An Algebraic Description of Iterative Decoding Schemes

2007 IEEE International Symposium on Signal Processing and Information Technology, 2007

Iterative threshold decoding of product and parallel concatenated block codes based on one step majority logic decodable (OSMLD) codes has proven to perform remarkably well on AWGN channels. For these codes to be applicable in wireless environment, their performance on fading channels must be examined. The purpose of this work is to study the performance of our iterative threshold decoding algorithm on the Rayleigh fading channel. Results have shown that the slope of the bit-error rate (BER) curve is as steep as for the Gaussian channel. We also present a comparison between our results and those for convolutional turbo code in terms of BER performance.

2009 IEEE International Conference on Acoustics, Speech and Signal Processing, 2009

Iterative decoding was not originally introduced as the solution to an optimization problem rendering the analysis of its convergence very difficult. In this paper, we investigate the link between iterative decoding and classical optimization techniques. We first show that iterative decoding can be rephrased as two embedded minimization processes involving the Fermi-Dirac distance. Based on this new formulation, an hybrid proximal point algorithm is first derived with the additional advantage of decreasing a desired criterion. In a second part, an hybrid minimum entropy algorithm is proposed with improved performance compared to the classical iterative decoding. Even if this paper focus on iterative decoding for BICM, the results can be applied to the large class of turbo-like decoders.

2009 IEEE International Conference on Communications, 2009

In this paper the issue of improving the performance of iterative decoders based on sub-optimal calculation of the messages exchanged during iterations (L-values) is addressed. It is well known in the literature that a simple-yet very effectiveway to improve the performance of suboptimal iterative decoders is based on applying a scaling factor to the L-values. In this paper, starting with a theoretical model based on the so-called consistency condition of a random variable, we propose a methodology for correcting the L-values that relies only on the distribution of the soft information exchanged in the iterative process. This methodology gives a clear explanation of why the well-known linear scaling factor provides a very good performance. Additionally, the proposed methodology allows us to avoid the exhaustive search required otherwise. Numerical simulations show that for turbo codes the scaling factors found closely follow the optimum values, which translates to a close-to-optimal BER performance. Moreover, for LDPC codes, the proposed methodology produces a better BER performance compared with the known method in the literature.

IEEE Transactions on Information Theory, 2005

We consider a class of message-passing decoders for low-density parity-check (LDPC) codes whose messages are binary valued. We prove that if the channel is symmetric and all codewords are equally likely to be transmitted, an optimum decoding rule (in the sense of minimizing message error rate) should satisfy certain symmetry and isotropy conditions. Using this result, we prove that Gallager's Algorithm B achieves the optimum decoding threshold among all binary message-passing decoding algorithms for regular codes. For irregular codes, we argue that when the nodes of the message-passing decoder do not exploit knowledge of their decoding neighborhood, optimality of Gallager's Algorithm B is preserved. We also consider the problem of designing irregular LDPC codes and find a bound on the achievable rates with Gallager's Algorithm B. Using this bound, we study the case of low error-rate channels and analytically find good degree distributions for them.

Journal of Advanced College of Engineering and Management, 2018

This paper presents a Thesis which consists of a study of turbo codes as an error-control Code and the software implementation of two different decoders, namely the Maximum a Posteriori (MAP), and soft-Output Viterbi Algorithm (SOVA) decoders. Turbo codes were introduced in 1993 by berrouet at [2] and are perhaps the most exciting and potentially important development in coding theory in recent years. They achieve near-Shannon-Limit error correction performance with relatively simple component codes and large interleavers. They can be constructed by concatenating at least two component codes in a parallel fashion, separated by an interleaver. The convolutional codes can achieve very good results. In order of a concatenated scheme such as a turbo codes to work properly, the decoding algorithm must affect an exchange of soft information between component decoders. The concept behind turbo decoding is to pass soft information from the output of one decoder to the input of the succeeding one, and to iterate this process several times to produce better decisions. Turbo codes are still in the process of standardization but future applications will include mobile communication systems, deep space communications, telemetry and multimedia. Finally, we will compare these two algorithms which have less complexity and which can produce better performance.

IEEE Transactions on Information Theory, 1994

To decode a long block code with a large minimum distance by maximum likelihood decoding is practically impossible because the decoding complexity is simply enormous. However, if a code can be decomposed into constituent codes with smaller dimensions and simpler structure, it is possible to devise a practical and yet efficient scheme to decode the code. This paper investigates a class of decomposable codes, their distance and structural properties. It is shown that this class includes several classes of well-known and efficient codes as subclasses. Several methods for constructing decomposable codes or decomposing codes are presented. A two-stage (soft-decision or hard-decision) decoding scheme for decomposable codes, their translates or unions of translates is devised, and its error performance is analyzed for an AWGN channel. The two-stage soft-decision decoding is suboptimum. Error performances of some specific decomposable codes based on the proposed twostage soft-decision decoding are evaluated. It is shown that the proposed two-stage suboptimum decoding scheme provides an excellent trade-off between the error performance and decoding complexity for codes of moderate and long block length.

IEEE Transactions on Information Theory, 2013

Known properties of cyclic codes are used to give a unified description of many classical decoding algorithms for Reed-Solomon codes up to half the minimum distance. This description allows also simplified proofs for these decoders. Further, a novel decoding algorithm is derived using these properties directly and variants of a new error/erasure decoding algorithm are given. For decoding beyond half the minimum distance, a basis of all solutions for decoding is derived. This basis allows to use side information in order to decode beyond half the minimum distance. Other methods where this basis can be used are power decoding, also known as virtual syndrome extension, where additional equations are generated by taking powers of the received symbols, and interleaved Reed-Solomon codes. The extended Euclidean algorithm, which calculates the greatest common divisor, plays an essential role in many presented methods.

