Iterative Decoding Performance Bounds for LDPC Codes on Noisy Channels

The asymptotic iterative decoding performances of low-density parity-check (LDPC) codes using min-sum (MS) and sum-product (SP) decoding algorithms on memoryless binary-input output-symmetric (MBIOS) channels are analyzed in this paper. For MS decoding, the analysis is done by upper bounding the bit error probability of the root bit of a tree code by the sequence error probability of a subcode of the tree code assuming the transmission of the allzero codeword. The result is a recursive upper bound on the bit error probability after each iteration. For SP decoding, we derive a tighter recursively determined upper bound on the bit error probability by tracking the evolution of the Bhattacharryya parameters associated with the decoding messages after each iteration. This recursive upper bound recovers the density evolution equation of LDPC codes on the binary erasure channel (BEC) with inequalities satisfied with equalities. A significant implication of this result is that the performance of LDPC codes under SP decoding on the BEC is a lower bound of the performance on all MBIOS channels with the same Bhattacharryya parameter. All results hold for the more general multi-edge type LDPC codes.