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Journal ArticleDOI

Spectral efficiency in the wideband regime

01 Jun 2002-IEEE Transactions on Information Theory (IEEE)-Vol. 48, Iss: 6, pp 1319-1343
TL;DR: The fundamental bandwidth-power tradeoff of a general class of channels in the wideband regime characterized by low, but nonzero, spectral efficiency and energy per bit close to the minimum value required for reliable communication is found.
Abstract: The tradeoff of spectral efficiency (b/s/Hz) versus energy-per-information bit is the key measure of channel capacity in the wideband power-limited regime. This paper finds the fundamental bandwidth-power tradeoff of a general class of channels in the wideband regime characterized by low, but nonzero, spectral efficiency and energy per bit close to the minimum value required for reliable communication. A new criterion for optimality of signaling in the wideband regime is proposed, which, in contrast to the traditional criterion, is meaningful for finite-bandwidth communication.

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Citations
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Book
01 Jan 2005

9,038 citations

Proceedings Article
01 Jan 2005
TL;DR: This book aims to provide a chronology of key events and individuals involved in the development of microelectronics technology over the past 50 years and some of the individuals involved have been identified and named.
Abstract: Alhussein Abouzeid Rensselaer Polytechnic Institute Raviraj Adve University of Toronto Dharma Agrawal University of Cincinnati Walid Ahmed Tyco M/A-COM Sonia Aissa University of Quebec, INRSEMT Huseyin Arslan University of South Florida Nallanathan Arumugam National University of Singapore Saewoong Bahk Seoul National University Claus Bauer Dolby Laboratories Brahim Bensaou Hong Kong University of Science and Technology Rick Blum Lehigh University Michael Buehrer Virginia Tech Antonio Capone Politecnico di Milano Javier Gómez Castellanos National University of Mexico Claude Castelluccia INRIA Henry Chan The Hong Kong Polytechnic University Ajit Chaturvedi Indian Institute of Technology Kanpur Jyh-Cheng Chen National Tsing Hua University Yong Huat Chew Institute for Infocomm Research Tricia Chigan Michigan Tech Dong-Ho Cho Korea Advanced Institute of Science and Tech. Jinho Choi University of New South Wales Carlos Cordeiro Philips Research USA Laurie Cuthbert Queen Mary University of London Arek Dadej University of South Australia Sajal Das University of Texas at Arlington Franco Davoli DIST University of Genoa Xiaodai Dong, University of Alberta Hassan El-sallabi Helsinki University of Technology Ozgur Ercetin Sabanci University Elza Erkip Polytechnic University Romano Fantacci University of Florence Frank Fitzek Aalborg University Mario Freire University of Beira Interior Vincent Gaudet University of Alberta Jairo Gutierrez University of Auckland Michael Hadjitheodosiou University of Maryland Zhu Han University of Maryland College Park Christian Hartmann Technische Universitat Munchen Hossam Hassanein Queen's University Soong Boon Hee Nanyang Technological University Paul Ho Simon Fraser University Antonio Iera University "Mediterranea" of Reggio Calabria Markku Juntti University of Oulu Stefan Kaiser DoCoMo Euro-Labs Nei Kato Tohoku University Dongkyun Kim Kyungpook National University Ryuji Kohno Yokohama National University Bhaskar Krishnamachari University of Southern California Giridhar Krishnamurthy Indian Institute of Technology Madras Lutz Lampe University of British Columbia Bjorn Landfeldt The University of Sydney Peter Langendoerfer IHP Microelectronics Technologies Eddie Law Ryerson University in Toronto

