Discrete-input two-dimensional (2-D) Gaussian channels with memory represent an important class of systems, which appears extensively in communications and storage. In spite of their widespread use, the workings of 2-D channels are still very much unknown. In this work, we try to explore their properties from the perspective of estimation theory and information theory. At the heart of our approach is a mapping of a 2-D channel to an undirected graphical model, and inferring its a posteriori probabilities (APPs) using generalized belief propagation GBP). The derived probabilities are shown to be practically accurate, thus enabling optimal maximum a posteriori (MAP) estimation of the transmitted symbols. Also, the Shannon-theoretic information rates are deduced either via the vector-wise Shannon-McMillan-Breiman (SMB) theorem, or via the recently derived symbol-wise Guo-Shamai-Verdú (GSV) theorem. Our approach is also described from the perspective of statistical mechanics, as the graphical model and inference algorithm have their analogues in physics. Our experimental study, based on common channel settings taken from cellular networks and magnetic recording devices, demonstrates that under nontrivial memory conditions, the performance of this fully tractable GBP estimator is almost identical to the performance of the optimal MAP estimator. It also enables practically accurate simulation-based estimate of the information rate. Rationalization of this excellent performance of GBP in 2-D Gaussian channel setting is addressed.
Bibliographical noteFunding Information:
Manuscript received September 18, 2006; revised September 19, 2007. The work of O. Shental was supported in part by the National Science Foundation under Grant CCF-0514859. The work of S. Shamai was supported by the U.S.-Israel, Binational Science Foundation. The work of I. Kanter was supported in part by the Israel Science Foundation. The first two authors were the lead authors for this paper. The material in this paper was presented in part at the IEEE Information Theory Workshop (ITW), San Antonio, TX, October 2004, and at the IEEE International Symposium on Information Theory (ISIT), Adelaide, Australia, September 2005.
- Cluster variation method
- Generalized belief propagation (GBP)
- Guo-Shamai-Verdú (GSV) theorem
- Information rate
- Intersymbol interference (ISI)
- Magnetic recording channels
- Maximum a posteriori (MAP) estimation
- Multiple-access (MA) channels
- Shannon-McMillan-Breiman (SMB) theorem
- Two-dimensional (2-D) channels