Information_Theory_and_Network_Coding__Information_Technology__Transmission__Processing_and_Storage_

Information_Theory_and_Network_Coding__Information_Technology__Transmission__Processing_and_Storage_

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Unformatted text preview: Raymond W. Yeung Information Theory and Network Coding SPIN Springer's internal project number, if known May 31, 2008 Springer To my parents and my family Preface This book is an evolution from my book A First Course in Information Theory published in 2002 when network coding was still at its infancy. The last few years have witnessed the rapid development of network coding into a research field of its own in information science. With its root in information theory, network coding not only has brought about a paradigm shift in network communications at large, but also has had significant influence on such specific research fields as coding theory, networking, switching, wireless communications, distributed data storage, cryptography, and optimization theory. While new applications of network coding keep emerging, the fundamental results that lay the foundation of the subject are more or less mature. One of the main goals of this book therefore is to present these results in a unifying and coherent manner. While the previous book focused only on information theory for discrete random variables, the current book contains two new chapters on information theory for continuous random variables, namely the chapter on differential entropy and the chapter on continuous-valued channels. With these topics included, the book becomes more comprehensive and is more suitable to be used as a textbook for a course in an electrical engineering department. What is in this book Out of the twenty-one chapters in this book, the first sixteen chapters belong to Part I, Components of Information Theory, and the last five chapters belong to Part II, Fundamentals of Network Coding. Part I covers the basic topics in information theory and prepare the reader for the discussions in Part II. A brief rundown of the chapters will give a better idea of what is in this book. Chapter 1 contains a high level introduction to the contents of this book. First, there is a discussion on the nature of information theory and the main results in Shannon's original paper in 1948 which founded the field. There are also pointers to Shannon's biographies and his works. VIII Preface Chapter 2 introduces Shannon's information measures for discrete random variables and their basic properties. Useful identities and inequalities in information theory are derived and explained. Extra care is taken in handling joint distributions with zero probability masses. There is a section devoted to the discussion of maximum entropy distributions. The chapter ends with a section on the entropy rate of a stationary information source. Chapter 3 is an introduction to the theory of I-Measure which establishes a one-to-one correspondence between Shannon's information measures and set theory. A number of examples are given to show how the use of information diagrams can simplify the proofs of many results in information theory. Such diagrams are becoming standard tools for solving information theory problems. Chapter 4 is a discussion of zero-error data compression by uniquely decodable codes, with prefix codes as a special case. A proof of the entropy bound for prefix codes which involves neither the Kraft inequality nor the fundamental inequality is given. This proof facilitates the discussion of the redundancy of prefix codes. Chapter 5 is a thorough treatment of weak typicality. The weak asymptotic equipartition property and the source coding theorem are discussed. An explanation of the fact that a good data compression scheme produces almost i.i.d. bits is given. There is also an introductory discussion of the ShannonMcMillan-Breiman theorem. The concept of weak typicality will be further developed in Chapter 10 for continuous random variables. Chapter 6 contains a detailed discussion of strong typicality which applies to random variables with finite alphabets. The results developed in this chapter will be used for proving the channel coding theorem and the rate-distortion theorem in the next two chapters. The discussion in Chapter 7 of the discrete memoryless channel is an enhancement of the discussion in the previous book. In particular, the new definition of the discrete memoryless channel enables rigorous formulation and analysis of coding schemes for such channels with or without feedback. The proof of the channel coding theorem uses a graphical model approach that helps explain the conditional independence of the random variables. Chapter 8 is an introduction to rate-distortion theory. The version of the rate-distortion theorem here, proved by using strong typicality, is a stronger version of the original theorem obtained by Shannon. In Chapter 9, the Blahut-Arimoto algorithms for computing the channel capacity and the rate-distortion function are discussed, and a simplified proof for convergence is given. Great care is taken in handling distributions with zero probability masses. Chapter 10 and Chapter 11 are the two chapters devoted to the discussion of information theory for continuous random variables. Chapter 10 introduces differential entropy and related information measures, and their basic properties are discussed. The asymptotic equipartion property for continuous random variables is proved. The last section on maximum differential entropy Preface IX distributions echos the section in Chapter 2 on maximum entropy distributions. Chapter 11 discusses a variety of continuous-valued channels, with the continuous memoryless channel being the basic building block. In proving the capacity of the memoryless Gaussian channel, a careful justification is given for the existence of the differential entropy of the output random variable. Based on this result, the capacity of a system of parallel/correlated Gaussian channels is obtained. Heuristic arguments leading to the formula for the capacity of the bandlimited white/colored Gaussian channel are given. The chapter ends with a proof of the fact that zero-mean Gaussian noise is the worst additive noise. Chapter 12 explores the structure of the I-Measure for Markov structures. Set-theoretic characterizations of full conditional independence and Markov random field are discussed. The treatment of Markov random field here maybe too specialized for the average reader, but the structure of the I-Measure and the simplicity of the information diagram for a Markov chain is best explained as a special case of a Markov random field. Information inequalities are sometimes called the laws of information theory because they govern the impossibilities in information theory. In Chapter 13, the geometrical meaning of information inequalities and the relation between information inequalities and conditional independence are explained in depth. The framework for information inequalities discussed here is the basis of the next two chapters. Chapter 14 explains how the problem of proving information inequalities can be formulated as a linear programming problem. This leads to a complete characterization of all information inequalities provable by conventional techniques. These inequalities, called Shannon-type inequalities, can be proved by the World Wide Web available software package ITIP. It is also shown how Shannon-type inequalities can be used to tackle the implication problem of conditional independence in probability theory. Shannon-type inequalities are all the information inequalities known during the first half century of information theory. In the late 1990's, a few new inequalities, called non-Shannon-type inequalities, were discovered. These inequalities imply the existence of laws in information theory beyond those laid down by Shannon. In Chapter 15, we discuss these inequalities and their applications. Chapter 16 explains an intriguing relation between information theory and group theory. Specifically, for every information inequality satisfied by any joint probability distribution, there is a corresponding group inequality satisfied by any finite group and its subgroups, and vice versa. Inequalities of the latter type govern the orders of any finite group and their subgroups. Group-theoretic proofs of Shannon-type information inequalities are given. At the end of the chapter, a group inequality is obtained from a non-Shannontype inequality discussed in Chapter 15. The meaning and the implication of this inequality are yet to be understood. X Preface Chapter 17 starts Part II of the book with a discussion of the butterfly network, the primary example in network coding. Variations of the butterfly network are analyzed in detail. The advantage of network coding over storeand-forward in wireless and satellite communications is explained through a simple example. We also explain why network coding with multiple information sources is substantially different from network coding with a single information source. In Chapter 18, the fundamental bound for single-source network coding, called the max-flow bound, is explained in detail. The bound is established for a general class of network codes. In Chapter 19, we discuss various classes of linear network codes on acyclic networks that achieve the max-flow bound to different extents. Static network codes, a special class of linear network codes that achieves the max-flow bound in the presence of channel failure, is also discussed. Polynomial-time algorithms for constructing these codes are presented. In Chapter 20, we formulate and analyze convolutional network codes on cyclic networks. The existence of such codes that achieve the max-flow bound is proved. Network coding theory is further developed in Chapter 21. The scenario when more than one information source are multicast in a point-to-point acyclic network is discussed. An implicit characterization of the achievable information rate region which involves the framework for information inequalities developed in Part I is proved. How to use this book 1 4 11 10 Part I 2 3 5 12 13 14 15 16 7 6 8 9 Part II 17 18 19 20 21 Preface XI Part I of this book by itself may be regarded as a comprehensive textbook in information theory. The main reason why the book is in the present form is because in my opinion, the discussion of network coding in Part II is incomplete without Part I. Nevertheless, except for Chapter 21 on multi-source network coding, Part II by itself may be used satisfactorily as a textbook on single-source network coding. An elementary course on probability theory and an elementary course on linear algebra are prerequisites to Part I and Part II, respectively. For Chapter 11, some background knowledge on digital communication systems would be helpful, and for Chapter 20, some prior exposure to discrete-time linear systems is necessary. The reader is recommended to read the chapters according to the above chart. However, one will not have too much difficulty jumping around in the book because there should be sufficient references to the previous relevant sections. This book inherits the writing style from the previous book, namely that all the derivations are from the first principle. The book contains a large number of examples, where important points are very often made. To facilitate the use of the book, there is a summary at the end of each chapter. This book can be used as a textbook or a reference book. As a textbook, it is ideal for a two-semester course, with the first and second semesters covering selected topics from Part I and Part II, respectively. A comprehensive instructor's manual is available upon request. Please contact the author at whyeung@ie.cuhk.edu.hk for information and access. Just like any other lengthy document, this book for sure contains errors and omissions. To alleviate the problem, an errata will be maintained at the book homepage http://www.ie.cuhk.edu.hk/IT book2/. Hong Kong, China December, 2007 Raymond W. Yeung Acknowledgments The current book, an expansion of my previous book A First Course in Information Theory, is written within the year 2007. Thanks to the generous support of the Friedrich Wilhelm Bessel Research Award from the Alexander von Humboldt Foundation of Germany, I had the luxury of working on the project full-time from January to April when I visited Munich University of Technology. I would like to thank Joachim Hagenauer and Ralf Koetter for nominating me for the award and for hosting my visit. I also would like to thank Department of Information Engineering, The Chinese University of Hong Kong, for making this arrangement possible. There are many individuals who have directly or indirectly contributed to this book. First, I am indebted to Toby Berger who taught me information theory and writing. I am most thankful to Zhen Zhang, Ning Cai, and Bob Li for their friendship and inspiration. Without the results obtained through our collaboration, the book cannot possibly be in its current form. I would also like to thank Venkat Anantharam, Vijay Bhargava, Dick Blahut, Agnes and Vincent Chan, Tom Cover, Imre Csiszr, Tony Ephremides, Bob Gallager, Bruce a Hajek, Te Sun Han, Jim Massey, Prakash Narayan, Alon Orlitsky, Shlomo Shamai, Sergio Verd, Victor Wei, Frans Willems, and Jack Wolf for their u support and encouragement throughout the years. I also would like to thank all the collaborators of my work for their contribution and all the anonymous reviewers for their useful comments. I would like to thank a number of individuals who helped in the project. I benefited tremendously from the discussions with David Tse who gave a lot of suggestions for writing the chapters on differential entropy and continuousvalued channels. Terence Chan, Ka Wo Cheung, Bruce Hajek, Siu-Wai Ho, Siu Ting Ho, Tat Ming Lok, Prakash Narayan, Will Ng, Sagar Shenvi, XiangGen Xia, Shaohua Yang, Ken Zeger, and Zhixue Zhang gave many valuable comments at different stages of the writing. My graduate students Silas Fong, Min Tan, and Shenghao Yang proofread the chapters on network coding in great detail. Silas Fong also helped compose the figures throughout the book. XIV Acknowledgments On the domestic side, I am most grateful to my wife Rebecca for her love. During our stay in Munich, she took good care of the whole family so that I was able to concentrate on my writing. We are most thankful to our family friend Ms. Pui Yee Wong for taking care of Rebecca when she was ill during the final stage of the project, and to my sister Georgiana for her moral support. In this regard, we are indebted to Dr. Yu Lap Yip for his timely diagnosis. I also would like to thank my sister-in-law Ophelia Tsang who comes over during the weekend to help taking of our daughter Shannon, who continues to the sweetheart of the family and was most supportive during the time her mom was ill. Contents 1 The Science of Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Part I Components of Information Theory 2 Information Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Independence and Markov Chains . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Shannon's Information Measures . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Continuity of Shannon's Information Measures for Fixed Finite Alphabets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Chain Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Informational Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 The Basic Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Some Useful Information Inequalities . . . . . . . . . . . . . . . . . . . . . . . 2.8 Fano's Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Maximum Entropy Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 Entropy Rate of a Stationary Source . . . . . . . . . . . . . . . . . . . . . . . Appendix 2.A: Approximation of Random Variables with Countably Infinite Alphabets by Truncation . . . . . . . . . . . . . . . . Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Historical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The I-Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The I-Measure for Two Random Variables . . . . . . . . . . . . . . . . . . 3.3 Construction of the I-Measure * . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 * Can be Negative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Information Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Examples of Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 3.A: A Variation of the Inclusion-Exclusion Formula . . . . . 7 7 12 18 21 23 26 28 32 36 38 41 43 45 49 51 52 53 55 59 61 67 74 3 XVI Contents Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 Historical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4 Zero-Error Data Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 The Entropy Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Prefix Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Definition and Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Huffman Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Redundancy of Prefix Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Historical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 82 86 86 88 93 97 98 99 5 Weak Typicality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.1 The Weak AEP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.2 The Source Coding Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.3 Efficient Source Coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.4 The Shannon-McMillan-Breiman Theorem . . . . . . . . . . . . . . . . . . 107 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 Historical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Strong Typicality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.1 Strong AEP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.2 Strong Typicality Versus Weak Typicality . . . . . . . . . . . . . . . . . . 121 6.3 Joint Typicality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 6.4 An Interpretation of the Basic Inequalities . . . . . . . . . . . . . . . . . . 131 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Historical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 Discrete Memoryless Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 7.1 Definition and Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 7.2 The Channel Coding Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 7.3 The Converse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 7.4 Achievability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 7.5 A Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 7.6 Feedback Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 7.7 Separation of Source and Channel Coding . . . . . . . . . . . . . . . . . . 172 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 Historical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 6 7 Contents XVII 8 Rate-Distortion Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 8.1 Single-Letter Distortion Measures . . . . . . . . . . . . . . . . . . . . . . . . . 184 8.2 The Rate-Distortion Function R(D) . . . . . . . . . . . . . . . . . . . . . . . 187 8.3 The Rate-Distortion Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 8.4 The Converse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 8.5 Achievability of RI (D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 Historical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 The Blahut-Arimoto Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 9.1 Alternating Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 9.2 The Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 9.2.1 Channel Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 9.2.2 The Rate-Distortion Function . . . . . . . . . . . . . . . . . . . . . . . 219 9.3 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 9.3.1 A Sufficient Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 9.3.2 Convergence to the Channel Capacity . . . . . . . . . . . . . . . . 225 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 Historical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 9 10 Differential Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 10.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 10.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 10.3 Joint and Conditional Differential Entropy . . . . . . . . . . . . . . . . . . 238 10.4 The AEP for Continuous Random Variables . . . . . . . . . . . . . . . . 245 10.5 Informational Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 10.6 Maximum Differential Entropy Distributions . . . . . . . . . . . . . . . . 248 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 Historical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 11 Continuous-Valued Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 11.1 Discrete-Time Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 11.2 The Channel Coding Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 11.3 Proof of the Channel Coding Theorem . . . . . . . . . . . . . . . . . . . . . 262 11.3.1 The Converse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 11.3.2 Achievability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 11.4 Memoryless Gaussian Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 11.5 Parallel Gaussian Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 11.6 Correlated Gaussian Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 11.7 The Bandlimited White Gaussian Channel . . . . . . . . . . . . . . . . . . 280 11.8 The Bandlimited Colored Gaussian Channel . . . . . . . . . . . . . . . . 287 11.9 Zero-Mean Gaussian Noise is the Worst Additive Noise . . . . . . . 289 XVIII Contents Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 Historical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 12 Markov Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 12.1 Conditional Mutual Independence . . . . . . . . . . . . . . . . . . . . . . . . . 300 12.2 Full Conditional Mutual Independence . . . . . . . . . . . . . . . . . . . . . 309 12.3 Markov Random Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 12.4 Markov Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 Historical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 13 Information Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 13.1 The Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 n 13.2 Information Expressions in Canonical Form . . . . . . . . . . . . . . . . . 326 13.3 A Geometrical Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 13.3.1 Unconstrained Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . 329 13.3.2 Constrained Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 13.3.3 Constrained Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 13.4 Equivalence of Constrained Inequalities . . . . . . . . . . . . . . . . . . . . 333 13.5 The Implication Problem of Conditional Independence . . . . . . . 336 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 Historical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 14 Shannon-Type Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 14.1 The Elemental Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 14.2 A Linear Programming Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 341 14.2.1 Unconstrained Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . 343 14.2.2 Constrained Inequalities and Identities . . . . . . . . . . . . . . . 344 14.3 A Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 14.4 Machine Proving ITIP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 14.5 Tackling the Implication Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 351 14.6 Minimality of the Elemental Inequalities . . . . . . . . . . . . . . . . . . . . 353 Appendix 14.A: The Basic Inequalities and the Polymatroidal Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358 Historical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360 15 Beyond Shannon-Type Inequalities . . . . . . . . . . . . . . . . . . . . . . . . 361 15.1 Characterizations of , , and . . . . . . . . . . . . . . . . . . . . . . 361 n 2 3 15.2 A Non-Shannon-Type Unconstrained Inequality . . . . . . . . . . . . . 369 15.3 A Non-Shannon-Type Constrained Inequality . . . . . . . . . . . . . . . 374 Contents XIX 15.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 Historical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 16 Entropy and Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 16.1 Group Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 16.2 Group-Characterizable Entropy Functions . . . . . . . . . . . . . . . . . . 393 16.3 A Group Characterization of . . . . . . . . . . . . . . . . . . . . . . . . . . 398 n 16.4 Information Inequalities and Group Inequalities . . . . . . . . . . . . . 401 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406 Historical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 Part II Fundamentals of Network Coding 17 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 17.1 The Butterfly Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 17.2 Wireless and Satellite Communications . . . . . . . . . . . . . . . . . . . . . 415 17.3 Source Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 Historical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 18 The Max-Flow Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 18.1 Point-to-Point Communication Networks . . . . . . . . . . . . . . . . . . . 421 18.2 Examples Achieving the Max-Flow Bound . . . . . . . . . . . . . . . . . . 424 18.3 A Class of Network Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 18.4 Proof of the Max-Flow Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 Historical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 19 Single-Source Linear Network Coding: Acyclic Networks . . 435 19.1 Acyclic Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 19.2 Linear Network Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 19.3 Desirable Properties of a Linear Network Code . . . . . . . . . . . . . . 442 19.4 Existence and Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449 19.5 Generic Network Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460 19.6 Static Network Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468 19.7 Random Network Coding: A Case Study . . . . . . . . . . . . . . . . . . . 473 19.7.1 How the System Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474 19.7.2 Model and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478 XX Contents Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479 Historical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482 20 Single-Source Linear Network Coding: Cyclic Networks . . . . 485 20.1 Delay-Free Cyclic Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 20.2 Convolutional Network Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488 20.3 Decoding of Convolutional Network Codes . . . . . . . . . . . . . . . . . . 498 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504 Historical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504 21 Multi-Source Network Coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505 21.1 The Max-Flow Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505 21.2 Examples of Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508 21.2.1 Multilevel Diversity Coding . . . . . . . . . . . . . . . . . . . . . . . . . 508 21.2.2 Satellite Communication Network . . . . . . . . . . . . . . . . . . . 510 21.3 A Network Code for Acyclic Networks . . . . . . . . . . . . . . . . . . . . . 510 21.4 The Achievable Information Rate Region . . . . . . . . . . . . . . . . . . . 512 21.5 Explicit Inner and Outer Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . 515 21.6 The Converse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516 21.7 Achievability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521 21.7.1 Random Code Construction . . . . . . . . . . . . . . . . . . . . . . . . 524 21.7.2 Performance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537 Historical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561 1 The Science of Information In a communication system, we try to convey information from one point to another, very often in a noisy environment. Consider the following scenario. A secretary needs to send facsimiles regularly and she wants to convey as much information as possible on each page. She has a choice of the font size, which means that more characters can be squeezed onto a page if a smaller font size is used. In principle, she can squeeze as many characters as desired on a page by using a small enough font size. However, there are two factors in the system which may cause errors. First, the fax machine has a finite resolution. Second, the characters transmitted may be received incorrectly due to noise in the telephone line. Therefore, if the font size is too small, the characters may not be recognizable on the facsimile. On the other hand, although some characters on the facsimile may not be recognizable, the recipient can still figure out the words from the context provided that the number of such characters is not excessive. In other words, it is not necessary to choose a font size such that all the characters on the facsimile are recognizable almost surely. Then we are motivated to ask: What is the maximum amount of meaningful information which can be conveyed on one page of facsimile? This question may not have a definite answer because it is not very well posed. In particular, we do not have a precise measure of meaningful information. Nevertheless, this question is an illustration of the kind of fundamental questions we can ask about a communication system. Information, which is not a physical entity but an abstract concept, is hard to quantify in general. This is especially the case if human factors are involved when the information is utilized. For example, when we play Beethoven's violin concerto from an audio compact disc, we receive the musical information from the loudspeakers. We enjoy this information because it arouses certain kinds of emotion within ourselves. While we receive the same information every time we play the same piece of music, the kinds of emotions aroused may be different from time to time because they depend on our mood at that particular moment. In other words, we can derive utility from the same information every time in a different way. For this reason, it is extremely 2 1 The Science of Information difficult to devise a measure which can quantify the amount of information contained in a piece of music. In 1948, Bell Telephone Laboratories scientist Claude E. Shannon (19162001) published a paper entitled "The Mathematical Theory of Communication" [322] which laid the foundation of an important field now known as information theory. In his paper, the model of a point-to-point communication system depicted in Figure 1.1 is considered. In this model, a message is generINFORMATION SOURCE TRANSMITTER RECEIVER DESTINATION SIGNAL MESSAGE RECEIVED SIGNAL MESSAGE NOISE SOURCE Fig. 1.1. Schematic diagram for a general point-to-point communication system. ated by the information source. The message is converted by the transmitter into a signal which is suitable for transmission. In the course of transmission, the signal may be contaminated by a noise source, so that the received signal may be different from the transmitted signal. Based on the received signal, the receiver then makes an estimate on the message and deliver it to the destination. In this abstract model of a point-to-point communication system, one is only concerned about whether the message generated by the source can be delivered correctly to the receiver without worrying about how the message is actually used by the receiver. In a way, Shannon's model does not cover all possible aspects of a communication system. However, in order to develop a precise and useful theory of information, the scope of the theory has to be restricted. In [322], Shannon introduced two fundamental concepts about "information" from the communication point of view. First, information is uncertainty. More specifically, if a piece of information we are interested in is deterministic, then it has no value at all because it is already known with no uncertainty. From this point of view, for example, the continuous transmission of a still picture on a television broadcast channel is superfluous. Consequently, an information source is naturally modeled as a random variable or a random process, and probability is employed to develop the theory of information. Second, information to be transmitted is digital. This means that the information source should first be converted into a stream of 0's and 1's called bits, 1 The Science of Information 3 and the remaining task is to deliver these bits to the receiver correctly with no reference to their actual meaning. This is the foundation of all modern digital communication systems. In fact, this work of Shannon appears to contain the first published use of the term "bit," which stands for binary digit. In the same work, Shannon also proved two important theorems. The first theorem, called the source coding theorem, introduces entropy as the fundamental measure of information which characterizes the minimum rate of a source code representing an information source essentially free of error. The source coding theorem is the theoretical basis for lossless data compression1 . The second theorem, called the channel coding theorem, concerns communication through a noisy channel. It was shown that associated with every noisy channel is a parameter, called the capacity, which is strictly positive except for very special channels, such that information can be communicated reliably through the channel as long as the information rate is less than the capacity. These two theorems, which give fundamental limits in point-to-point communication, are the two most important results in information theory. In science, we study the laws of Nature which must be obeyed by any physical systems. These laws are used by engineers to design systems to achieve specific goals. Therefore, science is the foundation of engineering. Without science, engineering can only be done by trial and error. In information theory, we study the fundamental limits in communication regardless of the technologies involved in the actual implementation of the communication systems. These fundamental limits are not only used as guidelines by communication engineers, but they also give insights into what optimal coding schemes are like. Information theory is therefore the science of information. Since Shannon published his original paper in 1948, information theory has been developed into a major research field in both communication theory and applied probability. For a non-technical introduction to information theory, we refer the reader to Encyclopedia Britannica [49]. In fact, we strongly recommend the reader to first read this excellent introduction before starting this book. For biographies of Claude Shannon, a legend of the 20th Century who had made fundamental contribution to the Information Age, we refer the readers to [56] and [340]. The latter is also a complete collection of Shannon's papers. Unlike most branches of applied mathematics in which physical systems are studied, abstract systems of communication are studied in information theory. In reading this book, it is not unusual for a beginner to be able to understand all the steps in a proof but has no idea what the proof is leading to. The best way to learn information theory is to study the materials first and come back at a later time. Many results in information theory are rather subtle, to the extent that an expert in the subject may from time to time realize that his/her 1 A data compression scheme is lossless if the data can be recovered with an arbitrarily small probability of error. 4 1 The Science of Information understanding of certain basic results has been inadequate or even incorrect. While a novice should expect to raise his/her level of understanding of the subject by reading this book, he/she should not be discouraged to find after finishing the book that there are actually more things yet to be understood. In fact, this is exactly the challenge and the beauty of information theory. Part I Components of Information Theory 2 Information Measures Shannon's information measures refer to entropy, conditional entropy, mutual information, and conditional mutual information. They are the most important measures of information in information theory. In this chapter, we introduce these measures and establish some basic properties they possess. The physical meanings of these measures will be discussed in depth in subsequent chapters. We then introduce the informational divergence which measures the "distance" between two probability distributions and prove some useful inequalities in information theory. The chapter ends with a section on the entropy rate of a stationary information source. 