of BP with infinite SNR over-estimation, the EXIT functions generated in this work can also be used for the efficient analysis and design of LDPC codes under the MS algorithm.

On Design of Optimized Low-Density Parity-Check Codes Starting From Random Constructions

2011

In this paper we present a novel two step design technique for Low Density Parity Check (LDPC) codes, which, among the others, have been exploited for performance enhancement of the second generation of Digital Video Broadcasting-Satellite (DVB-S2). In the first step we develop an efficient algorithm for construction of quasi-random LDPC codes via minimization of a cost function related to the distribution of the length of cycles in the Tanner graph of the code. The cost function aims at constructing high girth bipartite graphs with reduced number of cycles of low length. In the second optimization step we aim at improving the asymptotic performance of the code via edge perturbation. The design philosophy is to avoid asymptotically weak LDPCs that have low minimum distance values and could potentially perform badly under iterative soft decoding at moderate to high Signal to Noise Ratio (SNR) values. Subsequently, we present sample results of our LDPC design strategy, present their simulated performance over an AWGN channel and make comparisons to some of the construction methods presented in the literature.

On the Design of Universal LDPC Codes

Computing Research Repository, 2008

Low-density parity-check (LDPC) coding for a multitude of equal-capacity channels is studied. First, based on numerous observations, a conjecture is stated that when the belief propagation decoder converges on a set of equal-capacity channels, it would also converge on any convex combination of those channels. Then, it is proved that when the stability condition is satisfied for a number of channels, it is also satisfied for any channel in their convex hull. For the purpose of code design, a method is proposed which can decompose every symmetric channel with capacity C into a set of identical-capacity basis channels. We expect codes that work on the basis channels to be suitable for any channel with capacity C. Such codes are found and in comparison with existing LDPC codes that are designed for specific channels, our codes obtain considerable coding gains when used across a multitude of channels.

Improved construction of LDPC convolutional codes with semi-random parity-check matrices

Science China Information Sciences, 2014

In this paper, we investigate the construction of time-varying convolutional low-density paritycheck (LDPC) codes derived from block LDPC codes based on improved progressive edge growth (PEG) method. Different from the conventional PEG algorithm, the parity-check matrix is initialized by inserting certain patterns. More specifically, the submatrices along the main diagonal are fixed to be the identity matrix that ensures the fast encoding feature of the LDPC convolutional codes. Second, we insert a nonzero pattern into the secondary diagonal submatrices that ensures the encoding memory length of the time-varying LDPC convolutional codes as large as possible. With this semi-random structure, we have analyzed the code performance by evaluating the number of short cycles as well as the the bound of free distance. Simulation results show that the constructed LDPC convolutional codes perform well over additive white Gaussian noise (AWGN) channels.

Design of capacity-approaching irregular low-density parity-check codes

IEEE Transactions on Information Theory, 2001

We design low-density parity-check (LDPC) codes that perform at rates extremely close to the Shannon capacity. The codes are built from highly irregular bipartite graphs with carefully chosen degree patterns on both sides. Our theoretical analysis of the codes is based on [1]. Assuming that the underlying communication channel is symmetric, we prove that the probability densities at the message nodes of the graph possess a certain symmetry. Using this symmetry property we then show that, under the assumption of no cycles, the message densities always converge as the number of iterations tends to infinity. Furthermore, we prove a stability condition which implies an upper bound on the fraction of errors that a belief-propagation decoder can correct when applied to a code induced from a bipartite graph with a given degree distribution.

Design of Low-Density Parity-Check Codes with Optimized Complexity-Rate Tradeoff

2009

The optimal complexity-rate tradeoff for errorcorrecting codes at rates strictly below the Shannon limit is a central question in coding theory. This paper proposes a numerical approach for the joint optimization of rate and decoding complexity for long-block-length irregular low-density parity-check (LDPC) codes. The proposed design methodology is applicable to any binary-input memoryless symmetric channel and any iterative message-passing decoding algorithm with a parallel-update schedule. A key feature of the proposed optimization method is a new complexity measure that incorporates both the number of operations required to carry out a single decoding iteration and the number of iterations required for convergence. This paper shows that the proposed complexity measure can be accurately estimated from a density-evolution and extrinsic-information transfer chart analysis of the code. Under certain mild conditions, the complexity measure is a convex function of the variable edge-degree distribution of the code, allowing an efficient design of complexity-optimized LDPC codes using convex optimization methods. The results presented herein show that when the decoding complexity is constrained, the complexity-optimized codes significantly outperform thresholdoptimized codes at long block lengths.