A mathematical theory of communication
TL;DR: This final installment of the paper considers the case where the signals or the messages or both are continuously variable, in contrast with the discrete nature assumed until now.
Abstract: In this final installment of the paper we consider the case where the signals or the messages or both are continuously variable, in contrast with the discrete nature assumed until now. To a considerable extent the continuous case can be obtained through a limiting process from the discrete case by dividing the continuum of messages and signals into a large but finite number of small regions and calculating the various parameters involved on a discrete basis. As the size of the regions is decreased these parameters in general approach as limits the proper values for the continuous case. There are, however, a few new effects that appear and also a general change of emphasis in the direction of specialization of the general results to particular cases.
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29 Jun 1997
TL;DR: It is proved that sequences of codes exist which, when optimally decoded, achieve information rates up to the Shannon limit, and experimental results for binary-symmetric channels and Gaussian channels demonstrate that practical performance substantially better than that of standard convolutional and concatenated codes can be achieved.
Abstract: We study two families of error-correcting codes defined in terms of very sparse matrices "MN" (MacKay-Neal (1995)) codes are recently invented, and "Gallager codes" were first investigated in 1962, but appear to have been largely forgotten, in spite of their excellent properties The decoding of both codes can be tackled with a practical sum-product algorithm We prove that these codes are "very good", in that sequences of codes exist which, when optimally decoded, achieve information rates up to the Shannon limit This result holds not only for the binary-symmetric channel but also for any channel with symmetric stationary ergodic noise We give experimental results for binary-symmetric channels and Gaussian channels demonstrating that practical performance substantially better than that of standard convolutional and concatenated codes can be achieved; indeed, the performance of Gallager codes is almost as close to the Shannon limit as that of turbo codes
3,842 citations
TL;DR: The standard similarity measure based on untransformed indices is shown to give misleading results, but transforming the indices or entropies to effective numbers of species produces a stable, easily interpreted, sensitive general similarity measure.
Abstract: Entropies such as the Shannon–Wiener and Gini–Simpson indices are not themselves diversities. Conversion of these to effective number of species is the key to a unified and intuitive interpretation of diversity. Effective numbers of species derived from standard diversity indices share a common set of intuitive mathematical properties and behave as one would expect of a diversity, while raw indices do not. Contrary to Keylock, the lack of concavity of effective numbers of species is irrelevant as long as they are used as transformations of concave alpha, beta, and gamma entropies. The practical importance of this transformation is demonstrated by applying it to a popular community similarity measure based on raw diversity indices or entropies. The standard similarity measure based on untransformed indices is shown to give misleading results, but transforming the indices or entropies to effective numbers of species produces a stable, easily interpreted, sensitive general similarity measure. General overlap measures derived from this transformed similarity measure yield the Jaccard index, Sorensen index, Horn index of overlap, and the Morisita–Horn index as special cases.
3,677 citations
TL;DR: The paper proves that, given any B-DMC W with I(W) > 0 and any target rate R< I( W) there exists a sequence of polar codes {Cfrn;nges1} such that Cfrn has block-length N=2n, rate ges R, and probability of block error under successive cancellation decoding bounded as Pe(N,R) les O(N-1/4) independently of the code rate.
Abstract: A method is proposed, called channel polarization, to construct code sequences that achieve the symmetric capacity I(W) of any given binary-input discrete memoryless channel (B-DMC) W. The symmetric capacity is the highest rate achievable subject to using the input letters of the channel with equal probability. Channel polarization refers to the fact that it is possible to synthesize, out of N independent copies of a given B-DMC W, a second set of N binary-input channels {WN(i)1 les i les N} such that, as N becomes large, the fraction of indices i for which I(WN(i)) is near 1 approaches I(W) and the fraction for which I(WN(i)) is near 0 approaches 1-I(W). The polarized channels {WN(i)} are well-conditioned for channel coding: one need only send data at rate 1 through those with capacity near 1 and at rate 0 through the remaining. Codes constructed on the basis of this idea are called polar codes. The paper proves that, given any B-DMC W with I(W) > 0 and any target rate R< I(W) there exists a sequence of polar codes {Cfrn;nges1} such that Cfrn has block-length N=2n , rate ges R, and probability of block error under successive cancellation decoding bounded as Pe(N,R) les O(N-1/4) independently of the code rate. This performance is achievable by encoders and decoders with complexity O(N logN) for each.
