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...