Signal-space characterization of iterative decoding

By tracing the flow of computations in the iterative decoders for low-density parity-check codes, we formulate a signal-space view for a finite number of iterations in a finite-length code. On a Gaussian channel, maximum a posteriori (MAP) codeword decoding (or "maximum-likelihood decoding") decodes to the codeword signal that is closest to the channel output in Euclidean distance. In contrast, we show that iterative decoding decodes to the "pseudosignal" that has highest correlation with the channel output. The set of pseudosignals corresponds to "pseudocodewords," only a vanishingly small number of which correspond to codewords, We show that some pseudocodewords cause decoding errors, but that there are also pseudocodewords that frequently correct the deleterious effects of other pseudocodewords.