  This paper considers low-density parity-check (LDPC) decoders affected by
deviations introduced by the electronic device on which the decoder is
implemented. Noisy density evolution (DE) that allows to theoretically study
the performance of these LDPC decoders can only consider symmetric deviation
models due to the all-zero codeword assumption. A novel DE method is proposed
that admits the use of asymmetric deviation models, thus widening the range of
faulty implementations that can be analyzed. DE equations are provided for
three noisy decoders: belief propagation, Gallager B, and quantized min-sum
(MS). Simulation results confirm that the proposed DE accurately predicts the
performance of LDPC decoders with asymmetric deviations. Furthermore,
asymmetric versions of the Gallager B and MS decoders are proposed to
compensate the effect of asymmetric deviations. The parameters of these
decoders are then optimized using the proposed DE, leading to better ensemble
thresholds and improved finite-length performance in the presence of asymmetric
deviations.
