2072 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 53, NO. 6, JUNE 2007 A Partial Ordering of General Finite-State Markov Channels Under LDPC Decoding Andrew W. Eckford, Member, IEEE, Frank R. Kschischang, Fellow, IEEE, and Subbarayan Pasupathy, Fellow, IEEE Abstract—A partial ordering on general finite-state Markov channels is given, which orders the channels in terms of proba- bility 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. Index Terms—Estimation-decoding, iterative decoding, low- density parity-check (LDPC) codes, Markov channels, partial ordering. I. INTRODUCTION F INITE-STATE Markov channels are binary-input chan- nels, each with a hidden channel state sequence that is generated by a finite-state Markov chain, where the state of the Markov chain determines the instantaneous behavior of the channel. (Throughout this paper, we will take the finite-state nature of the channel to be implicit, and simply refer to Markov channels.) These channels have many applications, such as approximating wireless channels with slow fading, or modeling other correlated noise effects. Capacity and coding for these channels was discussed in [1]. A low-density parity-check (LDPC) code is a type of block code with a very sparse parity-check matrix [2]. Using the sum–product algorithm (SPA) [3] for decoding, it is well Manuscript received May 14, 2003; revised March 4, 2007. The material in this paper was presented at the 2003 IEEE International Symposium on Infor- mation Theory, Yokohama, Japan, June/July 2003. A. W. Eckford was with The Edward S. Rogers Sr. Department of Electrical and Computer Engineering, University of Toronto, Toronto, ON, Canada. He is now with the Department of Computer Science and Engineering, York Univer- sity, Toronto, ON M3J 1P3, Canada (e-mail: aeckford@yorku.ca). F. R. Kschischang and S. Pasupathy are with The Edward S. Rogers Sr. Department of Electrical and Computer Engineering, University of Toronto, Toronto, ON M5S 3G4, Canada (e-mail: frank@comm.utoronto.ca; pas@comm.utoronto.ca). Communicated by A. Kavˇ cic ´, Associate Editor for Detection and Estimation. Digital Object Identifier 10.1109/TIT.2007.896877 known that LDPC codes have excellent performance in mem- oryless channels. The SPA can be extended to obtain natural estimation-decoding strategies in channels with memory, which from recent work are known to have excellent performance in, for example, partial response channels [4], [5], and the Gilbert–Elliott (GE) channel, the simplest type of binary-output Markov channel [6], [7]. Recent work has also focused on the applicability of LDPC codes for source compression, especially for Slepian–Wolf encoding (see, e.g., [8], [9]). Markov sources, which are analogous to Markov channels, have been proposed to model temporal correlations in data, requiring many of the same approaches as for Markov channels [10]–[12]. Partial orderings of communication channels can be traced back to Shannon [13], who described a partial ordering of memoryless channels using general codes. For Markov chan- nels, some analytical performance results exist for decoding using classical codes, such as burst error correcting codes [14] and convolutional codes [15]. Knowledge of the performance of LDPC decoding may be obtained using Monte Carlo sim- ulation, or using some analytical technique such as density evolution [16]. Markov channels have large parameter spaces—the GE channel, with two channel states, is characterized by four parameters, and, as we discuss in Section II, parameters are required to completely describe a -state Markov channel. Since contemporary analysis techniques can only examine one channel at a time, it is complicated to analyze large classes or families of Markov channels. By contrast, many memoryless channels, such as the binary symmetric channel (BSC) and the additive Gaussian channel, are characterized by a single parameter. The analysis of these single-parameter memoryless channels is simplified because some channels are known to be degraded with respect to others. For instance, for a memoryless Gaussian channel, we immediately know that channels with a given noise variance are better than any channel with greater variance. Analogously, to simplify the analysis of Markov channels, if the performance of an LDPC code (in terms of probability of symbol error) using some channel is known, that knowledge should immediately imply something about the performance for a “region” of channels neighboring . To that end, in this paper we give a method of recursively constructing general Markov channels, and show that this construction results in a partial ordering in terms of probability of symbol error under iterative LDPC decoding. The results in the present paper are largely a generalization of previous results from [6]. In that paper, three results were given to order GE channels in terms of their iterative decoding 0018-9448/$25.00 © 2007 IEEE