Ludovic Danjean

Finite Alphabet Iterative Decoders—Part I: Decoding Beyond Belief Propagation on the Binary Symmetric Channel

We introduce a new paradigm for finite precision iterative decoding on low-density parity-check codes over the binary symmetric channel. The messages take values from a finite alphabet, and unlike traditional quantized decoders which are quantized versions of the belief propagation (BP) decoder, the proposed finite alphabet iterative decoders (FAIDs) do not propagate quantized probabilities or log-likelihoods and the variable node update functions do not mimic the BP decoder. Rather, the update functions are maps designed using the knowledge of potentially harmful subgraphs that could be present in a given code, thereby rendering these decoders capable of outperforming the BP in the error floor region. On certain column-weight-three codes of practical interest, we show that there exist {FAIDs that surpass the floating-point BP decoder in the error floor region while requiring only three bits of precision for the representation of the messages}. Hence, FAIDs are able to achieve a superior performance at much lower complexity. We also provide a methodology for the selection of FAIDs that is not code-specific, but gives a set of candidate FAIDs containing potentially good decoders in the error floor region for any column-weight-three code. We validate the code generality of our methodology by providing particularly good three-bit precision FAIDs for a variety of codes with different rates and lengths.

On the selection of finite alphabet iterative decoders for LDPC codes on the BSC

Recently new message passing decoders for LDPC codes, called finite alphabet iterative decoders (FAIDs) were proposed. The messages belong to a finite alphabet and the update functions are simple boolean maps different from the functions used for the belied propagation (BP) decoder. The maps can be chosen using the knowledge of potential trapping sets such that the decoders surpass the BP decoder in the error floor. In this paper, we address the issue of selecting good FAIDs which perform well in the error floor for column weight three codes. We introduce the notion of noisy trapping set which is a generalization based on analyzing the local dynamic behaviour of a given FAID on a trapping set. Using this notion as the core, we provide an iterative greedy algorithm that outputs a set of candidate FAIDs containing potentially good decoders for any given code. To illustrate the appliance of the methodology on several codes, we show that the set of candidate FAIDs contains particularly good FAIDs for different codes with different rates and lengths.

Iterative decoding beyond belief propagation

At the heart of modern coding theory lies the fact that low-density parity-check (LDPC) codes can be efficiently decoded by belief propagation (BP). The BP is an inference algorithm which operates on a graphical model of a code, and lends itself to low-complexity and high-speed implementations, making it the algorithm of choice in many applications. It has unprecedentedly good error rate performance, so good that when decoded by the BP, LDPC codes approach theoretical limits of channel capacity. However, this capacity approaching property holds only in the asymptotic limit of code length, while codes of practical lengths suffer abrupt performance degradation in the low noise regime known as the error floor phenomenon. Our study of error floor has led to an interesting and surprising finding that it is possible to design iterative decoders which are much simpler yet better than belief propagation! These decoders do not propagate beliefs but a rather different kind of messages that reflect the local structure of the code graph. This has opened a plethora of exciting theoretical problems and applications. This paper introduces this new paradigm.

Finite alphabet iterative decoding (FAID) of the (155,64,20) Tanner code

It is now well established that iterative decoding approaches the performance of Maximum Likelihood Decoding of sparse graph codes, asymptotically in the block length. For a finite length sparse code, iterative decoding fails on specific subgraphs generically termed as trapping sets. Trapping sets give rise to error floor, an abrupt degradation of the code error performance in the high signal to noise ratio regime. In this paper, we will study a recently introduced class of quantized iterative decoders, for which the messages are defined on a finite alphabet and which successfully decode errors on subgraphs that are uncorrectable by conventional decoders such as the min-sum or the belief propagation. We will especially study the performance of the proposed finite alphabet iterative decoders on the famous (155,64,20) Tanner code.

