Analysis of sum-product decoding of low-density parity-check codes using aGaussian approximation

Density evolution is an algorithm for computing the capacity of low-density parity-check (LDPC) codes under message-passing decoding, For memoryless b inary-input continuous-output additive white Gaussian noise (AWGN) channels and sum-product decoders, we use a Gaussian approximation for message dens ities under density evolution to simplify the analysis of the decoding algo rithm. We convert the infinite-dimensional problem of iteratively calculati ng message densities, which is needed to find the exact threshold, to a one -dimensional problem of updating means of Gaussian densities, This simplifi cation not only allows us to calculate the threshold quickly and to underst and the behavior of the decoder better, but also makes it easier to design good irregular LDPC codes for AWGN channels. For various regular LDPC codes we have examined, thresholds can be estimate d within 0.1 dB of the exact value, For rates between 0.5 and 0.9, codes de signed using the Gaussian approximation perform within 0.02 dB of the best performing codes found so Far by using density evolution when the maximum v ariable degree is 10, We show that by using the Gaussian approximation, we can visualize the sum-product decoding algorithm. We also show that the opt imization of degree distributions can be understood and done graphically us ing the visualization.