7,826 citations

Journal ArticleDOI
TL;DR: Under certain mild conditions, this scheme is found to be throughput-wise asymptotically optimal for both high and low signal-to-noise ratio (SNR), and some numerical results are provided for the ergodic throughput of the simplified zero-forcing scheme in independent Rayleigh fading.
Abstract: A Gaussian broadcast channel (GBC) with r single-antenna receivers and t antennas at the transmitter is considered. Both transmitter and receivers have perfect knowledge of the channel. Despite its apparent simplicity, this model is, in general, a nondegraded broadcast channel (BC), for which the capacity region is not fully known. For the two-user case, we find a special case of Marton's (1979) region that achieves optimal sum-rate (throughput). In brief, the transmitter decomposes the channel into two interference channels, where interference is caused by the other user signal. Users are successively encoded, such that encoding of the second user is based on the noncausal knowledge of the interference caused by the first user. The crosstalk parameters are optimized such that the overall throughput is maximum and, surprisingly, this is shown to be optimal over all possible strategies (not only with respect to Marton's achievable region). For the case of r>2 users, we find a somewhat simpler choice of Marton's region based on ordering and successively encoding the users. For each user i in the given ordering, the interference caused by users j>i is eliminated by zero forcing at the transmitter, while interference caused by users j

2,616 citations


Cites methods from "Spectral efficiency in the wideband..."

  • ...Following [49], we define for the GBC as...

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Journal ArticleDOI
TL;DR: An overview of the extensive results on the Shannon capacity of single-user and multiuser multiple-input multiple-output (MIMO) channels is provided and it is shown that the capacity region of the MIMO multiple access and the largest known achievable rate region (called the dirty-paper region) for the M IMO broadcast channel are intimately related via a duality transformation.
Abstract: We provide an overview of the extensive results on the Shannon capacity of single-user and multiuser multiple-input multiple-output (MIMO) channels. Although enormous capacity gains have been predicted for such channels, these predictions are based on somewhat unrealistic assumptions about the underlying time-varying channel model and how well it can be tracked at the receiver, as well as at the transmitter. More realistic assumptions can dramatically impact the potential capacity gains of MIMO techniques. For time-varying MIMO channels there are multiple Shannon theoretic capacity definitions and, for each definition, different correlation models and channel information assumptions that we consider. We first provide a comprehensive summary of ergodic and capacity versus outage results for single-user MIMO channels. These results indicate that the capacity gain obtained from multiple antennas heavily depends on the available channel information at either the receiver or transmitter, the channel signal-to-noise ratio, and the correlation between the channel gains on each antenna element. We then focus attention on the capacity region of the multiple-access channels (MACs) and the largest known achievable rate region for the broadcast channel. In contrast to single-user MIMO channels, capacity results for these multiuser MIMO channels are quite difficult to obtain, even for constant channels. We summarize results for the MIMO broadcast and MAC for channels that are either constant or fading with perfect instantaneous knowledge of the antenna gains at both transmitter(s) and receiver(s). We show that the capacity region of the MIMO multiple access and the largest known achievable rate region (called the dirty-paper region) for the MIMO broadcast channel are intimately related via a duality transformation. This transformation facilitates finding the transmission strategies that achieve a point on the boundary of the MIMO MAC capacity region in terms of the transmission strategies of the MIMO broadcast dirty-paper region and vice-versa. Finally, we discuss capacity results for multicell MIMO channels with base station cooperation. The base stations then act as a spatially diverse antenna array and transmission strategies that exploit this structure exhibit significant capacity gains. This section also provides a brief discussion of system level issues associated with MIMO cellular. Open problems in this field abound and are discussed throughout the paper.

2,480 citations


Cites background from "Spectral efficiency in the wideband..."

  • ...Matrix channels describe not only multiantenna systems but also channels with crosstalk [85] and wideband channels [72]....

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Book
28 Jun 2004
TL;DR: A tutorial on random matrices is provided which provides an overview of the theory and brings together in one source the most significant results recently obtained.
Abstract: Random matrix theory has found many applications in physics, statistics and engineering since its inception. Although early developments were motivated by practical experimental problems, random matrices are now used in fields as diverse as Riemann hypothesis, stochastic differential equations, condensed matter physics, statistical physics, chaotic systems, numerical linear algebra, neural networks, multivariate statistics, information theory, signal processing and small-world networks. This article provides a tutorial on random matrices which provides an overview of the theory and brings together in one source the most significant results recently obtained. Furthermore, the application of random matrix theory to the fundamental limits of wireless communication channels is described in depth.

2,308 citations


Cites methods from "Spectral efficiency in the wideband..."

