Channel Detection in Coded Communication

We consider the problem of block-coded communication, where in each block, the channel law belongs to one of two disjoint sets. The decoder is aimed to decode only messages that have undergone a channel from one of the sets, and thus has to detect the set which contains the prevailing channel. We begin with the simplified case where each of the sets is a singleton. For any given code, we derive the optimum detection/decoding rule in the sense of the best trade-off among the probabilities of decoding error, false alarm, and misdetection, and also introduce sub-optimal detection/decoding rules which are simpler to implement. Then, various achievable bounds on the error exponents are derived, including the exact single-letter characterization of the random coding exponents for the optimal detector/decoder. We then extend the random coding analysis to general sets of channels, and show that there exists a universal detector/decoder which performs asymptotically as well as the optimal detector/decoder, when tuned to detect a channel from a specific pair of channels. The case of a pair of binary symmetric channels is discussed in detail.Comment: Submitted to IEEE Transactions on Information Theor

Asymptotic Estimates in Information Theory with Non-Vanishing Error Probabilities

This monograph presents a unified treatment of single- and multi-user problems in Shannon's information theory where we depart from the requirement that the error probability decays asymptotically in the blocklength. Instead, the error probabilities for various problems are bounded above by a non-vanishing constant and the spotlight is shone on achievable coding rates as functions of the growing blocklengths. This represents the study of asymptotic estimates with non-vanishing error probabilities. In Part I, after reviewing the fundamentals of information theory, we discuss Strassen's seminal result for binary hypothesis testing where the type-I error probability is non-vanishing and the rate of decay of the type-II error probability with growing number of independent observations is characterized. In Part II, we use this basic hypothesis testing result to develop second- and sometimes, even third-order asymptotic expansions for point-to-point communication. Finally in Part III, we consider network information theory problems for which the second-order asymptotics are known. These problems include some classes of channels with random state, the multiple-encoder distributed lossless source coding (Slepian-Wolf) problem and special cases of the Gaussian interference and multiple-access channels. Finally, we discuss avenues for further research.Comment: Further comments welcom

Low-complexity approaches to distributed data dissemination

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2006.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Includes bibliographical references (p. 145-153).In this thesis we consider practical ways of disseminating information from multiple senders to multiple receivers in an optimal or provably close-to-optimal fashion. The basis for our discussion of optimal transmission of information is mostly information theoretic - but the methods that we apply to do so in a low-complexity fashion draw from a number of different engineering disciplines. The three canonical multiple-input, multiple-output problems we focus our attention upon are: * The Slepian-Wolf problem where multiple correlated sources must be distributedly compressed and recovered with a common receiver. * The discrete memoryless multiple access problem where multiple senders communicate across a common channel to a single receiver. * The deterministic broadcast channel problem where multiple messages are sent from a common sender to multiple receivers through a deterministic medium. Chapter 1 serves as an introduction and provides models, definitions, and a discussion of barriers between theory and practice for the three canonical data dissemination problems we will discuss. Here we also discuss how these three problems are all in different senses 'dual' to each other, and use this as a motivating force to attack them with unifying themes.(cont.) Chapter 2 discusses the Slepian-Wolf problem of distributed near-lossless compression of correlated sources. Here we consider embedding any achievable rate in an M-source problem to a corner point in a 2M - 1-source problem. This allows us to employ practical iterative decoding techniques and achieve rates near the boundary with legitimate empirical performance. Both synthetic data and real correlated data from sensors at the International Space Station are used to successfully test our approach. Chapter 3 generalizes the investigation of practical and provably good decoding algorithms for multiterminal systems to the case where the statistical distribution of the memoryless system is unknown. It has been well-established in the theoretical literature that such 'universal' decoders exist and do not suffer a performance penalty, but their proposed structure is highly nonlinear and therefore believed to be complex. For this reason, most discussion of such decoders has been limited to the realm of ontology and proof of existence. By exploiting recently derived results in other engineering disciplines (i.e. expander graphs, linear programming relaxations, etc), we discuss a code construction and two decoding algorithms that have polynomial complexity and admit provably good performance (exponential error probability decay).(cont.) Because there is no need for a priori statistical knowledge in decoding (which in many settings - for instance a sensor network - might be difficult to repeatedly acquire without significant cost), this approach has very attractive robustness, energy efficiency, and stand-alone practical implications. Finally, Chapter 4 walks away from the multiple-sender, single-receiver setting and steps into the single-sender-multiple receiver setting. We focus our attention here on the deterministic broadcast channel, which is dual to the Slepian-Wolf and multiple access problems in a number of ways - including how the difficulty of practical implementation lies in the encoding rather than decoding. Here we illustrate how again a splitting approach can be applied, and how the same properties from the Slepian-Wolf and multiple access splitting settings remain. We also discuss practical coding strategies for some problems motivated by wireless, and show how by properly 'dualizing' provably good decoding strategies for some channel coding problems, we admit provably good encoding for this setting.by Todd Prentice Coleman.Ph.D

Asymmetric Evaluations of Erasure and Undetected Error Probabilities

The problem of channel coding with the erasure option is revisited for discrete memoryless channels. The interplay between the code rate, the undetected and total error probabilities is characterized. Using the information spectrum method, a sequence of codes of increasing blocklengths $n$ is designed to illustrate this tradeoff. Furthermore, for additive discrete memoryless channels with uniform input distribution, we establish that our analysis is tight with respect to the ensemble average. This is done by analysing the ensemble performance in terms of a tradeoff between the code rate, the undetected and the total errors. This tradeoff is parametrized by the threshold in a generalized likelihood ratio test. Two asymptotic regimes are studied. First, the code rate tends to the capacity of the channel at a rate slower than $n^{-1/2}$ corresponding to the moderate deviations regime. In this case, both error probabilities decay subexponentially and asymmetrically. The precise decay rates are characterized. Second, the code rate tends to capacity at a rate of $n^{-1/2}$. In this case, the total error probability is asymptotically a positive constant while the undetected error probability decays as $\exp(- b n^{ 1/2})$ for some $b>0$. The proof techniques involve applications of a modified (or "shifted") version of the G\"artner-Ellis theorem and the type class enumerator method to characterize the asymptotic behavior of a sequence of cumulant generating functions.Comment: 28 pages, no figures in IEEE Transactions on Information Theory, 201