Generalized EXIT Charts for Irregular LDPC Codes Hossein Mamani and Hamid Saeedi Department of Electrical and Computer Engineering Tarbiat Modares University Tehran, Iran Email: {h.mamani, hsaeedi}@ieee.org Abstract—It is well-known that irregular Low-Density Parity-Check (LDPC) code ensembles can have performance approaching the capacity of binary memoryless channels as opposed to regular LDPC codes. Recently, an asymptotic analysis tool named Generalized Extrinsic Information Transfer (GEXIT) chart has been proposed for regular LDPC codes that can be used as a substitution to density evolution (DE), the main asymptotic analysis tool for LDPC codes. GEXIT charts can be seen as an extension of formerly introduced EXIT charts which have an extensive use in analysis and design of many iterative schemes including LDPC codes. EXIT charts provide exactly the same analysis results as DE for the Binary Erasure Channel. However, they can only be seen as an approximation to DE for other channel types. GEXIT charts have only been derived for regular LDPC codes. To be able to use GEXIT charts in code design and in particular, to devise capacity approaching ensembles, they have to be obtained for irregular codes. In this paper we derive GEXIT charts for irregular LDPC codes and show that a similar relationship which exists between EXIT charts of irregular and regular LDPC code also holds for the case of GEXIT charts. Keywords-LDPC codes; density evolution; entropy; mutual information; EXIT function; area theoram; Genaral EXIT function; irregular LDPC codes. I. INTRODUCTION Low-Density Parity-Check (LDPC) codes have been an active area of research in the past decade due to their good performance under iterative message passing algorithms [1, 2]. For regular LDPC codes, there is a fixed but different number of ones in each column and row of the parity check matrix. This constraint makes it impossible to design codes with performance approaching the channel capacity. Therefore, irregular LDPC codes were introduced [3] for which this constraint is not enforced any more. In both cases the number of ones is in such a way that the parity check matrix is sparse, i.e. the density of ones is low. For large block size, the ensembles of LDPC codes are considered and presented by their degree distribution pairs (λ(x),ρ(x)). Concentration results [4, 5] indicate that for sufficiently long block sizes, the performance of any ensemble over a binary input symmetric channel tends to the average performance of the ensembles. This important result makes it possible to use asymptotic analysis tools such as density This work was supported by the Iran Telecommunication Research Center (ITRC) under project number 90-01-04. evolution (DE) [4] to analyze and design LDPC code ensem- bles with performance approaching the capacity. In density evolution technique, assuming that the all-one codeword is transmitted and starting from initial channel density, the evo- lution of this density is tracked throughout iterations. At each iteration, we compute the probability of error of the density, which is defined as the probability that the value of random variable corresponding to the density is negative. Depending on the initial density which itself directly depends on the channel parameter 1 , two cases can happen. The probability of error of the evolving density either tends to zeros (DE con- verges) or is bounded away from zero after infinite number of iterations. It can be proved that for a large set of channels types including Binary Erasure Channel (BEC), Binary Symmetric Channel (BSC) and Binary Input Additive White Gaussian Noise (BIAWGN) channel, there is a boundary value on the channel parameter such that if the channel parameter is less than that value, the DE converges and if it is more, DE does not converge. We call this boundary value the threshold of the ensemble over the given channel parameter [4]. It can be seen that in each iteration of the DE, a possibly large set of values representing the density, should be updated which can be computationally expensive especially when DE is used to design ensembles. Consequently, many approxima- tion techniques have been developed to reduce the complexity of DE such as Extrinsic Information Transfer (EXIT) charts [6, 7] and Gaussian approximation [8]. The aim of all these methods is to map a given density to a scalar. This not only reduces the complexity, but can provide a better insight into the dynamic of iterative decoding. Concentrating now on the EXIT chart method, it can be shown that for the BEC, EXIT chart is not an approximation to DE anymore and is actually able to provide exactly the same threshold predicted by DE. Moreover an area theorem is proved for the EXIT charts of BEC indicating that the area between the EXIT curve of variable node and inverse EXIT curve of check node can be translated into the distance to capacity. In other words, to obtain capacity achieving codes over the BEC, it is enough to find a degree distribution for which the area between the curves tends to zero. This is consistent with the previously obtained results by Shokrollahi [9] and also in [10], known as flatness 1 In this paper we consider channels that are parameterized by one variable ϵ and we assume that the quality of channel degrades as ϵ increases