New explicitly invertible approximation of the function involved in LDPC codes density evolution analysis using a Gaussian approximation

Abstract

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.

A Survey on Old and New Approximations to the Function ϕ(x) Involved in LDPC Codes Density Evolution Analysis Using a Gaussian Approximation

“…As noted in [10], the fact that this new approximation has always an univocal analytical expression (i.e., it is not piecewise defined) implies its easy mathematical handling, in contrast to formerly defined piecewise functions, defined by distinct analytical expressions being dependent on the range of values taken by the argument x. Moreover, its explicit invertibility implies that no interpolated functions approximating it by points (x k , φ(x k )) and its inverse by points (φ(x k ), x k ) are required.…”

“…This is analogous but improved with respect to the one in [2], in the sense that it allows to obtain GA thresholds closer to the ones obtained applying density evolution than the GA thresholds given in [2]. In [10], a new approximation of the function φ(x) was given, which, unlike the other approximations found in the literature (see [2,3,6]), 1.…”

“…for the message probability distribution, simplifying the evolution of the infinite dimensional density space to the evolution of a single parameter. They mathematically described the passage of messages between variable and check nodes of the Tanner graph by means of a recurring succession of the mean values of the check nodes' output messages, having shown that regular LDPC codes present message densities that are roughly Gaussian, whereas irregular ones present message densities that are Gaussian combinations [10]. In this way, instead of describing the variable and check node output message as a function of the incoming messages from their respective neighbors, a recursive sequence is obtained by expressing a check node output message mean value at the l-th iteration as a function of the same mean value at the (l − 1)-th iteration [4].…”

“…In this paper, a review of the abovementioned old approximations (published in [2,6]) and new approximations (published in [3,10]) to the function φ(x) is performed, so that to analyze the differences among them and their characteristics in terms of accuracy and computational complexity. The paper is organized as follows.…”

A joint design method of frame synchronization code and LDPC code

“…With each iteration of the decoding algorithm, ) (v P l and ) (u Q l will change accordingly. The density evolution is to track the values of ) (v P l and ) (u Q l in each iteration to determine the bit error rate [10]. According to probability theory, if multiple independent random variables are added together, the probability density of the sum is equal to the convolution of the probability density function of the original random variable.…”