2206 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 4, APRIL 2010 Markov Chain Monte Carlo Detectors for Channels With Intersymbol Interference Rong-Hui Peng, Rong-Rong Chen, Member, IEEE, and Behrouz Farhang-Boroujeny, Senior Member, IEEE Abstract—In this paper, we propose novel low-complexity soft-in soft-out (SISO) equalizers using the Markov chain Monte Carlo (MCMC) technique. We develop a bitwise MCMC equal- izer (b-MCMC) that adopts a Gibbs sampler to update one bit at a time, as well as a group-wise MCMC (g-MCMC) equal- izer where multiple symbols are updated simultaneously. The g-MCMC equalizer is shown to outperform both the b-MCMC and the linear minimum mean square error (MMSE) equalizer significantly for channels with severe amplitude distortion. Direct application of MCMC to channel equalization requires sequential processing which leads to long processing delay. We develop a parallel processing algorithm that reduces the processing delay by orders of magnitude. Numerical results show that both the sequential and parallel processing MCMC equalizers perform similarly well and achieve a performance that is only slightly worse than the optimum maximum a posteriori (MAP) equalizer. The MAP equalizer, on the other hand, has a complexity that grows exponentially with the size of the memory of the channel, while the complexity of the proposed MCMC equalizers grows linearly. Index Terms—Equalization, intersymbol interference, Markov chain Monte Carlo, soft-in soft-out detection. I. INTRODUCTION T HE increasing demand for high speed wireless products has motivated a significant amount of research to combat the intersymbol interference (ISI) resulting from multipath transmission. Early developments date back to the 1960s and 1970s when symbol-by-symbol linear and decision feedback equalizers were developed [1], [2]. These traditional methods face the problem of noise enhancement, in the case of linear equalizers, or error propagation, in the case of decision feed- back equalizers. Furthermore, these methods are based on hard decisions and thus cannot benefit from the modern coding techniques where by making use of soft information one can approach the channel capacity. Significant improvement in system performance can be achieved by joint processing of equalization and channel decoding. Such systems operate based on turbo principles where soft information are exchanged between a soft-in soft-out (SISO) equalizer and a channel decoder. The optimal maximum a posteriori (MAP) equalizer can be used to find the best bit/symbol stream that matches the Manuscript received June 30, 2009; accepted November 14, 2009. First pub- lished December 18, 2009; current version published March 10, 2010. The as- sociate editor coordinating the review of this manuscript and approving it for publication was Dr. Milica Stojanovic. This work is supported in part by the NSF under Grants ECS-0547433 and ECS-0524720. The material in this paper was presented in part at the IEEE International Conference on Communications (ICC), Dresden, Germany, June 14–19, 2009. The authors are with the Department of Electrical and Computer Engineering, University of Utah, Salt Lake City, UT 84112 USA (e-mail: peng@ece.utah.edu; rchen@ece.utah.edu; farhang@ece.utah.edu). Digital Object Identifier 10.1109/TSP.2009.2038958 received signal in the presence of prior information provided by the channel decoder. However, the computational complexity of the MAP equalizer, even with the use of efficient imple- mentations such as the BCJR algorithm [3] is exponential with respect to the length of the channel impulse response and the constellation size and thus may be prohibitive in many cases. To resolve this problem, low complexity equalizers have been developed and remain an active area of research. The first work to reduce the complexity of the MAP equalizer for a turbo equaliza- tion system is due to [4], where the soft-output Viterbi algorithm (SOVA) is used for the implementation of the SISO equalizer. In [5], a low-complexity SISO equalizer is implemented with an adaptive soft interference canceler based on linear filters. A SISO turbo equalizer based on the minimum mean square error (MMSE) criteria is proposed [6] and [7]. This turbo MMSE equalizer performs better than the SIC approach of [5] and is widely used due to its excellent performance. Trellis-based approaches that prune the insignificant branches of trellis to reduce the complexity of the BCJR algorithm have also been suc- cessfully developed. Examples of such algorithms are breadth- first algorithms such as the -best BCJR (M-BCJR) and the threshold-based BCJR (T-BCJR) of [8], and the depth-first algo- rithm such as the list-sequential (LISS) algorithm of [9] and [10]. In this paper, we develop a novel low complexity approxima- tion to the MAP equalizer based on the Markov chain Monte Carlo (MCMC) simulation principles [11]. The MCMC simu- lation is a mathematical tool that may be used to draw samples from an arbitrary and possibly unknown distribution. The key point that makes the MCMC attractive to SISO equalization is the fact that, unlike the BCJR algorithm, its computational com- plexity does not grow exponentially with the channel memory. The BCJR algorithm uses a complete sample set of trellis states to achieve optimal detection, while the other trellis-based ap- proaches [8] trade the use of an incomplete set of trellis states for suboptimal performance. As noted in [8], to ensure good detection performance, the trellis states chosen should be those that have significant contribution to the soft information that one seeks. The SISO MCMC equalizer proposed in this paper fol- lows a similar strategy. It uses an statistical method to search for a small (to keep the complexity low) but important (to achieve good performance) sample set containing important samples that match the best with received signals. The contributions of this work are summarized as follows: 1) We develop new MCMC detectors for ISI channels that can achieve near optimal MAP performance at low com- plexity. Such detectors operate either bit-wise (b-MCMC), or group-wise (g-MCMC) over groups of symbols. Pre- vious work in the literature considers only b-MCMC [12], or symbol-wise MCMC (s-MCMC) [13], and the latter is 1053-587X/$26.00 © 2010 IEEE Authorized licensed use limited to: The University of Utah. Downloaded on March 13,2010 at 10:06:35 EST from IEEE Xplore. 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