  • ...Asymptotic spectrum results have also been used in [161] to characterize the wideband capacity of correlated multi-antenna channel using the tools of [274]....

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References
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Journal ArticleDOI
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.

65,425 citations

Book
01 Jan 1991
TL;DR: The author examines the role of entropy, inequality, and randomness in the design of codes and the construction of codes in the rapidly changing environment.
Abstract: Preface to the Second Edition. Preface to the First Edition. Acknowledgments for the Second Edition. Acknowledgments for the First Edition. 1. Introduction and Preview. 1.1 Preview of the Book. 2. Entropy, Relative Entropy, and Mutual Information. 2.1 Entropy. 2.2 Joint Entropy and Conditional Entropy. 2.3 Relative Entropy and Mutual Information. 2.4 Relationship Between Entropy and Mutual Information. 2.5 Chain Rules for Entropy, Relative Entropy, and Mutual Information. 2.6 Jensen's Inequality and Its Consequences. 2.7 Log Sum Inequality and Its Applications. 2.8 Data-Processing Inequality. 2.9 Sufficient Statistics. 2.10 Fano's Inequality. Summary. Problems. Historical Notes. 3. Asymptotic Equipartition Property. 3.1 Asymptotic Equipartition Property Theorem. 3.2 Consequences of the AEP: Data Compression. 3.3 High-Probability Sets and the Typical Set. Summary. Problems. Historical Notes. 4. Entropy Rates of a Stochastic Process. 4.1 Markov Chains. 4.2 Entropy Rate. 4.3 Example: Entropy Rate of a Random Walk on a Weighted Graph. 4.4 Second Law of Thermodynamics. 4.5 Functions of Markov Chains. Summary. Problems. Historical Notes. 5. Data Compression. 5.1 Examples of Codes. 5.2 Kraft Inequality. 5.3 Optimal Codes. 5.4 Bounds on the Optimal Code Length. 5.5 Kraft Inequality for Uniquely Decodable Codes. 5.6 Huffman Codes. 5.7 Some Comments on Huffman Codes. 5.8 Optimality of Huffman Codes. 5.9 Shannon-Fano-Elias Coding. 5.10 Competitive Optimality of the Shannon Code. 5.11 Generation of Discrete Distributions from Fair Coins. Summary. Problems. Historical Notes. 6. Gambling and Data Compression. 6.1 The Horse Race. 6.2 Gambling and Side Information. 6.3 Dependent Horse Races and Entropy Rate. 6.4 The Entropy of English. 6.5 Data Compression and Gambling. 6.6 Gambling Estimate of the Entropy of English. Summary. Problems. Historical Notes. 7. Channel Capacity. 7.1 Examples of Channel Capacity. 7.2 Symmetric Channels. 7.3 Properties of Channel Capacity. 7.4 Preview of the Channel Coding Theorem. 7.5 Definitions. 7.6 Jointly Typical Sequences. 7.7 Channel Coding Theorem. 7.8 Zero-Error Codes. 7.9 Fano's Inequality and the Converse to the Coding Theorem. 7.10 Equality in the Converse to the Channel Coding Theorem. 7.11 Hamming Codes. 7.12 Feedback Capacity. 7.13 Source-Channel Separation Theorem. Summary. Problems. Historical Notes. 8. Differential Entropy. 8.1 Definitions. 8.2 AEP for Continuous Random Variables. 8.3 Relation of Differential Entropy to Discrete Entropy. 8.4 Joint and Conditional Differential Entropy. 8.5 Relative Entropy and Mutual Information. 8.6 Properties of Differential Entropy, Relative Entropy, and Mutual Information. Summary. Problems. Historical Notes. 9. Gaussian Channel. 9.1 Gaussian Channel: Definitions. 9.2 Converse to the Coding Theorem for Gaussian Channels. 9.3 Bandlimited Channels. 