The behavior of iteratively decoded low-density parity-check (LDPC) codes over the binary erasure channel in the so-called "waterfall region" is investigated and shows that the performance curves in this region follow a very basic scaling law. This scaling law, combined with previously known expressions for the error floor, yields a promising direction for analyzing the performance of irregular LDPC codes of practical lengths.
The behavior of iteratively decoded low-density parity-check (LDPC) codes over the binary erasure channel in the so-called "waterfall region" is investigated and shows that the performance curves in this region follow a very basic scaling law. This scaling law, combined with previously known expressions for the error floor, yields a promising direction for analyzing the performance of irregular LDPC codes of practical lengths.
We investigate the behavior of iteratively decoded low-density parity-check (LDPC) codes over the binary erasure channel in the so-called ldquowaterfall region.rdquo We show that the performance curves in this region follow a simple scaling law. We conjecture that essentially the same scaling behavior applies in a much more general setting and we provide some empirical evidence to support this conjecture. The scaling law, together with the error floor expressions developed previously, can be used for a fast finite-length optimization.
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 thesis addresses the topic of Low-Density Parity-Check (LDPC) code analysis, both asymptotically and for finite block lengths. Since in general it is a difficult problem to analyze individual code instances, ensemble averages are studied by following Gallager's original idea and results of Luby et al. Often, one can relate the insights gained by studying ensemble averages to statements regarding individual codes by proving that most elements in the ensemble behave "close" to the average. One important such average is the average weight distribution, another is the average pseudo-codeword distribution, where pseudo-codewords play the same role under iterative decoding as codewords do for maximum likelihood decoding. One of the main contributions of this thesis is the calculation of such averages for fully irregular LDPC code ensembles. Much of classical coding theory is aimed at the construction of codes with large minimum distance. This is so since away from capacity accurate bounds on the performance of a code can be given in terms of its minimum distance (or more generally, in terms of its weight distribution). Therefore, it is of interest to investigate what role the minimum distance plays for iteratively decoded LDPC code ensembles. In particular, sequences of capacity-achieving LDPC code ensembles of increasing length when transmission takes place over the Binary Erasure Channel (BEC) are investigated. It is shown that, under certain technical conditions, the minimum distance of such ensembles grows sub-linearly in the block length, a result which is somewhat surprising from a classical point of view. Specific attention is also given to the design of LDPC code ensembles, where an inherent trade-off is observed between achieving large minimum distance (which is relevant for the so called error-floor regime) and achieving a large threshold (which in the limit of long block lengths determines the worst channel on which transmission can be accomplished reliably). Most results on iterative coding systems to date address their asymptotic performance, i.e., their performance when the block length tends to infinity. For small and moderate block lengths the behavior of a code can deviate significantly from its asymptotic limit. It is therefore of high practical value to be able to analyze the finite-length performance of LDPC code ensembles. In this thesis, an exact such analysis is presented for iteratively decoded LDPC code ensembles over the BEC. In particular, expressions for the exact average bit and block erasure probabilities are computed by solving a set of recursions. Such an analysis is the starting point for a finite-length optimization, a topic which is slated for future work. The methods used in this thesis include a combinatorial approach (familiar to the coding society) as well as the powerful techniques developed in the statistical physics community, for example, the replica method or the mean-field approximation. Although the statistical physics techniques are ideally suited for the analysis of iterative coding systems, they are to date only accessible to a relatively small community. Therefore, an overview of these techniques is first presented using a language familiar to the coding theory community. Next, in order to highlight the differences and common points between the combinatorial and the statistical physics approaches, both techniques are applied to the weight distribution problem. It turns out that for regular ensembles, both methods yield the same result, while for irregular ensembles, they do differ in general.
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.
Assuming iterative decoding for binary erasure channels (BECs), a novel tree-based technique for upper bounding the bit error rates (BERs) of arbitrary, finite low-density parity-check (LDPC) codes is provided and the resulting bound can be evaluated for all operating erasure probabilities, including both the waterfall and the error floor regions. This upper bound can also be viewed as a narrowing search of stopping sets, which is an approach different from the stopping set enumeration used for lower bounding the error floor. When combined with optimal leaf-finding modules, this upper bound is guaranteed to be tight in terms of the asymptotic order. The Boolean framework proposed herein further admits a composite search for even tighter results. For comparison, a refinement of the algorithm is capable of exhausting all stopping sets of size les 13 for irregular LDPC codes of length n ap 500, which requires (13500) ap 1.67 times 1025trials if a brute force approach is taken. These experiments indicate that this upper bound can be used both as an analytical tool and as a deterministic worst-performance (error floor) guarantee, the latter of which is crucial to optimizing LDPC codes for extremely low BER applications, e.g., optical/satellite communications
Low-density parity-check (LDPC) codes in conjunction with iterative decoding based on message-passing algorithms have been shown to achieve excellent performance over a variety of communication channels. In this dissertation, we investigate LDPC codes under two scenarios which present different challenges for providing robust communications. First, LDPC codes are analyzed and designed for time-selective, frequency-nonselective complex-fading channels where both the amplitude and the phase of the transmitted signal are altered by the channel. To combat the fading, we use a pilot symbol-assisted scheme where known pilot symbols are added to the LDPC code and used to help estimate the channel at the receiver. Using asymptotic, infinite-length analysis, we investigate several iterative, message-passing, joint-estimation-and-decoding strategies; optimal energy distribution between pilot and data symbols; and LDPC code design. Several interesting results are obtained regarding unification of analysis and code design. Next, we address the issue of achieving very low error rates with practical, finite-length LDPC codes for binary erasure channels. Finite-length LDPC codes suffer from error floors which limit the achievable error rate, and this error-floor performance is determined by stopping sets. Asymptotic analysis of weight and stopping-set enumerators, for codewords and stopping sets which grow linearly with codelength, has aided in designing codes with lower error floors but does not reflect the behavior of sublinearly-sized stopping sets, which can dominate the iterative-decoding, error-floor performance. Thus, we provide a perspective on protograph-based and standard LDPC ensemble enumerators, based on analysis of stopping sets with sublinear growth, which brings new insight into sublinear stopping-set behavior, advantages of protograph structure, and effects of precoding. We show for stopping sets that grow at most logarithmically with codelength, the enumerators follow a polynomial relationship with codelength, unlike the exponential relationship for linearly-growing stopping sets, and this polynomial relationship can be approximately captured by a single parameter. Further, we begin to address the question, Given finite stopping-set sizes and finite codelengths, do the stopping sets follow the behavior predicted by linear or sublinear analysis?