Gallager's decoding algorithm Α over high order modulations

Gallager's decoding algorithm A (GDA) for low density parity check (LDPC) codes is of interest mainly due to its good performance with extremely low complexity. In this paper, we analyze GDA over 2
^m
-ary pulse-amplitude modulation (2
^m
-PAM). Firstly, we represent its average error probability by means of a recursion formula. We then define its threshold for additive white Gaussian noise (AWGN) channels and derive its stability condition. The bit-to-symbol mapping strategies have a strong influence on the performance of LDPC-coded modulation systems. Finally, we show that the bit-reliability mapping strategy proposed by Li and Ryan maximizes the threshold of the LDPC coded modulation system with GDA over Gray-mapped 2
^m
-PAM in the AWGN channel.