9.4 Parallel Gaussian Channels. 9.5 Channels with Colored Gaussian Noise. 9.6 Gaussian Channels with Feedback. Summary. Problems. Historical Notes. 10. Rate Distortion Theory. 10.1 Quantization. 10.2 Definitions. 10.3 Calculation of the Rate Distortion Function. 10.4 Converse to the Rate Distortion Theorem. 10.5 Achievability of the Rate Distortion Function. 10.6 Strongly Typical Sequences and Rate Distortion. 10.7 Characterization of the Rate Distortion Function. 10.8 Computation of Channel Capacity and the Rate Distortion Function. Summary. Problems. Historical Notes. 11. Information Theory and Statistics. 11.1 Method of Types. 11.2 Law of Large Numbers. 11.3 Universal Source Coding. 11.4 Large Deviation Theory. 11.5 Examples of Sanov's Theorem. 11.6 Conditional Limit Theorem. 11.7 Hypothesis Testing. 11.8 Chernoff-Stein Lemma. 11.9 Chernoff Information. 11.10 Fisher Information and the Cram-er-Rao Inequality. Summary. Problems. Historical Notes. 12. Maximum Entropy. 12.1 Maximum Entropy Distributions. 12.2 Examples. 12.3 Anomalous Maximum Entropy Problem. 12.4 Spectrum Estimation. 12.5 Entropy Rates of a Gaussian Process. 12.6 Burg's Maximum Entropy Theorem. Summary. Problems. Historical Notes. 13. Universal Source Coding. 13.1 Universal Codes and Channel Capacity. 13.2 Universal Coding for Binary Sequences. 13.3 Arithmetic Coding. 13.4 Lempel-Ziv Coding. 13.5 Optimality of Lempel-Ziv Algorithms. Compression. Summary. Problems. Historical Notes. 14. Kolmogorov Complexity. 14.1 Models of Computation. 14.2 Kolmogorov Complexity: Definitions and Examples. 14.3 Kolmogorov Complexity and Entropy. 14.4 Kolmogorov Complexity of Integers. 14.5 Algorithmically Random and Incompressible Sequences. 14.6 Universal Probability. 14.7 Kolmogorov complexity. 14.9 Universal Gambling. 14.10 Occam's Razor. 14.11 Kolmogorov Complexity and Universal Probability. 14.12 Kolmogorov Sufficient Statistic. 14.13 Minimum Description Length Principle. Summary. Problems. Historical Notes. 15. Network Information Theory. 15.1 Gaussian Multiple-User Channels. 15.2 Jointly Typical Sequences. 15.3 Multiple-Access Channel. 15.4 Encoding of Correlated Sources. 15.5 Duality Between Slepian-Wolf Encoding and Multiple-Access Channels. 15.6 Broadcast Channel. 15.7 Relay Channel. 15.8 Source Coding with Side Information. 15.9 Rate Distortion with Side Information. 15.10 General Multiterminal Networks. Summary. Problems. Historical Notes. 16. Information Theory and Portfolio Theory. 16.1 The Stock Market: Some Definitions. 16.2 Kuhn-Tucker Characterization of the Log-Optimal Portfolio. 16.3 Asymptotic Optimality of the Log-Optimal Portfolio. 16.4 Side Information and the Growth Rate. 16.5 Investment in Stationary Markets. 16.6 Competitive Optimality of the Log-Optimal Portfolio. 16.7 Universal Portfolios. 16.8 Shannon-McMillan-Breiman Theorem (General AEP). Summary. Problems. Historical Notes. 17. Inequalities in Information Theory. 17.1 Basic Inequalities of Information Theory. 17.2 Differential Entropy. 17.3 Bounds on Entropy and Relative Entropy. 17.4 Inequalities for Types. 17.5 Combinatorial Bounds on Entropy. 17.6 Entropy Rates of Subsets. 17.7 Entropy and Fisher Information. 17.8 Entropy Power Inequality and Brunn-Minkowski Inequality. 17.9 Inequalities for Determinants. 17.10 Inequalities for Ratios of Determinants. Summary. Problems. Historical Notes. Bibliography. List of Symbols. Index.