2.1 Independence and Markov Chains We begin our discussion in this chapter by reviewing two basic concepts in probability: independence of random variables and Markov chain. All the random variables in this book except for Chapters 10 and 11 are assumed to be discrete unless otherwise specified. Let X be a random variable taking values in an alphabet X . The probability distribution for X is denoted as {pX (x), x X }, with pX (x) = Pr{X = x}. When there is no ambiguity, pX (x) will be abbreviated as p(x), and {p(x)} will be abbreviated as p(x). The support of X, denoted by SX , is the set of all x X such that p(x) > 0. If SX = X , we say that p is strictly positive. Otherwise, we say that p is not strictly positive, or p contains zero probability masses. All the above notations naturally extend to two or more random variables. As we will see, probability distributions with zero probability masses are very delicate, and they need to be handled with great care. Definition 2.1. Two random variables X and Y are independent, denoted by X Y , if p(x, y) = p(x)p(y) (2.1) for all x and y (i.e., for all (x, y) X Y). 8 2 Information Measures For more than two random variables, we distinguish between two types of independence. Definition 2.2 (Mutual Independence). For n 3, random variables X1 , X2 , , Xn are mutually independent if p(x1 , x2 , , xn ) = p(x1 )p(x2 ) p(xn ) for all x1 , x2 , , xn . Definition 2.3 (Pairwise Independence). For n 3, random variables X1 , X2 , , Xn are pairwise independent if Xi and Xj are independent for all 1 i < j n. Note that mutual independence implies pairwise independence. We leave it as an exercise for the reader to show that the converse is not true. Definition 2.4 (Conditional Independence). For random variables X, Y , and Z, X is independent of Z conditioning on Y , denoted by X Z|Y , if p(x, y, z)p(y) = p(x, y)p(y, z) for all x, y, and z, or equivalently, p(x, y, z) = p(x,y)p(y,z) p(y) (2.2) (2.3) 0 = p(x, y)p(z|y) if p(y) > 0 otherwise. (2.4) The first definition of conditional independence above is sometimes more convenient to use because it is not necessary to distinguish between the cases p(y) > 0 and p(y) = 0. However, the physical meaning of conditional independence is more explicit in the second definition. Proposition 2.5. For random variables X, Y , and Z, X Z|Y if and only if p(x, y, z) = a(x, y)b(y, z) (2.5) for all x, y, and z such that p(y) > 0. Proof. The `only if' part follows immediately from the definition of conditional independence in (2.4), so we will only prove the `if' part. Assume p(x, y, z) = a(x, y)b(y, z) (2.6) for all x, y, and z such that p(y) > 0. Then for such x, y, and z, we have p(x, y) = z p(x, y, z) = z a(x, y)b(y, z) = a(x, y) z b(y, z) (2.7) 2.1 Independence and Markov Chains 9 and p(y, z) = x p(x, y, z) = x a(x, y)b(y, z) = b(y, z) x a(x, y). (2.8) Furthermore, p(y) = z p(y, z) = x a(x, y) z b(y, z) > 0. (2.9) Therefore, a(x, y) p(x, y)p(y, z) = p(y) z b(y, z) a(x, y) x b(y, z) x a(x, y) (2.10) b(y, z) z = a(x, y)b(y, z) = p(x, y, z). For x, y, and z such that p(y) = 0, since 0 p(x, y, z) p(y) = 0, we have p(x, y, z) = 0. Hence, X Z|Y according to (2.4). The proof is accomplished. (2.11) (2.12) (2.13) (2.14) Definition 2.6 (Markov Chain). For random variables X1 , X2 , , Xn , where n 3, X1 X2 Xn forms a Markov chain if p(x1 , x2 , , xn )p(x2 )p(x3 ) p(xn-1 ) = p(x1 , x2 )p(x2 , x3 ) p(xn-1 , xn ) for all x1 , x2 , , xn , or equivalently, (2.15) p(x1 , x2 , , xn ) = p(x1 , x2 )p(x3 |x2 ) p(xn |xn-1 ) if p(x2 ), p(x3 ), , p(xn-1 ) > 0 0 otherwise. (2.16) We note that X Z|Y is equivalent to the Markov chain X Y Z. Proposition 2.7. X1 X2 Xn forms a Markov chain if and only if Xn Xn-1 X1 forms a Markov chain. 10 2 Information Measures Proof. This follows directly from the symmetry in the definition of a Markov chain in (2.15). In the following, we state two basic properties of a Markov chain. The proofs are left as an exercise. Proposition 2.8. X1 X2 Xn forms a Markov chain if and only if X1 X2 X3 (X1 , X2 ) X3 X4 . . . (X1 , X2 , , Xn-2 ) Xn-1 Xn form Markov chains. Proposition 2.9. X1 X2 Xn forms a Markov chain if and only if p(x1 , x2 , , xn ) = f1 (x1 , x2 )f2 (x2 , x3 ) fn-1 (xn-1 , xn ) (2.18) for all x1 , x2 , , xn such that p(x2 ), p(x3 ), , p(xn-1 ) > 0. Note that Proposition 2.9 is a generalization of Proposition 2.5. From Proposition 2.9, one can prove the following important property of a Markov chain. Again, the details are left as an exercise. Proposition 2.10 (Markov subchains). Let Nn = {1, 2, , n} and let X1 X2 Xn form a Markov chain. For any subset of Nn , denote (Xi , i ) by X . Then for any disjoint subsets 1 , 2 , , m of Nn such that k1 < k2 < < km (2.19) for all kj j , j = 1, 2, , m, X1 X2 Xm (2.20) (2.17) forms a Markov chain. That is, a subchain of X1 X2 Xn is also a Markov chain. Example 2.11. Let X1 X2 X10 form a Markov chain and 1 = {1, 2}, 2 = {4}, 3 = {6, 8}, and 4 = {10} be subsets of N10 . Then Proposition 2.10 says that (X1 , X2 ) X4 (X6 , X8 ) X10 also forms a Markov chain. (2.21) 2.1 Independence and Markov Chains 11 We have been very careful in handling probability distributions with zero probability masses. In the rest of the section, we show that such distributions are very delicate in general. We first prove the following property of a strictly positive probability distribution involving four random variables1 . Proposition 2.12. Let X1 , X2 , X3 , and X4 be random variables such that p(x1 , x2 , x3 , x4 ) is strictly positive. Then X1 X4 |(X2 , X3 ) X1 X3 |(X2 , X4 ) X1 (X3 , X4 )|X2 . (2.22) Proof. If X1 X4 |(X2 , X3 ), then p(x1 , x2 , x3 , x4 ) = p(x1 , x2 , x3 )p(x2 , x3 , x4 ) . p(x2 , x3 ) (2.23) On the other hand, if X1 X3 |(X2 , X4 ), then p(x1 , x2 , x3 , x4 ) = p(x1 , x2 , x4 )p(x2 , x3 , x4 ) . p(x2 , x4 ) (2.24) Equating (2.23) and (2.24), we have p(x1 , x2 , x3 ) = Therefore, p(x1 , x2 ) = x3 p(x2 , x3 )p(x1 , x2 , x4 ) . p(x2 , x4 ) (2.25) p(x1 , x2 , x3 ) p(x2 , x3 )p(x1 , x2 , x4 ) p(x2 , x4 ) (2.26) (2.27) (2.28) = x3 = or p(x2 )p(x1 , x2 , x4 ) , p(x2 , x4 ) p(x1 , x2 , x4 ) p(x1 , x2 ) = . p(x2 , x4 ) p(x2 ) (2.29) Hence from (2.24), p(x1 , x2 , x3 , x4 ) = p(x1 , x2 , x4 )p(x2 , x3 , x4 ) p(x1 , x2 )p(x2 , x3 , x4 ) = (2.30) p(x2 , x4 ) p(x2 ) for all x1 , x2 , x3 , and x4 , i.e., X1 (X3 , X4 )|X2 . 1 Proposition 2.12 is called the intersection axiom in Bayesian networks. See [287]. 12 2 Information Measures If p(x1 , x2 , x3 , x4 ) = 0 for some x1 , x2 , x3 , and x4 , i.e., p is not strictly positive, the arguments in the above proof are not valid. In fact, the proposition may not hold in this case. For instance, let X1 = Y , X2 = Z, and X3 = X4 = (Y, Z), where Y and Z are independent random variables. Then X1 X4 |(X2 , X3 ), X1 X3 |(X2 , X4 ), but X1 (X3 , X4 )|X2 . Note that for this construction, p is not strictly positive because p(x1 , x2 , x3 , x4 ) = 0 if x3 = (x1 , x2 ) or x4 = (x1 , x2 ). The above example is somewhat counter-intuitive because it appears that Proposition 2.12 should hold for all probability distributions via a continuity argument2 which would go like this. For any distribution p, let {pk } be a sequence of strictly positive distributions such that pk p and pk satisfies (2.23) and (2.24) for all k, i.e., pk (x1 , x2 , x3 , x4 )pk (x2 , x3 ) = pk (x1 , x2 , x3 )pk (x2 , x3 , x4 ) and pk (x1 , x2 , x3 , x4 )pk (x2 , x4 ) = pk (x1 , x2 , x4 )pk (x2 , x3 , x4 ). Then by the proposition, pk also satisfies (2.30), i.e., pk (x1 , x2 , x3 , x4 )pk (x2 ) = pk (x1 , x2 )pk (x2 , x3 , x4 ). Letting k , we have p(x1 , x2 , x3 , x4 )p(x2 ) = p(x1 , x2 )p(x2 , x3 , x4 ) (2.34) (2.33) (2.32) (2.31) for all x1 , x2 , x3 , and x4 , i.e., X1 (X3 , X4 )|X2 . Such an argument would be valid if there always exists a sequence {pk } as prescribed. However, the existence of the distribution p(x1 , x2 , x3 , x4 ) constructed immediately after Proposition 2.12 simply says that it is not always possible to find such a sequence {pk }. Therefore, probability distributions which are not strictly positive can be very delicate. For strictly positive distributions, we see from Proposition 2.5 that their conditional independence structures are closely related to the factorization problem of such distributions, which has been investigated by Chan [60]. 2.2 Shannon's Information Measures We begin this section by introducing the entropy of a random variable. As we will see shortly, all Shannon's information measures can be expressed as linear combinations of entropies. 2 See Section 2.3 for a more detailed discussion on continuous functionals. 2.2 Shannon's Information Measures 13 Definition 2.13. The entropy H(X) of a random variable X is defined as H(X) = - x p(x) log p(x). (2.35) In the definitions of all information measures, we adopt the convention that summation is taken over the corresponding support. Such a convention is necessary because p(x) log p(x) in (2.35) is undefined if p(x) = 0. The base of the logarithm in (2.35) can be chosen to be any convenient real number greater than 1. We write H(X) as H (X) when the base of the logarithm is . When the base of the logarithm is 2, the unit for entropy is the bit. When the base of the logarithm is e, the unit for entropy is the nat. When the base of the logarithm is an integer D 2, the unit for entropy is the D-it (D-ary digit). In the context of source coding, the base is usually taken to be the size of the code alphabet. This will be discussed in Chapter 4. In computer science, a bit means an entity which can take the value 0 or 1. In information theory, the entropy of a random variable is measured in bits. The reader should distinguish these two meanings of a bit from each other carefully. Let g(X) be any function of a random variable X. We will denote the expectation of g(X) by Eg(X), i.e., Eg(X) = x p(x)g(x), (2.36) where the summation is over SX . Then the definition of the entropy of a random variable X can be written as H(X) = -E log p(X). (2.37) Expressions of Shannon's information measures in terms of expectations will be useful in subsequent discussions. The entropy H(X) of a random variable X is a functional of the probability distribution p(x) which measures the average amount of information contained in X, or equivalently, the average amount of uncertainty removed upon revealing the outcome of X. Note that H(X) depends only on p(x), not on the actual values in X . Occasionally, we also denote H(X) by H(p). For 0 1, define hb () = - log - (1 - ) log(1 - ) (2.38) with the convention 0 log 0 = 0, so that hb (0) = hb (1) = 0. With this convention, hb () is continuous at = 0 and = 1. hb is called the binary entropy function. For a binary random variable X with distribution {, 1 - }, H(X) = hb (). (2.39) Figure 2.1 is the plot of hb () versus in the base 2. Note that hb () achieves 1 the maximum value 1 when = 2 . 14 2 Information Measures Fig. 2.1. hb () versus in the base 2. The definition of the joint entropy of two random variables is similar to the definition of the entropy of a single random variable. Extension of this definition to more than two random variables is straightforward. Definition 2.14. The joint entropy H(X, Y ) of a pair of random variables X and Y is defined as H(X, Y ) = - x,y p(x, y) log p(x, y) = -E log p(X, Y ). (2.40) For two random variables, we define in the following the conditional entropy of one random variable when the other random variable is given. Definition 2.15. For random variables X and Y , the conditional entropy of Y given X is defined as H(Y |X) = - x,y p(x, y) log p(y|x) = -E log p(Y |X). (2.41) From (2.41), we can write H(Y |X) = x p(x) - y p(y|x) log p(y|x) . (2.42) The inner sum is the entropy of Y conditioning on a fixed x SX . Thus we are motivated to express H(Y |X) as H(Y |X) = x p(x)H(Y |X = x), (2.43) where 2.2 Shannon's Information Measures 15 H(Y |X = x) = - y p(y|x) log p(y|x). (2.44) Observe that the right hand sides of (2.35) and (2.44) have exactly the same form. Similarly, for H(Y |X, Z), we write H(Y |X, Z) = z p(z)H(Y |X, Z = z), (2.45) where H(Y |X, Z = z) = - x,y p(x, y|z) log p(y|x, z). (2.46) Proposition 2.16. H(X, Y ) = H(X) + H(Y |X) and H(X, Y ) = H(Y ) + H(X|Y ). (2.48) (2.47) Proof. Consider H(X, Y ) = -E log p(X, Y ) = -E log[p(X)p(Y |X)] = -E log p(X) - E log p(Y |X) = H(X) + H(Y |X). (2.49) (2.50) (2.51) (2.52) Note that (2.50) is justified because the summation of the expectation is over SXY , and we have used the linearity of expectation3 to obtain (2.51). This proves (2.47), and (2.48) follows by symmetry. This proposition has the following interpretation. Consider revealing the outcome of a pair of random variables X and Y in two steps: first the outcome of X and then the outcome of Y . Then the proposition says that the total amount of uncertainty removed upon revealing both X and Y is equal to the sum of the uncertainty removed upon revealing X (uncertainty removed in the first step) and the uncertainty removed upon revealing Y once X is known (uncertainty removed in the second step). Definition 2.17. For random variables X and Y , the mutual information between X and Y is defined as I(X; Y ) = x,y p(x, y) log p(X, Y ) p(x, y) = E log . p(x)p(y) p(X)p(Y ) (2.53) 3 See Problem 5 at the end of the chapter. 16 2 Information Measures Remark I(X; Y ) is symmetrical in X and Y . Proposition 2.18. The mutual information between a random variable X and itself is equal to the entropy of X, i.e., I(X; X) = H(X). Proof. This can be seen by considering I(X; X) = E log p(X) p(X)2 = -E log p(X) = H(X). (2.54) (2.55) (2.56) The proposition is proved. Remark The entropy of X is sometimes called the self-information of X. Proposition 2.19. I(X; Y ) = H(X) - H(X|Y ), I(X; Y ) = H(Y ) - H(Y |X), and I(X; Y ) = H(X) + H(Y ) - H(X, Y ), (2.59) provided that all the entropies and conditional entropies are finite (see Example 2.46 in Section 2.8). The proof of this proposition is left as an exercise. From (2.57), we can interpret I(X; Y ) as the reduction in uncertainty about X when Y is given, or equivalently, the amount of information about X provided by Y . Since I(X; Y ) is symmetrical in X and Y , from (2.58), we can as well interpret I(X; Y ) as the amount of information about Y provided by X. The relations between the (joint) entropies, conditional entropies, and mutual information for two random variables X and Y are given in Propositions 2.16 and 2.19. These relations can be summarized by the diagram in Figure 2.2 which is a variation of the Venn diagram4 . One can check that all the relations between Shannon's information measures for X and Y which are shown in Figure 2.2 are consistent with the relations given in Propositions 2.16 and 2.19. This one-to-one correspondence between Shannon's information measures and set theory is not just a coincidence for two random variables. We will discuss this in depth when we introduce the I-Measure in Chapter 3. Analogous to entropy, there is a conditional version of mutual information called conditional mutual information. 4 (2.57) (2.58) The rectangle representing the universal set in a usual Venn diagram is missing in Figure 2.2. 2.2 Shannon's Information Measures 17 H(X,Y) H(X|Y) H ( Y|X ) H(X) I(X;Y) H(Y) Fig. 2.2. Relationship between entropies and mutual information for two random variables. Definition 2.20. For random variables X, Y and Z, the mutual information between X and Y conditioning on Z is defined as I(X; Y |Z) = x,y,z p(x, y, z) log p(x, y|z) p(X, Y |Z) = E log . (2.60) p(x|z)p(y|z) p(X|Z)p(Y |Z) Remark I(X; Y |Z) is symmetrical in X and Y . Analogous to conditional entropy, we write I(X; Y |Z) = z p(z)I(X; Y |Z = z), (2.61) where I(X; Y |Z = z) = x,y p(x, y|z) log p(x, y|z) . p(x|z)p(y|z) (2.62) Similarly, when conditioning on two random variables, we write I(X; Y |Z, T ) = t p(t)I(X; Y |Z, T = t) (2.63) where I(X; Y |Z, T = t) = x,y,z p(x, y, z|t) log p(x, y|z, t) . p(x|z, t)p(y|z, t) (2.64) Conditional mutual information satisfies the same set of relations given in Propositions 2.18 and 2.19 for mutual information except that all the terms are now conditioned on a random variable Z. We state these relations in the next two propositions. The proofs are omitted. 18 2 Information Measures Proposition 2.21. The mutual information between a random variable X and itself conditioning on a random variable Z is equal to the conditional entropy of X given Z, i.e., I(X; X|Z) = H(X|Z). Proposition 2.22. I(X; Y |Z) = H(X|Z) - H(X|Y, Z), I(X; Y |Z) = H(Y |Z) - H(Y |X, Z), and I(X; Y |Z) = H(X|Z) + H(Y |Z) - H(X, Y |Z), provided that all the conditional entropies are finite. Remark All Shannon's information measures are finite if the random variables involved have finite alphabets. Therefore, Propositions 2.19 and 2.22 apply provided that all the random variables therein have finite alphabets. To conclude this section, we show that all Shannon's information measures are special cases of conditional mutual information. Let be a degenerate random variable, i.e., takes a constant value with probability 1. Consider the mutual information I(X; Y |Z). When X = Y and Z = , I(X; Y |Z) becomes the entropy H(X). When X = Y , I(X; Y |Z) becomes the conditional entropy H(X|Z). When Z = , I(X; Y |Z) becomes the mutual information I(X; Y ). Thus all Shannon's information measures are special cases of conditional mutual information. (2.67) (2.65) (2.66) 2.3 Continuity of Shannon's Information Measures for Fixed Finite Alphabets In this section, we prove that for fixed finite alphabets, all Shannon's information measures are continuous functionals of the joint distribution of the random variables involved. To formulate the notion of continuity, we first introduce the variational distance5 as a distance measure between two probability distributions on a common alphabet. Definition 2.23. Let p and q be two probability distributions on a common alphabet X . The variational distance between p and q is defined as V (p, q) = xX |p(x) - q(x)|. (2.68) 5 The variational distance is also referred to as the L1 distance in mathematics. 2.3 Continuity of Shannon's Information Measures for Fixed Finite Alphabets 19 For a fixed finite alphabet X , let PX be the set of all distributions on X . Then according to (2.35), the entropy of a distribution p on an alphabet X is defined as p(x) log p(x) (2.69) H(p) = - xSp where Sp denotes the support of p and Sp X . In order for H(p) to be continuous with respect to convergence in variational distance6 at a particular distribution p PX , for any > 0, there exists > 0 such that |H(p) - H(q)| < for all q PX satisfying V (p, q) < , or equivalently, p p (2.70) (2.71) lim H(p ) = H p p lim p = H(p), (2.72) where the convergence p p is in variational distance. Since a log a 0 as a 0, we define a function l : [0, ) l(a) = a log a if a > 0 0 if a = 0, by (2.73) i.e., l(a) is a continuous extension of a log a. Then (2.69) can be rewritten as H(p) = - xX l(p(x)), (2.74) where the summation above is over all x in X instead of Sp . Upon defining a function lx : PX for all x X by lx (p) = l(p(x)), (2.74) becomes H(p) = - xX (2.75) lx (p). (2.76) Evidently, lx (p) is continuous in p (with respect to convergence in variational distance). Since the summation in (2.76) involves a finite number of terms, we conclude that H(p) is a continuous functional of p. We now proceed to prove the continuity of conditional mutual information which covers all cases of Shannon's information measures. Consider I(X; Y |Z) and let pXY Z be the joint distribution of X, Y , and Z, where the alphabets X , Y, and Z are assumed to be finite. From (2.47) and (2.67), we obtain I(X; Y |Z) = H(X, Z) + H(Y, Z) - H(X, Y, Z) - H(Z). 6 (2.77) Convergence in variational distance is the same as L1 -convergence. 20 2 Information Measures Note that each term on the right hand side above is the unconditional entropy of the corresponding marginal distribution. Then (2.77) can be rewritten as IX;Y |Z (pXY Z ) = H(pXZ ) + H(pY Z ) - H(pXY Z ) - H(pZ ), (2.78) where we have used IX;Y |Z (pXY Z ) to denote I(X; Y |Z). It follows that pXY Z pXY Z lim IX;Y |Z (pXY Z ) [H(pXZ ) + H(pY Z ) - H(pXY Z ) - H(pZ )] H(pXZ ) + pXY Z pXY Z = = pXY Z pXY Z pXY Z pXY Z lim (2.79) lim lim H(pY Z ) H(pZ ). (2.80) - pXY Z pXY Z lim H(pXY Z ) - pXY Z pXY Z lim It can readily be proved, for example, that pXY Z pXY Z lim pXZ = pXZ , (2.81) so that pXY Z pXY Z lim H(pXZ ) = H pXY Z pXY Z lim pXZ = H(pXZ ) (2.82) by the continuity of H() when the alphabets involved are fixed and finite. The details are left as an exercise. Hence, we conclude that pXY Z pXY Z lim IX;Y |Z (pXY Z ) (2.83) (2.84) = H(pXZ ) + H(pY Z ) - H(pXY Z ) - H(pZ ) = IX;Y |Z (pXY Z ), i.e., IX;Y |Z (pXY Z ) is a continuous functional of pXY Z . Since conditional mutual information covers all cases of Shannon's information measures, we have proved that all Shannon's information measures are continuous with respect to convergence in variational distance under the assumption that the alphabets are fixed and finite. It is not difficult to show that under this assumption, convergence in variational distance is equivalent to L2 -convergence, i.e., convergence in Euclidean distance (see Problem 8). It follows that Shannon's information measures are also continuous with respect to L2 -convergence. The variational distance, however, is more often used as a distance measure between two probability distributions because it can be directly related with the informational divergence to be discussed in Section 2.5. The continuity of Shannon's information measures we have proved in this section is rather restrictive and need to be applied with caution. In fact, if the alphabets are not fixed, Shannon's information measures are everywhere discontinuous with respect to convergence in a number of commonly used distance measures. We refer the readers to Problems 28 to 31 for a discussion of these issues. 2.4 Chain Rules 21 2.4 Chain Rules In this section, we present a collection of information identities known as the chain rules which are often used in information theory. Proposition 2.24 (Chain Rule for Entropy). n H(X1 , X2 , , Xn ) = i=1 H(Xi |X1 , , Xi-1 ). (2.85) Proof. The chain rule for n = 2 has been proved in Proposition 2.16. We prove the chain rule by induction on n. Assume (2.85) is true for n = m, where m 2. Then H(X1 , , Xm , Xm+1 ) = H(X1 , , Xm ) + H(Xm+1 |X1 , , Xm ) m (2.86) (2.87) = i=1 H(Xi |X1 , , Xi-1 ) + H(Xm+1 |X1 , , Xm ) m+1 = i=1 H(Xi |X1 , , Xi-1 ), (2.88) where in (2.86) we have used (2.47) by letting X = (X1 , , Xm ) and Y = Xm+1 , and in (2.87) we have used (2.85) for n = m. This proves the chain rule for entropy. The chain rule for entropy has the following conditional version. Proposition 2.25 (Chain Rule for Conditional Entropy). n H(X1 , X2 , , Xn |Y ) = i=1 H(Xi |X1 , , Xi-1 , Y ). (2.89) Proof. The proposition can be proved by considering H(X1 , X2 , , Xn |Y ) = H(X1 , X2 , , Xn , Y ) - H(Y ) = H((X1 , Y ), X2 , , Xn ) - H(Y ) n (2.90) (2.91) (2.92) (2.93) (2.94) = H(X1 , Y ) + i=2 n H(Xi |X1 , , Xi-1 , Y ) - H(Y ) H(Xi |X1 , , Xi-1 , Y ) i=2 = H(X1 |Y ) + n = i=1 H(Xi |X1 , , Xi-1 , Y ), 22 2 Information Measures where (2.90) and (2.93) follow from Proposition 2.16, while (2.92) follows from Proposition 2.24. Alternatively, the proposition can be proved by considering H(X1 , X2 , , Xn |Y ) = y n p(y)H(X1 , X2 , , Xn |Y = y) p(y) y n i=1 (2.95) = H(Xi |X1 , , Xi-1 , Y = y) (2.96) = i=1 n y p(y)H(Xi |X1 , , Xi-1 , Y = y) H(Xi |X1 , , Xi-1 , Y ), i=1 (2.97) = (2.98) where (2.95) and (2.98) follow from (2.43) and (2.45), respectively, and (2.96) follows from an application of Proposition 2.24 to the joint distribution of X1 , X2 , , Xn conditioning on {Y = y}. This proof offers an explanation to the observation that (2.89) can be obtained directly from (2.85) by conditioning on Y in every term. Proposition 2.26 (Chain Rule for Mutual Information). n I(X1 , X2 , , Xn ; Y ) = i=1 I(Xi ; Y |X1 , , Xi-1 ). (2.99) Proof. Consider I(X1 , X2 , , Xn ; Y ) = H(X1 , X2 , , Xn ) - H(X1 , X2 , , Xn |Y ) n (2.100) (2.101) (2.102) = i=1 n [H(Xi |X1 , , Xi-1 ) - H(Xi |X1 , , Xi-1 , Y )] I(Xi ; Y |X1 , , Xi-1 ), i=1 = where in (2.101), we have invoked both Propositions 2.24 and 2.25. The chain rule for mutual information is proved. Proposition 2.27 (Chain Rule for Conditional Mutual Information). For random variables X1 , X2 , , Xn , Y , and Z, n I(X1 , X2 , , Xn ; Y |Z) = i=1 I(Xi ; Y |X1 , , Xi-1 , Z). (2.103) 2.5 Informational Divergence 23 Proof. This is the conditional version of the chain rule for mutual information. The proof is similar to that for Proposition 2.25. The details are omitted. 2.5 Informational Divergence Let p and q be two probability distributions on a common alphabet X . We very often want to measure how much p is different from q, and vice versa. In order to be useful, this measure must satisfy the requirements that it is always nonnegative and it takes the zero value if and only if p = q. We denote the support of p and q by Sp and Sq , respectively. The informational divergence defined below serves this purpose. Definition 2.28. The informational divergence between two probability distributions p and q on a common alphabet X is defined as D(p q) = x p(x) log p(X) p(x) = Ep log , q(x) q(X) (2.104) where Ep denotes expectation with respect to p. In the above definition, in addition to the convention that the summation c is taken over Sp , we further adopt the convention c log 0 = for c > 0. With this convention, if D(p q) < , then p(x) = 0 whenever q(x) = 0, i.e., Sp Sq . In the literature, the informational divergence is also referred to as relative entropy or the Kullback-Leibler distance. We note that D(p q) is not symmetrical in p and q, so it is not a true metric or "distance." Moreover, D( ) does not satisfy the triangular inequality (see Problem 14). In the rest of the book, the informational divergence will be referred to as divergence for brevity. Before we prove that divergence is always nonnegative, we first establish the following simple but important inequality called the fundamental inequality in information theory. Lemma 2.29 (Fundamental Inequality). For any a > 0, ln a a - 1 with equality if and only if a = 1. Proof. Let f (a) = ln a - a + 1. Then f (a) = 1/a - 1 and f (a) = -1/a2 . Since f (1) = 0, f (1) = 0, and f (1) = -1 < 0, we see that f (a) attains its maximum value 0 when a = 1. This proves (2.105). It is also clear that equality holds in (2.105) if and only if a = 1. Figure 2.3 is an illustration of the fundamental inequality. (2.105) 24 2 Information Measures 1.5 a !1 ln a 1 0.5 0 1 a !0.5 !1 !1.5 !0.5 0 0.5 1 1.5 2 2.5 3 Fig. 2.3. The fundamental inequality ln a a - 1. Corollary 2.30. For any a > 0, ln a 1 - with equality if and only if a = 1. Proof. This can be proved by replacing a by 1/a in (2.105). We can see from Figure 2.3 that the fundamental inequality results from the convexity of the logarithmic function. In fact, many important results in information theory are also direct or indirect consequences of the convexity of the logarithmic function! Theorem 2.31 (Divergence Inequality). For any two probability distributions p and q on a common alphabet X , D(p q) 0 with equality if and only if p = q. Proof. If q(x) = 0 for some x Sp , then D(p q) = and the theorem is trivially true. Therefore, we assume that q(x) > 0 for all x Sp . Consider D(p q) = (log e) xSp 1 a (2.106) (2.107) p(x) ln p(x) q(x) q(x) p(x) q(x) xSp (2.108) (2.109) (log e) xSp p(x) 1 - = (log e) xSp p(x) - (2.110) (2.111) 0, 2.5 Informational Divergence 25 where (2.109) results from an application of (2.106), and (2.111) follows from q(x) 1 = xSp xSp p(x). (2.112) This proves (2.107). For equality to hold in (2.107), equality must hold in (2.109) for all x Sp and also in (2.111). For the former, we see from Lemma 2.29 that this is the case if and only if p(x) = q(x) for all x Sp , (2.113) which implies q(x) = xSp xSp p(x) = 1, (2.114) i.e., (2.111) holds with equality. Thus (2.113) is a necessary and sufficient condition for equality to hold in (2.107). It is immediate that p = q implies (2.113), so it remains to prove the converse. Since x q(x) = 1 and q(x) 0 for all x, p(x) = q(x) for all x Sp implies q(x) = 0 for all x Sp , and therefore p = q. The theorem is proved. We now prove a very useful consequence of the divergence inequality called the log-sum inequality. Theorem 2.32 (Log-Sum Inequality). For positive numbers a1 , a2 , and nonnegative numbers b1 , b2 , such that i ai < and 0 < i bi < , ai log i ai bi ai i log i ai i bi (2.115) with the convention that log ai bi = constant for all i. ai 0 = . Moreover, equality holds if and only if The log-sum inequality can easily be understood by writing it out for the case when there are two terms in each of the summations: a1 a2 a1 + a2 a1 log + a2 log (a1 + a2 ) log . (2.116) b1 b2 b1 + b2 Proof of Theorem 2.32. Let ai = ai / j aj and bi = bi / j bj . Then {ai } and {bi } are probability distributions. Using the divergence inequality, we have 0 i ai log ai bi j (2.117) aj aj (2.118) j = i ai / ai log bi / j aj 1 j j bj = aj ai log i ai - bi ai i log j bj , (2.119) 26 2 Information Measures ai bi which implies (2.115). Equality holds if and only if ai = bi for all i, or constant for all i. The theorem is proved. = One can also prove the divergence inequality by using the log-sum inequality (see Problem 20), so the two inequalities are in fact equivalent. The log-sum inequality also finds application in proving the next theorem which gives a lower bound on the divergence between two probability distributions on a common alphabet in terms of the variational distance between them. We will see further applications of the log-sum inequality when we discuss the convergence of some iterative algorithms in Chapter 9. Theorem 2.33 (Pinsker's Inequality). D(p q) 1 V 2 (p, q). 2 ln 2 (2.120) Both divergence and the variational distance can be used as measures of the difference between two probability distributions defined on the same alphabet. Pinsker's inequality has the important implication that for two probability distributions p and q defined on the same alphabet, if D(p q) or D(q p) is small, then so is V (p, q). Furthermore, for a sequence of probability distributions qk , as k , if D(p qk ) 0 or D(qk p) 0, then V (p, qk ) 0. In other words, convergence in divergence is a stronger notion of convergence than convergence in variational distance. The proof of Pinsker's inequality as well as its consequence discussed above is left as an exercise (see Problems 23 and 24). 2.6 The Basic Inequalities In this section, we prove that all Shannon's information measures, namely entropy, conditional entropy, mutual information, and conditional mutual information are always nonnegative. By this, we mean that these quantities are nonnegative for all joint distributions for the random variables involved. Theorem 2.34. For random variables X, Y , and Z, I(X; Y |Z) 0, (2.121) with equality if and only if X and Y are independent when conditioning on Z. Proof. Observe that 2.6 The Basic Inequalities 27 I(X; Y |Z) = x,y,z p(x, y, z) log p(z) z x,y p(x, y|z) p(x|z)p(y|z) p(x, y|z) p(x|z)p(y|z) (2.122) (2.123) (2.124) = = z p(x, y|z) log p(z)D(pXY |z pX|z pY |z ), where we have used pXY |z to denote {p(x, y|z), (x, y) X Y}, etc. Since for a fixed z, both pXY |z and pX|z pY |z are joint probability distributions on X Y, we have D(pXY |z pX|z pY |z ) 0. (2.125) Therefore, we conclude that I(X; Y |Z) 0. Finally, we see from Theorem 2.31 that I(X; Y |Z) = 0 if and only if for all z Sz , p(x, y|z) = p(x|z)p(y|z), or p(x, y, z) = p(x, z)p(y|z) (2.127) for all x and y. Therefore, X and Y are independent conditioning on Z. The proof is accomplished. As we have seen in Section 2.2 that all Shannon's information measures are special cases of conditional mutual information, we already have proved that all Shannon's information measures are always nonnegative. The nonnegativity of all Shannon's information measures is called the basic inequalities. For entropy and conditional entropy, we offer the following more direct proof for their nonnegativity. Consider the entropy H(X) of a random variable X. For all x SX , since 0 < p(x) 1, log p(x) 0. It then follows from the definition in (2.35) that H(X) 0. For the conditional entropy H(Y |X) of random variable Y given random variable X, since H(Y |X = x) 0 for each x SX , we see from (2.43) that H(Y |X) 0. Proposition 2.35. H(X) = 0 if and only if X is deterministic. Proof. If X is deterministic, i.e., there exists x X such that p(x ) = 1 and p(x) = 0 for all x = x , then H(X) = -p(x ) log p(x ) = 0. On the other hand, if X is not deterministic, i.e., there exists x X such that 0 < p(x ) < 1, then H(X) -p(x ) log p(x ) > 0. Therefore, we conclude that H(X) = 0 if and only if X is deterministic. Proposition 2.36. H(Y |X) = 0 if and only if Y is a function of X. Proof. From (2.43), we see that H(Y |X) = 0 if and only if H(Y |X = x) = 0 for each x SX . Then from the last proposition, this happens if and only if Y is deterministic for each given x. In other words, Y is a function of X. (2.126) 28 2 Information Measures Proposition 2.37. I(X; Y ) = 0 if and only if X and Y are independent. Proof. This is a special case of Theorem 2.34 with Z being a degenerate random variable. One can regard (conditional) mutual information as a measure of (conditional) dependency between two random variables. When the (conditional) mutual information is exactly equal to 0, the two random variables are (conditionally) independent. We refer to inequalities involving Shannon's information measures only (possibly with constant terms) as information inequalities. The basic inequalities are important examples of information inequalities. Likewise, we refer to identities involving Shannon's information measures only as information identities. From the information identities (2.47), (2.57), and (2.65), we see that all Shannon's information measures can be expressed as linear combinations of entropies provided that the latter are all finite. Specifically, H(Y |X) = H(X, Y ) - H(X), I(X; Y ) = H(X) + H(Y ) - H(X, Y ), and I(X; Y |Z) = H(X, Z) + H(Y, Z) - H(X, Y, Z) - H(Z). (2.130) (2.128) (2.129) Therefore, an information inequality is an inequality which involves only entropies. As we will see later in the book, information inequalities form the most important set of tools for proving converse coding theorems in information theory. Except for a number of so-called non-Shannon-type inequalities, all known information inequalities are implied by the basic inequalities. Information inequalities will be studied systematically in Chapters 13, 14, and 15. In the next section, we will prove some consequences of the basic inequalities which are often used in information theory. 