A Partial Ordering of General Finite-State Markov Channels Under LDPC Decoding

A partial ordering on general finite-state Markov channels is given, which orders the channels in terms of probability of symbol error under iterative estimation decoding of a low-density parity-check (LDPC) code. This result is intended to mitigate the complexity of characterizing the performance of general finite-state Markov channels, which is difficult due to the large parameter space of this class of channel. An analysis tool, originally developed for the Gilbert-Elliott channel, is extended and generalized to general finite-state Markov channels. In doing so, an operator is introduced for combining finite-state Markov channels to create channels with larger state alphabets, which are then subject to the partial ordering. As a result, the probability of symbol error performance of finite-state Markov channels with different numbers of states and wide ranges of parameters can be directly compared. Several examples illustrating the use of the techniques are provided, focusing on binary finite-state Markov channels and Gaussian finite-state Markov channels. Furthermore, this result is used to order Gilbert-Elliott channels with different marginal state probabilities, which was left as an open problem by previous work.