Low-density parity-check codes over Gaussian channels with erasures

We consider low-density parity-check code (LDPCC) design for additive white Gaussian noise (AWGN) channels with erasures. This model, for example, represents a common situation in magnetic and optical recording where defects or thermal asperities in the system are detected and presented to the decoder as erasures. We give thresholds of regular and irregular LDPCCs and discuss practical code design over the mixed Gaussian/erasures channel. The analysis is an extension of the Gaussian approximation work of Chung et al. In the two limiting cases of no erasures and large signal-to-noise ratio (SNR), the analysis tends to the results of Chung et al. (see ibid., vol. 47, p.657-670, Feb. 2001) and Luby et al. (1997), respectively, giving a general tool for a class of mixed channels. We derive a steady-state equation which gives a graphical interpretation of decoder convergence. This allows one to estimate the maximum erasure capability on the mixture channel, or conversely, to estimate the additional signal power required to compensate for the loss due to erasures. We see that a good (capacity-approaching) LDPCC over an AWGN channel is also good over the mixed channel up to a moderate erasure probability. We also investigate practical issues such as the maximum number of iterations of message-passing decoders, the coded block length, and types of erasure patterns (random/block erasures). Finally, we design an optimized LDPCC for the mixed channel, which shows better performance if the erasure probability is larger than a certain value (0.1 in our simulation) at the expense of performance degradation at unerased (AWGN channel) and lower erasure probability regions (less than 0.1 in our simulation).