2.7 Some Useful Information Inequalities In this section, we prove some useful consequences of the basic inequalities introduced in the last section. Note that the conditional versions of these inequalities can be proved by techniques similar to those used in the proof of Proposition 2.25. Theorem 2.38 (Conditioning Does Not Increase Entropy). H(Y |X) H(Y ) with equality if and only if X and Y are independent. (2.131) 2.7 Some Useful Information Inequalities 29 Proof. This can be proved by considering H(Y |X) = H(Y ) - I(X; Y ) H(Y ), (2.132) where the inequality follows because I(X; Y ) is always nonnegative. The inequality is tight if and only if I(X; Y ) = 0, which is equivalent by Proposition 2.37 to X and Y being independent. Similarly, it can be shown that H(Y |X, Z) H(Y |Z), (2.133) which is the conditional version of the above proposition. These results have the following interpretation. Suppose Y is a random variable we are interested in, and X and Z are side-information about Y . Then our uncertainty about Y cannot be increased on the average upon receiving side-information X. Once we know X, our uncertainty about Y again cannot be increased on the average upon further receiving side-information Z. Remark Unlike entropy, the mutual information between two random variables can be increased by conditioning on a third random variable. We refer the reader to Section 3.4 for a discussion. Theorem 2.39 (Independence Bound for Entropy). n H(X1 , X2 , , Xn ) i=1 H(Xi ) (2.134) with equality if and only if Xi , i = 1, 2, , n are mutually independent. Proof. By the chain rule for entropy, n H(X1 , X2 , , Xn ) = i=1 n H(Xi |X1 , , Xi-1 ) H(Xi ), i=1 (2.135) (2.136) where the inequality follows because we have proved in the last theorem that conditioning does not increase entropy. The inequality is tight if and only if it is tight for each i, i.e., H(Xi |X1 , , Xi-1 ) = H(Xi ) (2.137) for 1 i n. From the last theorem, this is equivalent to Xi being independent of X1 , X2 , , Xi-1 for each i. Then 30 2 Information Measures p(x1 , x2 , , xn ) = p(x1 , x2 , , xn-1 )p(xn ) = p(p(x1 , x2 , , xn-2 )p(xn-1 )p(xn ) . . . = p(x1 )p(x2 ) p(xn ) for all x1 , x2 , , xn , i.e., X1 , X2 , , Xn are mutually independent. Alternatively, we can prove the theorem by considering n (2.138) (2.139) (2.140) H(Xi ) - H(X1 , X2 , , Xn ) i=1 n =- i=1 E log p(Xi ) + E log p(X1 , X2 , , Xn ) (2.141) (2.142) (2.143) (2.144) (2.145) = -E log[p(X1 )p(X2 ) p(Xn )] + E log p(X1 , X2 , , Xn ) p(X1 , X2 , , Xn ) = E log p(X1 )p(X2 ) p(Xn ) = D(pX1 X2 Xn pX1 pX2 pXn ) 0, where equality holds if and only if p(x1 , x2 , , xn ) = p(x1 )p(x2 ) p(xn ) for all x1 , x2 , , xn , i.e., X1 , X2 , , Xn are mutually independent. Theorem 2.40. I(X; Y, Z) I(X; Y ), with equality if and only if X Y Z forms a Markov chain. Proof. By the chain rule for mutual information, we have I(X; Y, Z) = I(X; Y ) + I(X; Z|Y ) I(X; Y ). (2.146) (2.147) (2.148) The above inequality is tight if and only if I(X; Z|Y ) = 0, or X Y Z forms a Markov chain. The theorem is proved. Lemma 2.41. If X Y Z forms a Markov chain, then I(X; Z) I(X; Y ) and I(X; Z) I(Y ; Z). (2.150) (2.149) 2.7 Some Useful Information Inequalities 31 Before proving this inequality, we first discuss its meaning. Suppose X is a random variable we are interested in, and Y is an observation of X. If we infer X via Y , our uncertainty about X on the average is H(X|Y ). Now suppose we process Y (either deterministically or probabilistically) to obtain a random variable Z. If we infer X via Z, our uncertainty about X on the average is H(X|Z). Since X Y Z forms a Markov chain, from (2.149), we have H(X|Z) = H(X) - I(X; Z) H(X) - I(X; Y ) = H(X|Y ), (2.151) (2.152) (2.153) i.e., further processing of Y can only increase our uncertainty about X on the average. Proof of Lemma 2.41. Assume X Y Z, i.e., X Z|Y . By Theorem 2.34, we have I(X; Z|Y ) = 0. (2.154) Then I(X; Z) = I(X; Y, Z) - I(X; Y |Z) I(X; Y, Z) = I(X; Y ) + I(X; Z|Y ) = I(X; Y ). (2.155) (2.156) (2.157) (2.158) In (2.155) and (2.157), we have used the chain rule for mutual information. The inequality in (2.156) follows because I(X; Y |Z) is always nonnegative, and (2.158) follows from (2.154). This proves (2.149). Since X Y Z is equivalent to Z Y X, we also have proved (2.150). This completes the proof of the lemma. From Lemma 2.41, we can prove the more general data processing theorem. Theorem 2.42 (Data Processing Theorem). If U X Y V forms a Markov chain, then I(U ; V ) I(X; Y ). (2.159) Proof. Assume U X Y V . Then by Proposition 2.10, we have U X Y and U Y V . From the first Markov chain and Lemma 2.41, we have I(U ; Y ) I(X; Y ). (2.160) From the second Markov chain and Lemma 2.41, we have I(U ; V ) I(U ; Y ). (2.161) Combining (2.160) and (2.161), we obtain (2.159), proving the theorem. 32 2 Information Measures 2.8 Fano's Inequality In the last section, we have proved a few information inequalities involving only Shannon's information measures. In this section, we first prove an upper bound on the entropy of a random variable in terms of the size of the alphabet. This inequality is then used in the proof of Fano's inequality, which is extremely useful in proving converse coding theorems in information theory. Theorem 2.43. For any random variable X, H(X) log |X |, (2.162) where |X | denotes the size of the alphabet X . This upper bound is tight if and only if X is distributed uniformly on X . Proof. Let u be the uniform distribution on X , i.e., u(x) = |X |-1 for all x X . Then log |X | - H(X) =- xSX p(x) log |X |-1 + xSX p(x) log p(x) p(x) log p(x) (2.163) (2.164) (2.165) (2.166) (2.167) =- xSX p(x) log u(x) + xSX = xSX p(x) log p(x) u(x) = D(p u) 0, proving (2.162). This upper bound is tight if and if only D(p u) = 0, which from Theorem 2.31 is equivalent to p(x) = u(x) for all x X , completing the proof. Corollary 2.44. The entropy of a random variable may take any nonnegative real value. Proof. Consider a random variable X defined on a fixed finite alphabet X . We see from the last theorem that H(X) = log |X | is achieved when X is distributed uniformly on X . On the other hand, H(X) = 0 is achieved when X is deterministic. For 0 a |X |-1 , let g(a) = H ({1 - (|X | - 1)a, a, , a}) = -l(1 - (|X | - 1)a) - (|X | - 1)l(a), (2.168) (2.169) where l() is defined in (2.73). Note that g(a) is continuous in a, with g(0) = 0 and g(|X |-1 ) = log |X |. For any value 0 < b < log |X |, by the intermediate 2.8 Fano's Inequality 33 value theorem of continuous functions, there exists a distribution for X such that H(X) = b. Then we see that H(X) can take any positive value by letting |X | be sufficiently large. This accomplishes the proof. Remark Let |X | = D, or the random variable X is a D-ary symbol. When the base of the logarithm is D, (2.162) becomes HD (X) 1. (2.170) Recall that the unit of entropy is the D-it when the logarithm is in the base D. This inequality says that a D-ary symbol can carry at most 1 D-it of information. This maximum is achieved when X has a uniform distribution. We already have seen the binary case when we discuss the binary entropy function hb (p) in Section 2.2. We see from Theorem 2.43 that the entropy of a random variable is finite as long as it has a finite alphabet. However, if a random variable has a countable alphabet7 , its entropy may or may not be finite. This will be shown in the next two examples. Example 2.45. Let X be a random variable such that Pr{X = i} = 2-i , i = 1, 2, . Then H2 (X) = i=1 (2.171) i2-i = 2, (2.172) which is finite. For a random variable X with a countable alphabet and finite entropy, we show in Appendix 2.A that the entropy of X can be approximated by the entropy of a truncation of the distribution of X. Example 2.46. Let Y be a random variable which takes values in the subset of pairs of integers (i, j) : 1 i < and 1 j such that for all i and j. First, we check that 22 2i i (2.173) Pr{Y = (i, j)} = 2-2 i (2.174) 7 An alphabet is countable means that it is either finite or countably infinite. 34 2 Information Measures 2 i 2 /2 i Pr{Y = (i, j)} = i=1 j=1 i=1 2-2 i 22 2i i = 1. (2.175) Then H2 (Y ) = - 2 i 2 /2 i 2 i=1 j=1 -2i log2 2 -2i = i=1 1, (2.176) which does not converge. ^ Let X be a random variable and X be an estimate on X which takes value in the same alphabet X . Let the probability of error Pe be ^ Pe = Pr{X = X}. (2.177) ^ ^ If Pe = 0, i.e., X = X with probability 1, then H(X|X) = 0 by Proposi^ tion 2.36. Intuitively, if Pe is small, i.e., X = X with probability close to ^ 1, then H(X|X) should be close to 0. Fano's inequality makes this intuition precise. ^ Theorem 2.47 (Fano's Inequality). Let X and X be random variables taking values in the same alphabet X . Then ^ H(X|X) hb (Pe ) + Pe log(|X | - 1), where hb is the binary entropy function. Proof. Define a random variable Y = ^ 0 if X = X ^ 1 if X = X. (2.179) (2.178) ^ The random variable Y is an indicator of the error event {X = X}, with ^ Pr{Y = 1} = Pe and H(Y ) = hb (Pe ). Since Y is a function X and X, ^ H(Y |X, X) = 0. Then ^ H(X|X) ^ ^ = H(X|X) + H(Y |X, X) ^ = H(X, Y |X) ^ ^ = H(Y |X) + H(X|X, Y ) ^ Y) H(Y ) + H(X|X, = H(Y ) + xX ^ (2.180) (2.181) (2.182) (2.183) (2.184) ^ ^ Pr{X = x, Y = 0}H(X|X = x, Y = 0) ^ ^ (2.185) ^ ^ +Pr{X = x, Y = 1}H(X|X = x, Y = 1) . ^ ^ 2.8 Fano's Inequality 35 In the above, (2.181) follows from (2.180), (2.184) follows because conditioning does not increase entropy, and (2.185) follows from an application of (2.43). ^ Now X must take the value x if X = x and Y = 0. In other words, X is ^ ^ ^ = x and Y = 0. Therefore, by Proposiconditionally deterministic given X ^ tion 2.35, ^ H(X|X = x, Y = 0) = 0. ^ (2.186) ^ = x and Y = 1, then X must take a value in the set {x X : x = x} If X ^ ^ which contains |X | - 1 elements. By Theorem 2.43, we have ^ H(X|X = x, Y = 1) log(|X | - 1), ^ where this upper bound does not depend on x. Hence, ^ ^ H(X|X) hb (Pe ) + xX ^ (2.187) ^ Pr{X = x, Y = 1} log(|X | - 1) ^ (2.188) (2.189) (2.190) = hb (Pe ) + Pr{Y = 1} log(|X | - 1) = hb (Pe ) + Pe log(|X | - 1), which completes the proof. Very often, we only need the following simplified version when we apply Fano's inequality. The proof is omitted. ^ Corollary 2.48. H(X|X) < 1 + Pe log |X |. Fano's inequality has the following implication. If the alphabet X is finite, ^ as Pe 0, the upper bound in (2.178) tends to 0, which implies H(X|X) also tends to 0. However, this is not necessarily the case if X is countable, which is shown in the next example. ^ Example 2.49. Let X take the value 0 with probability 1. Let Z be an independent binary random variable taking values in {0, 1}. Define the random variable X by 0 if Z = 0 X= (2.191) Y if Z = 1, where Y is the random variable in Example 2.46 whose entropy is infinity. Let ^ Pe = Pr{X = X} = Pr{Z = 1}. Then ^ H(X|X) = H(X) H(X|Z) = Pr{Z = 0}H(X|Z = 0) + Pr{Z = 1}H(X|Z = 1) = (1 - Pe ) 0 + Pe H(Y ) = (2.193) (2.194) (2.195) (2.196) (2.197) (2.198) (2.192) 36 2 Information Measures ^ for any Pe > 0. Therefore, H(X|X) does not tend to 0 as Pe 0. 2.9 Maximum Entropy Distributions In Theorem 2.43, we have proved that for any random variable X, H(X) log |X |, (2.199) with equality when X is distributed uniformly over X . In this section, we revisit this result in the context that X is a real random variable. To simplify our discussion, all the logarithms are in the base e. Consider the following problem: Maximize H(p) over all probability distributions p defined on a countable subset S of the set of real numbers, subject to p(x)ri (x) = ai xSp for 1 i m, (2.200) where Sp S and ri (x) is defined for all x S. The following theorem renders a solution to this problem. Theorem 2.50. Let p (x) = e-0 - m i=1 i ri (x) (2.201) for all x S, where 0 , 1 , , m are chosen such that the constraints in (2.200) are satisfied. Then p maximizes H(p) over all probability distribution p on S, subject to the constraints in (2.200). Proof. For any p satisfying the constraints in (2.200), consider H(p ) - H(p) =- xS p (x) ln p (x) + xSp p(x) ln p(x) (2.202) =- xS p (x) -0 - i i ri (x) + xSp p(x) ln p(x) (2.203) = 0 xS p (x) + i i xS p (x)ri (x) + xSp p(x) ln p(x) (2.204) (2.205) = 0 1 + i i ai + xSp p(x) ln p(x) i i xSp = 0 xSp p(x) + p(x)ri (x) + xSp p(x) ln p(x) (2.206) 2.9 Maximum Entropy Distributions 37 =- xSp p(x) -0 - i i ri (x) + xSp p(x) ln p(x) (2.207) (2.208) (2.209) (2.210) (2.211) =- xSp p(x) ln p (x) + xSp p(x) ln p(x) = xSp p(x) ln p(x) p (x) = D(p p ) 0. In the above, (2.207) is obtained from (2.203) by replacing p (x) by p(x) and x S by x Sp in the first summation, while the intermediate steps (2.204) to (2.206) are justified by noting that both p and p satisfy the constraints in (2.200). The last step is an application of the divergence inequality (Theorem 2.31). The proof is accomplished. Remark For all x S, p (x) > 0, so that Sp = S. The following corollary of Theorem 2.50 is rather subtle. Corollary 2.51. Let p be a probability distribution defined on S with p (x) = e-0 - m i=1 i ri (x) (2.212) for all x S. Then p maximizes H(p) over all probability distribution p defined on S, subject to the constraints p(x)ri (x) = xSp xS p (x)ri (x) for 1 i m. (2.213) Example 2.52. Let S be finite and let the set of constraints in (2.200) be empty. Then p (x) = e-0 , (2.214) a constant that does not depend on x. Therefore, p is simply the uniform distribution over S, i.e., p (x) = |S|-1 for all x S. This is consistent with Theorem 2.43. Example 2.53. Let S = {0, 1, 2, }, and let the set of constraints in (2.200) be p(x)x = a, (2.215) x where a 0, i.e., the mean of the distribution p is fixed at some nonnegative value a. We now determine p using the prescription in Theorem 2.50. Let qi = e-i (2.216) 38 2 Information Measures for i = 0, 1. Then by (2.201), x p (x) = q0 q1 . (2.217) Evidently, p is a geometric distribution, so that q0 = 1 - q1 . (2.218) Finally, we invoke the constraint (2.200) on p to obtain q1 = (a + 1)-1 . The details are omitted. 2.10 Entropy Rate of a Stationary Source In the previous sections, we have discussed various properties of the entropy of a finite collection of random variables. In this section, we discuss the entropy rate of a discrete-time information source. A discrete-time information source {Xk , k 1} is an infinite collection of random variables indexed by the set of positive integers. Since the index set is ordered, it is natural to regard the indices as time indices. We will refer to the random variables Xk as letters. We assume that H(Xk ) < for all k. Then for any finite subset A of the index set {k : k 1}, we have H(Xk , k A) kA H(Xk ) < . (2.219) However, it is not meaningful to discuss H(Xk , k 1) because the joint entropy of an infinite collection of letters is infinite except for very special cases. On the other hand, since the indices are ordered, we can naturally define the entropy rate of an information source, which gives the average entropy per letter of the source. Definition 2.54. The entropy rate of an information source {Xk } is defined as 1 (2.220) HX = lim H(X1 , X2 , , Xn ) n n when the limit exists. We show in the next two examples that the entropy rate of a source may or may not exist. Example 2.55. Let {Xk } be an i.i.d. source with generic random variable X. Then 1 nH(X) lim H(X1 , X2 , , Xn ) = lim (2.221) n n n n = lim H(X) (2.222) n = H(X), (2.223) i.e., the entropy rate of an i.i.d. source is the entropy of any of its single letters. 2.10 Entropy Rate of a Stationary Source 39 Example 2.56. Let {Xk } be a source such that Xk are mutually independent and H(Xk ) = k for k 1. Then 1 1 H(X1 , X2 , , Xn ) = n n = n k k=1 (2.224) (2.225) (2.226) 1 n(n + 1) n 2 1 = (n + 1), 2 which does not converge as n although H(Xk ) < for all k. Therefore, the entropy rate of {Xk } does not exist. Toward characterizing the asymptotic behavior of {Xk }, it is natural to consider the limit HX = lim H(Xn |X1 , X2 , , Xn-1 ) n (2.227) if it exists. The quantity H(Xn |X1 , X2 , , Xn-1 ) is interpreted as the conditional entropy of the next letter given that we know all the past history of the source, and HX is the limit of this quantity after the source has been run for an indefinite amount of time. Definition 2.57. An information source {Xk } is stationary if X1 , X2 , , Xm and X1+l , X2+l , , Xm+l have the same joint distribution for any m, l 1. In the rest of the section, we will show that stationarity is a sufficient condition for the existence of the entropy rate of an information source. Lemma 2.58. Let {Xk } be a stationary source. Then HX exists. Proof. Since H(Xn |X1 , X2 , , Xn-1 ) is lower bounded by zero for all n, it suffices to prove that H(Xn |X1 , X2 , , Xn-1 ) is non-increasing in n to conclude that the limit HX exists. Toward this end, for n 2, consider H(Xn |X1 , X2 , , Xn-1 ) H(Xn |X2 , X3 , , Xn-1 ) = H(Xn-1 |X1 , X2 , , Xn-2 ), (2.230) (2.231) (2.229) (2.228) where the last step is justified by the stationarity of {Xk }. The lemma is proved. 40 2 Information Measures Lemma 2.59 (Cesro Mean). Let ak and bk be real numbers. If an a as a n 1 n and bn = n k=1 ak , then bn a as n . Proof. The idea of the lemma is the following. If an a as n , then the average of the first n terms in {ak }, namely bn , also tends to a as n . The lemma is formally proved as follows. Since an a as n , for every > 0, there exists N ( ) such that |an - a| < for all n > N ( ). For n > N ( ), consider |bn - a| = = 1 n 1 n 1 n n ai - a i=1 n (2.232) (ai - a) i=1 n (2.233) (2.234) n |ai - a| i=1 N( ) 1 = |ai - a| + n i=1 < 1 n 1 n N( ) |ai - a| (2.235) i=N ( )+1 |ai - a| + i=1 N( ) (n - N ( )) n (2.236) < |ai - a| + . i=1 (2.237) The first term tends to 0 as n . Therefore, for any > 0, by taking n to be sufficiently large, we can make |bn - a| < 2 . Hence bn a as n , proving the lemma. We now prove that HX is an alternative definition/interpretation of the entropy rate of {Xk } when {Xk } is stationary. Theorem 2.60. The entropy rate HX of a stationary source {Xk } exists and is equal to HX . Proof. Since we have proved in Lemma 2.58 that HX always exists for a stationary source {Xk }, in order to prove the theorem, we only have to prove that HX = HX . By the chain rule for entropy, 1 1 H(X1 , X2 , , Xn ) = n n Since k n H(Xk |X1 , X2 , , Xk-1 ). k=1 (2.238) lim H(Xk |X1 , X2 , , Xk-1 ) = HX (2.239) Appendix 2.A: Approximation of Random Variables by Truncation 41 from (2.227), it follows from Lemma 2.59 that HX = lim The theorem is proved. In this theorem, we have proved that the entropy rate of a random source {Xk } exists under the fairly general assumption that {Xk } is stationary. However, the entropy rate of a stationary source {Xk } may not carry any physical meaning unless {Xk } is also ergodic. This will be explained when we discuss the Shannon-McMillan-Breiman Theorem in Section 5.4. 1 H(X1 , X2 , , Xn ) = HX . n (2.240) n Appendix 2.A: Approximation of Random Variables with Countably Infinite Alphabets by Truncation Let X be a random variable with a countable alphabet X such that H(X) < . Without loss of generality, X is taken to be the set of positive integers. Define a random variable X(m) which takes values in Nm = {1, 2, , m} such that Pr{X(m) = k} = Pr{X = k} Pr{X Nm } (2.241) (2.242) for all k Nm , i.e., the distribution of X(m) is the truncation of the distribution of X up to m. It is intuitively correct that H(X(m)) H(X) as m , which we formally prove in this appendix. For every m 1, define the binary random variable 1 if X m B(m) = (2.243) 0 if X > m. Consider m H(X) = - - Pr{X = k} log Pr{X = k} k=1 Pr{X = k} log Pr{X = k}. k=m+1 (2.244) As m , m - k=1 Pr{X = k} log Pr{X = k} H(X). (2.245) Since H(X) < , 42 2 Information Measures - k=m+1 Pr{X = k} log Pr{X = k} 0 (2.246) as k . Now consider H(X) = H(X|B(m)) + I(X; B(m)) = H(X|B(m) = 1)Pr{B(m) = 1} + H(X|B(m) = 0) Pr{B(m) = 0} + I(X; B(m)) = H(X(m))Pr{B(m) = 1} + H(X|B(m) = 0) Pr{B(m) = 0} + I(X; B(m)). (2.249) (2.248) (2.247) As m , H(B(m)) 0 since Pr{B(m) = 1} 1. This implies I(X; B(m)) 0 because I(X; B(m)) H(B(m)). In (2.249), we further consider H(X|B(m) = 0)Pr{B(m) = 0} (2.250) =- k=m+1 Pr{X = k} log Pr{X = k} Pr{B(m) = 0} (2.251) =- k=m+1 Pr{X = k}(log Pr{X = k} (2.252) - log Pr{B(m) = 0}) =- k=m+1 (Pr{X = k} log Pr{X = k}) + k=m+1 Pr{X = k} log Pr{B(m) = 0} (2.253) =- k=m+1 Pr{X = k} log Pr{X = k} (2.254) +Pr{B(m) = 0} log Pr{B(m) = 0}. As m , the summation above tends to 0 by (2.246). Since Pr{B(m) = 0} 0, Pr{B(m) = 0} log Pr{B(m) = 0} 0. Therefore, H(X|B(m) = 0)Pr{B(m) = 0} 0, and we see from (2.249) that H(X(m)) H(X) as m . (2.255) Chapter Summary 43 Chapter Summary Markov Chain: X Y Z forms a Markov chain if and only if p(x, y, z) = a(x, y)b(y, z) for all x, y, and z such that p(y) > 0. Shannon's Information Measures: H(X) = - x p(x) log p(x) = -E log p(X) p(x, y) log p(x, y) p(X, Y ) = E log p(x)p(y) p(X)p(Y ) I(X; Y ) = x,y H(Y |X) = - x,y p(x, y) log p(y|x) = -E log p(Y |X) p(x, y, z) log p(X, Y |Z) p(x, y|z) = E log . p(x|z)p(y|z) p(X|Z)p(Y |Z) I(X; Y |Z) = x,y,z Some Useful Identitites: H(X) = I(X; X) H(Y |X) = H(X, Y ) - H(X) I(X; Y ) = H(X) - H(X|Y ) I(X; Y |Z) = H(X|Z) - H(X|Y, Z). Chain Rule for Entropy: n H(X1 , X2 , , Xn ) = i=1 H(Xi |X1 , , Xi-1 ). Chain Rule for Mutual Information: n I(X1 , X2 , , Xn ; Y ) = i=1 I(Xi ; Y |X1 , , Xi-1 ). Informational Divergence: For two probability distributions p and q on a common alphabet X , D(p q) = x p(x) log p(x) p(X) = Ep log . q(x) q(X) Fundamental Inequality: For any a > 0, ln a a - 1, with equality if and only if a = 1. 44 2 Information Measures Divergence Inequality: D(p q) 0, with equality if and only if p = q. Log-Sum Inequality: For positive numbers a1 , a2 , and nonnegative numbers b1 , b2 , such that i ai < and 0 < i bi < , ai log i ai bi ai i log i ai . i bi Equality holds if and only if ai bi = constant for all i. The Basic Inequalities: All Shannon's information measures are nonnegative. Some Useful Properties of Shannon's Information Measures: H(X) log |X | with equality if and only if X is uniform. H(X) = 0 if and only if X is deterministic. H(Y |X) = 0 if and only if Y is a function of X. I(X; Y ) = 0 if and only X and Y are independent. ^ Fano's Inequality: Let X and X be random variables taking values in the same alphabet X . Then 1. 2. 3. 4. ^ H(X|X) hb (Pe ) + Pe log(|X | - 1). Conditioning Does Not Increase Entropy: H(Y |X) H(Y ), with equality if and only if X and Y are independent. Independence Bound for Entropy: n H(X1 , X2 , , Xn ) i=1 H(Xi ) with equality if and only if Xi , i = 1, 2, , n are mutually independent. Data Processing Theorem: If U X Y V forms a Markov chain, then I(U ; V ) I(X; Y ). Maximum Entropy Distributions: Let p (x) = e-0 - m i=1 i ri (x) for all x S, where 0 , 1 , , m are chosen such that the constraints p(x)ri (x) = ai xSp for 1 i m are satisfied. Then p maximizes H(p) over all probability distributions p on S subject to the above constraints. Entropy Rate of a Stationary Source: Problems 45 1. The entropy rate of an information source {Xk } is defined as HX = lim n 1 H(X1 , X2 , , Xn ) n when the limit exists. 2. The entropy rate HX of a stationary source {Xk } exists and is equal to HX = lim H(Xn |X1 , X2 , , Xn-1 ). n Problems 1. Let X and Y be random variables with alphabets X = Y = {1, 2, 3, 4, 5} and joint distribution p(x, y) given by 11111 2 1 2 0 0 1 2 0 1 1 1. 25 0 3 0 2 0 00113 Calculate H(X), H(Y ), H(X|Y ), H(Y |X), and I(X; Y ). 2. Prove Propositions 2.8, 2.9, 2.10, 2.19, 2.21, and 2.22. 3. Give an example which shows that pairwise independence does not imply mutual independence. 4. Verify that p(x, y, z) as defined in Definition 2.4 is a probability distribution. You should exclude all the zero probability masses from the summation carefully. 5. Linearity of expectation It is well-known that expectation is linear, i.e., E[f (X) + g(Y )] = Ef (X) + Eg(Y ), where the summation in an expectation is taken over the corresponding alphabet. However, we adopt in information theory the convention that the summation in an expectation is taken over the corresponding support. Justify carefully the linearity of expectation under this convention. 6. The identity I(X; Y ) = H(X) - H(X|Y ) is invalid if H(X|Y ) (and hence H(X)) is equal to infinity. Give an example such that I(X; Y ) has a finite value but both H(X) and H(Y |X) are equal to infinity. 7. Let pXY and pXY be probability distributions defined on X Y, where X and Y are fixed finite alphabets. Prove that pXY pXY lim px = pX , where the limit is taken with respect to the variational distance. 8. Let pk and p be probability distributions defined on a common finite alphabet. Show that as k , if pk p in variational distance, then pk p in L2 , and vice versa. 46 2 Information Measures 9. Consider any probability distribution p(x, y, z) and let q(x, y, z) = p(x)p(y)p(z|x, y) if p(x, y) > 0 0 otherwise. a) Show that q(x, y, z) is in general not a probability distribution. b) By ignoring the fact that q(x, y, z) may not be a probability distribution, application of the divergence inequality D(p q) 0 would yield the inequality H(X) + H(Y ) + H(Z|X, Y ) H(X, Y, Z), which indeed holds for all jointly distributed random variables X, Y , and Z. Explain. 1 10. Let C = n=2 n(log n) . a) Prove that < if > 1 C = if 0 1. Then p (n) = [C n(log n) ]-1 , n = 2, 3, is a probability distribution for > 1. b) Prove that < if > 2 H(p ) = if 1 < 2. 11. Prove that H(p) is concave in p, i.e., for 0 1 and = 1 - , H(p1 ) + H(p2 ) H(p1 + p2 ). 12. Let (X, Y ) p(x, y) = p(x)p(y|x). a) Prove that for fixed p(x), I(X; Y ) is a convex functional of p(y|x). b) Prove that for fixed p(y|x), I(X; Y ) is a concave functional of p(x). 13. Do I(X; Y ) = 0 and I(X; Y |Z) = 0 imply each other? If so, give a proof. If not, give a counterexample. 14. Give an example for which D( ) does not satisfy the triangular inequality. 15. Let X be a function of Y . Prove that H(X) H(Y ). Interpret this result. 16. Prove that for any n 2, n H(X1 , X2 , , Xn ) i=1 H(Xi |Xj , j = i). 17. Prove that H(X1 , X2 ) + H(X2 , X3 ) + H(X1 , X3 ) 2H(X1 , X2 , X3 ). Hint: Sum the identities H(X1 , X2 , X3 ) = H(Xj , j = i) + H(Xi |Xj , j = i) for i = 1, 2, 3 and apply the result in Problem 16. Problems 47 18. For a subset of Nn = {1, 2, , n}, denote (Xi , i ) by X . For 1 k n, let H(X ) 1 . Hk = n k k :||=k Here Hk is interpreted as the average entropy per random variable when k random variables are taken from X1 , X2 , , Xn at a time. Prove that H1 H2 Hn . This sequence of inequalities, due to Han [147], is a generalization of the independence bound for entropy (Theorem 2.39). See Problem 6 in Chapter 21 for an application of these inequalities. 19. For a subset of Nn = {1, 2, , n}, let = Nn \ and denote (Xi , i ) by X . For 1 k n, let Hk = Prove that H1 H2 Hn . Note that Hn is equal to Hn in the last problem. This sequence of inequalities is again due to Han [147]. See Yeung and Cai [406] for an application of these inequalities. 20. Prove the divergence inequality by using the log-sum inequality. 21. Prove that D(p q) is convex in the pair (p, q), i.e., if (p1 , q1 ) and (p2 , q2 ) are two pairs of probability distributions on a common alphabet, then D(p1 + p2 q1 + q2 ) D(p1 q1 ) + D(p2 q2 ) for all 0 1, where = 1 - . 22. Let pXY and qXY be two probability distributions on X Y. Prove that D(pXY qXY ) D(pX qX ). 23. Pinsker's inequality Let V (p, q) denotes the variational distance between two probability distributions p and q on a common alphabet X . We will determine the largest c which satisfies D(p q) cd2 (p, q). a) Let A = {x : p(x) q(x)}, p = {p(A), 1 - p(A)}, and q = {q(A), 1 - ^ ^ q(A)}. Show that D(p q) D(^ q ) and V (p, q) = V (^, q ). p ^ p ^ b) Show that toward determining the largest value of c, we only have to consider the case when X is binary. c) By virtue of b), it suffices to determine the largest c such that p log p 1-p + (1 - p) log - 4c(p - q)2 0 q 1-q 1 n k :||=k H(X |X ) . k 48 2 Information Measures 24. 25. 26. 27. 28. for all 0 p, q 1, with the convention that 0 log 0 = 0 for b 0 b and a log a = for a > 0. By observing that equality in the above 0 holds if p = q and considering the derivative of the left hand side with respect to q, show that the largest value of c is equal to (2 ln 2)-1 . Let p and qk , k 1 be probability distributions on a common alphabet. Show that if qk converges to p in divergence, then it also converges to p in variational distance. Find a necessary and sufficient condition for Fano's inequality to be tight. Determine the probability distribution defined on {0, 1, , n} that maximizes the entropy subject to the constraint that the mean is equal to m, where 0 m n. 1 Show that for a stationary source {Xk }, n H(X1 , X2 , , Xn ) is nonincreasing in n. For real numbers > 1 and > 0 and an integer n , define the probability distribution 1 log 1 log log (,) , ,, , 0, 0, . Dn = 1- log n n log n n log n n Let = {1, 0, 0, . . .} be the deterministic distribution. a) Show that limn D ||Dn (,) = 0. . b) Determine limn H 29. Discontinuity of entropy with respect to convergence in divergence Let P be the set of all probability distributions on a countable alphabet. A function f : P is continuous with respect to convergence in divergence at P P if for any > 0, there exists > 0 such that |f (P ) - f (Q)| < for all Q P satisfying D(P Q) < ; otherwise, f is discontinuous at P . a) Let H : P be the entropy function. Show that H is discontinuous at the deterministic distribution = {1, 0, 0, , }. Hint: Use the results in Problem 28. b) Show that H is discontinuous at P = {p0 , p1 , p2 , } for all P such that H(P ) < . Hint: Consider the probability distribution Qn = p0 p0 p0 , p1 + , p2 + ,, log n n log n n log n p0 , pn+1 , pn+2 , . . . pn + n log n p0 - (,) Dn for large n. 30. Discontinuity of entropy with respect to convergence in variational distance Refer to Problem 29. The continuity of a function f : P with respect to convergence in variational distance can be defined similarly. Historical Notes 49 a) Show that if a function f is continuous with respect to convergence in variational distance, then it is also continuous with respect to convergence in divergence. Hint: Use Pinsker's inequality. b) Repeat b) in Problem 29 with continuity defined with respect to convergence in variational distance. 31. Continuity of the entropy function for a fixed finite alphabet Refer to Problems 29 and 30. Suppose the domain of H is confined to P , the set of all probability distributions on a fixed finite alphabet. Show that H is continuous with respect to convergence in divergence. 32. Let p = {p1 , p2 , , pn } and q = {q1 , q2 , , qn } be two sets of real numbers such that pi pi and qi qi for all i < i . We say that p is majorized m m by q if i=1 pi j=1 qj for all m = 1, 2, . . . , n, where equality holds when m = n. A function f : n is Schur-concave if f (p) f (q) whenever p is majorized by q. Now let p and q be probability distributions. We will show in the following steps that H() is Schur-concave. a) Show that for p = q, there exist 1 j < k n which satisfy the following: i) j is the largest index i such that pi < qi ii) k is the smallest index i such that i > j and pi > qi iii) pi = qi for all j < i < k. b) Consider the distribution q = {q1 , q2 , , qn } defined by qi = qi for i = j, k and (qj , qk ) = (pj , qk + (qj - pj )) if pk - qk qj - pj (qj - (pk - qk ), pk ) if pk - qk < qj - pj . Note that either qj = pj or qk = pk . Show that i) qi qi for all i i m m ii) i=1 pi i=1 qi for all m = 1, 2, , n iii) H(q ) H(q). c) Prove that H(p) H(q) by induction on the Hamming distance between p and q, i.e., the number of places where p and q differ. In general, if a concave function f is symmetric, i.e., f (p) = f (p ) where p is a permutation of p, then f is Schur-concave. We refer the reader to [246] for the theory of majorization. (Hardy, Littlewood, and Plya o [154].) Historical Notes The concept of entropy has its root in thermodynamics. Shannon [322] was the first to use entropy as a measure of information. Informational divergence was introduced by Kullback and Leibler [214], and it has been studied extensively by Csiszr [81] and Amari [14]. a Most of the materials in this chapter can be found in standard textbooks in information theory. The main concepts and results are due to Shannon 50 2 Information Measures [322]. Pinsker's inequality is due to Pinsker [292]. Fano's inequality has its origin in the converse proof of the channel coding theorem (to be discussed in Chapter 7) by Fano [107]. Generalizations of Fano's inequality which apply to random variables with countable alphabets have been obtained by Han and Verd [153] and by Ho [165] (see also [168]). Maximum entropy, a concept in u statistical mechanics, was expounded in Jaynes [186]. 3 The I-Measure In Chapter 2, we have illustrated the relationship between Shannon's information measures for two random variables by the diagram in Figure 2.2. For convenience, Figure 2.2 is reproduced in Figure 3.1 with the random variables X and Y replaced by X1 and X2 , respectively. This diagram suggests that Shannon's information measures for any n 2 random variables may have a set-theoretic structure. In this chapter, we develop a theory which establishes a one-to-one correspondence between Shannon's information measures and set theory in full generality. With this correspondence, manipulations of Shannon's information measures can be viewed as set operations, thus allowing the rich suite of tools in set theory to be used in information theory. Moreover, the structure of Shannon's information measures can easily be visualized by means of an information diagram if four or fewer random variables are involved. The use of information diagrams simplifies many difficult proofs in information theory H ( X1 ,X 2) H ( X1 X 2) H ( X 2 X1) H ( X1) I ( X1; X 2) H (X 2) Fig. 3.1. Relationship between entropies and mutual information for two random variables. 52 3 The I-Measure problems. More importantly, these results, which may be difficult to discover in the first place, can easily be obtained by inspection of an information diagram. The main concepts to be used in this chapter are from measure theory. However, it is not necessary for the reader to know measure theory to read this chapter. 3.1 Preliminaries In this section, we introduce a few basic concepts in measure theory which will be used subsequently. These concepts will be illustrated by simple examples. ~ ~ ~ Definition 3.1. The field Fn generated by sets X1 , X2 , , Xn is the collection of sets which can be obtained by any sequence of usual set operations ~ ~ ~ (union, intersection, complement, and difference) on X1 , X2 , , Xn . Definition 3.2. The atoms of Fn are sets of the form n Yi , where Yi is i=1 ~ ~c ~ either Xi or Xi , the complement of Xi . There are 2n atoms and 22 sets in Fn . Evidently, all the atoms in Fn are disjoint, and each set in Fn can be expressed uniquely as the union of a subset ~ ~ ~ of the atoms of Fn 1 . We assume that the sets X1 , X2 , , Xn intersect with each other generically, i.e., all the atoms of Fn are nonempty unless otherwise specified. ~ ~ Example 3.3. The sets X1 and X2 generate the field F2 . The atoms of F2 are ~ ~ ~c ~ ~ ~c ~c ~c X1 X2 , X1 X2 , X1 X2 , X1 X2 , (3.1) n which are represented by the four distinct regions in the Venn diagram in Figure 3.2. The field F2 consists of the unions of subsets of the atoms in (3.1). There are a total of 16 sets in F2 , which are precisely all the sets which can ~ ~ be obtained from X1 and X2 by the usual set operations. Definition 3.4. A real function defined on Fn is called a signed measure if it is set-additive, i.e., for disjoint A and B in Fn , (A B) = (A) + (B). For a signed measure , we have () = 0, which can be seen as follows. For any A in Fn , 1 (3.2) (3.3) We adopt the convention that the union of the empty subset of the atoms of Fn is the empty set. 3.2 The I-Measure for Two Random Variables 53 X1 X2 ~ ~ Fig. 3.2. The Venn diagram for X1 and X2 . (A) = (A ) = (A) + () (3.4) by set-additivity because A and are disjoint, which implies (3.3). A signed measure on Fn is completely specified by its values on the atoms of Fn . The values of on the other sets in Fn can be obtained via set-additivity. Example 3.5. A signed measure on F2 is completely specified by the values ~ ~ ~c ~ ~ ~c ~c ~c (X1 X2 ), (X1 X2 ), (X1 X2 ), (X1 X2 ). ~ The value of on X1 , for example, can be obtained as ~ ~ ~ ~ ~c (X1 ) = ((X1 X2 ) (X1 X2 )) ~ ~ ~ ~ = (X1 X2 ) + (X1 X c ). 2 (3.5) (3.6) (3.7) 3.2 The I-Measure for Two Random Variables To fix ideas, we first formulate in this section the one-to-one correspondence between Shannon's information measures and set theory for two random vari~ ~ ables. For random variables X1 and X2 , let X1 and X2 be sets corresponding ~ 1 and X2 generates the field F2 whose ~ to X1 and X2 , respectively. The sets X atoms are listed in (3.1). In our formulation, we set the universal set to ~ ~ X1 X2 for reasons which will become clear later. With this choice of , the ~ ~ Venn diagram for X1 and X2 is represented by the diagram in Figure 3.3. For ~ 1 and X2 are respectively labeled by X1 and X2 in the ~ simplicity, the sets X diagram. We call this the information diagram for the random variables X1 ~ ~ and X2 . In this diagram, the universal set, which is the union of X1 and X2 , is not shown explicitly just as in a usual Venn diagram. Note that with our ~c ~c choice of the universal set, the atom X1 X2 degenerates to the empty set, because 54 3 The I-Measure X1 X2 Fig. 3.3. The generic information diagram for X1 and X2 . ~c ~c ~ ~ X1 X2 = (X1 X2 )c = c = . (3.8) Thus this atom is not shown in the information diagram in Figure 3.3. For random variables X1 and X2 , the Shannon's information measures are H(X1 ), H(X2 ), H(X1 |X2 ), H(X2 |X1 ), H(X1 , X2 ), I(X1 ; X2 ). Writing A B c as A - B, we now define a signed measure2 by ~ ~ (X1 - X2 ) = H(X1 |X2 ) ~ ~ (X2 - X1 ) = H(X2 |X1 ), and ~ ~ (X1 X2 ) = I(X1 ; X2 ). (3.12) These are the values of on the nonempty atoms of F2 (i.e., atoms of F2 ~c ~c other than X1 X2 ). The values of on the other sets in F2 can be obtained via set-additivity. In particular, the relations ~ ~ (X1 X2 ) = H(X1 , X2 ) ~ (X1 ) = H(X1 ), and ~ (X2 ) = H(X2 ) ~ ~ (X1 X2 ) ~ ~ ~ ~ ~ ~ = (X1 - X2 ) + (X2 - X1 ) + (X1 X2 ) = H(X1 |X2 ) + H(X2 |X1 ) + I(X1 ; X2 ) = H(X1 , X2 ). (3.15) can readily be verified. For example, (3.13) is seen to be true by considering (3.13) (3.14) (3.10) (3.11) (3.9) (3.16) (3.17) (3.18) The right hand sides of (3.10) to (3.15) are the six Shannon's information measures for X1 and X2 in (3.9). Now observe that (3.10) to (3.15) are consistent with how the Shannon's information measures on the right hand side 2 It happens that defined here for n = 2 assumes only nonnegative values, but we will see in Section 3.4 that can assume negative values for n 3. 