Carpathian Journal of Electronic and Computer Engineering, 2017

Several modern error-correcting codes can perform close to the Shannon limit thanks to the turbo principle applied in their iterative decoding algorithms. In this paper the principle is discussed in relation to LDPC codes and turbo codes. Methods for improvement of both these codes are described, namely removal of short cycles for LDPC codes and trellis termination for turbo codes. Performance of reference LDPC and turbo codes with and without these improvements is simulated and compared.

IEEE Communications Letters, 2004

The sum-product iterative decoder, conventionally used for low-density parity-check (LDPC) codes, hold promise as a decoder for general linear block code decoding. However, the promise is only partly fulfilled because, as we show experimentally, the decoder performance degrades rapidly as a function of parity check matrix weight. Even in the case of decoder failure, however, we demonstrate that there is information present in the decoder output probabilities that can still help with the decoding problem.

IEEE Journal of Selected Topics in Signal Processing, 2000

This paper considers the joint-decoding problem for finite-state channels (FSCs) and low-density parity-check (LDPC) codes. In the first part, the linear-programming (LP) decoder for binary linear codes is extended to joint-decoding of binary-input FSCs. In particular, we provide a rigorous definition of LP joint-decoding pseudo-codewords (JD-PCWs) that enables evaluation of the pairwise error probability between codewords and JD-PCWs in AWGN. This leads naturally to a provable upper bound on decoder failure probability. If the channel is a finite-state intersymbol interference channel, then the joint LP decoder also has the maximum-likelihood (ML) certificate property and all integer-valued solutions are codewords. In this case, the performance loss relative to ML decoding can be explained completely by fractional-valued JD-PCWs. After deriving these results, we discovered some elements were equivalent to earlier work by Flanagan on linearprogramming receivers.

2003

Iterative decoding of block codes is a rather old subject that regained much interest recently. The main idea behind iterative decoding is to break up the decoding problem into a sequence of stages, iterations, such that each stage utilizes the output from the previous stages to formulate its own result. In order for the iterative decoding algorithms to be practically feasible, the complexity in each stage, in terms of number of operations and hardware complexity, should be much less than that for the original non-iterative decoding problem. At the same time, the performance should approach the optimum, maximum likelihood decoding performance in terms of bit error rate.

IEEE Transactions on Information Theory, 1998

IEEE Journal on Selected Areas in Communications, 2001

Iterative decoding is used to achieve backward compatible performance improvement in several existing systems. Concatenated coding and iterative decoding are first set up using composite mappings, so that various applications in digital communication and recording can be described in a concise and uniform manner. An ambiguity zone detection (AZD) based iterative decoder, operating on generalized erasures, is described as an alternative for concatenated systems where turbo decoding cannot be performed. Described iterative decoding techniques are then applied to selected wireless communication and digital recording systems. Simulation results and utilization of decoding gains are briefly discussed.

Electronics Letters, 2005

We describe a Turbo-like iterative decoding scheme for analog product codes and prove that the iterative decoding method is an iterative projection in Euclidean space and converges to the least-squares solution. Using this geometric point of view, any block analog codes can be decoded by a similar iterative method. The described procedure may serve as a step towards a more intuitive understanding of Turbo decoding.

IEEE Transactions on Signal Processing, 2006

A turbo-decoding message-passing (TDMP) algorithm for sparse parity-check matrix (SPCM) codes such as low-density parity-check, repeat-accumulate, and turbo-like codes is presented. The main advantages of the proposed algorithm over the standard decoding algorithm are 1) its faster convergence speed by a factor of two in terms of decoding iterations, 2) improvement in coding gain by an order of magnitude at high signal-to-noise ratio (SNR), 3) reduced memory requirements, and 4) reduced decoder complexity. In addition, an efficient algorithm for message computation using simple "max" operations is also presented. Analysis using EXIT charts shows that the TDMP algorithm offers a better performance-complexity tradeoff when the number of decoding iterations is small, which is attractive for high-speed applications. A parallel version of the TDMP algorithm in conjunction with architecture-aware (AA) SPCM codes, which have embedded structure that enables efficient high-throughput decoder implementation, are presented. Design examples of AA-SPCM codes based on graphs with large girth demonstrate that AA-SPCM codes have very good error-correcting capability using the TDMP algorithm. Index Terms-Iterative decoding, low-density parity-check (LDPC) codes, repeat-accumulate (RA) codes, Ramanujan graphs, turbo decoding algorithm, very large scale integration (VLSI) decoder architectures. I. INTRODUCTION I TERATIVE decoding techniques have gained widespread attention in advanced communications systems after the seminal work of Berrou et al. on turbo codes in 1993 [2]. These techniques have proven very effective in extending the limits and services of wireless communications, improving the throughput of broadband systems over long ranges, and expanding the areal density of magnetic recording systems. The underlying principle of operation of turbo codes dates back to Gallager in his work on low-density parity-check (LDPC) codes in 1963 [3]. It is based on iterative suboptimal decoding by message-passing as an alternative to one-shot maximum-likelihood decoding. Codeword symbols are processed iteratively in an attempt to improve their reliability based on Manuscript