3,554 citations
TL;DR: A minimum-redundancy code is one constructed in such a way that the average number of coding digits per message is minimized.
Abstract: An optimum method of coding an ensemble of messages consisting of a finite number of members is developed. A minimum-redundancy code is one constructed in such a way that the average number of coding digits per message is minimized.
3,528 citations
TL;DR: This survey characterizes an emerging research area, sometimes called coordination theory, that focuses on the interdisciplinary study of coordination, that uses and extends ideas about coordination from disciplines such as computer science, organization theory, operations research, economics, linguistics, and psychology.
Abstract: This survey characterizes an emerging research area, sometimes called coordination theory, that focuses on the interdisciplinary study of coordination. Research in this area uses and extends ideas about coordination from disciplines such as computer science, organization theory, operations research, economics, linguistics, and psychology.A key insight of the framework presented here is that coordination can be seen as the process of managing dependencies among activities. Further progress, therefore, should be possible by characterizing different kinds of dependencies and identifying the coordination processes that can be used to manage them. A variety of processes are analyzed from this perspective, and commonalities across disciplines are identified. Processes analyzed include those for managing shared resources, producer/consumer relationships, simultaneity constraints, and task/subtask dependencies.Section 3 summarizes ways of applying a coordination perspective in three different domains:(1) understanding the effects of information technology on human organizations and markets, (2) designing cooperative work tools, and (3) designing distributed and parallel computer systems. In the final section, elements of a research agenda in this new area are briefly outlined.
3,447 citations
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TL;DR: In this article, a quantitative measure of information is developed which is based on physical as contrasted with psychological considerations, and how the rate of transmission of this information over a system is limited by the distortion resulting from storage of energy is discussed from the transient viewpoint.
Abstract: A quantitative measure of “information” is developed which is based on physical as contrasted with psychological considerations. How the rate of transmission of this information over a system is limited by the distortion resulting from storage of energy is discussed from the transient viewpoint. The relation between the transient and steady state viewpoints is reviewed. It is shown that when the storage of energy is used to restrict the steady state transmission to a limited range of frequencies the amount of information that can be transmitted is proportional to the product of the width of the frequency-range by the time it is available. Several illustrations of the application of this principle to practical systems are included. In the case of picture transmission and television the spacial variation of intensity is analyzed by a steady state method analogous to that commonly used for variations with time.
1,741 citations
TL;DR: In this paper, it was shown that the wave shape depends somewhat on the type of circuit over which intelligence is to be transmitted and that for most cases the optimum wave is neither rectangular nor a half cycle sine wave as is frequently used but a wave of special form produced by sending a simple rectangular wave through a suitable network.
Abstract: This paper considers two fundamental factors entering into the maximum speed of transmission of intelligence by telegraph. These factors are signal shaping and choice of codes. The first is concerned with the best wave shape to be impressed on the transmitting medium so as to permit of greater speed without undue interference either in the circuit under consideration or in those adjacent, while the latter deals with the choice of codes which will permit of transmitting a maximum amount of intelligence with a given number of signal elements. It is shown that the wave shape depends somewhat on the type of circuit over which intelligence is to be transmitted and that for most cases the optimum wave is neither rectangular nor a half cycle sine wave as is frequently used but a wave of special form produced by sending a simple rectangular wave through a suitable network. The impedances usually associated with telegraph circuits are such as to produce a fair degree of signal shaping when a rectangular voltage wave is impressed. Consideration of the choice of codes show that while it is desirable to use those involving more than two current values, there are limitations which prevent a large number of current values being used. A table of comparisons shows the relative speed efficiencies of various codes proposed. It is shown that no advantages result from the use of a sine wave for telegraph transmission as proposed by Squier and others2 and that their arguments are based on erroneous assumptions.
459 citations