3.3 Construction of the I-Measure * 55 are identified in Figure 3.1, with the left circle and the right circle represent~ ~ ing the sets X1 and X2 , respectively. Specifically, in each of these equations, the left hand side and the right hand side correspond to each other via the following substitution of symbols: H/I , ; | -. Note that we make no distinction between the symbols H and I in this substitution. Thus, for two random variables X1 and X2 , Shannon's information measures can be regarded formally as a signed measure on F2 . We will refer to as the I-Measure for the random variables X1 and X2 3 . Upon realizing that Shannon's information measures can be viewed as a signed measure, we can apply the rich family of operations in set theory to information theory. This explains why Figure 3.1 or Figure 3.3 represents the relationships among all Shannon's information measures for two random variables correctly. As an example, consider the following set identity which is readily identified in Figure 3.3: ~ ~ ~ ~ ~ ~ (X1 X2 ) = (X1 ) + (X2 ) - (X1 X2 ). (3.20) (3.19) This identity is a special case of the inclusion-exclusion formula in set theory. By means of the substitution of symbols in (3.19), we immediately obtain the information identity H(X1 , X2 ) = H(X1 ) + H(X2 ) - I(X1 ; X2 ). (3.21) ~c ~c We end this section with a remark. The value of on the atom X1 X2 has no apparent information-theoretic meaning. In our formulation, we set the ~ ~ ~c ~c universal set to X1 X2 so that the atom X1 X2 degenerates to the empty ~c ~ c ) naturally vanishes because is a measure, so that set. Then (X1 X2 is completely specified by all Shannon's information measures involving the random variables X1 and X2 . 3.3 Construction of the I-Measure * We have constructed the I-Measure for two random variables in the last section. We now construct the I-Measure for any n 2 random variables. Consider n random variables X1 , X2 , , Xn . For any random variable X, ~ let X be a set corresponding to X. Let 3 The reader should not confuse with the probability measure defining the random variables X1 and X2 . The former, however, is determined by the latter. 56 3 The I-Measure Nn = {1, 2, , n}. (3.22) ~ ~ ~ Define the universal set to be the union of the sets X1 , X2 , , Xn , i.e., = iNn ~ Xi . (3.23) ~ ~ ~ We use Fn to denote the field generated by X1 , X2 , , Xn . The set A0 = iNn ~c Xi (3.24) is called the empty atom of Fn because c ~c Xi = iNn iNn ~ Xi = c = . (3.25) All the atoms of Fn other than A0 are called nonempty atoms. Let A be the set of all nonempty atoms of Fn . Then |A|, the cardinality of A, is equal to 2n - 1. A signed measure on Fn is completely specified by the values of on the nonempty atoms of Fn . ~ To simplify notation, we will use XG to denote (Xi , i G) and XG to ~ denote iG Xi for any nonempty subset G of Nn . Theorem 3.6. Let ~ B = XG : G is a nonempty subset of Nn . (3.26) Then a signed measure on Fn is completely specified by {(B), B B}, which can be any set of real numbers. Proof. The number of elements in B is equal to the number of nonempty subsets of Nn , which is 2n - 1. Thus |A| = |B| = 2n - 1. Let k = 2n - 1. Let u be a column k-vector of (A), A A, and h be a column k-vector of (B), B B. Since all the sets in B can expressed uniquely as the union of some nonempty atoms in A, by the set-additivity of , for each B B, (B) can be expressed uniquely as the sum of some components of u. Thus h = Cn u, (3.27) where Cn is a unique k k matrix. On the other hand, it can be shown (see Appendix 3.A) that for each A A, (A) can be expressed as a linear combination of (B), B B by applications, if necessary, of the following two identities: (A B - C) = (A - C) + (B - C) - (A B - C) (A - B) = (A B) - (B). (3.28) (3.29) 3.3 Construction of the I-Measure * 57 However, the existence of the said expression does not imply its uniqueness. Nevertheless, we can write u = Dn h (3.30) for some k k matrix Dn . Upon substituting (3.27) into (3.30), we obtain u = (Dn Cn )u, (3.31) which implies that Dn is the inverse of Cn as (3.31) holds regardless of the choice of . Since Cn is unique, so is Dn . Therefore, (A), A A are uniquely determined once (B), B B are specified. Hence, a signed measure on Fn is completely specified by {(B), B B}, which can be any set of real numbers. The theorem is proved. We now prove the following two lemmas which are related by the substitution of symbols in (3.19). Lemma 3.7. (A B - C) = (A C) + (B C) - (A B C) - (C). (3.32) Proof. From (3.28) and (3.29), we have (A B - C) = (A - C) + (B - C) - (A B - C) = ((A C) - (C)) + ((B C) - (C)) -((A B C) - (C)) = (A C) + (B C) - (A B C) - (C). The lemma is proved. Lemma 3.8. I(X; Y |Z) = H(X, Z) + H(Y, Z) - H(X, Y, Z) - H(Z). (3.36) (3.34) (3.35) (3.33) Proof. Consider I(X; Y |Z) = H(X|Z) - H(X|Y, Z) = H(X, Z) - H(Z) - (H(X, Y, Z) - H(Y, Z)) = H(X, Z) + H(Y, Z) - H(X, Y, Z) - H(Z). The lemma is proved. (3.37) (3.38) (3.39) 58 3 The I-Measure We now construct the I-Measure on Fn using Theorem 3.6 by defining ~ (XG ) = H(XG ) (3.40) for all nonempty subsets G of Nn . In order for to be meaningful, it has to be consistent with all Shannon's information measures (via the substitution of symbols in (3.19)). In that case, the following must hold for all (not necessarily disjoint) subsets G, G , G of Nn where G and G are nonempty: ~ ~ ~ (XG XG - XG ) = I(XG ; XG |XG ). When G = , (3.41) becomes ~ ~ (XG XG ) = I(XG ; XG ). When G = G , (3.41) becomes ~ ~ (XG - XG ) = H(XG |XG ). When G = G and G = , (3.41) becomes ~ (XG ) = H(XG ). (3.44) (3.43) (3.42) (3.41) Thus (3.41) covers all the four cases of Shannon's information measures, and it is the necessary and sufficient condition for to be consistent with all Shannon's information measures. Theorem 3.9. is the unique signed measure on Fn which is consistent with all Shannon's information measures. Proof. Consider ~ ~ ~ (XG XG - XG ) ~ ~ ~ ~ = (XGG ) + (XG G ) - (XGG G ) - (XG ) = H(XGG ) + H(XG G ) - H(XGG G ) - H(XG ) = I(XG ; XG |XG ), (3.45) (3.46) (3.47) where (3.45) and (3.47) follow from Lemmas 3.7 and 3.8, respectively, and (3.46) follows from (3.40), the definition of . Thus we have proved (3.41), i.e., is consistent with all Shannon's information measures. In order that is consistent with all Shannon's information measures, for all nonempty subsets G of Nn , has to satisfy (3.44), which in fact is the definition of in (3.40). Therefore, is the unique signed measure on Fn which is consistent with all Shannon's information measures. 3.4 * Can be Negative 59 3.4 * Can be Negative In the previous sections, we have been cautious in referring to the I-Measure as a signed measure instead of a measure4 . In this section, we show that in fact can take negative values for n 3. For n = 2, the three nonempty atoms of F2 are ~ ~ ~ ~ ~ ~ X1 X2 , X1 - X2 , X2 - X1 . The values of on these atoms are respectively I(X1 ; X2 ), H(X1 |X2 ), H(X2 |X1 ). (3.49) (3.48) These quantities are Shannon's information measures and hence nonnegative by the basic inequalities. Therefore, is always nonnegative for n = 2. For n = 3, the seven nonempty atoms of F3 are ~ ~ ~ ~ ~ ~ ~ ~ Xi - X{j,k} , Xi Xj - Xk , X1 X2 X3 , (3.50) where 1 i < j < k 3. The values of on the first two types of atoms are ~ ~ (Xi - X{j,k} ) = H(Xi |Xj , Xk ) and ~ ~ ~ (Xi Xj - Xk ) = I(Xi ; Xj |Xk ), (3.52) respectively, which are Shannon's information measures and therefore non~ ~ ~ negative. However, (X1 X2 X3 ) does not correspond to a Shannon's ~ ~ ~ information measure. In the next example, we show that (X1 X2 X3 ) can actually be negative. Example 3.10. In this example, all entropies are in the base 2. Let X1 and X2 be independent binary random variables with Pr{Xi = 0} = Pr{Xi = 1} = 0.5, i = 1, 2. Let X3 = (X1 + X2 ) mod 2. (3.54) It is easy to check that X3 has the same marginal distribution as X1 and X2 . Thus, H(Xi ) = 1 (3.55) for i = 1, 2, 3. Moreover, X1 , X2 , and X3 are pairwise independent. Therefore, H(Xi , Xj ) = 2 and 4 (3.51) (3.53) (3.56) A measure can assume only nonnegative values. 60 3 The I-Measure I(Xi ; Xj ) = 0 (3.57) for 1 i < j 3. We further see from (3.54) that each random variable is a function of the other two random variables. Then by the chain rule for entropy, we have H(X1 , X2 , X3 ) = H(X1 , X2 ) + H(X3 |X1 , X2 ) = 2+0 = 2. Now for 1 i < j < k 3, I(Xi ; Xj |Xk ) = H(Xi , Xk ) + H(Xj , Xk ) - H(X1 , X2 , X3 ) - H(Xk ) = 2+2-2-1 = 1, where we have invoked Lemma 3.8. It then follows that ~ ~ ~ ~ ~ ~ ~ ~ (X1 X2 X3 ) = (X1 X2 ) - (X1 X2 - X3 ) = I(X1 ; X2 ) - I(X1 ; X2 |X3 ) = 0-1 = -1. ~ ~ ~ Thus takes a negative value on the atom X1 X2 X3 . Motivated by the substitution of symbols in (3.19) for Shannon's informa~ ~ ~ tion measures, we will write (X1 X2 X3 ) as I(X1 ; X2 ; X3 ). In general, we will write ~ ~ ~ ~ (XG1 XG2 XGm - XF ) (3.68) as I(XG1 ; XG2 ; ; XGm |XF ) (3.69) and refer to it as the mutual information between XG1 , XG2 , , XGm conditioning on XF . Then (3.64) in the above example can be written as I(X1 ; X2 ; X3 ) = I(X1 ; X2 ) - I(X1 ; X2 |X3 ). For this example, I(X1 ; X2 ; X3 ) < 0, which implies I(X1 ; X2 |X3 ) > I(X1 ; X2 ). (3.71) (3.70) (3.64) (3.65) (3.66) (3.67) (3.61) (3.62) (3.63) (3.58) (3.59) (3.60) Therefore, unlike entropy, the mutual information between two random variables can be increased by conditioning on a third random variable. Also, we note in (3.70) that although the expression on the right hand side is not symbolically symmetrical in X1 , X2 , and X3 , we see from the left hand side that it is in fact symmetrical in X1 , X2 , and X3 . 3.5 Information Diagrams 61 3.5 Information Diagrams We have established in Section 3.3 a one-to-one correspondence between Shannon's information measures and set theory. Therefore, it is valid to use an information diagram, which is a variation of a Venn diagram, to represent the relationship between Shannon's information measures. ~ For simplicity, a set Xi will be labeled by Xi in an information diagram. We have seen the generic information diagram for n = 2 in Figure 3.3. A generic information diagram for n = 3 is shown in Figure 3.4. The informationtheoretic labeling of the values of on some of the sets in F3 is shown in the diagram. As an example, the information diagram for the I-Measure for random variables X1 , X2 , and X3 discussed in Example 3.10 is shown in Figure 3.5. For n 4, it is not possible to display an information diagram perfectly in two dimensions. In general, an information diagram for n random variables needs n - 1 dimensions to be displayed perfectly. Nevertheless, for n = 4, an information diagram can be displayed in two dimensions almost perfectly as shown in Figure 3.6. This information diagram is correct in that the region ~ representing the set X4 splits each atom in Figure 3.4 into two atoms. However, the adjacency of certain atoms are not displayed correctly. For example, the ~ ~ ~c ~ ~ ~ ~c ~ set X1 X2 X4 , which consists of the atoms X1 X2 X3 X4 and X1 ~ 2 X c X c , is not represented by a connected region because the two atoms ~ ~ X 3 4 are not adjacent to each other. When takes the value zero on an atom A of Fn , we do not need to display the atom A in an information diagram because the atom A does not contribute to (B) for any set B containing the atom A. As we will see shortly, this can happen if certain Markov constraints are imposed on the random variables involved, and the information diagram can be simplified accordingly. In a generic information diagram (i.e., when there is no constraint I ( X 1 ; X 2 ; X 3) I ( X1 ; X 2 X 3) H ( X 1) X2 H ( X 2 X 1) X1 H ( X 1 X 2 , X 3) X3 I ( X 1 ; X 3) Fig. 3.4. The generic information diagram for X1 , X2 , and X3 . 62 3 The I-Measure X2 0 1 0 1 1 1 0 X1 X3 Fig. 3.5. The information diagram for X1 , X2 , and X3 in Example 3.10. on the random variables), however, all the atoms have to be displayed, as is implied by the next theorem. Theorem 3.11. If there is no constraint on X1 , X2 , , Xn , then can take any set of nonnegative values on the nonempty atoms of Fn . Proof. We will prove the theorem by constructing an I-Measure which can take any set of nonnegative values on the nonempty atoms of Fn . Recall that A is the set of all nonempty atoms of Fn . Let YA , A A be mutually independent random variables. Now define the random variables Xi , i = 1, 2, , n by ~ Xi = (YA : A A and A Xi ). (3.72) We determine the I-Measure for X1 , X2 , , Xn so defined as follows. Since YA are mutually independent, for any nonempty subsets G of Nn , we have H(XG ) = H(Xi , i G) ~ = H((YA : A A and A Xi ), i G) (3.73) (3.74) X2 X1 X3 X4 Fig. 3.6. The generic information diagram for X1 , X2 , X3 , and X4 . 3.5 Information Diagrams 63 ~ = H(YA : A A and A XG ) = ~ AA:AXG (3.75) (3.76) H(YA ). On the other hand, ~ H(XG ) = (XG ) = ~ AA:AXG (A). (3.77) Equating the right hand sides of (3.76) and (3.77), we have H(YA ) = ~ AA:AXG ~ AA:AXG (A). (3.78) Evidently, we can make the above equality hold for all nonempty subsets G of Nn by taking (A) = H(YA ) (3.79) for all A A. By the uniqueness of , this is also the only possibility for . Since H(YA ) can take any nonnegative value by Corollary 2.44, can take any set of nonnegative values on the nonempty atoms of Fn . The theorem is proved. In the rest of this section, we explore the structure of Shannon's information measures when X1 X2 Xn forms a Markov chain. To start with, we consider n = 3, i.e., X1 X2 X3 forms a Markov chain. Since ~ ~c ~ (X1 X2 X3 ) = I(X1 ; X3 |X2 ) = 0, (3.80) ~ ~c ~ the atom X1 X2 X3 does not have to be displayed in an information diagram. Therefore, in constructing the information diagram, the regions representing the random variables X1 , X2 , and X3 should overlap with each other such ~ ~c ~ that the region corresponding to the atom X1 X2 X3 is empty, while the regions corresponding to all other nonempty atoms are nonempty. Figure 3.7 shows such a construction, in which each random variable is represented by a ~ ~ ~ "mountain." From Figure 3.7, we see that X1 X2 X3 , as the only atom on which may take a negative value, now becomes identical to the atom ~ ~ X1 X3 . Therefore, we have ~ ~ ~ I(X1 ; X2 ; X3 ) = (X1 X2 X3 ) ~ ~3) = (X1 X = I(X1 ; X3 ) 0. (3.81) (3.82) (3.83) (3.84) Hence, we conclude that when X1 X2 X3 forms a Markov chain, is always nonnegative. Next, we consider n = 4, i.e., X1 X2 X3 X4 forms a Markov chain. With reference to Figure 3.6, we first show that under this Markov constraint, always vanishes on certain nonempty atoms: 64 3 The I-Measure X1 X2 X3 Fig. 3.7. The information diagram for the Markov chain X1 X2 X3 . 1. The Markov chain X1 X2 X3 implies I(X1 ; X3 ; X4 |X2 ) + I(X1 ; X3 |X2 , X4 ) = I(X1 ; X3 |X2 ) = 0. 2. The Markov chain X1 X2 X4 implies I(X1 ; X3 ; X4 |X2 ) + I(X1 ; X4 |X2 , X3 ) = I(X1 ; X4 |X2 ) = 0. 3. The Markov chain X1 X3 X4 implies I(X1 ; X2 ; X4 |X3 ) + I(X1 ; X4 |X2 , X3 ) = I(X1 ; X4 |X3 ) = 0. 4. The Markov chain X2 X3 X4 implies I(X1 ; X2 ; X4 |X3 ) + I(X2 ; X4 |X1 , X3 ) = I(X2 ; X4 |X3 ) = 0. 5. The Markov chain (X1 , X2 ) X3 X4 implies I( X2 ; X4 |X3 ) + I(X1 ; X4 |X2 , X3 ) + I(X2 ; X4 |X1 , X3 ) X 1; = I(X1 , X2 ; X4 |X3 ) = 0. Now (3.85) and (3.86) imply I(X1 ; X4 |X2 , X3 ) = I(X1 ; X3 |X2 , X4 ), (3.87) and (3.91) imply I(X1 ; X2 ; X4 |X3 ) = -I(X1 ; X3 |X2 , X4 ), and (3.88) and (3.92) imply I(X2 ; X4 |X1 , X3 ) = I(X1 ; X3 |X2 , X4 ). (3.93) (3.92) (3.91) (3.89) (3.90) (3.88) (3.87) (3.86) (3.85) The terms on the left hand sides of (3.91), (3.92), and (3.93) are the three terms on the left hand side of (3.90). Then we substitute (3.91), (3.92), and (3.93) in (3.90) to obtain 3.5 Information Diagrams 65 X2 * * X1 * * X3 * X4 Fig. 3.8. The atoms of F4 on which vanishes when X1 X2 X3 X4 forms a Markov chain. ~ ~c ~ ~c (X1 X2 X3 X4 ) = I(X1 ; X3 |X2 , X4 ) = 0. From (3.85), (3.91), (3.92), and (3.93), (3.94) implies ~ ~c ~ ~ (X1 X2 X3 X4 ) = I(X1 ; X3 ; X4 |X2 ) = 0 ~ c c ~ ~ ~ (X1 X2 X3 X4 ) = I(X1 ; X4 |X2 , X3 ) = 0 ~ ~ ~c ~ (X1 X2 X3 X4 ) = I(X1 ; X2 ; X4 |X3 ) = 0 ~c ~ ~c ~ (X1 X2 X3 X4 ) = I(X2 ; X4 |X1 , X3 ) = 0. (3.94) (3.95) (3.96) (3.97) (3.98) From (3.94) to (3.98), we see that always vanishes on the atoms ~ ~c ~ ~c X1 X2 X3 X4 c ~ ~ ~ ~ X1 X2 X3 X4 ~ ~c ~c ~ X1 X2 X3 X4 ~ ~ ~c ~ X1 X2 X3 X4 ~c ~ ~c ~ X1 X2 X3 X4 of F4 , which we mark by an asterisk in the information diagram in Figure 3.8. In fact, the reader can gain a lot of insight by letting I(X1 ; X3 |X2 , X4 ) = a 0 in (3.85) and tracing the subsequent steps leading to the above conclusion in the information diagram in Figure 3.6. It is not necessary to display the five atoms in (3.99) in an information diagram because always vanishes on these atoms. Therefore, in constructing the information diagram, the regions representing the random variables should overlap with each other such that the regions corresponding to these five nonempty atoms are empty, while the regions corresponding to the other ten (3.99) 66 3 The I-Measure nonempty atoms, namely ~ ~c ~c ~c ~ ~ ~c ~c X1 X2 X3 X4 , X1 X2 X3 X4 ~ ~ ~ ~ ~ ~ ~ ~ X1 X2 X3 X c , X1 X2 X3 X4 , 4 ~c ~ ~c ~c ~c ~ ~ ~c X1 X2 X3 X4 , X1 X2 X3 X4 , c c c ~ ~ ~ ~ ~ ~ ~ ~ X X2 X3 X4 , X X X3 X c 1 1 2 4 (3.100) ~c ~c ~ ~ ~c ~c ~c ~ X1 X2 X3 X4 , X1 X2 X3 X4 , are nonempty. Figure 3.9 shows such a construction. The reader should compare the information diagrams in Figures 3.7 and 3.9 and observe that the latter is an extension of the former. From Figure 3.9, we see that the values of on the ten nonempty atoms in (3.100) are equivalent to H(X1 |X2 , X3 , X4 ), I(X1 ; X2 |X3 , X4 ) I(X1 ; X3 |X4 ), I(X1 ; X4 ) H(X2 |X1 , X3 , X4 ), I(X2 ; X3 |X1 ; X4 ) I(X2 ; X4 |X1 ), H(X3 |X1 , X2 , X4 ) I(X3 ; X4 |X1 , X2 ), H(X4 |X1 , X2 , X3 ), respectively5 . Since these are all Shannon's information measures and thus nonnegative, we conclude that is always nonnegative. When X1 X2 Xn forms a Markov chain, for n = 3, there is only ~ ~c ~ one nonempty atom, namely X1 X2 X3 , on which always vanishes. This atom can be determined directly from the Markov constraint I(X1 ; X3 |X2 ) = 0. For n = 4, the five nonempty atoms on which always vanishes are listed in (3.99). The determination of these atoms, as we have seen, is not straightforward. We have also shown that for n = 3 and n = 4, is always nonnegative. (3.101) X1 X2 X3 X4 Fig. 3.9. The information diagram for the Markov chain X1 X2 X3 X4 . 5 A formal proof will be given in Theorem 12.30. 3.6 Examples of Applications 67 We will extend this theme in Chapter 12 to finite Markov random fields with Markov chains being a special case. For a Markov chain, the information diagram can always be displayed in two dimensions as in Figure 3.10, and is always nonnegative. These will be explained in Chapter 12. 3.6 Examples of Applications In this section, we give a few examples of applications of information diagrams. These examples show how information diagrams can help solving information theory problems. The use of an information diagram is highly intuitive. To obtain an information identity from an information diagram is WYSIWYG6 . However, how to obtain an information inequality from an information diagram needs some explanation. Very often, we use a Venn diagram to represent a measure which takes nonnegative values. If we see in the Venn diagram two sets A and B such that A is a subset of B, then we can immediately conclude that (A) (B) because (B) - (A) = (B - A) 0. (3.102) However, an I-Measure can take negative values. Therefore, when we see in an information diagram that A is a subset of B, we cannot conclude from this fact alone that (A) (B) unless we know from the setup of the problem that is nonnegative. (For example, is nonnegative if the random variables involved form a Markov chain.) Instead, information inequalities can be obtained from an information diagram in conjunction with the basic inequalities. The following examples illustrate how it works. Example 3.12 (Concavity of Entropy). Let X1 p1 (x) and X2 p2 (x). Let X p(x) = p1 (x) + p2 (x), (3.103) X1 X2 ... Xn-1 Xn Fig. 3.10. The information diagram for the Markov chain X1 X2 Xn . 6 What you see is what you get. 68 3 The I-Measure Z=1 X1 X X2 Z=2 Fig. 3.11. The schematic diagram for Example 3.12. where 0 1 and = 1 - . We will show that H(X) H(X1 ) + H(X2 ). (3.104) Consider the system in Figure 3.11 in which the position of the switch is determined by a random variable Z with Pr{Z = 1} = and Pr{Z = 2} = , (3.105) where Z is independent of X1 and X2 . The switch takes position i if Z = i, i = 1, 2. The random variable Z is called a mixing random variable for the probability distributions p1 (x) and p2 (x). Figure 3.12 shows the information ~ ~ ~ diagram for X and Z. From the diagram, we see that X - Z is a subset of X. Since is nonnegative for two random variables, we can conclude that ~ ~ ~ (X) (X - Z), which is equivalent to H(X) H(X|Z). Then H(X) H(X|Z) = Pr{Z = 1}H(X|Z = 1) + Pr{Z = 2}H(X|Z = 2) = H(X1 ) + H(X2 ), (3.108) (3.109) (3.110) (3.107) (3.106) proving (3.104). This shows that H(X) is a concave functional of p(x). X Z Fig. 3.12. The information diagram for Example 3.12. 3.6 Examples of Applications 69 Z=1 p1 ( y x ) X p2 ( y x ) Z=2 Y Fig. 3.13. The schematic diagram for Example 3.13. Example 3.13 (Convexity of Mutual Information). Let (X, Y ) p(x, y) = p(x)p(y|x). (3.111) We will show that for fixed p(x), I(X; Y ) is a convex functional of p(y|x). Let p1 (y|x) and p2 (y|x) be two transition matrices. Consider the system in Figure 3.13 in which the position of the switch is determined by a random variable Z as in the last example, where Z is independent of X, i.e., I(X; Z) = 0. In the information diagram for X, Y , and Z in Figure 3.14, let I(X; Z|Y ) = a 0. Since I(X; Z) = 0, we see that I(X; Y ; Z) = -a, because I(X; Z) = I(X; Z|Y ) + I(X; Y ; Z). Then (3.115) (3.114) (3.113) (3.112) Y -a a X Z Fig. 3.14. The information diagram for Example 3.13. 70 3 The I-Measure Z =1 p1(x) X p(y|x) Y p2(x) Z =2 Fig. 3.15. The schematic diagram for Example 3.14. I(X; Y ) = I(X; Y |Z) + I(X; Y ; Z) = I(X; Y |Z) - a I(X; Y |Z) = Pr{Z = 1}I(X; Y |Z = 1) + Pr{Z = 2}I(X; Y |Z = 2) = I(p(x), p1 (y|x)) + I(p(x), p2 (y|x)), (3.116) (3.117) (3.118) (3.119) (3.120) where I(p(x), pi (y|x)) denotes the mutual information between the input and output of a channel with input distribution p(x) and transition matrix pi (y|x). This shows that for fixed p(x), I(X; Y ) is a convex functional of p(y|x). Example 3.14 (Concavity of Mutual Information). Let (X, Y ) p(x, y) = p(x)p(y|x). (3.121) We will show that for fixed p(y|x), I(X; Y ) is a concave functional of p(x). Consider the system in Figure 3.15, where the position of the switch is determined by a random variable Z as in the last example. In this system, when X is given, Y is independent of Z, or Z X Y forms a Markov chain. Then is nonnegative, and the information diagram for X, Y , and Z is shown in Figure 3.16. ~ ~ ~ ~ ~ From Figure 3.16, since XY -Z is a subset of XY and is nonnegative, we immediately see that Z X Y Fig. 3.16. The information diagram for Example 3.14. 3.6 Examples of Applications 71 I(X; Y ) I(X; Y |Z) = Pr{Z = 1}I(X; Y |Z = 1) + Pr{Z = 2}I(X; Y |Z = 2) = I(p1 (x), p(y|x)) + I(p2 (x), p(y|x)). (3.122) (3.123) (3.124) This shows that for fixed p(y|x), I(X; Y ) is a concave functional of p(x). Example 3.15 (Imperfect Secrecy Theorem). Let X be the plain text, Y be the cipher text, and Z be the key in a secret key cryptosystem. Since X can be recovered from Y and Z, we have H(X|Y, Z) = 0. We will show that this constraint implies I(X; Y ) H(X) - H(Z). (3.126) (3.125) The quantity I(X; Y ) is a measure of the security level of the cryptosystem. In general, we want to make I(X; Y ) small so that the eavesdropper cannot obtain too much information about the plain text X by observing the cipher text Y . The inequality in (3.126) says that the system can attain a certain level of security only if H(Z) (often called the key length) is sufficiently large. In particular, if perfect secrecy is required, i.e., I(X; Y ) = 0, then H(Z) must be at least equal to H(X). This special case is known as Shannon's perfect secrecy theorem [323]7 . We now prove (3.126). Let I(X; Y |Z) = a 0 I(Y ; Z|X) = b 0 H(Z|X, Y ) = c 0, and I(X; Y ; Z) = d. (See Figure 3.17.) Since I(Y ; Z) 0, b + d 0. (3.131) (3.130) (3.127) (3.128) (3.129) In comparing H(X) with H(Z), we do not have to consider I(X; Z|Y ) and I(X; Y ; Z) since they belong to both H(X) and H(Z). Then we see from Figure 3.17 that H(X) - H(Z) = a - b - c. (3.132) 7 Shannon used a combinatorial argument to prove this theorem. An informationtheoretic proof can be found in Massey [251]. 72 3 The I-Measure Y a 0 X d b c Z Fig. 3.17. The information diagram for Example 3.15. Therefore, I(X; Y ) = a + d a-b a-b-c = H(X) - H(Z), (3.133) (3.134) (3.135) (3.136) where (3.134) and (3.135) follow from (3.131) and (3.129), respectively, proving (3.126). Note that in deriving our result, the assumptions that H(Y |X, Z) = 0, i.e., the cipher text is a function of the plain text and the key, and I(X; Z) = 0, i.e., the plain text and the key are independent, are not necessary. Example 3.16. Figure 3.18 shows the information diagram for the Markov chain X Y Z. From this diagram, we can identify the following two information identities: I(X; Y ) = I(X; Y, Z) H(X|Y ) = H(X|Y, Z). ~ ~ ~ ~ Since is nonnegative and X Z is a subset of X Y , we have I(X; Z) I(X; Y ), (3.139) (3.137) (3.138) X Y Z Fig. 3.18. The information diagram for the Markov chain X Y Z. 3.6 Examples of Applications 73 X Y Z T Fig. 3.19. The information diagram for the Markov chain X Y Z T . which has already been obtained in Lemma 2.41. Similarly, we can also obtain H(X|Y ) H(X|Z). (3.140) Example 3.17 (Data Processing Theorem). Figure 3.19 shows the information diagram for the Markov chain X Y Z T . Since is nonnegative and ~ ~ ~ ~ X T is a subset of Y Z, we have I(X; T ) I(Y ; Z), which is the data processing theorem (Theorem 2.42). We end this chapter by giving an application of the information diagram for a Markov chain with five random variables. Example 3.18. In this example, we prove with the help of an information diagram that for five random variables X, Y, Z, T , and U such that X Y Z T U forms a Markov chain, H(Y ) + H(T ) = I(Z; X, Y, T, U ) + I(X, Y ; T, U ) + H(Y |Z) + H(T |Z). (3.142) (3.141) In the information diagram for X, Y, Z, T , and U in Figure 3.20, we first identify the atoms of H(Y ) and then the atoms of H(T ) by marking each of X Y . . . Z . .. . . T . . U . . . . . . Fig. 3.20. The atoms of H(Y ) + H(T ). 74 3 The I-Measure X Y . . . Z . .. . . T . . U . . . . . . Fig. 3.21. The atoms of I(Z; X, Y, T, U ) + I(X, Y ; T, U ) + H(Y |Z) + H(T |Z). them by a dot. If an atom belongs to both H(Y ) and H(T ), it receives two dots. The resulting diagram represents H(Y ) + H(T ). By repeating the same procedure for I(Z; X, Y, T, U ) + I(X, Y ; T, U ) + H(Y |Z) + H(T |Z), (3.144) (3.143) we obtain the information diagram in Figure 3.21. Comparing these two information diagrams, we find that they are identical. Hence, the information identity in (3.142) always holds conditioning on the Markov chain X Y Z T U . This identity is critical in proving an outer bound on the achievable coding rate region of the multiple descriptions problem in Fu et al. [125]. It is virtually impossible to discover this identity without the help of an information diagram! Appendix 3.A: A Variation of the Inclusion-Exclusion Formula In this appendix, we show that for each A A, (A) can be expressed as a linear combination of (B), B B via applications of (3.28) and (3.29). We first prove by using (3.28) the following variation of the inclusive-exclusive formula. Theorem 3.19. For a set-additive function , n k=1 Ak - B = 1in (Ai - B) - 1i<jn (Ai Aj - B) (3.145) + + (-1)n+1 (A1 A2 An - B). Appendix 3.A: A Variation of the Inclusion-Exclusion Formula 75 Proof. The theorem will be proved by induction on n. First, (3.145) is obviously true for n = 1. Assume (3.145) is true for some n 1. Now consider n+1 k=1 Ak - B n = k=1 n Ak An+1 - B n (3.146) = k=1 Ak - B + (An+1 - B) - k=1 Ak An+1 - B (3.147) = 1in (Ai - B) - 1i<jn (Ai Aj - B) + + (-1)n+1 (A1 A2 An - B) n + (An+1 - B) (3.148) - k=1 (Ak An+1 ) - B (Ai - B) - (Ai Aj - B) + 1i<jn = 1in + (-1)n+1 (A1 A2 An - B) - 1in + (An+1 - B) (Ai Aj An+1 - B) (Ai An+1 - B) - 1i<jn + + (-1)n+1 (A1 A2 An An+1 - B) = 1in+1 (3.149) (Ai - B) - 1i<jn+1 (Ai Aj - B) + (3.150) +(-1)n+2 (A1 A2 An+1 - B). In the above, (3.28) was used in obtaining (3.147), and the induction hypothesis was used in obtaining (3.148) and (3.149). The theorem is proved. Now a nonempty atom of Fn has the form n Yi , i=1 (3.151) 76 3 The I-Measure ~ ~c ~ where Yi is either Xi or Xi , and there exists at least one i such that Yi = Xi . Then we can write the atom in (3.151) as ~ Xi - ~ i:Yi =Xi ~c j:Yj =Xj ~ Xj . (3.152) Note that the intersection above is always nonempty. Then using (3.145) and (3.29), we see that for each A A, (A) can be expressed as a linear combination of (B), B B. Chapter Summary ~ ~ ~ Definition: The field Fn generated by sets X1 , X2 , , Xn is the collection of sets which can be obtained by any sequence of usual set operations (union, ~ ~ ~ intersection, complement, and difference) on X1 , X2 , , Xn . Definition: A real function defined on Fn is called a signed measure if it is set-additive, i.e., for disjoint A and B in Fn , (A B) = (A) + (B). I-Measure : There exists a unique signed measure on Fn which is consistent with all Shannon's information measures. Can be Negative: Let X1 and X2 be i.i.d. uniform on {0, 1}, and let X3 = X1 + X2 mod 2. Then ~ ~ ~ (X1 X2 X3 ) = I(X1 ; X2 ; X3 ) = -1. Information Diagrams for Two, Three, and Four Random Variables: H ( X1 ,X 2) H ( X1 X 2) H ( X 2 X1) H ( X1) I ( X1; X 2) H (X 2) Chapter Summary 77 I ( X 1 ; X 2 ; X 3) I ( X1 ; X 2 X 3) H ( X 1) X2 H ( X 2 X 1) X1 H ( X 1 X 2 , X 3) X3 I ( X 1 ; X 3) X2 X1 X3 X4 Information Diagram for Markov Chain X1 X2 Xn : X1 X2 ... Xn-1 Xn 78 3 The I-Measure Problems 1. Show that I(X; Y ; Z) = E log p(X, Y )p(Y, Z)p(X, Z) p(X)p(Y )p(Z)p(X, Y, Z) and obtain a general formula for I(X1 ; X2 , ; ; Xn ). 2. Suppose X Y and X Z. Does X (Y, Z) hold in general? 3. Show that I(X; Y ; Z) vanishes if at least one of the following conditions hold: a) X, Y , and Z are mutually independent; b) X Y Z forms a Markov chain and X and Z are independent. 4. a) Verify that I(X; Y ; Z) vanishes for the distribution p(x, y, z) given by p(0, 0, 0) = 0.0625, p(0, 0, 1) = 0.0772, p(0, 1, 0) = 0.0625 p(0, 1, 1) = 0.0625, p(1, 0, 0) = 0.0625, p(1, 0, 1) = 0.1103 p(1, 1, 0) = 0.1875, p(1, 1, 1) = 0.375. b) Verify that the distribution in part a) does not satisfy the conditions in Problem 3. Weak independence X is weakly independent of Y if the rows of the transition matrix [p(x|y)] are linearly dependent. a) Show that if X and Y are independent, then X is weakly independent of Y . b) Show that for random variables X and Y , there exists a random variable Z satisfying i) X Y Z ii) X and Z are independent iii) Y and Z are not independent if and only if X is weakly independent of Y . (Berger and Yeung [29].) Prove that a) I(X; Y ; Z) - min{I(X; Y |Z), I(Y ; Z|X), I(X, Z|Y )} b) I(X; Y ; Z) min{I(X; Y ), I(Y ; Z), I(X; Z)}. a) Prove that if X and Y are independent, then I(X, Y ; Z) I(X; Y |Z). b) Show that the inequality in part a) is not valid in general by giving a counterexample. In Example 3.15, it was shown that I(X; Y ) H(X) - H(Z), where X is the plain text, Y is the cipher text, and Z is the key in a secret key cryptosystem. Give an example of a secret key cryptosystem such that this inequality is tight. a) Prove that under the constraint that X Y Z forms a Markov chain, X Y |Z and X Z imply X Y . b) Prove that the implication in a) continues to be valid without the Markov chain constraint. 5. 6. 7. 8. 9. Problems 79 10. a) Show that Y Z|T does not imply Y Z|(X, T ) by giving a counterexample. b) Prove that Y Z|T implies Y Z|(X, T ) conditioning on X Y Z T. 11. a) Let X Y (Z, T ) form a Markov chain. Prove that I(X; Z) + I(X; T ) I(X; Y ) + I(Z; T ). b) Let X Y Z T form a Markov chain. Determine which of the following inequalities always hold: i) I(X; T ) + I(Y ; Z) I(X; Z) + I(Y ; T ) ii) I(X; T ) + I(Y ; Z) I(X; Y ) + I(Z; T ) iii) I(X; Y ) + I(Z; T ) I(X; Z) + I(Y ; T ). 12. Secret sharing For a given finite set P and a collection A of subsets of P, a secret sharing scheme is a random variable S and a family of random variables {Xp : p P} such that for all A A, H(S|XA ) = 0, and for all B A, H(S|XB ) = H(S). Here, S is the secret and P is the set of participants of the scheme. A participant p of the scheme possesses a share Xp of the secret. The set A specifies the access structure of the scheme: For a subset A of P, by pooling their shares, if A A, the participants in A can reconstruct S, otherwise they can know nothing about S. a) i) Prove that for A, B P, if B A and A B A, then H(XA |XB ) = H(S) + H(XA |XB , S). ii) Prove that if B A, then H(XA |XB ) = H(XA |XB , S). (Capocelli et al. [57].) b) Prove that for A, B, C P such that A C A, B C A, and C A, then I(XA ; XB |XC ) H(S). (van Dijk [363].) 13. Consider four random variables X, Y, Z, and T which satisfy the following constraints: H(T |X) = H(T ), H(T |X, Y ) = 0, H(T |Y ) = H(T ), H(Y |Z) = 0, and H(T |Z) = 0. Prove that a) H(T |X, Y, Z) = I(Z; T |X, Y ) = 0. b) I(X; T |Y, Z) = I(X; Y ; T |Z) = I(Y ; T |X, Z) = 0. c) I(X; Z; T ) = I(Y ; Z; T ) = 0. d) H(Y |X, Z, T ) = I(X; Y |Z, T ) = 0. e) I(X; Y ; Z) 0. 80 3 The I-Measure f) I(X; Z) H(T ). The inequality in f) finds application in a secret sharing problem studied by Blundo et al. [43]. 14. Prove that for random variables X, Y , Z, and T , X Z|Y (X, Y ) T |Z Y Z|T Y Z. Y Z|X XT Hint: Observe that X Z|Y and (X, Y ) T |Z are equivalent to X Y Z T and use an information diagram. 15. Prove that XY Z T X Y |(Z, T ) Z T |(X, Y ) Z T |X X Y |Z Z T |Y X Y |T. (Studen [346].) y Historical Notes The original work on the set-theoretic structure of Shannon's information measures is due to Hu [173]. It was established in this paper that every information identity implies a set identity via a substitution of symbols. This allows the tools for proving information identities to be used in proving set identities. Since the paper was published in Russian, it was largely unknown to the West until it was described in Csiszr and Krner [84]. Throughout the years, a o the use of Venn diagrams to represent the structure of Shannon's information measures for two or three random variables has been suggested by various authors, for example, Reza [302], Abramson [2], and Papoulis [286], but no formal justification was given until Yeung [398] introduced the I-Measure. McGill [265] proposed a multiple mutual information for any number of random variables which is equivalent to the mutual information between two or more random variables discussed here. Properties of this quantity have been investigated by Kawabata [196] and Yeung [398]. Along a related direction, Han [146] viewed the linear combination of entropies as a vector space and developed a lattice-theoretic description of Shannon's information measures. 