Focus of this letter is the oldest class of codes\ud that can approach the Shannon limit quite closely, i.e., lowdensity\ud parity-check (LDPC) codes, and two mathematical tools\ud that can make their design an easier job under appropriate\ud assumptions. In particular, we present a simple algorithmic\ud method to estimate the threshold for regular and irregular LDPC\ud codes on memoryless binary-input continuous-output AWGN\ud channels with sum-product decoding, and, to determine how close\ud are the obtained thresholds to the theoretical maximum, i.e., to\ud the Shannon limit, we give a simple and invertible expression\ud of the AWGN channel capacity in the binary input - soft output\ud case. For these codes, the thresholds are defined as the maximum\ud noise level such that an arbitrarily small bit-error probability\ud can be achieved as the block length tends to infinity. We assume\ud a Gaussian approximation for message densities under density\ud evolution, a widely used simplification of the decoding algorithm
Since irregular low-density parity-check (LDPC) codes are known to perform better than regular ones, and to exhibit, like them, the so called "threshold phenomenon", this letter investigates a low complexity upper bound on belief-propagation decoding thresholds for this class of codes on memoryless BI-AWGN (Binary Input -Additive White Gaussian Noise) channels, with sum-product decoding. We use a simplified analysis of the belief-propagation decoding algorithm, i.e., consider a Gaussian approximation for message densities under density evolution, and a simple algorithmic method, defined recently, to estimate the decoding thresholds for regular and irregular LDPC codes.Introduction: As first noticed by Gallager in his introductory work to regular LDPC codes [1], these exhibit the so called "threshold phenomenon". Namely, an upper bound for the channel noise can be defined by the noise threshold so that, if the channel noise is maintained below this threshold, the probability of lost information can be made as small as desired. Later it was shown in [2] that irregular LDPC codes perform better than regular ones, and exhibit this phenomenon, too.LDPC codes are capacity-approaching codes, which means that practical constructions exist that allow the noise threshold to be set very close to the theoretical maximum (the Shannon limit) for a symmetric memoryless channel. Thus, the problem of an easy evaluation of the threshold, and, in general, of the performance of belief propagation decoding (see, e.g., [3] and [4]) is important to allow the design of capacity-approaching codes, based on noise threshold maximization.Maximum Likelihood decoding of LDPC codes is in general not feasible [3]. Instead, Gallager proposed an iterative soft decoding algorithm, also called belief propagation [5]. Gallager also noted that, for any given channel conditions, it is possible to evaluate the performance of belief propagation by following the evolution of the distribution of the messages. This idea was extended in [6], where it was shown how to apply density evolution efficiently. One difficulty encountered when applying density evolution is given by the continuous nature of the messages which makes them hard to analyze. As an alternative, in [7] a Gaussian approximation for the message distribution was proposed, reducing the evolution of the infinite dimensional density space to the evolution of a single parameter. In this way, the mean value of a generic check node output message at the l-th iteration is simply described as a function of the check node output message mean value at the (l − 1)-th iteration, thus obtaining a recurrent sequence. With this simplified description, the threshold can be calculated as the last value such that the recurrent sequence converges but no mathematical methods were provided in [7] to determine it.In [8] it was presented a mathematical method to allow the noise thresholds evaluation using the quadratic degeneracy theory, thus transforming a recurrence relation convergence problem in a problem of ma...
This letter presents a more accurate mathematical analysis, with respect to the one performed in Chung et al.'s 2001 paper, of belief-propagation decoding for Low-Density Parity-Check (LDPC) codes on memoryless Binary Input-Additive White Gaussian Noise (BI-AWGN) channels, when considering a Gaussian Approximation (GA) for message densities under density evolution. The recurrent sequence, defined in Chung et al.'s 2001 paper, describing the message passing between variable and check nodes, follows from the GA approach and involves the function φ(x), therein defined, and its inverse. The analysis of this function is here resumed and studied in depth, to obtain tighter upper and lower bounds on it. Moreover, unlike the upper bound given in the above cited paper, the tighter upper bound on φ(x) is invertible. This allows a more accurate evaluation of the asymptotical performance of sum-product decoding of LDPC codes when a GA is assumed.
Low density parity check (LDPC) codes are still intensively studied investigating their iterative decoding convergence performance. Since the probability distribution function of the decoder's log‐likelihood ratio messages was observed to be approximately Gaussian, a variety of low‐complexity approaches to this investigation were proposed. One of them was presented in Chung et al.'s 2001 paper, involving the function ϕfalse(xfalse), therein specified, and its inverse. In this Letter, a new approximation of the function ϕfalse(xfalse) is given, such that, unlike the other approximations found in the literature, it is defined by a single expression (i.e. it is not piecewise defined), it is explicitly invertible, and it has less relative error in any x than the other approximations.
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