Discrete-input two-dimensional Gaussian channels with memory: Estimation and information rates via graphical models and statistical mechanics

Ori Shental*, Noam Shental, Shlomo Shamai, Ido Kanter, Anthony J. Weiss, Yair Weiss

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

52 Scopus citations


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.

Original languageEnglish
Pages (from-to)1500-1513
Number of pages14
JournalIEEE Transactions on Information Theory
Issue number4
StatePublished - Apr 2008


FundersFunder number
National Science FoundationCCF-0514859
National Science Foundation
United States-Israel Binational Science Foundation
Israel Science Foundation


    • 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


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