4 Zero-Error Data Compression In a random experiment, a coin is tossed n times. Let Xi be the outcome of the ith toss, with Pr{Xi = HEAD} = p and Pr{Xi = TAIL} = 1 - p, (4.1) where 0 p 1. It is assumed that Xi are i.i.d., and the value of p is known. We are asked to describe the outcome of the random experiment without error (with zero error) by using binary symbols. One way to do this is to encode a HEAD by a `0' and a TAIL by a `1.' Then the outcome of the random experiment is encoded into a binary codeword of length n. When the coin is fair, i.e., p = 0.5, this is the best we can do because the probability of every outcome of the experiment is equal to 2-n . In other words, all the outcomes are equally likely. However, if the coin is biased, i.e., p = 0.5, the probability of an outcome of the experiment depends on the number of HEADs and the number of TAILs in the outcome. In other words, the probabilities of the outcomes are no longer uniform. It turns out that we can take advantage of this by encoding more likely outcomes into shorter codewords and less likely outcomes into longer codewords. By doing so, it is possible to use fewer than n bits on the average to describe the outcome of the random experiment. In particular, in the extreme case when p = 0 or 1, we actually do not need to describe the outcome of the experiment because it is deterministic. At the beginning of Chapter 2, we mentioned that the entropy H(X) measures the amount of information contained in a random variable X. In this chapter, we substantiate this claim by exploring the role of entropy in the context of zero-error data compression. 82 4 Zero-Error Data Compression 4.1 The Entropy Bound In this section, we establish that H(X) is a fundamental lower bound on the expected length of the number of symbols needed to describe the outcome of a random variable X with zero error. This is called the entropy bound. Definition 4.1. A D-ary source code C for a source random variable X is a mapping from X to D , the set of all finite length sequences of symbols taken from a D-ary code alphabet. Consider an information source {Xk , k 1}, where Xk are discrete random variables which take values in the same alphabet. We apply a source code C to each Xk and concatenate the codewords. Once the codewords are concatenated, the boundaries of the codewords are no longer explicit. In other words, when the code C is applied to a source sequence, a sequence of code symbols are produced, and the codewords may no longer be distinguishable. We are particularly interested in uniquely decodable codes which are defined as follows. Definition 4.2. A code C is uniquely decodable if for any finite source sequence, the sequence of code symbols corresponding to this source sequence is different from the sequence of code symbols corresponding to any other (finite) source sequence. Suppose we use a code C to encode a source file into a coded file. If C is uniquely decodable, then we can always recover the source file from the coded file. An important class of uniquely decodable codes, called prefix codes, are discussed in the next section. But we first look at an example of a code which is not uniquely decodable. Example 4.3. Let X = {A, B, C, D}. Consider the code C defined by x C(x) A 0 B 1 C 01 D 10 Then all the three source sequences AAD, ACA, and AABA produce the code sequence 0010. Thus from the code sequence 0010, we cannot tell which of the three source sequences it comes from. Therefore, C is not uniquely decodable. In the next theorem, we prove that for any uniquely decodable code, the lengths of the codewords have to satisfy an inequality called the Kraft inequality. 4.1 The Entropy Bound 83 Theorem 4.4 (Kraft Inequality). Let C be a D-ary source code, and let l1 , l2 , , lm be the lengths of the codewords. If C is uniquely decodable, then m D-lk 1. k=1 (4.2) Proof. Let N be an arbitrary positive integer, and consider m N m m m D k=1 -lk = k1 =1 k2 =1 kN =1 D-(lk1 +lk2 ++lkN ) . (4.3) By collecting terms on the right-hand side, we write m N N lmax D k=1 -lk = i=1 Ai D-i (4.4) where lmax = max lk 1km m N (4.5) -lk and Ai is the coefficient of D-i in . Now observe that Ai gives k=1 D the total number of sequences of N codewords with a total length of i code symbols (see Example 4.5 below). Since the code is uniquely decodable, these code sequences must be distinct, and therefore Ai D i (4.6) because there are Di distinct sequences of i code symbols. Substituting this inequality into (4.4), we have m N N lmax D k=1 -lk i=1 1 = N lmax , (4.7) or m D-lk (N lmax )1/N . k=1 (4.8) Since this inequality holds for any N , upon letting N , we obtain (4.2), completing the proof. Example 4.5. In this example, we illustrate the quantity Ai in the proof of Theorem 4.4 for the code C in Example 4.3. Let l1 = l2 = 1 and l3 = l4 = 2. Let N = 2 and consider 4 2 2 k=1 -lk = (2 2-1 + 2 2-2 )2 = 4 2-2 + 8 2-3 + 4 2-4 . (4.9) (4.10) 84 4 Zero-Error Data Compression Then A2 = 4, A3 = 8, and A4 = 8, i.e., the total number of sequences of 2 codewords with a total length of 2, 3, and 4 code symbols are 4, 8, and 4, respectively. For a total length of 3, for instance, the 8 code sequences are 001(AC), 010(AD), 101(BC), 110(BD), 010(CA), 011(CB), 100(DA), and 101(DB). Let X be a source random variable with probability distribution {p1 , p2 , , pm }, (4.11) where m 2. When we use a uniquely decodable code C to encode the outcome of X, we are naturally interested in the expected length of a codeword, which is given by L= p i li . (4.12) i We will also refer to L as the expected length of the code C. The quantity L gives the average number of symbols we need to describe the outcome of X when the code C is used, and it is a measure of the efficiency of the code C. Specifically, the smaller the expected length L is, the better the code C is. In the next theorem, we will prove a fundamental lower bound on the expected length of any uniquely decodable D-ary code. We first explain why this is the lower bound we should expect. In a uniquely decodable code, we use L D-ary symbols on the average to describe the outcome of X. Recall from the remark following Theorem 2.43 that a D-ary symbol can carry at most one D-it of information. Then the maximum amount of information which can be carried by the codeword on the average is L 1 = L D-its. Since the code is uniquely decodable, the amount of entropy carried by the codeword on the average is HD (X). Therefore, we have HD (X) L. (4.13) In other words, the expected length of a uniquely decodable code is at least the entropy of the source. This argument is rigorized in the proof of the next theorem. Theorem 4.6 (Entropy Bound). Let C be a D-ary uniquely decodable code for a source random variable X with entropy HD (X). Then the expected length of C is lower bounded by HD (X), i.e., L HD (X). This lower bound is tight if and only if li = - logD pi for all i. Proof. Since C is uniquely decodable, the lengths of its codewords satisfy the Kraft inequality. Write L= pi logD Dli (4.15) i (4.14) 4.1 The Entropy Bound 85 and recall from Definition 2.35 that HD (X) = - i pi logD pi . (4.16) Then L - HD (X) = i pi logD (pi Dli ) pi ln(pi Dli ) i (4.17) (4.18) (4.19) = (ln D)-1 (ln D)-1 i pi 1 - pi - i i 1 pi Dli D-li = (ln D)-1 (4.20) (4.21) (4.22) (ln D)-1 (1 - 1) = 0, where we have invoked the fundamental inequality in (4.19) and the Kraft inequality in (4.21). This proves (4.14). In order for this lower bound to be tight, both (4.19) and (4.21) have to be tight simultaneously. Now (4.19) is tight if and only if pi Dli = 1, or li = - logD pi for all i. If this holds, we have D-li = i i pi = 1, (4.23) i.e., (4.21) is also tight. This completes the proof of the theorem. The entropy bound can be regarded as a generalization of Theorem 2.43, as is seen from the following corollary. Corollary 4.7. H(X) log |X |. Proof. Considering encoding each outcome of a random variable X by a distinct symbol in {1, 2, , |X |}. This is obviously a |X |-ary uniquely decodable code with expected length 1. Then by the entropy bound, we have H|X | (X) 1, which becomes H(X) log |X | when the base of the logarithm is not specified. Motivated by the entropy bound, we now introduce the redundancy of a uniquely decodable code. Definition 4.8. The redundancy R of a D-ary uniquely decodable code is the difference between the expected length of the code and the entropy of the source. We see from the entropy bound that the redundancy of a uniquely decodable code is always nonnegative. (4.25) (4.24) 86 4 Zero-Error Data Compression 4.2 Prefix Codes 4.2.1 Definition and Existence Definition 4.9. A code is called a prefix-free code if no codeword is a prefix of any other codeword. For brevity, a prefix-free code will be referred to as a prefix code. Example 4.10. The code C in Example 4.3 is not a prefix code because the codeword 0 is a prefix of the codeword 01, and the codeword 1 is a prefix of the codeword 10. It can easily be checked that the following code C is a prefix code. x C (x) A 0 B 10 C 110 D 1111 A D-ary tree is a graphical representation of a collection of finite sequences of D-ary symbols. In a D-ary tree, each node has at most D children. If a node has at least one child, it is called an internal node, otherwise it is called a leaf. The children of an internal node are labeled by the D symbols in the code alphabet. A D-ary prefix code can be represented by a D-ary tree with the leaves of the tree being the codewords. Such a tree is called the code tree for the prefix code. Figure 4.1 shows the code tree for the prefix code C in Example 4.10. As we have mentioned in Section 4.1, once a sequence of codewords are concatenated, the boundaries of the codewords are no longer explicit. Prefix codes have the desirable property that the end of a codeword can be recognized instantaneously so that it is not necessary to make reference to the future codewords during the decoding process. For example, for the source sequence BCDAC , the code C in Example 4.10 produces the code sequence 0 10 110 1111 Fig. 4.1. The code tree for the code C . 4.2 Prefix Codes 87 1011011110110 . Based on this binary sequence, the decoder can reconstruct the source sequence as follows. The first bit 1 cannot form the first codeword because 1 is not a valid codeword. The first two bits 10 must form the first codeword because it is a valid codeword and it is not the prefix of any other codeword. The same procedure is repeated to locate the end of the next codeword, and the code sequence is parsed as 10, 110, 1111, 0, 110, . Then the source sequence BCDAC can be reconstructed correctly. Since a prefix code can always be decoded correctly, it is a uniquely decodable code. Therefore, by Theorem 4.4, the codeword lengths of a prefix code also satisfies the Kraft inequality. In the next theorem, we show that the Kraft inequality fully characterizes the existence of a prefix code. Theorem 4.11. There exists a D-ary prefix code with codeword lengths l1 , l2 , , lm if and only if the Kraft inequality m D-lk 1 k=1 (4.26) is satisfied. Proof. We only need to prove the existence of a D-ary prefix code with codeword lengths l1 , l2 , , lm if these lengths satisfy the Kraft inequality. Without loss of generality, assume that l1 l2 lm . Consider all the D-ary sequences of lengths less than or equal to lm and regard them as the nodes of the full D-ary tree of depth lm . We will refer to a sequence of length l as a node of order l. Our strategy is to choose nodes as codewords in nondecreasing order of the codeword lengths. Specifically, we choose a node of order l1 as the first codeword, then a node of order l2 as the second codeword, so on and so forth, such that each newly chosen codeword is not prefixed by any of the previously chosen codewords. If we can successfully choose all the m codewords, then the resultant set of codewords forms a prefix code with the desired set of lengths. There are Dl1 > 1 (since l1 1 ) nodes of order l1 which can be chosen as the first codeword. Thus choosing the first codeword is always possible. Assume that the first i codewords have been chosen successfully, where 1 i m-1, and we want to choose a node of order li+1 as the (i+1)st codeword such that it is not prefixed by any of the previously chosen codewords. In other words, the (i + 1)st node to be chosen cannot be a descendant of any of the previously chosen codewords. Observe that for 1 j i, the codeword with length lj has Dli+1 -lj descendents of order li+1 . Since all the previously chosen codewords are not prefeces of each other, their descendents of order li+1 do not overlap. Therefore, upon noting that the total number of nodes of order li+1 is Dli+1 , the number of nodes which can be chosen as the (i + 1)st codeword is Dli+1 - Dli+1 -l1 - - Dli+1 -li . (4.27) 88 4 Zero-Error Data Compression If l1 , l2 , , lm satisfy the Kraft inequality, we have D-l1 + + D-li + D-li+1 1. Multiplying by Dli+1 and rearranging the terms, we have Dli+1 - Dli+1 -l1 - - Dli+1 -li 1. (4.29) (4.28) The left hand side is the number of nodes which can be chosen as the (i + 1)st codeword as given in (4.27). Therefore, it is possible to choose the (i + 1)st codeword. Thus we have shown the existence of a prefix code with codeword lengths l1 , l2 , , lm , completing the proof. A probability distribution {pi } such that for all i, pi = D-ti , where ti is a positive integer, is called a D-adic distribution. When D = 2, {pi } is called a dyadic distribution. From Theorem 4.6 and the above theorem, we can obtain the following result as a corollary. Corollary 4.12. There exists a D-ary prefix code which achieves the entropy bound for a distribution {pi } if and only if {pi } is D-adic. Proof. Consider a D-ary prefix code which achieves the entropy bound for a distribution {pi }. Let li be the length of the codeword assigned to the probability pi . By Theorem 4.6, for all i, li = - logD pi , or pi = D-li . Thus {pi } is D-adic. Conversely, suppose {pi } is D-adic, and let pi = D-ti for all i. Let li = ti for all i. Then by the Kraft inequality, there exists a prefix code with codeword lengths {li }, because D-li = i i D-ti = i pi = 1. (4.30) Assigning the codeword with length li to the probability pi for all i, we see from Theorem 4.6 that this code achieves the entropy bound. 4.2.2 Huffman Codes As we have mentioned, the efficiency of a uniquely decodable code is measured by its expected length. Thus for a given source X, we are naturally interested in prefix codes which have the minimum expected length. Such codes, called optimal codes, can be constructed by the Huffman procedure, and these codes are referred to as Huffman codes. In general, there exists more than one optimal code for a source, and some optimal codes cannot be constructed by the Huffman procedure. For simplicity, we first discuss binary Huffman codes. A binary prefix code for a source X with distribution {pi } is represented by a binary code tree, with each leaf in the code tree corresponding to a codeword. The Huffman procedure is to form a code tree such that the expected length is minimum. The procedure is described by a very simple rule: 4.2 Prefix Codes 89 Keep merging the two smallest probability masses until one probability mass (i.e., 1) is left. The merging of two probability masses corresponds to the formation of an internal node of the code tree. We now illustrate the Huffman procedure by the following example. Example 4.13. Let X be the source with X = {A, B, C, D, E}, and the probabilities are 0.35, 0.1, 0.15, 0.2, 0.2, respectively. The Huffman procedure is shown in Figure 4.2. In the first step, we merge probability masses 0.1 and 0.15 pi 0.6 0.25 1 0.4 0.35 0.1 0.15 0.2 0.2 Fig. 4.2. The Huffman procedure. codeword 00 010 011 10 11 into a probability mass 0.25. In the second step, we merge probability masses 0.2 and 0.2 into a probability mass 0.4. In the third step, we merge probability masses 0.35 and 0.25 into a probability mass 0.6. Finally, we merge probability masses 0.6 and 0.4 into a probability mass 1. A code tree is then formed. Upon assigning 0 and 1 (in any convenient way) to each pair of branches at an internal node, we obtain the codeword assigned to each source symbol. In the Huffman procedure, sometimes there is more than one choice of merging the two smallest probability masses. We can take any one of these choices without affecting the optimality of the code eventually obtained. For an alphabet of size m, it takes m - 1 steps to complete the Huffman procedure for constructing a binary code, because we merge two probability masses in each step. In the resulting code tree, there are m leaves and m - 1 internal nodes. In the Huffman procedure for constructing a D-ary code, the smallest D probability masses are merged in each step. If the resulting code tree is formed in k + 1 steps, where k 0, then there will be k + 1 internal nodes and D + k(D - 1) leaves, where each leaf corresponds to a source symbol in the alphabet. If the alphabet size m has the form D + k(D - 1), then we can apply the Huffman procedure directly. Otherwise, we need to add a few dummy symbols with probability 0 to the alphabet in order to make the total number of symbols have the form D + k(D - 1). 90 4 Zero-Error Data Compression Example 4.14. If we want to construct a quaternary Huffman code (D = 4) for the source in the last example, we need to add 2 dummy symbols so that the total number of symbols becomes 7 = 4 + (1)3, where k = 1. In general, we need to add at most D - 2 dummy symbols. In Section 4.1, we have proved the entropy bound for a uniquely decodable code. This bound also applies to a prefix code since a prefix code is uniquely decodable. In particular, it applies to a Huffman code, which is a prefix code by construction. Thus the expected length of a Huffman code is at least the entropy of the source. In Example 4.13, the entropy H(X) is 2.202 bits, while the expected length of the Huffman code is 0.35(2) + 0.1(3) + 0.15(3) + 0.2(2) + 0.2(2) = 2.25. (4.31) We now turn to proving the optimality of a Huffman code. For simplicity, we will only prove the optimality of a binary Huffman code. Extension of the proof to the general case is straightforward. Without loss of generality, assume that p1 p2 pm . (4.32) Denote the codeword assigned to pi by ci , and denote its length by li . To prove that a Huffman code is actually optimal, we make the following observations. Lemma 4.15. In an optimal code, shorter codewords are assigned to larger probabilities. Proof. Consider 1 i < j m such that pi > pj . Assume that in a code, the codewords ci and cj are such that li > lj , i.e., a shorter codeword is assigned to a smaller probability. Then by exchanging ci and cj , the expected length of the code is changed by (pi lj + pj li ) - (pi li + pj lj ) = (pi - pj )(lj - li ) < 0 (4.33) since pi > pj and li > lj . In other words, the code can be improved and therefore is not optimal. The lemma is proved. Lemma 4.16. There exists an optimal code in which the codewords assigned to the two smallest probabilities are siblings, i.e., the two codewords have the same length and they differ only in the last symbol. Proof. The reader is encouraged to trace the steps in this proof by drawing a code tree. Consider any optimal code. From the last lemma, the codeword cm assigned to pm has the longest length. Then the sibling of cm cannot be the prefix of another codeword. We claim that the sibling of cm must be a codeword. To see this, assume that it is not a codeword (and it is not the prefix of another codeword). Then we can replace cm by its parent to improve the code because the length of 4.2 Prefix Codes 91 the codeword assigned to pm is reduced by 1, while all the other codewords remain unchanged. This is a contradiction to the assumption that the code is optimal. Therefore, the sibling of cm must be a codeword. If the sibling of cm is assigned to pm-1 , then the code already has the desired property, i.e., the codewords assigned to the two smallest probabilities are siblings. If not, assume that the sibling of cm is assigned to pi , where i < m - 1. Since pi pm-1 , lm-1 li = lm . On the other hand, by Lemma 4.15, lm-1 is always less than or equal to lm , which implies that lm-1 = lm = li . Then we can exchange the codewords for pi and pm-1 without changing the expected length of the code (i.e., the code remains optimal) to obtain the desired code. The lemma is proved. Suppose ci and cj are siblings in a code tree. Then li = lj . If we replace ci and cj by a common codeword at their parent, call it cij , then we obtain a reduced code tree, and the probability of cij is pi + pj . Accordingly, the probability set becomes a reduced probability set with pi and pj replaced by a probability pi + pj . Let L and L be the expected lengths of the original code and the reduced code, respectively. Then L - L = (pi li + pj lj ) - (pi + pj )(li - 1) = (pi li + pj li ) - (pi + pj )(li - 1) = pi + pj , which implies L = L + (pi + pj ). (4.37) This relation says that the difference between the expected length of the original code and the expected length of the reduced code depends only on the values of the two probabilities merged but not on the structure of the reduced code tree. Theorem 4.17. The Huffman procedure produces an optimal prefix code. Proof. Consider an optimal code in which cm and cm-1 are siblings. Such an optimal code exists by Lemma 4.16. Let {pi } be the reduced probability set obtained from {pi } by merging pm and pm-1 . From (4.37), we see that L is the expected length of an optimal code for {pi } if and only if L is the expected length of an optimal code for {pi }. Therefore, if we can find an optimal code for {pi }, we can use it to construct an optimal code for {pi }. Note that by merging pm and pm-1 , the size of the problem, namely the total number of probability masses, is reduced by one. To find an optimal code for {pi }, we again merge the two smallest probability in {pi }. This is repeated until the size of the problem is eventually reduced to 2, which we know that an optimal code has two codewords of length 1. In the last step of the Huffman procedure, two probability masses are merged, which corresponds to the formation of a code with two codewords of length 1. Thus the Huffman procedure indeed produces an optimal code. (4.34) (4.35) (4.36) 92 4 Zero-Error Data Compression We have seen that the expected length of a Huffman code is lower bounded by the entropy of the source. On the other hand, it would be desirable to obtain an upper bound in terms of the entropy of the source. This is given in the next theorem. Theorem 4.18. The expected length of a Huffman code, denoted by LHuff , satisfies LHuff < HD (X) + 1. (4.38) This bound is the tightest among all the upper bounds on LHuff which depend only on the source entropy. Proof. We will construct a prefix code with expected length less than H(X) + 1. Then, because a Huffman code is an optimal prefix code, its expected length LHuff is upper bounded by H(X) + 1. Consider constructing a prefix code with codeword lengths {li }, where li = - logD pi . Then - logD pi li < - logD pi + 1, or pi D-li > D-1 pi . Thus D-li i i (4.39) (4.40) (4.41) (4.42) pi = 1, i.e., {li } satisfies the Kraft inequality, which implies that it is possible to construct a prefix code with codeword lengths {li }. It remains to show that L, the expected length of this code, is less than H(X) + 1. Toward this end, consider L= i pi li pi (- logD pi + 1) i (4.43) (4.44) pi i < =- pi logD pi + i (4.45) (4.46) = H(X) + 1, where (4.44) follows from the upper bound in (4.40). Thus we conclude that LHuff L < H(X) + 1. (4.47) To see that this upper bound is the tightest possible, we have to show that there exists a sequence of distributions Pk such that LHuff approaches H(X)+1 as k . This can be done by considering the sequence of D-ary distributions 4.3 Redundancy of Prefix Codes 93 Pk = 1- D-1 1 1 , ,, k k k , (4.48) where k D. The Huffman code for each Pk consists of D codewords of length 1. Thus LHuff is equal to 1 for all k. As k , H(X) 0, and hence LHuff approaches H(X) + 1. The theorem is proved. The code constructed in the above proof is known as the Shannon code. The idea is that in order for the code to be near-optimal, we should choose li close to - log pi for all i. When {pi } is D-adic, li can be chosen to be exactly - log pi because the latter are integers. In this case, the entropy bound is tight. From the entropy bound and the above theorem, we have H(X) LHuff < H(X) + 1. (4.49) Now suppose we use a Huffman code to encode X1 , X2 , , Xn which are n i.i.d. copies of X. Let us denote the length of this Huffman code by Ln . Huff Then (4.49) becomes nH(X) Ln < nH(X) + 1. Huff Dividing by n, we obtain H(X) 1 1 n L < H(X) + . n Huff n (4.51) (4.50) As n , the upper bound approaches the lower bound. Therefore, n-1 Ln , the coding rate of the code, namely the average number of code Huff symbols needed to encode a source symbol, approaches H(X) as n . But of course, as n becomes large, constructing a Huffman code becomes very complicated. Nevertheless, this result indicates that entropy is a fundamental measure of information. 4.3 Redundancy of Prefix Codes The entropy bound for a uniquely decodable code has been proved in Section 4.1. In this section, we present an alternative proof specifically for prefix codes which offers much insight into the redundancy of such codes. Let X be a source random variable with probability distribution {p1 , p2 , , pm }, (4.52) where m 2. A D-ary prefix code for X can be represented by a D-ary code tree with m leaves, where each leaf corresponds to a codeword. We denote the leaf corresponding to pi by ci and the order of ci by li , and assume that the alphabet is 94 4 Zero-Error Data Compression {0, 1, , D - 1}. (4.53) Let I be the index set of all the internal nodes (including the root) in the code tree. Instead of matching codewords by brute force, we can use the code tree of a prefix code for more efficient decoding. To decode a codeword, we trace the path specified by the codeword from the root of the code tree until it terminates at the leaf corresponding to that codeword. Let qk be the probability of reaching an internal node k I during the decoding process. The probability qk is called the reaching probability of internal node k. Evidently, qk is equal to the sum of the probabilities of all the leaves descending from node k. Let pk,j be the probability that the jth branch of node k is taken during ~ the decoding process. The probabilities pk,j , 0 j D - 1, are called the ~ branching probabilities of node k, and qk = j pk,j . ~ (4.54) Once node k is reached, the conditional branching distribution is ~ pk,D-1 ~ pk,0 pk,1 ~ , ,, qk qk qk Then define the conditional entropy of node k by hk = HD pk,0 pk,1 ~ ~ pk,D-1 ~ , ,, qk qk qk , (4.56) . (4.55) where with a slight abuse of notation, we have used HD () to denote the entropy in the base D of the conditional branching distribution in the parenthesis. By Theorem 2.43, hk 1. The following lemma relates the entropy of X with the structure of the code tree. Lemma 4.19. HD (X) = kI qk hk . Proof. We prove the lemma by induction on the number of internal nodes of the code tree. If there is only one internal node, it must be the root of the tree. Then the lemma is trivially true upon observing that the reaching probability of the root is equal to 1. Assume the lemma is true for all code trees with n internal nodes. Now consider a code tree with n + 1 internal nodes. Let k be an internal node such that k is the parent of a leaf c with maximum order. Each sibling of c may or may not be a leaf. If it is not a leaf, then it cannot be the ascendent of another leaf because we assume that c is a leaf with maximum order. Now consider revealing the outcome of X in two steps. In the first step, if the outcome of X is not a leaf descending from node k, we identify the outcome exactly, otherwise we identify the outcome to be a child of node k. We call this random 4.3 Redundancy of Prefix Codes 95 variable V . If we do not identify the outcome exactly in the first step, which happens with probability qk , we further identify in the second step which of the children (child) of node k the outcome is (there is only one child of node k which can be the outcome if all the siblings of c are not leaves). We call this random variable W . If the second step is not necessary, we assume that W takes a constant value with probability 1. Then X = (V, W ). The outcome of V can be represented by a code tree with n internal nodes which is obtained by pruning the original code tree at node k. Then by the induction hypothesis, H(V ) = qk hk . (4.57) k I\{k} By the chain rule for entropy, we have H(X) = H(V ) + H(W |V ) = k I\{k} (4.58) (4.59) (4.60) qk hk + (1 - qk ) 0 + qk hk qk hk . k I = The lemma is proved. The next lemma expresses the expected length L of a prefix code in terms of the reaching probabilities of the internal nodes of the code tree. Lemma 4.20. L = Proof. Define aki = Then li = kI kI qk . 1 if leaf ci is a descendent of internal node k 0 otherwise. aki , (4.61) (4.62) because there are exactly li internal nodes of which ci is a descendent if the order of ci is li . On the other hand, qk = i aki pi . (4.63) Then L= i pi li pi i kI (4.64) aki (4.65) = 96 4 Zero-Error Data Compression = kI i pi aki qk , kI (4.66) (4.67) = proving the lemma. Define the local redundancy of an internal node k by rk = qk (1 - hk ). (4.68) This quantity is local to node k in the sense that it depends only on the branching probabilities of node k, and it vanishes if and only if pk,j = qk /D ~ for all j, i.e., if and only if the node is balanced. Note that rk 0 because hk 1. The next theorem says that the redundancy R of a prefix code is equal to the sum of the local redundancies of all the internal nodes of the code tree. Theorem 4.21 (Local Redundancy Theorem). Let L be the expected length of a D-ary prefix code for a source random variable X, and R be the redundancy of the code. Then R= kI rk . (4.69) Proof. By Lemmas 4.19 and 4.20, we have R = L - HD (X) = kI (4.70) qk hk (4.71) (4.72) (4.73) qk - k = kI qk (1 - hk ) rk . kI = The theorem is proved. We now present an slightly different version of the entropy bound. Corollary 4.22 (Entropy Bound). Let R be the redundancy of a prefix code. Then R 0 with equality if and only if all the internal nodes in the code tree are balanced. Proof. Since rk 0 for all k, it is evident from the local redundancy theorem that R 0. Moreover R = 0 if and only if rk = 0 for all k, which means that all the internal nodes in the code tree are balanced. Chapter Summary 97 Remark Before the entropy bound was stated in Theorem 4.6, we gave the intuitive explanation that the entropy bound results from the fact that a D-ary symbol can carry at most one D-it of information. Therefore, when the entropy bound is tight, each code symbol has to carry exactly one D-it of information. Now consider revealing a random codeword one symbol after another. The above corollary states that in order for the entropy bound to be tight, all the internal nodes in the code tree must be balanced. That is, as long as the codeword is not completed, the next code symbol to be revealed always carries one D-it of information because it is distributed uniformly on the alphabet. This is consistent with the intuitive explanation we gave for the entropy bound. Example 4.23. The local redundancy theorem allows us to lower bound the redundancy of a prefix code based on partial knowledge on the structure of the code tree. More specifically, R kI rk (4.74) for any subset I of I. Let pm-1 , pm be the two smallest probabilities in the source distribution. In constructing a binary Huffman code, pm-1 and pm are merged. Then the redundancy of a Huffman code is lower bounded by (pm-1 + pm ) 1 - H2 pm pm-1 , pm-1 + pm pm-1 + pm , (4.75) the local redundancy of the parent of the two leaves corresponding to pm-1 and pm . See Yeung [399] for progressive lower and upper bounds on the redundancy of a Huffman code. Chapter Summary Kraft Inequality: For a D-ary uniquely decodable source code, m D-lk 1. k=1 Entropy Bound: L= k pk lk HD (X), with equality if and only if the distribution of X is D-adic. 98 4 Zero-Error Data Compression Definition: A code is called a prefix code if no codeword is a prefix of any other codeword. Existence of Prefix Code: A D-ary prefix code with codeword lengths l1 , l2 , , lm exists if and only if the Kraft inequality is satisfied. Huffman Code: 1. A Huffman code is a prefix code with the shortest expected length for a given source. 2. HD (X) LHuff < HD (X) + 1. Huffman Procedure: Keep merging the D smallest probability masses. Redundancy of Prefix Code: L - HD (X) = R = kI rk , where rk = qk (1 - hk ) is the local redundancy of an internal node k. Problems 1. Construct a binary Huffman code for the distribution {0.25, 0.05, 0.1, 0.13, 0.2, 0.12, 0.08, 0.07}. 2. Construct a ternary Huffman code for the source distribution in Problem 1. 3. Show that a Huffman code is an optimal uniquely decodable code for a given source distribution. 4. Construct an optimal binary prefix code for the source distribution in Problem 1 such that all the codewords have even lengths. 5. Prove directly that the codeword lengths of a prefix code satisfy the Kraft inequality without using Theorem 4.4. 6. Prove that if p1 > 0.4, then the shortest codeword of a binary Huffman code has length equal to 1. Then prove that the redundancy of such a Huffman code is lower bounded by 1 - hb (p1 ). (Johnsen [192].) 7. Suffix codes A code is a suffix code if no codeword is a suffix of any other codeword. Show that a suffix code is uniquely decodable. 8. Fix-free codes A code is a fix-free code if it is both a prefix code and a suffix code. Let l1 , l2 , , lm be m positive integers. Prove that if m 2-lk k=1 1 , 2 then there exists a binary fix-free code with codeword lengths l1 , l2 , , lm . (Ahlswede et al. [5].) Historical Notes 99 9. Random coding for prefix codes Construct a binary prefix code with codeword lengths l1 l2 lm as follows. For each 1 k m, the codeword with length lk is chosen independently from the set of all 2lk possible binary strings with length lk according the uniform distribution. Let Pm (good) be the probability that the code so constructed is a prefix code. a) Prove that P2 (good) = (1 - 2-l1 )+ , where (x)+ = b) Prove by induction on m that m x if x 0 0 if x < 0. 1 - k-1 + s-lj . Pm (good) = k=1 j=1 c) Observe that there exists a prefix code with codeword lengths l1 , l2 , , lm if and only if Pm (good) > 0. Show that Pm (good) > 0 is equivalent to the Kraft inequality. By using this random coding method, one can derive the Kraft inequality without knowing the inequality ahead of time. (Ye and Yeung [395].) 10. Let X be a source random variable. Suppose a certain probability mass pk in the distribution of X is given. Let lj = where xj = p j for a) b) c) - log pj if j = k - log(pj + xj ) if j = k, pk - 2- - log pk 1 - pk all j = k. Show that 1 lj - log pj for all j. Show that {lj } satisfies the Kraft inequality. Obtain an upper bound on LHuff in terms of H(X) and pk which is tighter than H(X)+1. This shows that when partial knowledge about the source distribution in addition to the source entropy is available, tighter upper bounds on LHuff can be obtained. (Ye and Yeung [396].) Historical Notes The foundation for the material in this chapter can be found in Shannon's original paper [322]. The Kraft inequality for uniquely decodable codes was first proved by McMillan [267]. The proof given here is due to Karush [195]. 100 4 Zero-Error Data Compression The Huffman coding procedure was devised and proved to be optimal by Huffman [175]. The same procedure was devised independently by Zimmerman [418]. Linder et al. [236] have proved the existence of an optimal prefix code for an infinite source alphabet which can be constructed from Huffman codes for truncations of the source distribution. The local redundancy theorem is due to Yeung [399]. A comprehensive survey of code trees for lossless data compression can be found in Abrahams [1]. 5 Weak Typicality In the last chapter, we have discussed the significance of entropy in the context of zero-error data compression. In this chapter and the next, we explore entropy in terms of the asymptotic behavior of i.i.d. sequences. Specifically, two versions of the asymptotic equipartition property (AEP), namely the weak AEP and the strong AEP, are discussed. The role of these AEP's in information theory is analogous to the role of the weak law of large numbers in probability theory. In this chapter, the weak AEP and its relation with the source coding theorem are discussed. All the logarithms are in the base 2 unless otherwise specified. 