Interval-Passing Algorithm for Non-Negative Measurement Matrices: Performance and Reconstruction Analysis

We consider the Interval-Passing Algorithm (IPA), an iterative reconstruction algorithm for reconstruction of non-negative sparse real-valued signals from noise-free measurements. We first generalize the IPA by relaxing the original constraint that the measurement matrix must be binary. The new algorithm operates on any non-negative sparse measurement matrix. We give a performance comparison of the generalized IPA with the reconstruction algorithms based on 1) linear programming and 2) verification decoding. Then we identify signals not recoverable by the IPA on a given measurement matrix, and show that these signals are related to stopping sets responsible to failures of iterative decoding algorithms on the binary erasure channel (BEC). Contrary to the results of the iterative decoding on the BEC, the smallest stopping set of a measurement matrix is not the smallest configuration on which the IPA fails. We analyze the recovery of sparse signals on subsets of stopping sets via the IPA and provide sufficient conditions on the exact recovery of sparse signals. Reconstruction performance of the IPA using the IEEE 802.16e LDPC codes as measurement matrices are given to show the effect of stopping sets in the performance of the IPA.

On iterative compressed sensing reconstruction of sparse non-negative vectors

We consider the iterative reconstruction of the Compressed Sensing (CS) problem over reals. The iterative reconstruction allows interpretation as a channel-coding problem, and it guarantees perfect reconstruction for properly chosen measurement matrices and sufficiently sparse error vectors. In this paper, we give a summary on reconstruction algorithms for compressed sensing and examine how the iterative reconstruction performs on quasi-cyclic low-density parity check (QC-LDPC) measurement matrices.

Iterative reconstruction algorithms in compressed sensing

In this paper we give an overview of current results in iterative reconstruction of sparse signals using parity check matrices of low-density parity check (LDPC) codes as measurement matrices in compressed sensing. We provide a detailed explanation of two iterative reconstruction algorithms, Interval Passing (IP) algorithm and verification algorithm. We then compare their performance using parity check matrices of quasi-cyclic low-density parity check (QC-LDPC) codes with different column-weights and rates.

Interval-Passing Algorithm for Chemical Mixture Estimation

In this letter, we propose a compressive sensing scheme for the mixture estimation problem in spectroscopy. We show that by applying an appropriate measurement matrix on the chemical mixture spectrum, we obtain an overall measurement matrix which is sparse. This enables the use of a low-complexity iterative reconstruction algorithm, called the interval-passing algorithm, to estimate the concentration of each chemical present in the mixture. Simulation results for the proportion of correct reconstructions show that chemical mixtures with a large number of chemicals present can be recovered.

Signal recovery performance of the interval-passing algorithm

This paper considers an iterative algorithm called the Interval-Passing Algorithm (IPA) which is used to reconstruct non-negative real signals using binary measurement matrices in compressed sensing (CS). The failures of the algorithm on stopping sets, also non-decodable configurations in iterative decoding of LDPC codes over the binary erasure channel (BEC), shows a connection between iterative reconstruction algorithm in CS and iterative decoding of LDPC codes over the BEC. In this paper, a stopping-set based approach is used to analyze the recovery of the IPA. We show that a smallest stopping set is not necessarily a smallest configuration on which the IPA fails and provide sufficient conditions under which the IPA recovers a sparse signal whose non-zero values lie on a subset of a stopping set. Reconstruction performance of the IPA using IEEE 802.16e LDPC measurement matrices are provided to show the effect of the stopping sets in the performance of the IPA.

An efficient exhaustive low-weight codeword search for structured LDPC codes

In this paper, we present an algorithm to find all low-weight codewords in a given quasi-cyclic (QC) low-density parity-check (LDPC) code with a fixed column-weight and girth. The main idea is to view a low-weight codeword as an (a, 0) trapping sets, and then show that each topologically different (a, 0) trapping set can be dissected into smaller trapping sets. The proposed search method relies on the knowledge of possible topologies of such smaller trapping sets present in a code ensemble, which enables an efficient search. Combined with the high-rate QC LDPC code construction which successively adds blocks of permutation matrices, the algorithm ensures that in the code construction procedure all codewords up to a certain weight are avoided, which leads to a code with the desired minimum distance. The algorithm can be also used to determine the multiplicity of the low-weight codewords with different trapping set structure.