More Accurate Analysis of Sum-Product Decoding of LDPC Codes Using a Gaussian Approximation

This letter presents a more accurate mathematical analysis, with respect to the one performed in Chung <italic>et al.</italic>’s 2001 paper, of belief-propagation decoding for low-density parity-check (LDPC) codes on memoryless binary input—additive white Gaussian noise channels, when considering a Gaussian approximation (GA) for message densities under density evolution. The recurrent sequence, defined in Chung <italic>et al.</italic>’s 2001 paper, describing the message passing between variable and check nodes follows from the GA approach and involves the function <inline-formula> <tex-math notation="LaTeX">$\boldsymbol {\phi (x)}$ </tex-math></inline-formula>, 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 <inline-formula> <tex-math notation="LaTeX">$\boldsymbol {\phi (x)}$ </tex-math></inline-formula> is invertible. This allows a more accurate evaluation of the asymptotical performance of sum-product decoding of LDPC codes when a GA is assumed.