5.1 The Weak AEP We consider an information source {Xk , k 1} where Xk are i.i.d. with distribution p(x). We use X to denote the generic random variable and H(X) to denote the common entropy for all Xk , where H(X) < . Let X = (X1 , X2 , , Xn ). Since Xk are i.i.d., p(X) = p(X1 )p(X2 ) p(Xn ). (5.1) Note that p(X) is a random variable because it is a function of the random variables X1 , X2 , , Xn . We now prove an asymptotic property of p(X) called the weak asymptotic equipartition property (weak AEP). Theorem 5.1 (Weak AEP I). - 1 log p(X) H(X) n > 0, for n sufficiently large, >1- . (5.3) (5.2) in probability as n , i.e., for any Pr - 1 log p(X) - H(X) n 102 5 Weak Typicality Proof. Since X1 , X2 , , Xn are i.i.d., by (5.1), - 1 1 log p(X) = - n n n log p(Xk ). k=1 (5.4) The random variables log p(Xk ) are also i.i.d. Then by the weak law of large numbers, the right hand side of (5.4) tends to -E log p(X) = H(X), in probability, proving the theorem. The weak AEP is nothing more than a straightforward application of the weak law of large numbers. However, as we will see shortly, this property has significant implications. n Definition 5.2. The weakly typical set W[X] with respect to p(x) is the set n of sequences x = (x1 , x2 , , xn ) X such that (5.5) - or equivalently, 1 log p(x) - H(X) , n (5.6) H(X) - - 1 log p(x) H(X) + , n (5.7) n where is an arbitrarily small positive real number. The sequences in W[X] are called weakly -typical sequences. The quantity 1 1 - log p(x) = - n n n log p(xk ) k=1 (5.8) is called the empirical entropy of the sequence x. The empirical entropy of a weakly typical sequence is close to the true entropy H(X). The important n properties of the set W[X] are summarized in the next theorem which we will see is equivalent to the weak AEP. Theorem 5.3 (Weak AEP II). The following hold for any n 1) If x W[X] , then > 0: 2-n(H(X)+ ) p(x) 2-n(H(X)- ) . 2) For n sufficiently large, n Pr{X W[X] } > 1 - . (5.9) (5.10) 5.1 The Weak AEP 103 3) For n sufficiently large, n (1 - )2n(H(X)- ) |W[X] | 2n(H(X)+ ) . (5.11) n Proof. Property 1 follows immediately from the definition of W[X] in (5.7). Property 2 is equivalent to Theorem 5.1. To prove Property 3, we use the lower bound in (5.9) and consider n n |W[X] |2-n(H(X)+ ) Pr{W[X] } 1, (5.12) (5.13) which implies n |W[X] | 2n(H(X)+ ) . Note that this upper bound holds for any n 1. On the other hand, using the upper bound in (5.9) and Theorem 5.1, for n sufficiently large, we have n n 1 - Pr{W[X] } |W[X] |2-n(H(X)- ) . (5.14) (5.15) Then n |W[X] | (1 - )2n(H(X)- ) . Combining (5.13) and (5.15) gives Property 3. The theorem is proved. Remark Theorem 5.3 is a consequence of Theorem 5.1. However, Property 2 in Theorem 5.3 is equivalent to Theorem 5.1. Therefore, Theorem 5.1 and Theorem 5.3 are equivalent, and they will both be referred to as the weak AEP. The weak AEP has the following interpretation. Suppose X = (X1 , X2 , , Xn ) is drawn i.i.d. according to p(x), where n is large. After the sequence is drawn, we ask what the probability of occurrence of the sequence is. The weak AEP says that the probability of occurrence of the sequence drawn is close to 2-nH(X) with very high probability. Such a sequence is called a weakly typical sequence. Moreover, the total number of weakly typical sequences is approximately equal to 2nH(X) . The weak AEP, however, does not say that most of the sequences in X n are weakly typical. In fact, the number of weakly typical sequences is in general insignificant compared with the total number of sequences, because n |W[X] | |X |n 2nH(X) = 2-n(log |X |-H(X)) 0 2n log |X | (5.16) as n as long as H(X) is strictly less than log |X |. The idea is that, although the size of the weakly typical set may be insignificant compared with the size of the set of all sequences, the former has almost all the probability. When n is large, one can almost think of the sequence X as being obtained by choosing a sequence from the weakly typical set according to the uniform 104 5 Weak Typicality distribution. Very often, we concentrate on the properties of typical sequences because any property which is proved to be true for typical sequences will then be true with high probability. This in turn determines the average behavior of a large sample. Remark The most likely sequence is in general not weakly typical although the probability of the weakly typical set is close to 1 when n is large. For example, for Xk i.i.d. with p(0) = 0.1 and p(1) = 0.9, (1, 1, , 1) is the most likely sequence, but it is not weakly typical because its empirical entropy is not close to the true entropy. The idea is that as n , the probability of every sequence, including that of the most likely sequence, tends to 0. Therefore, it is not necessary for a weakly typical set to include the most likely sequence in order to possess a probability close to 1. 5.2 The Source Coding Theorem To encode a random sequence X = (X1 , X2 , , Xn ) drawn i.i.d. according to p(x) by a block code, we construct a one-to-one mapping from a subset A of X n to an index set I = {1, 2, , M }, (5.17) where |A| = M |X |n . We do not have to assume that |X | is finite. The indices in I are called codewords, and the integer n is called the block length of the code. If a sequence x A occurs, the encoder outputs the corresponding codeword which is specified by approximately log M bits. If a sequence x A occurs, the encoder outputs the constant codeword 1. In either case, the codeword output by the encoder is decoded to the sequence in A corresponding to that codeword by the decoder. If a sequence x A occurs, then x is decoded correctly by the decoder. If a sequence x A occurs, then x is not decoded correctly by the decoder. For such a code, its performance is measured by the coding rate defined as n-1 log M (in bits per source symbol), and the probability of error is given by Pe = Pr{X A}. (5.18) If the code is not allowed to make any error, i.e., Pe = 0, it is clear that M must be taken to be |X |n , or A = X n . In that case, the coding rate is equal to log |X |. However, if we allow Pe to be any small quantity, Shannon [322] showed that there exists a block code whose coding rate is arbitrarily close to H(X) when n is sufficiently large. This is the direct part of Shannon's source coding theorem, and in this sense the source sequence X is said to be reconstructed almost perfectly. We now prove the direct part of the source coding theorem by constructing a desired code. First, we fix > 0 and take 5.2 The Source Coding Theorem n A = W[X] 105 (5.19) and M = |A|. For sufficiently large n, by the weak AEP, n (1 - )2n(H(X)- ) M = |A| = |W[X] | 2n(H(X)+ ) . (5.20) (5.21) Therefore, the coding rate n-1 log M satisfies 1 1 log(1 - ) + H(X) - log M H(X) + . n n Also by the weak AEP, n Pe = Pr{X A} = Pr{X W[X] } < . (5.22) (5.23) Letting 0, the coding rate tends to H(X), while Pe tends to 0. This proves the direct part of the source coding theorem. The converse part of the source coding theorem says that if we use a block code with block length n and coding rate less than H(X) - , where > 0 does not change with n, then Pe 1 as n . To prove this, consider any code with block length n and coding rate less than H(X) - , so that M , the total number of codewords, is at most 2n(H(X)-) . We can use some of these n codewords for the typical sequences x W[X] , and some for the non-typical n sequences x W[X] . The total probability of the typical sequences covered by the code, by the weak AEP, is upper bounded by 2n(H(X)-) 2-n(H(X)- ) = 2-n(- ) . (5.24) Therefore, the total probability covered by the code is upper bounded by n 2-n(- ) + Pr{X W[X] } < 2-n(- ) + (5.25) for n sufficiently large, again by the weak AEP. This probability is equal to 1 - Pe because Pe is the probability that the source sequence X is not covered by the code. Thus 1 - Pe < 2-n(- ) + , (5.26) or Pe > 1 - (2-n(- ) + ). (5.27) This inequality holds when n is sufficiently large for any > 0, in particular for < . Then for any < , Pe > 1 - 2 when n is sufficiently large. Hence, Pe 1 as n and then 0. This proves the converse part of the source coding theorem. 106 5 Weak Typicality 5.3 Efficient Source Coding Theorem 5.4. Let Y = (Y1 , Y2 , , Ym ) be a random binary sequence of length m. Then H(Y) m with equality if and only if Yi are drawn i.i.d. according to the uniform distribution on {0, 1}. Proof. By the independence bound for entropy, m H(Y) i=1 H(Yi ) (5.28) with equality if and only if Yi are mutually independent. By Theorem 2.43, H(Yi ) log 2 = 1 (5.29) with equality if and only if Yi is distributed uniformly on {0, 1}. Combining (5.28) and (5.29), we have m H(Y) i=1 H(Yi ) m, (5.30) where this upper bound is tight if and only if Yi are mutually independent and each of them is distributed uniformly on {0, 1}. The theorem is proved. Let Y = (Y1 , Y2 , , Yn ) be a sequence of length n such that Yi are drawn i.i.d. according to the uniform distribution on {0, 1}, and let Y denote the generic random variable. Then H(Y ) = 1. According to the source coding theorem, for almost perfect reconstruction of Y, the coding rate of the source code must be at least 1. It turns out that in this case it is possible to use a source code with coding rate exactly equal to 1 while the source sequence Y can be reconstructed with zero error. This can be done by simply encoding all the 2n possible binary sequences of length n, i.e., by taking M = 2n . Then the coding rate is given by n-1 log M = n-1 log 2n = 1. (5.31) Since each symbol in Y is a bit and the rate of the best possible code describing Y is 1 bit per symbol, Y1 , Y2 , , Yn are called fair bits, with the connotation that they are incompressible. It turns out that the whole idea of efficient source coding by a block code is to describe the information source by a binary sequence consisting of "almost fair" bits. Consider a sequence of block codes which encode X = (X1 , X2 , , Xn ) into Y = (Y1 , Y2 , , Ym ), where Xk are i.i.d. with generic random variable X, Y is a binary sequence with length m nH(X), (5.32) 5.4 The Shannon-McMillan-Breiman Theorem 107 and n . For simplicity, we assume that the common alphabet X is fi^ nite. Let X X n be the reconstruction of X by the decoder and Pe be the probability of error, i.e., ^ Pe = Pr{X = X}. (5.33) Further assume Pe 0 as n . We will show that Y consists of almost fair bits. By Fano's inequality, ^ H(X|X) 1 + Pe log |X |n = 1 + nPe log |X |. ^ Since X is a function of Y, ^ ^ H(Y) = H(Y, X) H(X). It follows that ^ H(Y) H(X) ^ I(X; X) ^ = H(X) - H(X|X) nH(X) - (1 + nPe log |X |) = n(H(X) - Pe log |X |) - 1. On the other hand, by Theorem 5.4, H(Y) m. Combining (5.40) and (5.41), we have n(H(X) - Pe log |X |) - 1 H(Y) m. (5.42) (5.41) (5.36) (5.37) (5.38) (5.39) (5.40) (5.35) (5.34) Since Pe 0 as n , the above lower bound on H(Y) is approximately equal to nH(X) m (5.43) when n is large (cf. (5.32)). Therefore, H(Y) m. (5.44) In light of Theorem 5.4, Y almost attains the maximum possible entropy. In this sense, we say that Y consists of almost fair bits. 5.4 The Shannon-McMillan-Breiman Theorem For an i.i.d. information source {Xk } with generic random variable X and generic distribution p(x), the weak AEP states that 108 5 Weak Typicality - 1 log p(X) H(X) n (5.45) in probability as n , where X = (X1 , X2 , , Xn ). Here H(X) is the entropy of the generic random variables X as well as the entropy rate of the source {Xk }. In Section 2.10, we showed that the entropy rate H of a source {Xk } exists if the source is stationary. The Shannon-McMillan-Breiman theorem states that if {Xk } is also ergodic, then Pr - lim 1 log Pr{X} = H n = 1. (5.46) n 1 This means that if {Xk } is stationary and ergodic, then - n log Pr{X} not only almost always converges, but it also almost always converges to H. For this reason, the Shannon-McMillan-Breiman theorem is also referred to as the weak AEP for ergodic stationary sources. The formal definition of an ergodic source and the statement of the Shannon-McMillan-Breiman theorem require the use of measure theory which is beyond the scope of this book. We point out that the event in (5.46) involves an infinite collection of random variables which cannot be described by a joint distribution except in very special cases. Without measure theory, the probability of this event in general cannot be properly defined. However, this does not prevent us from developing some appreciation of the ShannonMcMillan-Breiman theorem. Let X be the common alphabet for a stationary source {Xk }. Roughly speaking, a stationary source {Xk } is ergodic if the time average exhibited by a single realization of the source is equal to the ensemble average with probability 1. More specifically, for any k1 , k2 , , km , Pr 1 n n lim n-1 f (Xk1 +l , Xk2 +l , , Xkm +l ) l=0 = Ef (Xk1 , Xk2 , , Xkm ) = 1, (5.47) where f is a function defined on X m which satisfies suitable conditions. For the special case that {Xk } satisfies 1 Pr lim n n n Xl = EXk l=1 = 1, (5.48) we say that {Xk } is mean ergodic. Example 5.5. The i.i.d. source {Xk } is mean ergodic under suitable conditions because the strong law of the large numbers states that (5.48) is satisfied. 5.4 The Shannon-McMillan-Breiman Theorem 109 Example 5.6. Consider the source {Xk } defined as follows. Let Z be a binary random variable uniformly distributed on {0, 1}. For all k, let Xk = Z. Then Pr and Pr Since EXk = 1 , 2 1 Pr lim n n 1 n n lim 1 n n lim n Xl = 0 l=1 = 1 2 (5.49) n Xl = 1 l=1 = 1 . 2 (5.50) n Xl = EXk l=1 = 0. (5.51) Therefore, {Xk } is not mean ergodic and hence not ergodic. If an information source {Xk } is stationary and ergodic, by the ShannonMcMillan-Breiman theorem, - 1 log Pr{X} H n (5.52) when n is large. That is, with probability close to 1, the probability of the sequence X which occurs is approximately equal to 2-nH . Then by means of arguments similar to the proof of Theorem 5.3, we see that there exist approximately 2nH sequences in X n whose probabilities are approximately equal to 2-nH , and the total probability of these sequences is almost 1. Therefore, by encoding these sequences with approximately nH bits, the source sequence X can be recovered with an arbitrarily small probability of error when the block length n is sufficiently large. This is a generalization of the direct part of the source coding theorem which gives a physical meaning to the entropy rate of an ergodic stationary sources. We remark that if a source is stationary but not ergodic, although the entropy rate always exists, it may not carry any physical meaning. As an example, by regarding printed English as a stationary ergodic process, Shannon [325] estimated by a guessing game that its entropy rate is about 1.3 bits per letter. Cover and King [78] described a gambling estimate of the entropy rate of printed English which gives 1.34 bits per letter. These results show that it is not possible to describe printed English accurately by using less than about 1.3 bits per letter. 110 5 Weak Typicality Chapter Summary Weak AEP I: - 1 log p(X) H(X) in probability. n Weakly Typical Set: n W[X] = x X n : -n-1 log p(x) - H(X) . Weak AEP II: n 1. 2-n(H(X)+ ) p(x) 2-n(H(X)- ) for x W[X] n 2. Pr{X W[X] } > 1 - for n sufficiently large n 3. (1 - )2n(H(X)- ) |W[X] | 2n(H(X)+ ) for n sufficiently large. Source Coding Theorem: An i.i.d. random sequence X1 , X2 , , Xn with generic random variable X can be compressed at rate H(X) + while Pe 0 as n . If a rate less than H(X) is used, then Pe 1 as n . Shannon-McMillan-Breiman Theorem: For a stationary source {Xk } with entropy rate H, Pr - lim 1 log Pr{X} = H n = 1. n Problems n 1. Show that for any > 0, W[X] is nonempty for sufficiently large n. 2. The source coding theorem with a general block code In proving the converse of the source coding theorem, we assume that each codeword in I corresponds to a unique sequence in X n . More generally, a block code with block length n is defined by an encoding function f : X n I and a decoding function g : I X n . Prove that Pe 1 as n even if we are allowed to use a general block code. 3. Following Problem 2, we further assume that we can use a block code with probabilistic encoding and decoding. For such a code, encoding is defined by a transition matrix F from X n to I and decoding is defined by a transition matrix G from I to X n . Prove that Pe 1 as n even if we are allowed to use such a code. 4. In the discussion in Section 5.3, we made the assumption that the common alphabet X is finite. Can you draw the same conclusion when X is countable but H(X) is finite? Hint: use Problem 2. Problems 111 5. Alternative definition of weak typicality an i.i.d. sequence whose generic random p(x). Let qx be the empirical distribution n-1 N (x; x) for all x X , where N (x; x) x in x. a) Show that for any x X n , - Let X = (X1 , X2 , , Xn ) be variable X is distributed with of the sequence x, i.e., qx (x) = is the number of occurrence of 1 log p(x) = D(qx p) + H(qx ). n n b) Show that for any > 0, the weakly typical set W[X] with respect to n p(x) is the set of sequences x X such that |D(qx p) + H(qx ) - H(p)| . c) Show that for sufficiently large n, Pr{|D(qx p) + H(qx ) - H(p)| } > 1 - . (Ho and Yeung [167].) 6. Verify that the empirical entropy of a sequence is different from the entropy of the empirical distribution of the sequence (see Problem 5 for definition). 7. Let p and q be two probability distributions on the same alphabet X such that H(p) = H(q). Show that there exists an > 0 such that pn x Xn : - 1 log pn (x) - H(q) < n 0 as n . Give an example that p = q but the above convergence does not hold. 8. Let p and q be two probability distributions on the same alphabet X with the same support. a) Prove that for any > 0, pn x Xn : - 1 log q n (x) - (H(p) + D(p q)) < n 1 as n . b) Prove that for any > 0, x Xn : - 1 log q n (x) - (H(p) + D(p q)) < n (s) 2n(H(p)+D(p q)+) . 9. Universal source coding Let F = {{Xk , k 1} : s S} be a family of i.i.d. information sources indexed by a finite set S with a common alphabet X . Define H = max H(X (s) ) sS where X (s) is the generic random variable for {Xk , k 1}, and (s) 112 5 Weak Typicality An (S) = sS n W[X (s) ] , where > 0. a) Prove that for all s S, Pr{X(s) An (S)} 1 as n , where X(s) = (X1 , X2 , , Xn ). b) Prove that for any > , |An (S)| 2n(H+ ) (s) (s) (s) for sufficiently large n. c) Suppose we know that an information source is in the family F but we do not know which one it is. Devise a compression scheme for the information source such that it is asymptotically optimal for every possible source in F. 10. Let {Xk , k 1} be an i.i.d. information source with generic random variable X and alphabet X . Assume p(x)[log p(x)]2 < x and define Zn = - log p(X) - nH(X) n for n = 1, 2, . Prove that Zn Z in distribution, where Z is a Gaussian random variable with mean 0 and variance x p(x)[log p(x)]2 - H(X)2 . Historical Notes The weak asymptotic equipartition property (AEP), which is instrumental in proving the source coding theorem, was first proved by Shannon in his original paper [322]. In this paper, he also stated that this property can be extended to an ergodic stationary source. Subsequently, McMillan [266] and Breiman [48] proved this property for an ergodic stationary source with a finite alphabet. Chung [71] extended the theme to a countable alphabet. 6 Strong Typicality Weak typicality requires that the empirical entropy of a sequence is close to the true entropy. In this chapter, we introduce a stronger notion of typicality which requires that the relative frequency of each possible outcome is close to the corresponding probability. As we will see later, strong typicality is more powerful and flexible than weak typicality as a tool for theorem proving for memoryless problems. However, strong typicality can be used only for random variables with finite alphabets. Throughout this chapter, typicality refers to strong typicality and all the logarithms are in the base 2 unless otherwise specified. 6.1 Strong AEP We consider an information source {Xk , k 1} where Xk are i.i.d. with distribution p(x). We use X to denote the generic random variable and H(X) to denote the common entropy for all Xk , where H(X) < . Let X = (X1 , X2 , , Xn ). n Definition 6.1. The strongly typical set T[X] with respect to p(x) is the set n of sequences x = (x1 , x2 , , xn ) X such that N (x; x) = 0 for x SX , and x 1 N (x; x) - p(x) , n (6.1) where N (x; x) is the number of occurrences of x in the sequence x, and is n an arbitrarily small positive real number. The sequences in T[X] are called strongly -typical sequences. Throughout this chapter, we adopt the convention that all the summations, products, unions, etc, are taken over the corresponding supports unless othn erwise specified. The strongly typical set T[X] shares similar properties with 114 6 Strong Typicality its weakly typical counterpart, which is summarized as the strong asymptotic equipartition property (strong AEP) below. The interpretation of the strong AEP is similar to that of the weak AEP. Theorem 6.2 (Strong AEP). There exists > 0 such that 0 as 0, and the following hold: n 1) If x T[X] , then 2-n(H(X)+) p(x) 2-n(H(X)-) . 2) For n sufficiently large, n Pr{X T[X] } > 1 - . (6.2) (6.3) 3) For n sufficiently large, n (1 - )2n(H(X)-) |T[X] | 2n(H(X)+) . (6.4) n Proof To prove Property 1, for x T[X] , we write p(x) = x p(x)N (x;x) . (6.5) Then log p(x) = x N (x; x) log p(x) (N (x; x) - np(x) + np(x)) log p(x) x (6.6) (6.7) (6.8) = =n p(x) log p(x) - n x x 1 N (x; x) - p(x) (- log p(x)) n = -n H(X) + x n Since x T[X] , 1 N (x; x) - p(x) (- log p(x)) . n (6.9) x 1 N (x; x) - p(x) , n (6.10) which implies 1 N (x; x) - p(x) (- log p(x)) n x 6.1 Strong AEP 115 x 1 N (x; x) - p(x) (- log p(x)) n x (6.11) (6.12) (6.13) (6.14) - log min p(x) x x 1 N (x; x) - p(x) n - log min p(x) = , where = - log min p(x) > 0. x (6.15) Therefore, - x 1 N (x; x) - p(x) (- log p(x)) . n (6.16) It then follows from (6.9) that -n(H(X) + ) log p(x) -n(H(X) - ), or 2-n(H(X)+) p(x) 2-n(H(X)-) , where 0 as 0, proving Property 1. To prove Property 2, we write n (6.17) (6.18) N (x; X) = k=1 Bk (x), (6.19) where Bk (x) = 1 if Xk = x 0 if Xk = x. (6.20) Then Bk (x), k = 1, 2, , n are i.i.d. random variables with Pr{Bk (x) = 1} = p(x) and Pr{Bk (x) = 0} = 1 - p(x). Note that EBk (x) = (1 - p(x)) 0 + p(x) 1 = p(x). By the weak law of large numbers, for any > 0 and for any x X , Pr 1 n n (6.21) (6.22) (6.23) Bk (x) - p(x) > k=1 |X | < |X | (6.24) for n sufficiently large. Then 116 6 Strong Typicality Pr 1 N (x; X) - p(x) > for some x n |X | 1 n n = Pr Bk (x) - p(x) > k=1 for some x |X | |X | (6.25) = Pr x 1 n 1 n n Bk (x) - p(x) > k=1 n (6.26) x Pr |X | Bk (x) - p(x) > k=1 |X | (6.27) (6.28) (6.29) < x = , where we have used the union bound1 to obtain (6.27). Since 1 N (x; x) - p(x) > n (6.30) x implies 1 N (x; x) - p(x) > n |X | we have n Pr X T[X] for some x X , (6.31) = Pr x 1 N (x; X) - p(x) n 1 N (x; X) - p(x) > n (6.32) = 1 - Pr x (6.33) (6.34) (6.35) 1 - Pr > 1 - , 1 N (x; X) - p(x) > for some x X n |X | proving Property 2. Finally, Property 3 follows from Property 1 and Property 2 in exactly the same way as in Theorem 5.3, so the proof is omitted. Remark Analogous to weak typicality, we note that the upper bound on n |T[X] | in Property 3 holds for all n 1, and for any > 0, there exists at least one strongly typical sequence when n is sufficiently large. See Problem 1 in Chapter 5. 1 The union bound refers to Pr{A B} Pr{A} + Pr{B}. 6.1 Strong AEP 117 In the rest of the section, we prove an enhancement of Property 2 of the strong AEP which gives an exponential bound on the probability of obtaining a non-typical vector2 . The reader may skip this part at the first reading. Theorem 6.3. For sufficiently large n, there exists () > 0 such that n Pr{X T[X] } < 2-n() . (6.36) The proof of this theorem is based on the Chernoff bound [67] which we prove in the next lemma. Lemma 6.4 (Chernoff Bound). Let Y be a real random variable and s be any nonnegative real number. Then for any real number a, log Pr{Y a} -sa + log E 2sY and log Pr{Y a} sa + log E 2-sY . Proof. Let u(y) = Then for any s 0, u(y - a) 2s(y-a) . This is illustrated in Fig. 6.1. Then E[u(Y - a)] E 2s(Y -a) = 2-sa E 2sY . Since E[u(Y - a)] = Pr{Y a} 1 + Pr{Y < a} 0 = Pr{Y a}, we see that sY Pr{Y a} 2-sa E 2sY = 2-sa+log E [2 ] . (6.37) (6.38) 1 if y 0 0 if y < 0. (6.39) (6.40) (6.41) (6.42) (6.43) Then (6.37) is obtained by taking logarithm in the base 2. Upon replacing Y by -Y and a by -a in (6.37), (6.38) is obtained. The lemma is proved. Proof of Theorem 6.3. We will follow the notation in the proof of Theorem 6.2. Consider x X such that p(x) > 0. Applying (6.37), we have 2 This result is due to Ning Cai and Raymond W. Yeung. An alternative proof based on Pinsker's inequality (Theorem 2.33) and the method of types has been given by Prakash Narayan (private communication). See also Proposition 1 in Weissman et al. [375]. 118 6 Strong Typicality n log Pr k=1 Bk (x) n (p(x) + ) n k=1 -sn (p(x) + ) + log E 2s = -sn (p(x) + ) + log k=1 b) a) n Bk (x) (6.44) (6.45) (6.46) (6.47) (6.48) E 2sBk (x) = -sn (p(x) + ) + n log(1 - p(x) + p(x)2s ) -sn (p(x) + ) + n(ln 2)-1 (-p(x) + p(x)2s ) = -n s (p(x) + ) + (ln 2) where -1 c) p(x)(1 - 2 ) , s a) follows because Bk (x) are mutually independent; b) is a direct evaluation of the expectation from the definition of Bk (x) in (6.20); c) follows from the fundamental inequality ln a a - 1. In (6.48), upon defining x (s, ) = s (p(x) + ) + (ln 2)-1 p(x)(1 - 2s ), we have log Pr k=1 n (6.49) Bk (x) n (p(x) + ) n -nx (s, ), (6.50) or Pr Bk (x) n (p(x) + ) k=1 2-nx (s,) . (6.51) 2 1.8 1.6 2 s ( y!a) 1.4 1.2 1 1 u ( y!a) 0.8 0.6 0.4 0.2 0 !1 0 1 2 a 3 4 5 y Fig. 6.1. An illustration of u(y - a) 2s(y-a) . 6.1 Strong AEP 119 It is readily seen that x (0, ) = 0. Regarding as fixed and differentiate with respect to s, we have x (s, ) = p(x)(1 - 2s ) + . Then x (0, ) = > 0 and it is readily verified that x (s, ) 0 for 0 s log 1 + p(x) (6.55) (6.54) (6.53) (6.52) . (6.56) Therefore, we conclude that x (s, ) is strictly positive for 0 < s log 1 + p(x) . (6.57) On the other hand, by applying (6.38), we can obtain in the same fashion the bound n log Pr k=1 Bk (x) n (p(x) - ) -nx (s, ), (6.58) or Pr n Bk (x) n (p(x) - ) k=1 2-nx (s,) , (6.59) where x (s, ) = -s (p(x) - ) + (ln 2)-1 p(x)(1 - 2-s ). Then x (0, ) = 0, and x (s, ) = p(x)(2-s - 1) + , which is nonnegative for 0 s - log 1 - In particular, x (0, ) = > 0. Therefore, we conclude that x (s, ) is strictly positive for (6.64) p(x) . (6.63) (6.62) (6.61) (6.60) 120 6 Strong Typicality 0 < s - log 1 - By choosing s satisfying 0 < s min log 1 + p(x) p(x) . (6.65) , - log 1 - p(x) , (6.66) both x (s, ) and x (s, ) are strictly positive. From (6.51) and (6.59), we have Pr 1 n n Bk (x) - p(x) k=1 n = Pr k=1 n Bk (x) - np(x) n Bk (x) n (p(x) + ) k=1 n (6.67) Pr +Pr 2 k=1 -nx (s,) Bk (x) n (p(x) - ) + 2-nx (s,) (6.68) (6.69) (6.70) (6.71) (6.72) 22 1 = 2-n[min(x (s,),x (s,))- n ] =2 where -nx () -n min(x (s,),x (s,)) , 1 . (6.73) n Then x () is strictly positive for sufficiently large n because both x (s, ) and x (s, ) are strictly positive. Finally, consider x () = min(x (s, ), x (s, )) - n Pr{X T[X] } = Pr x 1 N (x; X) - p(x) n (6.74) (6.75) (6.76) (6.77) 1 N (x; X) - p(x) for all x X n |X | 1 = 1 - Pr N (x; X) - p(x) > for some x X n |X | 1 1- Pr N (x; X) - p(x) > n |X | x Pr 6.2 Strong Typicality Versus Weak Typicality 121 = 1- x Pr 1 n Pr n Bk (x) - p(x) > k=1 |X | |X | (6.78) = 1- x:p(x)>0 1 n n Bk (x) - p(x) > k=1 |X | (6.79) (6.80) 1- x:p(x)>0 2 -nx , where the last step follows from (6.72). Define () = Then for sufficiently large n, n Pr{X T[X] } > 1 - 2-n() , 1 2 x:p(x)>0 min x |X | . (6.81) (6.82) (6.83) or n Pr{X T[X] } < 2-n() , where () is strictly positive. The theorem is proved. 6.2 Strong Typicality Versus Weak Typicality As we have mentioned at the beginning of the chapter, strong typicality is more powerful and flexible than weak typicality as a tool for theorem proving for memoryless problems, but it can be used only for random variables with finite alphabets. We will prove in the next proposition that strong typicality is stronger than weak typicality in the sense that the former implies the latter. n n Proposition 6.5. For any x X n , if x T[X] , then x W[X] , where 0 as 0. n Proof. By Property 1 of strong AEP (Theorem 6.2), if x T[X] , then 2-n(H(X)+) p(x) 2-n(H(X)-) , or (6.84) 1 log p(x) H(X) + , (6.85) n n where 0 as 0. Then x W[X] by Definition 5.2. The proposition is proved. H(X) - - We have proved in this proposition that strong typicality implies weak typicality, but the converse is not true. This idea can be explained without 122 6 Strong Typicality any detailed analysis. Let X be distributed with p such that p(0) = 0.5, p(1) = 0.25, and p(2) = 0.25. Consider a sequence x of length n and let q(i) 1 be the relative frequency of occurrence of symbol i in x, i.e., n N (i; x), where i = 0, 1, 2. In order for the sequence x to be weakly typical, we need - 1 log p(x) n = -q(0) log 0.5 - q(1) log 0.25 - q(2) log 0.25 H(X) = -(0.5) log 0.5 - (0.25) log 0.25 - (0.25) log 0.25. (6.86) (6.87) (6.88) Obviously, this can be satisfied by choosing q(i) = p(i) for all i. But alternatively, we can choose q(0) = 0.5, q(1) = 0.5, and q(2) = 0. With such a choice of {q(i)}, the sequence x is weakly typical with respect to p but obviously not strongly typical with respect to p, because the relative frequency of occurrence of each symbol i is q(i), which is not close to p(i) for i = 1, 2. Therefore, we conclude that strong typicality is indeed stronger than weak typicality. However, as we have pointed out at the beginning of the chapter, strong typicality can only be used for random variables with finite alphabets. 6.3 Joint Typicality In this section, we discuss strong joint typicality with respect to a bivariate distribution. Generalization to a multivariate distribution is straightforward. Consider a bivariate information source {(Xk , Yk ), k 1} where (Xk , Yk ) are i.i.d. with distribution p(x, y). We use (X, Y ) to denote the pair of generic random variables. n Definition 6.6. The strongly jointly typical set T[XY ] with respect to p(x, y) n n is the set of (x, y) X Y such that N (x, y; x, y) = 0 for (x, y) SXY , and 1 (6.89) N (x, y; x, y) - p(x, y) , n x y where N (x, y; x, y) is the number of occurrences of (x, y) in the pair of sequences (x, y), and is an arbitrarily small positive real number. A pair of n sequences (x, y) is called strongly jointly -typical if it is in T[XY ] . Strong typicality satisfies the following consistency property. n n Theorem 6.7 (Consistency). If (x, y) T[XY ] , then x T[X] and y n T[Y ] . n Proof. If (x, y) T[XY ] , then 6.3 Joint Typicality 123 x y 1 N (x, y; x, y) - p(x, y) . n (6.90) Upon observing that N (x; x) = y N (x, y; x, y), (6.91) we have 1 N (x; x) - p(x) n 1 n N (x, y; x, y) - y y x = x p(x, y) (6.92) = x y 1 N (x, y; x, y) - p(x, y) n 1 N (x, y; x, y) - p(x, y) n (6.93) (6.94) (6.95) x y . n n Therefore, x T[X] . Similarly, y T[Y ] . The theorem is proved. The following thoerem asserts that strong typicality is preserved when a function is applied to a vector componentwise. Theorem 6.8 (Preservation). Let Y = f (X). If n x = (x1 , x2 , , xn ) T[X] , (6.96) then n f (x) = (y1 , y2 , , yn ) T[Y ] , (6.97) where yi = f (xi ) for 1 i n. n Proof. Consider x T[X] , i.e., x 1 N (x; x) - p(x) < . n (6.98) Since Y = f (X), p(y) = xf -1 (y) p(x) (6.99) for all y Y. On the other hand, 124 6 Strong Typicality N (y; f (x)) = xf -1 (y) N (x; x) (6.100) for all y Y. Then 1 N (y; f (x)) - p(y) n 1 N (x; x) - p(x) n 1 N (x; x) - p(x) n (6.101) y = y xf -1 (y) y xf -1 (y) (6.102) (6.103) (6.104) = x 1 N (x; x) - p(x) n < . n Therefore, f (x) T[Y ] , proving the lemma. For a bivariate i.i.d. source {(Xk , Yk )}, we have the strong joint asymptotic equipartition property (strong JAEP), which can readily be obtained by applying the strong AEP to the source {(Xk , Yk )}. Theorem 6.9 (Strong JAEP). Let (X, Y) = ((X1 , Y1 ), (X2 , Y2 ), , (Xn , Yn )), (6.105) where (Xi , Yi ) are i.i.d. with generic pair of random variables (X, Y ). Then there exists > 0 such that 0 as 0, and the following hold: n 1) If (x, y) T[XY ] , then 2-n(H(X,Y )+) p(x, y) 2-n(H(X,Y )-) . 2) For n sufficiently large, n Pr{(X, Y) T[XY ] } > 1 - . (6.106) (6.107) 3) For n sufficiently large, n (1 - )2n(H(X,Y )-) |T[XY ] | 2n(H(X,Y )+) . (6.108) From the strong JAEP, we can see the following. Since there are approximately 2nH(X,Y ) typical (x, y) pairs and approximately 2nH(X) typical x, for a typical x, the number of y such that (x, y) is jointly typical is approximately 6.3 Joint Typicality 125 2nH(X,Y ) = 2nH(Y |X) 2nH(X) (6.109) on the average. The next theorem reveals that this is not only true on the average, but it is in fact true for every typical x as long as there exists at least one y such that (x, y) is jointly typical. n Theorem 6.10 (Conditional Strong AEP). For any x T[X] , define n n n T[Y |X] (x) = {y T[Y ] : (x, y) T[XY ] }. n If |T[Y |X] (x)| 1, then n 2n(H(Y |X)-) |T[Y |X] (x)| 2n(H(Y |X)+) , (6.110) (6.111) where 0 as n and 0. We first prove the following lemma which is along the line of Stirling's approximation [113]. Lemma 6.11. For any n > 0, n ln n - n < ln n! < (n + 1) ln(n + 1) - n. (6.112) Proof. First, we write ln n! = ln 1 + ln 2 + + ln n. Since ln x is a monotonically increasing function of x, we have k k+1 (6.113) ln x dx < ln k < k-1 k ln x dx. (6.114) Summing over 1 k n, we have n n+1 ln x dx < ln n! < 0 1 ln x dx, (6.115) or n ln n - n < ln n! < (n + 1) ln(n + 1) - n. The lemma is proved. Proof of Theorem 6.10. Let be a small positive real number and n be a n large positive integer to be specified later. Fix an x T[X] , so that 1 N (x; x) - p(x) . n (6.117) (6.116) x 126 6 Strong Typicality This implies that for all x X , 1 N (x; x) - p(x) , n or p(x) - (6.118) 1 N (x; x) p(x) + . (6.119) n n We first prove the upper bound on |T[Y |X] (x)|. For any > 0, consider 2-n(H(X)-/2) p(x) = yY n a) (6.120) p(x, y) p(x, y) (6.121) (6.122) n yT[Y |X] (x) b) n yT[Y |X] (x) 2-n(H(X,Y )+/2) (6.123) (6.124) n = |T[Y |X] (x)|2-n(H(X,Y )+/2) , where a) and b) follow from the strong AEP (Theorem 6.2) and the strong joint AEP (Theorem 6.9), respectively. Then we obtain n |T[Y |X] (x)| 2n(H(Y |X)+) , (6.125) which is the upper bound to be proved. n n Assume that |T[Y |X] (x)| 1. We now prove the lower bound on |T[Y |X] (x)|. Let {K(x, y), (x, y) X Y} (6.126) be any set of nonnegative integers such that 1. K(x, y) = N (x; x) y (6.127) for all x X , and 2. for any y Y n , if N (x, y; x, y) = K(x, y) n for all (x, y) X Y, then (x, y) T[XY ] . (6.128) Then by Definition 6.6, {K(x, y)} satisfy 1 K(x, y) - p(x, y) , n (6.129) x y 6.3 Joint Typicality 127 which implies that for all (x, y) X Y, 1 K(x, y) - p(x, y) , n or p(x, y) - (6.130) 1 K(x, y) p(x, y) + . (6.131) n n Such a set {K(x, y)} exists because T[Y |X] (x) is assumed to be nonempty. Straightforward combinatorics reveals that the number of y which satisfy the constraints in (6.128) is equal to M (K) = x N (x; x)! , y K(x, y)! (6.132) and it is readily seen that n |T[Y |X] (x)| M (K). (6.133) Using Lemma 6.11, we can lower bound ln M (K) as follows. ln M (K) x N (x; x) ln N (x; x) - N (x; x) - y [(K(x, y) + 1) ln(K(x, y) + 1) - K(x, y)] N (x; x) ln N (x; x) (6.134) = x a) - y b) (K(x, y) + 1) ln(K(x, y) + 1) {N (x; x) ln(n (p(x) - )) (6.135) x - y (K(x, y) + 1) ln n p(x, y) + + 1 n . (6.136) In the above, a) follows from (6.127), and b) is obtained by applying the lower bound on n-1 N (x; x) in (6.119) and the upper bound on n-1 K(x, y) in (6.131). Also from (6.127), the coefficient of ln n in (6.136) is given by N (x; x) - x y (K(x, y) + 1) = -|X ||Y|. (6.137) Let be sufficiently small and n be sufficiently large so that 128 6 Strong Typicality 0 < p(x) - < 1 1 <1 n for all x and y. Then in (6.136), both the logarithms p(x, y) + + ln(p(x) - ) and ln p(x, y) + + 1 n and (6.138) (6.139) (6.140) (6.141) are negative. Note that the logarithm in (6.140) is well-defined by virtue of (6.138). Rearranging the terms in (6.136), applying the upper bound in (6.119) and the lower bound3 in (6.131), and dividing by n, we have n-1 ln M (K) x (p(x) + ) ln (p(x) - ) - x y p(x, y) - + 1 n (6.142) (6.143) (6.144) 1 |X ||Y| ln n - n n = -He (X) + He (X, Y ) + Ll (n, ) ln p(x, y) + + = He (Y |X) + Ll (n, ), where Ll (n, ) denotes a function of n and which tends to 0 as n and 0. Changing the base of the logarithm to 2, we have n-1 log M (K) H(Y |X) + Ll (n, ). Then it follows from (6.133) that n n-1 log |T[Y |X] (x)| H(Y |X) + Ll (n, ). (6.145) (6.146) Upon replacing Ll (n, ) by , we obtain n |T[Y |X] (x)| 2n(H(Y |X)-) , (6.147) where 0 as n and 0 as required. The theorem is proved. The above theorem says that for any typical x, as long as there is one typical y such that (x, y) is jointly typical, there are approximately 2nH(Y |X) y such that (x, y) is jointly typical. This theorem has the following corollary that the number of such typical x grows with n at almost the same rate as the total number of typical x. 3 1 For the degenerate case when p(x, y) = 1 for some x and y, p(x, y) + + n > 1, and the logarithm in (6.141) is in fact positive. Then the upper bound instead of the lower bound should be applied. The details are omitted. 