45,034 citations

Journal ArticleDOI
Gerard J. Foschini1
TL;DR: This paper addresses digital communication in a Rayleigh fading environment when the channel characteristic is unknown at the transmitter but is known (tracked) at the receiver with the aim of leveraging the already highly developed 1-D codec technology.
Abstract: This paper addresses digital communication in a Rayleigh fading environment when the channel characteristic is unknown at the transmitter but is known (tracked) at the receiver. Inventing a codec architecture that can realize a significant portion of the great capacity promised by information theory is essential to a standout long-term position in highly competitive arenas like fixed and indoor wireless. Use (n T , n R ) to express the number of antenna elements at the transmitter and receiver. An (n, n) analysis shows that despite the n received waves interfering randomly, capacity grows linearly with n and is enormous. With n = 8 at 1% outage and 21-dB average SNR at each receiving element, 42 b/s/Hz is achieved. The capacity is more than 40 times that of a (1, 1) system at the same total radiated transmitter power and bandwidth. Moreover, in some applications, n could be much larger than 8. In striving for significant fractions of such huge capacities, the question arises: Can one construct an (n, n) system whose capacity scales linearly with n, using as building blocks n separately coded one-dimensional (1-D) subsystems of equal capacity? With the aim of leveraging the already highly developed 1-D codec technology, this paper reports just such an invention. In this new architecture, signals are layered in space and time as suggested by a tight capacity bound.

6,812 citations


"Spectral efficiency in the wideband..." refers background in this paper

  • ...The low- slope (215) is greater than or equal to the highslope, [13], with equality if and only if ....

    [...]

  • ...Recent results on the slope b/s/Hz/(3 dB) of various fading channels in the region of high spectral efficiency can be found in [13], [15], and [16]....

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  • ...Proof: In this case, the optimum input distribution is an -dimensional Gaussian vector with independent and identically distributed components achieving capacity [33], [13] (209) Formula (208) follows from (140) upon taking the first and second derivatives of (209)....

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  • ..., [13]), the whole -dimensional vector occupies 1 s hertz, in which case, the units of are b/s/Hz/(antenna element)....

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Journal ArticleDOI
Claude E. Shannon1
01 Jan 1949
TL;DR: A method is developed for representing any communication system geometrically and a number of results in communication theory are deduced concerning expansion and compression of bandwidth and the threshold effect.
Abstract: A method is developed for representing any communication system geometrically Messages and the corresponding signals are points in two "function spaces," and the modulation process is a mapping of one space into the other Using this representation, a number of results in communication theory are deduced concerning expansion and compression of bandwidth and the threshold effect Formulas are found for the maximum rate of transmission of binary digits over a system when the signal is perturbed by various types of noise Some of the properties of "ideal" systems which transmit at this maximum rate are discussed The equivalent number of binary digits per second for certain information sources is calculated

6,712 citations


"Spectral efficiency in the wideband..." refers background in this paper

  • ...INTRODUCTION SHORTLY after “A Mathematical Theory of Communication,” Claude Shannon [1] pointed out that as the bandwidth tends to infinity, the channel capacity of an ideal bandlimited additive white Gaussian noise (AWGN) channel approaches...

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Book
01 Jan 2004
TL;DR: The book gives many numerical illustrations expressed in large collections of system performance curves, allowing the researchers or system designers to perform trade-off studies of the average bit error rate and symbol error rate.
Abstract: noncoherent communication systems, as well as a large variety of fading channel models typical of communication links often found in the real world, including single- and multichannel reception with a large variety of types. The book gives many numerical illustrations expressed in large collections of system performance curves, allowing the researchers or system designers to perform trade-off studies of the average bit error rate and symbol error rate. This book is a very good reference book for researchers and communication engineers and may also be a source for supplementary material of a graduate course on communication or signal processing. Nowadays, many new books attach a CD-ROM for more supplementary material. With the many numerical examples in this book, it appears that an attached CD-ROM would be ideal for this book. It would be even better to present the computer program in order to be interactive so that the readers can plug in their arbitrary parameters for the performance evaluation. —H. Hsu

6,469 citations


"Spectral efficiency in the wideband..." refers background in this paper

  • ...10The “amount of fading” defined in [30] is equal to the kurtosis minus 1....

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  • ...See [30], [15] for tables of standard fading distributions....

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