This paper presents a reduced-complexity approximate density evolution (DE) scheme for low-density parity-check (LDPC) codes in channels with memory in the form of a hidden Markov chain. This approximation is used to design degree sequences representing some of the best known LDPC code ensembles for the Gilbert-Elliott channel, and example optimizations are also given for other Markov channels. The problem of approximating the channel estimation is addressed by obtaining a specially constructed message-passing schedule in which the channel messages all approach their stable densities. It is shown that this new schedule is much easier to approximate than the standard schedule, but has the same ultimate performance in the limits of long block length and many decoding iterations. This result is extended to show that all message-passing schedules that satisfy mild conditions will have the same threshold under density evolution
This paper presents a reduced-complexity approximate density evolution (DE) scheme for low-density parity-check (LDPC) codes in channels with memory in the form of a hidden Markov chain. This approximation is used to design degree sequences representing some of the best known LDPC code ensembles for the Gilbert-Elliott channel, and example optimizations are also given for other Markov channels. The problem of approximating the channel estimation is addressed by obtaining a specially constructed message-passing schedule in which the channel messages all approach their stable densities. It is shown that this new schedule is much easier to approximate than the standard schedule, but has the same ultimate performance in the limits of long block length and many decoding iterations. This result is extended to show that all message-passing schedules that satisfy mild conditions will have the same threshold under density evolution
The problem of low-density parity-check (LDPC) decoding in channels with memory has been attracting increasing attention in the literature. In this thesis, we use LDPC codes in an estimation-decoding scheme for the Gilbert-Elliott (GE) channel and more general Markov-modulated channels. The major accomplishments of this thesis include both analysis and design components. To analyze our estimation-decoding scheme, we derive density evolution for the GE channel, which has previously been used largely in memoryless channels. Furthermore, we develop techniques that use density evolution results to more efficiently characterize the space of Markov-modulated parameters. We begin by applying a characterization to the GE channel, following which we generalize this characterization into a partial ordering of Markov-modulated channels in terms of probability of symbol error. We also consider the design problem of developing LDPC degree sequences that are optimized for the GE channel. We obtain a novel design tool that approximates density evolution for our estimation-decoding algorithm, and present degree sequences that represent the best known codes in the GE channel. We also present a method of generalizing this tool to Markov-modulated channels, and give some of the first optimized degree sequences ever obtained for these channels.
It has been shown using density evolution (DE) that joint estimation-decoding strategies have excellent performance for regular lowdensity parity-check (LDPC) codes in the GilbertElliott (GE) channel. However, DE is computationally intensive, and is therefore not suitable as a tool in searching for good degree sequences for irregular LDPC codes. In this paper, we present a reduced-complexity approximate DE scheme that is particularly suitable for the GE channel, based on the expected value approximation from [7]. The proposed technique takes advantage of a specially constructed decoder for which the ultimate performance is the same as the standard decoder, but for which DE is easier to approximate. We show that this straightforward approximation allows the channel messages to be precalculated, resulting in computational complexity comparable to the memoryless case.
In this paper, we investigate the design of low-density parity-check (LDPC) codes for the Gilbert-Elliott (GE) channel. Recently, Eckford et al. proposed a design method of irregular LDPC codes using approximate density-evolution (DE) for Markov channels. In the design method proposed by Eckford et al., the probability density function (PDF) of the messages from variable nodes to check nodes is approximated by the Gaussian distribution. In this paper, we first show the method to obtain the accurate PDF of the messages from variable nodes to check nodes by utilizing two DE steps for the Gaussian distribution. We call this method the iterative density approximation (IDA). Using this method, we can design the good LDPC codes. Next, we propose an efficient design method of irregular LDPC codes by using Beta approximation to the PDF of the channel state probability for the GE channel. Consequently, we show that the complexity to calculate PDFs of the channel messages is considerably reduced though the rates of LDPC codes obtained by using the proposed approximation are almost the same as that of the IDA method.
Low-density parity-check (LDPC) codes perform very close to capacity for long lengths on several channels. However, the amount of memory (fixed-point numbers that need to be stored) required for implementing the message-passing algorithm increases linearly as the number of edges in the graph increases. In this letter, we propose a decoding algorithm for decoding LDPC codes that reduces the memory requirement at the decoder. The proposed decoding algorithm can be analyzed using density evolution; further, we show how to design good LDPC codes using this. Results show that this algorithm provides almost the same performance as the conventional sum-product decoding of LDPC codes.
Error control coding is an essential part of modern communications systems. LDPC codes have been demonstrated to offer performance near the fundamental limits of channels corrupted by random noise. Optimal maximum likelihood decoding of LDPC codes is too complex to be practically useful even at short block lengths and so a graph-based message passing decoder known as the belief propagation algorithm is used instead. In fact, on graphs without closed paths known as cycles the iterative message passing decoding is known to be optimal and may converge in a single iteration, although identifying the message update schedule which allows single-iteration convergence is not trivial. At finite block lengths graphs without cycles have poor minimum distance properties and perform poorly even under optimal decoding. LDPC codes with large block length have been demonstrated to offer performance close to that predicted for codes of infinite length, as the cycles present in the graph are quite long. In this thesis, LDPC codes of shorter length are considered as they offer advantages in terms of latency and complexity, at the cost of performance degradation from the increased number of short cycles in the graph. For these shorter LDPC codes, the problems considered are: \n \nFirst, improved construction of structured and unstructured LDPC code graphs of short length with a view to reducing the harmful effects of the cycles on error rate performance, based on knowledge of the decoding process. Structured code graphs are particularly interesting as they allow benefits in encoding and decoding complexity and speed. Secondly, the design and construction of LDPC codes for the block fading channel, a particularly challenging scenario from the point of view of error control code design. Both established and novel classes of codes for the channel are considered. Finally the decoding of LDPC codes by the belief propagation algorithm is considered, in particular the scheduling of messages passed in the iterative decoder. A knowledge-aided approach is developed based on message reliabilities and residuals to allow fast convergence and significant improvements in error rate performance.
An efficient decoding schedule for low-density parity-check (LDPC) codes that outperforms the conventional approach, in terms of both complexity and performance, is presented. Conventionally, in each iteration, all symbol nodes and, subsequently, all the check nodes, send messages to their neighbors ("flooding schedule"). In contrast, in the proposed method, the updating of nodes is performed according to a serial schedule which propagates the information twice as fast. A density evolution (DE) algorithm for asymptotic analysis of the new schedule is derived, showing that, when working near the code's capacity, the decoder converges in approximately half the number of iterations. In addition, a concentration theorem is proved, showing that, for a randomly chosen serial schedule, code graph, and decoder input, the decoder's performance approaches its expected one as predicted by the DE algorithm, when the code length increases.