6.3 Joint Typicality 129 2nH(Y) y 2nH(X) x S[n] X S[n] Y . . . . . . . . . . . . . . . 2nH(X,Y) n (x,y) T[XY] Fig. 6.2. A two-dimensional strong joint typicality array. n Corollary 6.12. For a joint distribution p(x, y) on X Y, let S[X] be the n n set of all sequences x T[X] such that T[Y |X] (x) is nonempty. Then n |S[X] | (1 - )2n(H(X)-) , (6.148) where 0 as n and 0. Proof. By the consistency of strong typicality (Theorem 6.7), if (x, y) n n n T[XY ] , then x T[X] . In particular, x S[X] . Then n T[XY ] = n xS[X] n {(x, y) : y T[Y |X] (x)}. (6.149) n Using the lower bound on |T[XY ] | in Theorem 6.9 and the upper bound on n |T[Y |X] (x)| in the last theorem, we have n n (1 - )2n(H(X,Y )-) |T[XY ] | |S[X] |2n(H(Y |X)+) (6.150) which implies n |S[X] | (1 - )2n(H(X)-(+)) . (6.151) The theorem is proved upon letting = + . We have established a rich set of structural properties for strong typicality with respect to a bivariate distribution p(x, y), which is summarized in the two-dimensional strong joint typicality array in Figure 6.2. In this array, the n n rows and the columns are the typical sequences x S[X] and y S[Y ] , respectively. The total number of rows and columns are approximately equal to 2nH(X) and 2nH(Y ) , respectively. An entry indexed by (x, y) receives a dot if (x, y) is strongly jointly typical. The total number of dots is approximately equal to 2nH(X,Y ) . The number of dots in each row is approximately equal to 2nH(Y |X) , while the number of dots in each column is approximately equal to 2nH(X|Y ) . 130 6 Strong Typicality 2nH(Z) z S [nZ] z0 2nH(Y) y S [nY] (x0,y0) 2nH(X) x S [nX] Fig. 6.3. A three-dimensional strong joint typicality array. For reasons which will become clear in Chapter 16, the strong joint typicality array in Figure 6.2 is said to exhibit an asymptotic quasi-uniform structure. By a two-dimensional asymptotic quasi-uniform structure, we mean that in the array all the columns have approximately the same number of dots, and all the rows have approximately the same number of dots. The strong joint typicality array for a multivariate distribution continues to exhibit an asymptotic quasi-uniform structure. The three-dimensional strong joint typicality array with respect to a distribution p(x, y, z) is illustrated in Figure 6.3. As before, an entry (x, y, z) receives a dot if (x, y, z) is strongly jointly typical. This is not shown in the figure otherwise it will be very confusing. The total number of dots in the whole array is approximately equal to 2nH(X,Y,Z) . These dots are distributed in the array such that all the planes parallel to each other have approximately the same number of dots, and all the cylinders parallel to each other have approximately the same number of dots. More specifically, n the total number of dots on the plane for any fixed z0 S[Z] (as shown) is approximately equal to 2nH(X,Y |Z) , and the total number of dots in the cylinn der for any fixed (x0 , y0 ) pair in S[XY ] (as shown) is approximately equal to 2nH(Z|X,Y ) , so on and so forth. n n We see from the strong AEP and Corollary 6.12 that S[X] and T[X] grow with n at approximately the same rate. We end this section by stating in the n next proposition that S[X] indeed contains almost all the probability when n is large. The proof is left as an exercise (see Problem 4). Proposition 6.13. With respect to a joint distribution p(x, y) on X Y, for any > 0, n Pr{X S[X] } > 1 - (6.152) for n sufficiently large. Chapter Summary 131 6.4 An Interpretation of the Basic Inequalities The asymptotic quasi-uniform structure exhibited in a strong joint typicality array discussed in the last section is extremely important in information theory. Later in the book, we will see how this structure is involved in proving results such as the channel coding theorem and the rate-distortion theorem. In this section, we show how the basic inequalities can be revealed by examining this structure. It has further been shown by Chan [59] that all unconstrained information inequalities can be obtained from this structure, thus giving a physical meaning to these inequalities. n Consider random variables X, Y , and Z and a fixed z S[Z] , so that n T[XY |Z] (z) is nonempty. By the consistency of strong typicality, if (x, y, z) n n n n T[XY Z] , then (x, z) T[XZ] and (y, z) T[Y Z] , or x T[X|Z] (z) and n y T[Y |Z] (z), respectively. Thus n n n T[XY |Z] (z) T[X|Z] (z) T[Y |Z] (z), (6.153) which implies n n n |T[XY |Z] (z)| |T[X|Z] (z)||T[Y |Z] (z)|. (6.154) n Applying the lower bound in Theorem 6.10 to T[XY |Z] (z) and the upper n n bound to T[X|Z] (z) and T[Y |Z] (z), we have 2n(H(X,Y |Z)-) 2n(H(X|Z)+) 2n(H(Y |Z)+) , (6.155) where , , 0 as n and 0. Taking logarithm to the base 2 and dividing by n, we obtain H(X, Y |Z) H(X|Z) + H(Y |Z) upon letting n and 0. This inequality is equivalent to I(X; Y |Z) 0. (6.157) (6.156) Thus we have proved the nonnegativity of conditional mutual information. Since all Shannon's information measures are special cases of conditional mutual information, we have proved the nonnegativity of all Shannon's information measures, namely the basic inequalities. Chapter Summary Strong typicality implies weak typicality but can be used only for random variables with finite alphabets. Strongly Typical Set: 132 6 Strong Typicality n T[X] = x Xn : x n-1 N (x; x) - p(x) . Strong AEP: n 1. 2-n(H(X)+) p(x) 2-n(H(X)-) for x T[X] n 2. Pr{X T[X] } > 1 - for n sufficiently large n 3. (1 - )2n(H(X)-) |T[X] | 2n(H(X)+) for n sufficiently large. Theorem: For sufficiently large n, n Pr{X T[X] } < 2-n() . n n n Consistency: If (x, y) T[XY ] , then x T[X] and y T[Y ] . n n Preservation: If x T[X] , then f (x) T[f (X)] . n Conditional Strong AEP: For x T[X] , let n n n T[Y |X] (x) = {y T[Y ] : (x, y) T[XY ] }. n If |T[Y |X] (x)| 1, then n 2n(H(Y |X)-) |T[Y |X] (x)| 2n(H(Y |X)+) . Problems n n 1. Show that (x, y) T[X,Y ] and (y, z) T[Y,Z] do not imply (x, z) n T[X,Z] . 2. Let X = (X1 , X2 , , Xn ), where Xk are i.i.d. with generic random variable X. Prove that n Pr{X T[X] } 1 - |X |3 n 2 n for any n and > 0. This shows that Pr{X T[X] } 1 as 0 and n if n . 3. Prove that for a random variable X with a countable alphabet, Property 2 of the strong AEP holds, while Properties 1 and 3 do not hold. n 4. Prove Proposition 6.13. Hint: Use the fact that if (X, Y) T[XY ] , then n X S[X] . 5. Let P(X ) be the set of all probability distributions over a finite alphabet X . Find a polynomial Q(n) such that for any integer n, there exists a subset Pn (X ) of P(X ) such that Problems 133 a) |Pn (X )| Q(n); b) for all P P(X ), there exists Pn Pn (X ) such that |Pn (x) - P (x)| < 1 n for all x X . Hint: Let Pn (X ) be the set of all probability distributions over X such that all the probability masses can be expressed as fractions with denominator n. 6. Let p be any probability distribution over a finite set X and be a real number in (0, 1). Prove that for any subset A of X n with pn (A) , n |A T[X] | 2n(H(p)- ) , where 0 as 0 and n . In the following problems, for a sequence x X n , let qx be the empirical distribution of x, i.e., qx (x) = n-1 N (x; x) for all x X . Similarly, for a pair of sequences (x, y) X n Y n , let qx,y be the joint empirical distribution of (x, y), i.e., qx,y (x, y) = n-1 N (x, y; x, y) for all (x, y) X Y. 7. Alternative definition of strong typicality Show that (6.1) is equivalent to V (qx , p) , where V (, ) denotes the variational distance. Thus strong typicality can be regarded as requiring the empirical distribution of a sequence to be close to the probability distribution of the generic random variable in variational distance. Also compare the result here with the alternative definition of weak typicality (Problem 5 in Chapter 5). 8. The empirical distribution qx of the sequence x is also called the type of x. Assuming that X is finite, show that there are a total of n+|X |-1 n distinct types qx . Hint: There are a+b-1 ways to distribute a identical a balls in b boxes. 9. Unified typicality Let X = (X1 , X2 , , Xn ) be an i.i.d. sequence whose generic random variable X is distributed with p(x), where the alpbabet n X is countable. For any > 0, the unified typical set U[X] with respect n to p(x) is the set of sequences x X such that D(qx p) + |H(qx ) - H(p)| . n n a) Show that for any x X n , if x U[X] , then x W[X] . n n b) Show that for any x X n , if x U[X] , then x T[X] , where = 2 ln 2. Therefore, unified typicality implies both weak typicality and strong typicality. 134 6 Strong Typicality 10. The AEP for unified typicality Unified typicality defined in Problem 9, unlike strong typicality, can be applied to random variables whose alphabets are countable . At the same time, it preserves the essential properties of strong typicality. The following outlines the proof of the AEP which has been discussed in Theorem 5.3 and Theorem 6.2 for weak typicality and strong typicality, respectively. a) Show that 2-n(H(X)+) p(x) 2-n(H(X)-) , i.e., Property 1 of the AEP. b) Show that for sufficiently large n, Pr{H(qx ) - H(p) > } < . Hint: Use the results in Problem 9 above and Problem 5 in Chapter 5. c) It can be proved by means of the result in Problem 9 that Pr{H(p) - H(qx ) > } < (see Ho and Yeung [167]). By assuming this inequality, prove that Pr{|H(qx ) - H(p)| } < 1 - 2 . d) Show that if |H(qx ) - H(p)| and |D(qx p) + H(qx ) - H(p)| , then D(qx p) + |H(qx ) - H(p)| 3 . e) Use the results in c) and d) above and the result in Problem 5, Part c) in Chapter 5 to show that Pr{D(qx p) + |H(qx ) - H(p)| } > 1 - . This proves Property 2 of the AEP. Property 3 of the AEP follows from Property 1 as in the proof of Theorem 5.3. 11. Consistency of unified typicality For any > 0, the unified jointly typical n set U[XY ] with respect to pXY (x, y) is the set of sequences (x, y) X n n Y such that D(qx,y pXY ) + |H(qx,y ) - H(pXY )| +|H(qx ) - H(pX )| + |H(qy ) - H(pY )| . n n n Show that if (x, y) U[XY ] , then x U[X] and y U[Y ] . Historical Notes Strong typicality was used by Wolfowitz [385] for proving channel coding theorems and by Berger [28] for proving the rate-distortion theorem and various Historical Notes 135 results in multiterminal source coding. The method of types, a refinement of the notion of strong typicality, was systematically developed in the book by Csiszr and Krner [84]. The interpretation of the basic inequalities in a o Section 6.4 is a preamble to the relation between entropy and groups to be discussed in Chapter 16. Recently, Ho and Yeung [167] introduced the notion of unified typicality which is stronger than both weak typicality and strong typicality. This notion of typicality can be applied to random variables with countable alphabets, while at the same time preserve the essential properties of strong typicality. See Problems 9, 10, and 11 for a discussion. 7 Discrete Memoryless Channels In all practical communication systems, when a signal is transmitted from one point to another point, the signal is inevitably contaminated by random noise, i.e., the signal received is correlated with but possibly different from the signal transmitted. We use a noisy channel to model such a situation. A noisy channel is a "system" which has one input terminal and one output terminal1 , with the input connected to the transmission point and the output connected to the receiving point. When the signal is transmitted through the channel, it is distorted in a random way which depends on the channel characteristics. As a consequence, the signal received may be different from the signal transmitted. In communication engineering, we are interested in conveying messages reliably through a noisy channel at the maximum possible rate. We first look at a simple channel called the binary symmetric channel (BSC), which is represented by the transition diagram in Figure 7.1. In this channel both the input X and the output Y take values in the set {0, 1}. There is a certain probability, denoted by , that the output is not equal to the input. That is, if the input is 0, then the output is 0 with probability 1 - , and is 1 with probability . Likewise, if the input is 1, then the output is 1 with probability 1 - , and is 0 with probability . The parameter is called the crossover probability of the BSC. Let {A, B} be the message set which contains two possible messages to be conveyed through a BSC with 0 < 0.5. We further assume that the two messages A and B are equally likely. If the message is A, we map it to the codeword 0, and if the message is B, we map it to the codeword 1. This is the simplest example of a channel code. The codeword is then transmitted through the channel. Our task is to decode the message based on the output of the channel, and an error is said to occur if the message is decoded incorrectly. Consider Pr{A|Y = 0} = Pr{X = 0|Y = 0} 1 (7.1) The discussion on noisy channels here is confined to point-to-point channels. 138 7 Discrete Memoryless Channels 0 1 0 X Y 1 1 1 Fig. 7.1. The transition diagram of a binary symmetric channel. = Pr{X = 0}Pr{Y = 0|X = 0} Pr{Y = 0} 0.5(1 - ) = . Pr{Y = 0} (7.2) (7.3) Since Pr{Y = 0} = Pr{Y = 1} = 0.5 by symmetry , it follows that Pr{A|Y = 0} = 1 - and Pr{B|Y = 0} = 1 - Pr{A|Y = 0} = . Since < 0.5, Pr{B|Y = 0} < Pr{A|Y = 0}. (7.7) (7.6) (7.5) 2 (7.4) Therefore, in order to minimize the probability of error, we decode a received 0 to the message A. By symmetry, we decode a received 1 to the message B. An error occurs if a 0 is received and the message is B, or if a 1 is received and the message is A. Therefore, the probability of error, denoted by Pe , is given by Pe = Pr{Y = 0}Pr{B|Y = 0} + Pr{Y = 1}Pr{A|Y = 1} = 0.5 + 0.5 = , 2 (7.8) (7.9) (7.10) More explicitly, Pr{Y = 0} = Pr{A}Pr{Y = 0|A} + Pr{B}Pr{Y = 0|B} = 0.5 Pr{Y = 0|X = 0} + 0.5 Pr{Y = 0|X = 1} = 0.5(1 - ) + 0.5 = 0.5. 7 Discrete Memoryless Channels 139 where (7.9) follows from (7.6) because Pr{A|Y = 1} = Pr{B|Y = 0} = (7.11) by symmetry. Let us assume that = 0. Then the above scheme obviously does not provide perfectly reliable communication. If we are allowed to use the channel only once, then this is already the best we can do. However, if we are allowed to use the same channel repeatedly, then we can improve the reliability by generalizing the above scheme. We now consider the following channel code which we refer to as the binary repetition code. Let n 1 be an odd positive integer which is called the block length of the code. In this code, the message A is mapped to the sequence of n 0's, and the message B is mapped to the sequence of n 1's. The codeword, which consists of a sequence of either n 0's or n 1's, is transmitted through the channel in n uses. Upon receiving a sequence of n bits at the output of the channel, we use the majority vote to decode the message, i.e., if there are more 0's than 1's in the sequence, we decode the sequence to the message A, otherwise we decode the sequence to the message B. Note that the block length is chosen to be odd so that there cannot be a tie. When n = 1, this scheme reduces to the previous scheme. For this more general scheme, we continue to denote the probability of error by Pe . Let N0 and N1 be the number of 0's and 1's in the received sequence, respectively. Clearly, N0 + N1 = n. (7.12) For large n, if the message is A, the number of 0's received is approximately equal to E[N0 |A] = n(1 - ) (7.13) and the number of 1's received is approximately equal to E[N1 |A] = n (7.14) with high probability by the weak law of large numbers. This implies that the probability of an error, namely the event {N0 < N1 }, is small because n(1 - ) > n with the assumption that < 0.5. Specifically, (7.16) (7.17) (7.18) (7.19) (7.15) Pr{error|A} = Pr{N0 < N1 |A} = Pr{n - N1 < N1 |A} = Pr{N1 > 0.5n|A} Pr{N1 > ( + )n|A}, 140 7 Discrete Memoryless Channels where 0 < < 0.5 - , so that is positive and + < 0.5. (7.21) Note that such a exists because < 0.5. Then by the weak law of large numbers, the upper bound in (7.19) tends to 0 as n . By symmetry, Pr{error|B} also tends to 0 as n . Therefore, Pe = Pr{A}Pr{error|A} + Pr{B}Pr{error|B} (7.22) (7.20) tends to 0 as n . In other words, by using a long enough repetition code, we can make Pe arbitrarily small. In this sense, we say that reliable communication is achieved asymptotically. We point out that for a BSC with > 0, for any given transmitted sequence of length n, the probability of receiving any given sequence of length n is nonzero. It follows that for any two distinct input sequences, there is always a nonzero probability that the same output sequence is produced so that the two input sequences become indistinguishable. Therefore, except for very special channels (e.g., the BSC with = 0), no matter how the encoding/decoding scheme is devised, a nonzero probability of error is inevitable, and asymptotically reliable communication is the best we can hope for. Though a rather naive approach, asymptotically reliable communication can be achieved by using the repetition code. The repetition code, however, is not without catch. For a channel code, the rate of the code in bit(s) per use, is defined as the ratio of the logarithm of the size of the message set in the base 2 to the block length of the code. Roughly speaking, the rate of a channel code is the average number of bits the channel code attempts to convey through the channel per use of the channel. For a binary repetition 1 1 code with block length n, the rate is n log 2 = n , which tends to 0 as n . Thus in order to achieve asymptotic reliability by using the repetition code, we cannot communicate through the noisy channel at any positive rate! In this chapter, we characterize the maximum rate at which information can be communicated through a discrete memoryless channel (DMC) with an arbitrarily small probability of error. This maximum rate, which is generally positive, is known as the channel capacity. Then we discuss the use of feedback in communicating through a DMC, and show that feedback does not increase the capacity. At the end of the chapter, we discuss transmitting an information source through a DMC, and we show that asymptotic optimality can be achieved by separating source coding and channel coding. 7.1 Definition and Capacity Definition 7.1. Let X and Y be discrete alphabets, and p(y|x) be a transition matrix from X to Y. A discrete channel p(y|x) is a single-input single-output 7.1 Definition and Capacity 141 system with input random variable X taking values in X and output random variable Y taking values in Y such that Pr{X = x, Y = y} = Pr{X = x}p(y|x) for all (x, y) X Y. Remark From (7.23), we see that if Pr{X = x} > 0, then Pr{Y = y|X = x} = Pr{X = x, Y = y} = p(y|x). Pr{X = x} (7.24) (7.23) Note that Pr{Y = y|X = x} is undefined if Pr{X = x} = 0. Nevertheless, (7.23) is valid for both cases. We now present an alternative description of a discrete channel. Let X and Y be discrete alphabets. Let X be a random variable taking values in X and p(y|x) be any transition matrix from X to Y. Define random variables Zx with Zx = Y for x X such that Pr{Zx = y} = p(y|x) (7.25) for all y Y. We assume that Zx , x X are mutually independent and also independent of X. Further define the random variable Z = (Zx : x X ), (7.26) called the noise variable. Note that Z is independent of X. Now define a random variable taking values in Y as Y = Zx if X = x. (7.27) Evidently, Y is a function of X and Z. Then for x X such that Pr{X = x} > 0, we have Pr{X = x, Y = y} = Pr{X = x}Pr{Y = y|X = x} = Pr{X = x}Pr{Zx = y|X = x} = Pr{X = x}Pr{Zx = y} = Pr{X = x}p(y|x), (7.28) (7.29) (7.30) (7.31) i.e., (7.23) in Definition 7.1, where (7.30) follows from the assumption that Zx is independent of X. For x X such that Pr{X = x} = 0, since Pr{X = x} = 0 implies Pr{X = x, Y = y} = 0, (7.23) continues to hold. Then by regarding X and Y as the input and output random variables, we have obtained an alternative description of the discrete channel p(y|x). Since Y is a function of X and Z, we can write Y = (X, Z). (7.32) Then we have the following equivalent definition for a discrete channel. 142 7 Discrete Memoryless Channels Z x p(y|x) (a) y X ! Y (b) Fig. 7.2. Illustrations of (a) a discrete channel p(y|x) and (b) a discrete channel (, Z). Definition 7.2. Let X , Y, and Z be discrete alphabets. Let : X Z Y, and Z be a random variable taking values in Z, called the noise variable. A discrete channel (, Z) is a single-input single-output system with input alphabet X and output alphabet Y. For any input random variable X, the noise variable Z is independent of X, and the output random variable Y is given by Y = (X, Z). (7.33) Figure 7.2 illustrates a discrete channel p(y|x) and a discrete channel (, Z). The next definition gives the condition for the equivalence of the two specifications of a discrete channel according to Definitions 7.1 and 7.2, respectively. Definition 7.3. Two discrete channels p(y|x) and (, Z) defined on the same input alphabet X and output alphabet Y are equivalent if Pr{(x, Z) = y} = p(y|x) for all x and y. We point out that the qualifier "discrete" in a discrete channel refers to the input and output alphabets of the channel being discrete. As part of a discretetime communication system, a discrete channel can be used repeatedly at every time index i = 1, 2, . As the simplest model, we may assume that the noise for the transmission over the channel at different time indices are independent of each other. In the next definition, we will introduce the discrete memoryless channel (DMC) as a discrete-time extension of a discrete channel that captures this modeling assumption. To properly formulate a DMC, we regard it as a subsystem of a discretetime stochastic system which will be referred to as "the system" in the sequel. In such a system, random variables are generated sequentially in discrete-time, and more than one random variable may be generated instantaneously but sequentially at a particular time index. (7.34) 7.1 Definition and Capacity 143 X1 X2 X3 p ( y x) p ( y x) p ( y x) Y1 Y2 Y3 Fig. 7.3. An illustration of a discrete memoryless channel p(y|x). Definition 7.4. A discrete memoryless channel (DMC) p(y|x) is a sequence of replicates of a generic discrete channel p(y|x). These discrete channels are indexed by a discrete-time index i, where i 1, with the ith channel being available for transmission at time i. Transmission through a channel is assumed to be instantaneous. Let Xi and Yi be respectively the input and the output of the DMC at time i, and let Ti- denote all the random variables that are generated in the system before Xi . The equality Pr{Yi = y, Xi = x, Ti- = t} = Pr{Xi = x, Ti- = t}p(y|x) holds for all (x, y, t) X Y Ti- . Remark Similar to the remark following Definition 7.1, if Pr{Xi = x, Ti- = t} > 0, then Pr{Yi = y|Xi = x, Ti- = t} = Pr{Yi = y, Xi = x, Ti- = t} Pr{Xi = x, Ti- = t} = p(y|x). (7.36) (7.37) (7.35) Note that Pr{Yi = y|Xi = x, Ti- = t} is undefined if Pr{Xi = x, Ti- = t} = 0. Nevertheless, (7.35) is valid for both cases. Invoking Proposition 2.5, we see from (7.35) that Ti- Xi Yi (7.38) forms a Markov chain, i.e., the output of the DMC at time i is independent of all the random variables that have already been generated in the system conditioning on the input at time i. This captures the memorylessness of a DMC. Figure 7.3 is an illustration of a DMC p(y|x). Paralleling Definition 7.2 for a discrete channel, we now present an alternative definition of a DMC. Definition 7.5. A discrete memoryless channel (, Z) is a sequence of replicates of a generic discrete channel (, Z). These discrete channels are indexed ... 144 7 Discrete Memoryless Channels Z1 X1 ! Y1 Z2 X2 ! Y2 Z3 X3 ! Y3 Fig. 7.4. An illustration of a discrete memoryless channel (, Z). by a discrete-time index i, where i 1, with the ith channel being available for transmission at time i. Transmission through a channel is assumed to be instantaneous. Let Xi and Yi be respectively the input and the output of the DMC at time i, and let Ti- denote all the random variables that are generated in the system before Xi . The noise variable Zi for the transmission at time i is a copy of the generic noise variable Z, and is independent of (Xi , Ti- ). The output of the DMC at time i is given by Yi = (Xi , Zi ). (7.39) Figure 7.4 is an illustration of a DMC (, Z). We now show that Definitions 7.4 and 7.5 specify the same DMC provided that the generic discrete channel p(y|x) in Definition 7.4 is equivalent to the generic discrete channel (, Z) in Definition 7.5, i.e., (7.34) holds. For the DMC (, Z) in Definition 7.5, consider 0 I(Ti- ; Yi |Xi ) I(Ti- ; Yi , Xi , Zi |Xi ) = I(Ti- ; Xi , Zi |Xi ) = I(Ti- ; Zi |Xi ) = 0, (7.40) (7.41) (7.42) (7.43) (7.44) where the first equality follows from (7.39) and the last equality follows from the assumption that Zi is independent of (Xi , Ti- ). Therefore, I(Ti- ; Yi |Xi ) = 0, (7.45) ... 7.1 Definition and Capacity 145 or Ti- Xi Yi forms a Markov chain. It remains to establish (7.35) for all (x, y, t) X Y Ti- . For x X such that Pr{Xi = x} = 0, both Pr{Yi = y, Xi = x, Ti- = t} and Pr{Xi = x, Ti- = t} vanish because they are upper bounded by Pr{Xi = x}. Therefore (7.35) holds. For x X such that Pr{Xi = x} > 0, Pr{Yi = y, Xi = x, Ti- = t} = Pr{Xi = x, Ti- = t}Pr{Yi = y|Xi = x} = Pr{Xi = x, Ti- = t}Pr{(Xi , Zi ) = y|Xi = x} = Pr{Xi = x, Ti- = t}Pr{(x, Zi ) = y|Xi = x} = Pr{Xi = x, Ti- = t}Pr{(x, Zi ) = y} = Pr{Xi = x, Ti- = t}Pr{(x, Z) = y} = Pr{Xi = x, Ti- = t}p(y|x), where a) follows from the Markov chain Ti- Xi Yi ; b) follows from (7.39); c) follows from Definition 7.5 that Zi is independent of Xi ; d) follows from Definition 7.5 that Zi and the generic noise variable Z have the same distribution; e) follows from (7.34). Hence, (7.35) holds for all (x, y, t) X Y Ti- , proving that the DMC (, Z) in Definition 7.4 is equivalent to the DMC (p(y|x) in Definition 7.5. Definition 7.5 renders the following physical conceptualization of a DMC. The DMC can be regarded as a "box" which has only two terminals, the input and the output. The box perfectly shields its contents from the rest of the system. At time i, upon the transmission of the input Xi , the noise variable Zi is generated inside the box according to the distribution of the generic noise variable Z. Since the box is perfectly shielded, the generation of the Zi is independent of Xi and any other random variable that has already been generated in the system. Then the function is applied to (Xi , Zi ) to produce the output Yi . In the rest of the section, we will define the capacity of a DMC and discuss some of its basic properties. The capacities of two simple DMCs will also be evaluated explicitly. To keep our discussion simple, we will assume that the alphabets X and Y are finite. Definition 7.6. The capacity of a discrete memoryless channel p(y|x) is defined as C = max I(X; Y ), (7.52) p(x) e) d) c) b) a) (7.46) (7.47) (7.48) (7.49) (7.50) (7.51) where X and Y are respectively the input and the output of the generic discrete channel, and the maximum is taken over all input distributions p(x). 146 7 Discrete Memoryless Channels Z X Y Fig. 7.5. An alternative representation for a binary symmetric channel. From the above definition, we see that C0 because I(X; Y ) 0 for all input distributions p(x). By Theorem 2.43, we have C = max I(X; Y ) max H(X) = log |X |. p(x) p(x) (7.53) (7.54) (7.55) Likewise, we have C log |Y|. Therefore, C min(log |X |, log |Y|). (7.57) Since I(X; Y ) is a continuous functional of p(x) and the set of all p(x) is a compact set (i.e., closed and bounded) in |X | , the maximum value of I(X; Y ) can be attained3 . This justifies taking the maximum rather than the supremum in the definition of channel capacity in (7.52). We will prove subsequently that C is in fact the maximum rate at which information can be communicated reliably through a DMC. We first give some examples of DMC's for which the capacities can be obtained in closed form. In the following, X and Y denote respectively the input and the output of the generic discrete channel, and all logarithms are in the base 2. Example 7.7 (Binary Symmetric Channel). The transition diagram of a BSC has been shown in Figure 7.1. Alternatively, a BSC can be represented by the system in Figure 7.5. Here, Z is a binary random variable representing the noise of the channel, with Pr{Z = 0} = 1 - and Z is independent of X. Then Y = X + Z mod 2. 3 (7.56) and Pr{Z = 1} = , (7.58) (7.59) The assumption that X is finite is essential in this argument. 7.1 Definition and Capacity 147 C 1 0 0.5 1 Fig. 7.6. The capacity of a binary symmetric channel. This representation for a BSC is in the form prescribed by Definition 7.2. In order to determine the capacity of the BSC, we first bound I(X; Y ) as follows. I(X; Y ) = H(Y ) - H(Y |X) = H(Y ) - x (7.60) (7.61) (7.62) (7.63) (7.64) p(x)H(Y |X = x) p(x)hb ( ) x = H(Y ) - = H(Y ) - hb ( ) 1 - hb ( ), where we have used hb to denote the binary entropy function in the base 2. In order to achieve this upper bound, we have to make H(Y ) = 1, i.e., the output distribution of the BSC is uniform. This can be done by letting p(x) be the uniform distribution on {0, 1}. Therefore, the upper bound on I(X; Y ) can be achieved, and we conclude that C = 1 - hb ( ) bit per use. (7.65) Figure 7.6 is a plot of the capacity C versus the crossover probability . We see from the plot that C attains the maximum value 1 when = 0 or = 1, and attains the minimum value 0 when = 0.5. When = 0, it is easy to see that C = 1 is the maximum rate at which information can be communicated through the channel reliably. This can be achieved simply by transmitting unencoded bits through the channel, and no decoding is necessary because all the bits are received unchanged. When = 1, the same can be achieved with the additional decoding step which complements all the received bits. By doing so, the bits transmitted through the channel can be recovered without error. Thus from the communication point of view, for binary channels, a channel 148 7 Discrete Memoryless Channels 0 1 0 X e Y 1 1 1 Fig. 7.7. The transition diagram of a binary erasure channel. which never makes error and a channel which always makes errors are equally good. When = 0.5, the channel output is independent of the channel input. Therefore, no information can possibly be communicated through the channel. Example 7.8 (Binary Erasure Channel). The transition diagram of a binary erasure channel is shown in Figure 7.7. In this channel, the input alphabet is {0, 1}, while the output alphabet is {0, 1, e}. With probability , the erasure symbol e is produced at the output, which means that the input bit is lost; otherwise the input bit is reproduced at the output without error. The parameter is called the erasure probability. To determine the capacity of this channel, we first consider C = max I(X; Y ) p(x) (7.66) (7.67) (7.68) = max(H(Y ) - H(Y |X)) p(x) = max H(Y ) - hb (). p(x) Thus we only have to maximize H(Y ). To this end, let Pr{X = 0} = a and define a binary random variable E by E= 0 if Y = e 1 if Y = e. (7.70) (7.69) The random variable E indicates whether an erasure has occurred, and it is a function of Y . Then H(Y ) = H(Y, E) = H(E) + H(Y |E) = hb () + (1 - )hb (a). (7.71) (7.72) (7.73) 7.2 The Channel Coding Theorem 149 Hence, C = max H(Y ) - hb () p(x) a (7.74) (7.75) (7.76) (7.77) = max[hb () + (1 - )hb (a)] - hb () = (1 - ) max hb (a) a = (1 - ) bit per use, where the capacity is achieved by letting a = 0.5, i.e., the input distribution is uniform. It is in general not possible to obtain the capacity of a DMC in closed form, and we have to resort to numerical computation. In Chapter 9 we will discuss the Blahut-Arimoto algorithm for computing the channel capacity. 7.2 The Channel Coding Theorem We will justify the definition of the capacity of a DMC by the proving the channel coding theorem. This theorem, which consists of two parts, will be formally stated at the end of the section. The direct part of the theorem asserts that information can be communicated through a DMC with an arbitrarily small probability of error at any rate less than the channel capacity. Here it is assumed that the decoder knows the transition matrix of the DMC. The converse part of the theorem asserts that if information is communicated through a DMC at a rate higher than the capacity, then the probability of error is bounded away from zero. For better appreciation of the definition of channel capacity, we will first prove the converse part in Section 7.3 and then prove the direct part in Section 7.4. Definition 7.9. An (n, M ) code for a discrete memoryless channel with input alphabet X and output alphabet Y is defined by an encoding function f : {1, 2, , M } X n and a decoding function g : Y n {1, 2, , M }. (7.79) (7.78) The set {1, 2, , M }, denoted by W, is called the message set. The sequences f (1), f (2), , f (M ) in X n are called codewords, and the set of codewords is called the codebook. In order to distinguish a channel code as defined above from a channel code with feedback which will be discussed in Section 7.6, we will refer to the former as a channel code without feedback. 150 7 Discrete Memoryless Channels W Message X Channel p(y|x) Y W Estimate of message Encoder Decoder Fig. 7.8. A channel code with block length n. We assume that a message W is randomly chosen from the message set W according to the uniform distribution. Therefore, H(W ) = log M. With respect to a channel code for a DMC, we let X = (X1 , X2 , , Xn ) and Y = (Y1 , Y2 , , Yn ) (7.82) be the input sequence and the output sequence of the channel, respectively. Evidently, X = f (W ). (7.83) We also let ^ W = g(Y) (7.84) be the estimate on the message W by the decoder. Figure 7.8 is the block diagram for a channel code. Definition 7.10. For all 1 w M , let ^ w = Pr{W = w|W = w} = yY n :g(y)=w (7.80) (7.81) Pr{Y = y|X = f (w)} (7.85) be the conditional probability of error given that the message is w. We now define two performance measures for a channel code. Definition 7.11. The maximal probability of error of an (n, M ) code is defined as max = max w . (7.86) w Definition 7.12. The average probability of error of an (n, M ) code is defined as ^ Pe = Pr{W = W }. (7.87) 7.3 The Converse 151 From the definition of Pe , we have ^ Pe = Pr{W = W } = w (7.88) (7.89) (7.90) (7.91) ^ Pr{W = w}Pr{W = W |W = w} 1 ^ Pr{W = w|W = w} M w , w = w = 1 M i.e., Pe is the arithmetic mean of w , 1 w M . It then follows that Pe max . (7.92) In fact, it can be readily seen that this inequality remains valid even without the assumption that W is distributed uniformly on the message set W. Definition 7.13. The rate of an (n, M ) channel code is n-1 log M in bits per use. Definition 7.14. A rate R is asymptotically achievable for a discrete ryless channel if for any > 0, there exists for sufficiently large n an code such that 1 log M > R - n and max < . memo(n, M ) (7.93) (7.94) For brevity, an asymptotically achievable rate will be referred to as an achievable rate. In other words, a rate R is achievable if there exists a sequence of codes whose rates approach R and whose probabilities of error approach zero. We end this section by stating the channel coding theorem, which gives a characterization of all achievable rates. This theorem will be proved in the next two sections. Theorem 7.15 (Channel Coding Theorem). A rate R is achievable for a discrete memoryless channel if and only if R C, the capacity of the channel. 