Binary-input Output-symmetric Channels Research Articles

Error Floor Estimation of LDPC Decoders — A Code Independent Approach to Measuring the Harmfulness of Trapping Sets

The linear state-space model is a well-known code-independent method to estimate the contribution of a trapping set (TS) structure to the error floor of low-density parity-check (LDPC) codes. In this paper, we provide an in-depth analysis of this method by incorporating a more accurate model for the incoming messages to the TS structure that takes into account the randomness and the correlation among such messages. Based on this analysis, we demonstrate that both randomness and correlation result in the over-estimation of the failure probability of the TS. We then propose an alternate code-independent technique for the error floor estimation of iterative LDPC decoders that can accurately estimate the contribution of different TS structures in the error floor. Compared to the linear state-space model, the proposed method is not only more accurate, but also more general, in that, it is applicable to any saturating iterative message-passing decoder, symmetrically quantized or unquantized, over any memoryless binary-input output-symmetric channel. The proposed technique can be viewed as the local application of importance sampling (IS) to the message-passing algorithm over the subgraph induced by the TS in the code’s Tanner graph. In the message-passing process, to account for the effect of the rest of the Tanner graph, density evolution along with a simple correlation model is used to generate the messages coming into the TS from the rest of the Tanner graph. Extensive simulations demonstrate that the proposed technique can accurately estimate the error floor of LDPC codes over both additive white Gaussian noise (AWGN) channel and binary symmetric channel (BSC), for a variety of iterative decoding algorithms and quantization schemes.

LP Decoding of Regular LDPC Codes in Memoryless Channels

We study error bounds for linear programming decoding of regular low-density parity-check (LDPC) codes. For memoryless binary-input output-symmetric channels, we prove bounds on the word error probability that are inverse doubly exponential in the girth of the factor graph. For memoryless binary-input AWGN channel, we prove lower bounds on the threshold for regular LDPC codes whose factor graphs have logarithmic girth under LP-decoding. Specifically, we prove a lower bound of σ = 0.735 (upper bound of [(Eb)/(N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> )]=2.67 dB) on the threshold of (3, 6)-regular LDPC codes whose factor graphs have logarithmic girth. Our proof is an extension of a recent paper of Arora, Daskalakis, and Steurer [STOC 2009] who presented a novel probabilistic analysis of LP decoding over a binary symmetric channel. Their analysis is based on the primal LP representation and has an explicit connection to message passing algorithms. We extend this analysis to any MBIOS channel.

A Linear Encoding Approach to Index Assignment in Lossy Source-Channel Coding

We present a general scheme for the lossy transmission of a source with arbitrary statistics through a noisy channel under the mean-square error fidelity criterion. Our approach is based on transform coding, scalar quantization of the transform coefficients and linear encoding of the quantization indices. Entropy coding and channel coding are merged into a single linear encoding function, such that the ¿catastrophic¿ behavior of conventional entropy coding is avoided and the full power of modern coding techniques and iterative ¿Belief-Propagation¿ decoding can be exploited. We show that this approach is asymptotically optimal in the limit of large block length, for arbitrary source statistics and binary-input output-symmetric channel. In the practical regime of finite block length and low decoding complexity, we show, through the explicit construction of codes for the lossy transmission of digital images over a binary symmetric channel, that our approach yields significant improvements with respect to previously proposed channel-optimized quantization schemes and also with respect to the conventional concatenation of state-of-the art image coding with state-of-the art channel coding. Although our constructive example focuses on a special case, the approach is general and can be applied to other classes of sources of practical interest.

Waterfall Performance Analysis of Finite-Length LDPC Codes on Symmetric Channels

An efficient method for analyzing the performance of finite-length low-density parity-check (LDPC) codes in the waterfall region, when transmission takes place on a memoryless binary-input output-symmetric channel is proposed. This method is based on studying the variations of the channel quality around its expected value when observed during the transmission of a finite-length codeword. We model these variations with a single parameter. This parameter is then viewed as a random variable and its probability distribution function is obtained. Assuming that a decoding failure is the result of an observed channel worse than the codeiquests decoding threshold, the block error probability of finite-length LDPC codes under different decoding algorithms is estimated. Using an extrinsic information transfer chart analysis, the bit error probability is obtained from the block error probability. Different parameters can be used for modeling the channel variations. In this work, two of such parameters are studied. Through examples, it is shown that this method can closely predict the performance of LDPC codes of a few thousand bits or longer in the waterfall region.

The equivalence between slepian-wolf coding and channel coding under density evolution

We consider Slepian-Wolf code design based on low density parity-check (LDPC) coset codes. The density evolution formula for Slepian-Wolf coding is derived. An intimate connection between Slepian-Wolf coding and channel coding is then established. Specifically we show that, under density evolution, each Slepian-Wolf coding problem is equivalent to a channel coding problem for a binary-input output-symmetric channel.

Performance analysis of fiber-optic BPPM CDMA systems with single parity-check product codes

The application of a family of single parity-check (SPC) product codes in the fiber-optic code-division multiple-access (CDMA) systems with binary pulse position modulation (BPPM) is considered in this paper. We evaluate the bit-error probabilities of the coded systems, taking into account the photodetector noise, thermal noise, and the multiple-access interference. In this paper, the performance of the employed coding schemes over a binary-input output-symmetric channel is analyzed, based on the density evolution approach. The validity of the analytically obtained bit-error rates (BERs) over binary symmetric channels (BSCs) for each of the decoding iterations is verified by the system simulation. It is then shown that the application of these coding schemes in the optical BPPM-CDMA communication system, which can be modeled as a BSC, significantly improves the system performance. Particularly, for a certain BER (less than 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-7</sup> ) and bandwidth, the coded systems not only permit a higher number (more than twice) of active users, compared with the uncoded systems, but also can operate at higher channel bit rate (up to four times) with more than 10% energy saving. Furthermore, by increasing the number of code dimensions, less power (less than 40%) is required to achieve the desired performance

Performance of low-density parity-check codes with linear minimum distance

This correspondence studies the performance of the iterative decoding of low-density parity-check (LDPC) code ensembles that have linear typical minimum distance and stopping set size. We first obtain a lower bound on the achievable rates of these ensembles over memoryless binary-input output-symmetric channels. We improve this bound for the binary erasure channel. We also introduce a method to construct the codes meeting the lower bound for the binary erasure channel. Then, we give upper bounds on the rate of LDPC codes with linear minimum distance when their right degree distribution is fixed. We compare these bounds to the previously derived upper bounds on the rate when there is no restriction on the code ensemble.

Tight Bounds for LDPC and LDGM Codes Under MAP Decoding

A new method for analyzing low density parity check (LDPC) codes and low density generator matrix (LDGM) codes under bit maximum a posteriori probability (MAP) decoding is introduced. The method is based on a rigorous approach to spin glasses developed by Francesco Guerra. It allows to construct lower bounds on the entropy of the transmitted message conditional to the received one. Based on heuristic statistical mechanics calculations, we conjecture such bounds to be tight. The result holds for standard irregular ensembles when used over binary input output symmetric channels. The method is first developed for Tanner graph ensembles with Poisson left degree distribution. It is then generalized to `multi-Poisson' graphs, and, by a completion procedure, to arbitrary degree distribution.