7.3 The Converse Let us consider a channel code with block length n. The random variables ^ involved in this code are W , Xi and Yi for 1 i n, and W . We see 152 7 Discrete Memoryless Channels X X1 X2 X3 p ( y | x) Y Y1 Y2 Y3 W W Xn Yn Fig. 7.9. The dependency graph for a channel code without feedback. from the definition of a channel code in Definition 7.9 that all the random variables are generated sequentially according to some deterministic or probabilistic rules. Specifically, the random variables are generated in the order ^ W, X1 , Y1 , X2 , Y2 , , Xn , Yn , W . The generation of these random variables can be represented by the dependency graph4 in Figure 7.9. In this graph, a node represents a random variable. If there is a (directed) edge from node X to node Y , then node X is called a parent of node Y . We further distinguish a solid edge and a dotted edge: a solid edge represents functional (deterministic) dependency, while a dotted edge represents the probabilistic dependency induced by the transition matrix p(y|x) of the generic discrete channel. For a node X, its parent nodes represent all the random variables on which random variable X depends when it is generated. We now explain the specific structure of the dependency graph. First, Xi is a function of W , so each Xi is connected to W by a solid edge. According to Definition 7.4, Ti- = (W, X1 , Y1 , , Xi-1 , Yi-1 ). By (7.35), the Markov chain (W, X1 , Y1 , , Xi-1 , Yi-1 ) Xi Yi (7.96) (7.95) prevails. Therefore, the generation of Yi depends only on Xi and not on W, X1 , Y1 , , Xi-1 , Yi-1 . So, Yi is connected to Xi by a dotted edge representing the discrete channel p(y|x) at time i, and there is no connection ^ between Yi and any of the nodes W, X1 , Y1 , , Xi-1 , Yi-1 . Finally, W is a ^ function of Y1 , Y2 , , Yn , so W is connected to each Yi by a solid edge. 4 A dependency graph can be regarded as a Bayesian network [287]. 7.3 The Converse 153 We will use q to denote the joint distribution of these random variables as well as all the marginals, and let xi denote the ith component of a sequence x. ^ From the dependency graph, we see that for all (w, x, y, w) W X n Y n W ^ such that q(x) > 0 and q(y) > 0, n n q(w, x, y w) = q(w) ^ i=1 q(xi |w) i=1 p(yi |xi ) q(w|y). ^ (7.97) Note that q(w) > 0 for all w so that q(xi |w) are well-defined, and q(xi |w) and q(w|y) are both deterministic. Denote the set of nodes X1 , X2 , , Xn by X ^ and the set of nodes Y1 , Y2 , , Yn by Y. We notice the following structure in the dependency graph: all the edges from W end in X, all the edges from X ^ end in Y, and all the edges from Y end in W . This suggests that the random ^ variables W, X, Y, and W form the Markov chain ^ W X Y W. (7.98) The validity of this Markov chain can be formally justified by applying Propo^ sition 2.9 to (7.97), so that for all (w, x, y, w) W X n Y n W such that ^ q(x) > 0 and q(y) > 0, we can write q(w, x, y, w) = q(w)q(x|w)q(y|x)q(w|y). ^ ^ (7.99) Now q(x, y) is obtained by summing over all w and w in (7.97), and q(x) is ^ obtained by further summing over all y. After some straightforward algebra and using q(x, y) (7.100) q(y|x) = q(x) for all x such that q(x) > 0, we obtain n q(y|x) = i=1 p(yi |xi ). (7.101) The Markov chain in (7.98) and the relation in (7.101) are apparent from the setup of the problem, and the above justification may seem superfluous. However, the methodology developed here is necessary for handling the more delicate situation which arises when the channel is used with feedback. This will be discussed in Section 7.6. Consider a channel code whose probability of error is arbitrarily small. ^ Since W, X, Y, and W form the Markov chain in (7.98), the information diagram for these four random variables is as shown in Figure 7.10. Moreover, ^ X is a function of W , and W is a function of Y. These two relations are equivalent to H(X|W ) = 0, (7.102) and 154 7 Discrete Memoryless Channels W 0 0 0 X 0 0 Y W 0 0 0 ^ Fig. 7.10. The information diagram for W X Y W . ^ H(W |Y) = 0, (7.103) ^ respectively. Since the probability of error is arbitrarily small, W and W are essentially identical. To gain insight into the problem, we assume for the time ^ being that W and W are equivalent, so that ^ ^ H(W |W ) = H(W |W ) = 0. (7.104) Since the I-Measure for a Markov chain is nonnegative, the constraints in (7.102) to (7.104) imply that vanishes on all the atoms in Figure 7.10 marked with a `0.' Immediately, we see that H(W ) = I(X; Y). (7.105) That is, the amount of information conveyed through the channel is essentially the mutual information between the input sequence and the output sequence of the channel. For a single transmission, we see from the definition of channel capacity that the mutual information between the input and the output cannot exceed the capacity of the channel, i.e., for all 1 i n, I(Xi ; Yi ) C. Summing i from 1 to n, we have n (7.106) I(Xi ; Yi ) nC. i=1 (7.107) Upon establishing in the next lemma that n I(X; Y) i=1 I(Xi ; Yi ), (7.108) the converse of the channel coding theorem then follows from 7.3 The Converse 155 1 1 log M = H(W ) n n 1 = I(X; Y) n n 1 I(Xi ; Yi ) n i=1 C. (7.109) (7.110) (7.111) (7.112) Lemma 7.16. For a discrete memoryless channel used with a channel code without feedback, for any n 1, n I(X; Y) i=1 I(Xi ; Yi ), (7.113) where Xi and Yi are, respectively, the input and the output of the channel at time i. Proof. For any (x, y) X n Y n , if q(x, y) > 0, then q(x) > 0 and (7.101) holds. Therefore, n q(Y|X) = i=1 p(Yi |Xi ) (7.114) holds for all (x, y) in the support of q(x, y). Then n n -E log q(Y|X) = -E log i=1 p(Yi |Xi ) = - i=1 n E log p(Yi |Xi ), (7.115) or H(Y|X) = H(Yi |Xi ). i=1 (7.116) Hence, I(X; Y) = H(Y) - H(Y|X) n n (7.117) (7.118) (7.119) i=1 n H(Yi ) - i=1 H(Yi |Xi ) = i=1 I(Xi ; Yi ). The lemma is proved. We now formally prove the converse of the channel coding theorem. Let R be an achievable rate, i.e., for any > 0, there exists for sufficiently large n an (n, M ) code such that 156 7 Discrete Memoryless Channels 1 log M > R - n and max < . Consider log M = H(W ) ^ ^ = H(W |W ) + I(W ; W ) ^ H(W |W ) + I(X; Y) ^ H(W |W ) + i=1 c) n b) a) (7.120) (7.121) (7.122) (7.123) (7.124) (7.125) (7.126) I(Xi ; Yi ) ^ H(W |W ) + nC, where a) b) c) d) d) follows from (7.80); ^ follows from the data processing theorem since W X Y W ; follows from Lemma 7.16; follows from (7.107). From (7.87) and Fano's inequality (cf. Corollary 2.48), we have ^ H(W |W ) < 1 + Pe log M. Therefore, from (7.126), log M < 1 + Pe log M + nC 1 + max log M + nC < 1 + log M + nC, (7.128) (7.129) (7.130) (7.127) where we have used (7.92) and (7.121), respectively, to obtain the last two inequalities. Dividing by n and rearranging the terms, we have 1 +C 1 log M < n , n 1- (7.131) and from (7.120), we obtain R- < 1 n +C . 1- (7.132) For any > 0, the above inequality holds for all sufficiently large n. Letting n and then 0, we conclude that R C. (7.133) 7.4 Achievability 157 Pe 1 C Fig. 7.11. An asymptotic upper bound on Pe . 1 log M n This completes the proof for the converse of the channel coding theorem. From the above proof, we can obtain an asymptotic bound on Pe when 1 the rate of the code n log M is greater than C. Consider (7.128) and obtain Pe 1 - Then Pe 1 - 1 + nC =1- log M 1 n +C 1 n log M 1 n +C 1 n log M . (7.134) 1- 1 n C log M (7.135) when n is large. This asymptotic bound on Pe , which is strictly positive if 1 n log M > C, is illustrated in Figure 7.11. 1 In fact, the lower bound in (7.134) implies that Pe > 0 for all n if n log M > (n ) C because if Pe 0 = 0 for some n0 , then for all k 1, by concatenating k copies of the code, we obtain a code with the same rate and block length equal (kn ) to kn0 such that Pe 0 = 0, which is a contradiction to our conclusion that Pe > 0 when n is large. Therefore, if we use a code whose rate is greater than the channel capacity, the probability of error is non-zero for all block lengths. The converse of the channel coding theorem we have proved is called the weak converse. A stronger version of this result called the strong converse can be proved, which says that Pe 1 as n if there exists an > 0 such 1 that n log M C + for all n. 7.4 Achievability We have shown in the last section that the channel capacity C is an upper bound on all the achievable rates for a DMC. In this section, we show that the rate C is achievable, which implies that any rate R C is achievable. 158 7 Discrete Memoryless Channels Consider a DMC p(y|x), and denote the input and the output of the generic discrete channel by X and Y , respectively. For every input distribution p(x), we will prove that the rate I(X; Y ) is achievable by showing for large n the existence of a channel code such that 1. the rate of the code is arbitrarily close to I(X; Y ); 2. the maximal probability of error max is arbitrarily small. Then by choosing the input distribution p(x) to be one that achieves the channel capacity, i.e., I(X; Y ) = C, we conclude that the rate C is achievable. Before we prove the achievability of the channel capacity, we first prove the following lemma. Lemma 7.17. Let (X , Y ) be n i.i.d. copies of a pair of generic random variables (X , Y ), where X and Y are independent and have the same marginal distributions as X and Y , respectively. Then n Pr{(X , Y ) T[XY ] } 2-n(I(X;Y )- ) , (7.136) where 0 as 0. Proof. Consider n Pr{(X , Y ) T[XY ] } = n (x,y)T[XY ] p(x)p(y). (7.137) n n By the consistency of strong typicality, for (x, y) T[XY ] , x T[X] and n y T[Y ] . By the strong AEP, all the p(x) and p(y) in the above summation satisfy p(x) 2-n(H(X)-) (7.138) and p(y) 2-n(H(Y )-) , where , 0 as 0. By the strong JAEP, n |T[XY ] | 2n(H(X,Y )+) , (7.139) (7.140) where 0 as 0. Then from (7.137), we have n Pr{(X , Y ) T[XY ] } 2n(H(X,Y )+) 2-n(H(X)-) 2-n(H(Y )-) =2 -n(H(X)+H(Y )-H(X,Y )---) -n(I(X;Y )---) (7.141) (7.142) (7.143) (7.144) (7.145) =2 = 2-n(I(X;Y )- ) , where =++ 0 7.4 Achievability 159 as 0. The lemma is proved. Fix any > 0 and let be a small positive quantity to be specified later. Toward proving the existence of a desired code, we fix an input distribution p(x) for the generic discrete channel p(y|x), and let M be an even integer satisfying 1 I(X; Y ) - < log M < I(X; Y ) - , (7.146) 2 n 4 where n is sufficiently large. We now describe a random coding scheme in the following steps: 1. Construct the codebook C of an (n, M ) code randomly by generating M codewords in X n independently and identically according to p(x)n . Denote ~ ~ ~ these codewords by X(1), X(2), , X(M ). 2. Reveal the codebook C to both the encoder and the decoder. 3. A message W is chosen from W according to the uniform distribution. ~ 4. The sequence X = X(W ), namely the W th codeword in the codebook C, is transmitted through the channel. 5. The channel outputs a sequence Y according to n ~ Pr{Y = y|X(W ) = x} = i=1 p(yi |xi ) (7.147) (cf. (7.101)). n ~ 6. The sequence Y is decoded to the message w if (X(w), Y) T[XY ] and n ~ there does not exists w = w such that (X(w ), Y) T[XY ] . Otherwise, ^ Y is decoded to a constant message in W. Denote by W the message to which Y is decoded. Remark 1 There are a total of |X |M n possible codebooks which can be constructed in Step 1 of the random coding scheme, where we regard two codebooks whose sets of codewords are permutations of each other as two different codebooks. Remark 2 Strong typicality is used in defining the decoding function in Step 6. This is made possible by the assumption that the alphabets X and Y are finite. We now analyze the performance of this random coding scheme. Let ^ Err = {W = W } (7.148) be the event of a decoding error. In the following, we analyze Pr{Err }, the probability of a decoding error for the random code constructed above. For all 1 w M , define the event 160 7 Discrete Memoryless Channels n ~ Ew = {(X(w), Y) T[XY ] }. (7.149) Now Pr{Err } = M Pr{Err |W = w}Pr{W = w}. w=1 (7.150) Since Pr{Err |W = w} are identical for all w by symmetry in the code construction, we have M Pr{Err } = Pr{Err |W = 1} w=1 Pr{W = w} (7.151) (7.152) = Pr{Err |W = 1}, i.e., we can assume without loss of generality that the message 1 is chosen. Then decoding is correct if the received sequence Y is decoded to the message 1. This is the case if E1 occurs but Ew does not occur for all 2 w M . It follows that5 c c c Pr{Err c |W = 1} Pr{E1 E2 E3 EM |W = 1}, (7.153) which implies Pr{Err |W = 1} = 1 - Pr{Err c |W = 1} 1 - Pr{E1 = Pr{(E1 c E2 c E2 (7.154) (7.155) (7.156) (7.157) c E3 c E3 c EM |W = 1} c EM )c |W = 1} c = Pr{E1 E2 E3 EM |W = 1}. By the union bound, we have M c Pr{Err |W = 1} Pr{E1 |W = 1} + w=2 Pr{Ew |W = 1}. (7.158) ~ First, conditioning on {W = 1}, (X(1), Y) are n i.i.d. copies of the pair of generic random variables (X, Y ). By the strong JAEP, for any > 0, c n ~ Pr{E1 |W = 1} = Pr{(X(1), Y) T[XY ] |W = 1} < (7.159) for sufficiently large n. This gives an upper bound on the first term on the right hand side of (7.158). ~ Second, conditioning on {W = 1}, for 2 w M , (X(w), Y) are n i.i.d. copies of the pair of generic random variables (X , Y ), where X and Y 5 If E1 does not occur or Ew occurs for some 1 w M , the received sequence Y is decoded to the constant message, which may happen to be the message 1. Therefore, the inequality in (7.153) is not an equality in general. 7.4 Achievability 161 have the same marginal distributions as X and Y , respectively. Furthermore, from the random coding scheme and the memorylessness of the DMC, it is ~ ~ intuitively correct that X and Y are independent because X(1) and X(w) ~ are independent and the generation of Y depends only on X(1). A formal proof of this claim requires a more detailed analysis. In our random coding scheme, the random variables are generated in the order ~ ~ ~ ^ X(1), X(2), , X(M ), W, X1 , Y1 , X2 , Y2 , , Xn , Yn , W . By considering the joint distribution of these random variables, similar to the discussion in Section 7.3, the Markov chain ~ ~ ~ ^ (X(1), X(2), , X(M ), W ) X Y W (7.160) can be established. See Problem 1 for the details. Then for any 2 w M , from the above Markov chain, we have ~ I(Y; X(w), W |X) = 0. (7.161) By the chain rule for mutual information, the left hand side can be written as ~ I(Y; W |X) + I(Y; X(w)|X, W ). By the nonnegativity of conditional mutual information, this implies ~ I(Y; X(w)|X, W ) = 0, or M (7.162) (7.163) ~ Pr{W = w}I(Y; X(w)|X, W = w) = 0. w=1 (7.164) ~ Since I(Y; X(w)|X, W = w) are all nonnegative, we see from the above that they must all vanish. In particular, ~ I(Y; X(w)|X, W = 1) = 0. Then ~ ~ ~ ~ I(Y; X(w)|X(1), W = 1) = I(Y; X(w)|X(W ), W = 1) ~ = I(Y; X(w)|X, W = 1) = 0. (7.166) (7.167) (7.168) (7.165) ~ ~ On the other hand, since X(1), X(w), and W are mutually independent, we have ~ ~ I(X(1); X(w)|W = 1) = 0. (7.169) Hence, 162 7 Discrete Memoryless Channels ~ I(Y; X(w)|W = 1) ~ ~ I(X(1), Y; X(w)|W = 1) ~ ~ ~ ~ = I(X(1); X(w)|W = 1) + I(Y; X(w)|X(1), W = 1) = 0+0 = 0, (7.170) (7.171) (7.172) (7.173) where (7.172) follows from (7.168) and (7.169), proving the claim. Let us now return to (7.158). For any 2 w M , it follows from the above claim and Lemma 7.17 that Pr{Ew |W = 1} ~ = Pr{(X(w), Y) T n 2 -n(I(X;Y )- ) [XY ] |W = 1} (7.174) (7.175) , where 0 as 0. From the upper bound in (7.146), we have M < 2n(I(X;Y )- 4 ) . (7.176) Using (7.159), (7.175), and the above upper bound on M , it follows from (7.152) and (7.158) that Pr{Err } < + 2n(I(X;Y )- 4 ) 2-n(I(X;Y )- ) = +2 -n( 4 - ) (7.177) (7.178) . Since 0 as 0, for sufficiently small , we have 4 - >0 (7.179) for any > 0, so that 2-n( 4 - ) 0 as n . Then by letting < 3 , it follows from (7.178) that (7.180) Pr{Err } < 2 for sufficiently large n. The main idea of the above analysis of Pr{Err } is the following. In constructing the codebook, we randomly generate M codewords in X n according to p(x)n , and one of the codewords is sent through the channel p(y|x). When n is large, with high probability, the received sequence is jointly typical with the codeword sent with respect to p(x, y). If the number of codewords M grows with n at a rate less than I(X; Y ), then the probability that the received sequence is jointly typical with a codeword other than the one sent through the channel is negligible. Accordingly, the message can be decoded correctly with probability arbitrarily close to 1. In constructing the codebook in Step 1 of the random coding scheme, we choose a codebook C with a certain probability Pr{C} from the ensemble of all possible codebooks. By conditioning on the codebook chosen, we have 7.4 Achievability 163 Pr{Err } = C Pr{C}Pr{Err |C}, (7.181) i.e., Pr{Err } is a weighted average of Pr{Err |C} over all C in the ensemble of all possible codebooks, where Pr{Err |C} is the average probability of error of the code, i.e., Pe , when the codebook C is chosen (cf. Definition 7.12). The reader should compare the two different expansions of Pr{Err } in (7.181) and (7.150). Therefore, there exists at least one codebook C such that . (7.182) 2 Thus we have shown that for any > 0, there exists for sufficiently large n an (n, M ) code such that 1 log M > I(X; Y ) - (7.183) n 2 (cf. (7.146)) and (7.184) Pe < . 2 We are still one step away from proving that the rate I(X; Y ) is achievable because we require that max instead of Pe is arbitrarily small. Toward this end, we write (7.184) as 1 M or M M Pr{Err |C } Pr{Err } < w < w=1 2 , (7.185) w < w=1 M 2 . (7.186) Upon ordering the codewords according to their conditional probabilities of error, we observe that the conditional probabilities of error of the better half of the M codewords are less than , otherwise the conditional probabilities of error of the worse half of the codewords are at least , and they contribute at least ( M ) to the summation in (7.186), which is a contradiction. 2 Thus by discarding the worse half of the codewords in C , for the resulting codebook, the maximal probability of error max is less than . Using (7.183) and considering 1 M 1 1 log = log M - (7.187) n 2 n n 1 > I(X; Y ) - - (7.188) 2 n > I(X; Y ) - (7.189) when n is sufficiently large, we see that the rate of the resulting code is greater than I(X; Y ) - . Hence, we conclude that the rate I(X; Y ) is achievable. Finally, upon letting the input distribution p(x) be one that achieves the channel capacity, i.e., I(X; Y ) = C, we have proved that the rate C is achievable. This completes the proof of the direct part of the channel coding theorem. 164 7 Discrete Memoryless Channels 7.5 A Discussion In the last two sections, we have proved the channel coding theorem which asserts that reliable communication through a DMC at rate R is possible if and only if R < C, the channel capacity. By reliable communication at rate R, we mean that the size of the message set grows exponentially with n at rate R, while the message can be decoded correctly with probability arbitrarily close to 1 as n . Therefore, the capacity C is a fundamental characterization of a DMC. The capacity of a noisy channel is analogous to the capacity of a water pipe in the following way. For a water pipe, if we pump water through the pipe at a rate higher than its capacity, the pipe would burst and water would be lost. For a communication channel, if we communicate through the channel at a rate higher than the capacity, the probability of error is bounded away from zero, i.e., information is lost. In proving the direct part of the channel coding theorem, we showed that there exists a channel code whose rate is arbitrarily close to C and whose probability of error is arbitrarily close to zero. Moreover, the existence of such a code is guaranteed only when the block length n is large. However, the proof does not indicate how we can find such a codebook. For this reason, the proof we gave is called an existence proof (as oppose to a constructive proof). For a fixed block length n, we in principle can search through the ensemble of all possible codebooks for a good one, but this is quite prohibitive even for small n because the number of all possible codebooks grows doubly exponentially with n. Specifically, the total number of all possible (n, M ) codebooks is equal to |X |M n . When the rate of the code is close to C, M is approximately equal to 2nC . Therefore, the number of codebooks we need to search through nC is about |X |n2 . Nevertheless, the proof of the direct part of the channel coding theorem does indicate that if we generate a codebook randomly as prescribed, the codebook is most likely to be good. More precisely, we now show that the probability of choosing a code C such that Pr{Err |C} is greater than any prescribed > 0 is arbitrarily small when n is sufficiently large. Consider Pr{Err } = C Pr{C}Pr{Err |C} Pr{C}Pr{Err |C} C:Pr{Err |C} (7.190) = + Pr{C}Pr{Err |C} C:Pr{Err |C}> (7.191) (7.192) (7.193) C:Pr{Err |C}> Pr{C}Pr{Err |C} Pr{C}, C:Pr{Err |C}> > 7.5 A Discussion 165 p ( y|x) 2 n H (Y | X ) 2 n I( X ; Y ) codewords in T[n ] X 2 n H (Y ) sequences in T[n ] Y Fig. 7.12. A channel code that achieves capacity. which implies Pr{C} < C:Pr{Err |C}> ... Pr{Err } . (7.194) From (7.182), we have 2 > 0 when n is sufficiently large. Then Pr{C} < C:Pr{Err |C}> Pr{Err } < (7.195) for any 2 . (7.196) Since is fixed, this upper bound can be made arbitrarily small by choosing a sufficiently small . Although the proof of the direct part of the channel coding theorem does not provide an explicit construction of a good code, it does give much insight into what a good code is like. Figure 7.12 is an illustration of a channel code that achieves the channel capacity. Here we assume that the input distribution p(x) is one that achieves the channel capacity, i.e., I(X; Y ) = C. The idea is that most of the codewords are typical sequences in X n with respect to p(x). (For this reason, the repetition code is not a good code.) When such a codeword is transmitted through the channel, the received sequence is likely to be one of about 2nH(Y |X) sequences in Y n which are jointly typical with the transmitted codeword with respect to p(x, y). The association between a codeword and the about 2nH(Y |X) corresponding sequences in Y n is shown as a cone in the figure. As we require that the probability of decoding error is small, the cones essentially do not overlap with each other. Since the number of typical sequences with respect to p(y) is about 2nH(Y ) , the number of codewords cannot exceed about 2nH(Y ) = 2nI(X;Y ) = 2nC . 2nH(Y |X) (7.197) 166 7 Discrete Memoryless Channels This is consistent with the converse of the channel coding theorem. The direct part of the channel coding theorem says that when n is large, as long as the number of codewords generated randomly is not more than about 2n(C- ) , the overlap among the cones is negligible with high probability. Therefore, instead of searching through the ensemble of all possible codebooks for a good one, we can generate a codebook randomly, and it is likely to be good. However, such a code is difficult to use due to the following implementation issues. A codebook with block length n and rate R consists of n2nR symbols from the input alphabet X . This means that the size of the codebook, i.e., the amount of storage required to store the codebook, grows exponentially with n. This also makes the encoding process inefficient. Another issue is regarding the computation required for decoding. Based on the sequence received at the output of the channel, the decoder needs to decide which of the about 2nR codewords was the one transmitted. This requires an exponential amount of computation. In practice, we are satisfied with the reliability of communication once it exceeds a certain level. Therefore, the above implementation issues may eventually be resolved with the advancement of microelectronics. But before then, we still have to deal with these issues. For this reason, the entire field of coding theory has been developed since the 1950's. Researchers in this field are devoted to searching for good codes and devising efficient decoding algorithms. In fact, almost all the codes studied in coding theory are linear codes. By taking advantage of the linear structures of these codes, efficient encoding and decoding can be achieved. In particular, Berrou et al. [33] proposed in 1993 a linear code called the turbo code6 that can practically achieve the channel capacity. Today, channel coding has been widely used in home entertainment systems (e.g., audio CD and DVD), computer storage systems (e.g., CD-ROM, hard disk, floppy disk, and magnetic tape), computer communication, wireless communication, and deep space communication. The most popular channel codes used in existing systems include the Hamming code, the Reed-Solomon code7 , the BCH code, and convolutional codes. We refer the interested reader to textbooks on coding theory [39] [234] [378] for discussions of this subject. 7.6 Feedback Capacity Feedback is common in practical communication systems for correcting possible errors which occur during transmission. As an example, during a telephone 6 7 The turbo code is a special case of the class of Low-density parity-check (LDPC) codes proposed by Gallager [127] in 1962 (see MacKay [240]). However, the performance of such codes was not known at that time due to lack of high speed computers for simulation. The Reed-Solomon code was independently discovered by Arimoto [18]. 7.6 Feedback Capacity X i =f i ( W, Y i- 1 ) W Message Channel p(y|x) Yi W Estimate of message 167 Encoder Decoder Fig. 7.13. A channel code with feedback. conversation, we often have to request the speaker to repeat due to poor voice quality of the telephone line. As another example, in data communication, the receiver may request a packet to be retransmitted if the parity check bits received are incorrect. In general, when feedback from the receiver is available at the transmitter, the transmitter can at any time decide what to transmit next based on the feedback so far, and can potentially transmit information through the channel reliably at a higher rate. In this section, we study a model in which a DMC is used with complete feedback. The block diagram for the model is shown in Figure 7.13. In this model, the symbol Yi received at the output of the channel at time i is available instantaneously at the encoder without error. Then depending on the message W and all the previous feedback Y1 , Y2 , , Yi , the encoder decides the value of Xi+1 , the next symbol to be transmitted. Such a channel code is formally defined below. Definition 7.18. An (n, M ) code with complete feedback for a discrete memoryless channel with input alphabet X and output alphabet Y is defined by encoding functions fi : {1, 2, , M } Y i-1 X (7.198) for 1 i n and a decoding function g : Y n {1, 2, , M }. (7.199) We will use Yi to denote (Y1 , Y2 , , Yi ) and Xi to denote fi (W, Yi-1 ). We note that a channel code without feedback is a special case of a channel code with complete feedback because for the latter, the encoder can ignore the feedback. Definition 7.19. A rate R is achievable with complete feedback for a discrete memoryless channel p(y|x) if for any > 0, there exists for sufficiently large n an (n, M ) code with complete feedback such that 1 log M > R - n and max < . (7.201) (7.200) 168 7 Discrete Memoryless Channels Definition 7.20. The feedback capacity, CFB , of a discrete memoryless channel is the supremum of all the rates achievable by codes with complete feedback. Proposition 7.21. The supremum in the definition of CFB in Definition 7.20 is the maximum. Proof. Consider rates R(k) which are achievable with complete feedback such that lim R(k) = R. (7.202) k Then for any > 0, for all k, there exists for sufficiently large n an (n, M (k) ) code with complete feedback such that 1 log M (k) > R(k) - n and (k) < . max By virtue of (7.202), let k( ) be an integer such that for all k > k( ), |R - R(k) | < , which implies R(k) > R - . Then for all k > k( ), 1 log M (k) > R(k) - > R - 2 . (7.207) n Therefore, it follows from (7.207) and (7.204) that R is achievable with complete feedback. This implies that the supremum in Definition 7.20, which can be achieved, is in fact the maximum. Since a channel code without feedback is a special case of a channel code with complete feedback, any rate R achievable by the former is also achievable by the latter. Therefore, CFB C. (7.208) A fundamental question is whether CFB is greater than C. The answer surprisingly turns out to be negative for a DMC, as we now show. From the description of a channel code with complete feedback, we obtain the depen^ dency graph for the random variables W, X, Y, W in Figure 7.14. From this dependency graph, we see that n n (7.203) (7.204) (7.205) (7.206) q(w, x, y, w) = q(w) ^ i=1 q(xi |w, yi-1 ) i=1 p(yi |xi ) q(w|y) ^ (7.209) for all (w, x, y, w) W X n Y n W such that q(w, yi-1 ), q(xi ) > 0 for ^ 1 i n and q(y) > 0, where yi = (y1 , y2 , , yi ). Note that q(xi |w, yi-1 ) and q(w|y) are deterministic. ^ 7.6 Feedback Capacity p ( y | x) X1 X2 X3 Y1 Y2 Y3 169 W W Yn-1 Xn Yn Fig. 7.14. The dependency graph for a channel code with feedback. Lemma 7.22. For all 1 i n, (W, Yi-1 ) Xi Yi forms a Markov chain. Proof. The dependency graph for the random variables W, Xi , and Yi is shown in Figure 7.15. Denote the set of nodes W, Xi-1 , and Yi-1 by Z. Then we see that all the edges from Z end at Xi , and the only edge from Xi ends at Yi . This means that Yi depends on (W, Xi-1 , Yi-1 ) only through Xi , i.e., (W, Xi-1 , Yi-1 ) Xi Yi (7.211) forms a Markov chain, or I(W, Xi-1 , Yi-1 ; Yi |Xi ) = 0. (7.212) (7.210) This can be formally justified by Proposition 2.9, and the details are omitted here. Since 0 = I(W, Xi-1 , Yi-1 ; Yi |Xi ) = I(W, Yi-1 ; Yi |Xi ) + I(Xi-1 ; Yi |W, Xi , Yi-1 ) and mutual information is nonnegative, we obtain I(W, Yi-1 ; Yi |Xi ) = 0, or (W, Yi-1 ) Xi Yi forms a Markov chain. The lemma is proved. (7.216) (7.215) (7.213) (7.214) 170 7 Discrete Memoryless Channels X1 X2 W X3 Y1 Y2 Y3 Z Xi-1 Xi Yi-1 Yi Fig. 7.15. The dependency graph for W, Xi , and Yi . From the definition of CFB and by virtue of Proposition 7.21, if R CFB , then R is a rate achievable with complete feedback. We will show that if a rate R is achievable with complete feedback, then R C. If so, then R CFB implies R C, which can be true if and only if CFB C. Then from (7.208), we can conclude that CFB = C. Let R be a rate achievable with complete feedback, i.e., for any > 0, there exists for sufficiently large n an (n, M ) code with complete feedback such that n-1 log M > R - (7.217) and max < . Consider log M = H(W ) = I(W ; Y) + H(W |Y) and bound I(W ; Y) and H(W |Y) as follows. First, I(W ; Y) = H(Y) - H(Y|W ) n (7.218) (7.219) (7.220) (7.221) (7.222) (7.223) (7.224) = H(Y) - = H(Y) - = H(Y) - i=1 n b) i=1 n a) i=1 n H(Yi |Yi-1 , W ) H(Yi |Yi-1 , W, Xi ) H(Yi |Xi ) n i=1 H(Yi ) - i=1 H(Yi |Xi ) 7.6 Feedback Capacity n 171 = i=1 I(Xi ; Yi ) (7.225) (7.226) nC, where a) follows because Xi is a function of W and Yi-1 and b) follows from Lemma 7.22. Second, ^ ^ H(W |Y) = H(W |Y, W ) H(W |W ). Thus ^ log M H(W |W ) + nC, (7.228) which is the same as (7.126). Then by (7.217) and an application of Fano's inequality, we conclude as in the proof for the converse of the channel coding theorem that R C. (7.229) Hence, we have proved that CFB = C. Remark 1 The proof for the converse of the channel coding theorem in Section 7.3 depends critically on the Markov chain ^ W XYW (7.230) (7.227) and the relation in (7.101) (the latter implies Lemma 7.16). Both of them do not hold in general in the presence of feedback. Remark 2 The proof for CFB = C in this section is also a proof for the converse of the channel coding theorem, so we actually do not need the proof in Section 7.3. However, the proof here and the proof in Section 7.3 have different spirits. Without comparing the two proofs, one cannot possibly understand the subtlety of the result that feedback does not increase the capacity of a DMC. Remark 3 Although feedback does not increase the capacity of a DMC, the availability of feedback often makes coding much simpler. For some channels, communication through the channel with zero probability of error can be achieved in the presence of feedback by using a variable-length channel code. These are discussed in the next example. Example 7.23. Consider the binary erasure channel in Example 7.8 whose capacity is 1 - , where is the erasure probability. In the presence of complete feedback, for every information bit to be transmitted, the encoder can transmit the same information bit through the channel until an erasure does not occur, i.e., the information bit is received correctly. Then the number of uses of the channel it takes to transmit an information bit through the channel correctly has a truncated geometrical distribution whose mean is (1 - )-1 . 172 7 Discrete Memoryless Channels U source W encoder channel encoder X p ( y| x ) Y channel decoder W source decoder U Fig. 7.16. Separation of source coding and channel coding. Therefore, the effective rate at which information can be transmitted through the channel is 1 - . In other words, the channel capacity is achieved by using a very simple variable-length code. Moreover, the channel capacity is achieved with zero probability of error. In the absence of feedback, the rate 1 - can also be achieved, but with an arbitrarily small probability of error and a much more complicated code. To conclude this section, we point out that the memoryless assumption of the channel is essential for drawing the conclusion that feedback does not increase the channel capacity not because the proof presented in this section does not go through without this assumption, but because if the channel has memory, feedback actually can increase the channel capacity. For an illustrating example, see Problem 12. 7.7 Separation of Source and Channel Coding We have so far considered the situation in which we want to convey a message through a DMC, where the message is randomly selected from a finite set according to the uniform distribution. However, in most situations, we want to convey an information source through a DMC. Let {Uk , k > -n} be an ergodic stationary information source with entropy rate H. Denote the common alphabet by U and assume that U is finite. To convey {Uk } through the channel, we can employ a source code with rate Rs and a channel code with rate Rc as shown in Figure 7.16 such that Rs < Rc . Let f s and g s be respectively the encoding function and the decoding function of the source code, and f c and g c be respectively the encoding function and the decoding function of the channel code. The block of n information symbols U = (U-(n-1) , U-(n-2) , , U0 ) is first encoded by the source encoder into an index W = f s (U), (7.231) called the source codeword. Then W is mapped by the channel encoder to a distinct channel codeword X = f c (W ), (7.232) where X = (X1 , X2 , , Xn ). This is possible because there are about 2nRs source codewords and about 2nRc channel codewords, and we assume that Rs < Rc . Then X is transmitted through the DMC p(y|x), and the sequence 7.7 Separation of Source and Channel Coding 173 Y = (Y1 , Y2 , , Yn ) is received. Based on Y, the channel decoder first estimates W as ^ W = g c (Y). (7.233) ^ Finally, the source decoder decodes W to ^ ^ U = g s (W ). (7.234) ^ For this scheme, an error occurs if U = U, and we denote the probability of error by Pe . We now show that if H < C, the capacity of the DMC p(y|x), then it is possible to convey U through the channel with an arbitrarily small probability of error. First, we choose Rs and Rc such that H < Rs < Rc < C. ^ Observe that if W = W and g s (W ) = U, then from (7.234), ^ ^ U = g s (W ) = g s (W ) = U, (7.236) (7.235) ^ i.e., an error does not occur. In other words, if an error occurs, either W = W s or g (W ) = U. Then by the union bound, we have ^ Pe Pr{W = W } + Pr{g s (W ) = U}. (7.237) For any > 0 and sufficiently large n, by the Shannon-McMillan-Breiman theorem, there exists a source code such that Pr{g s (W ) = U} . (7.238) By the channel coding theorem, there exists a channel code such that max , where max is the maximal probability of error. This implies ^ Pr{W = W } = w ^ Pr{W = W |W = w}Pr{W = w} Pr{W = w} w (7.239) (7.240) (7.241) (7.242) max = max . Combining (7.238) and (7.242), we have Pe 2 . (7.243) Therefore, we conclude that as long as H < C, it is possible to convey {Uk } through the DMC reliably. In the scheme we have discussed, source coding and channel coding are separated. In general, source coding and channel coding can be combined. 174 7 Discrete Memoryless Channels X i =f i sc( W, Y i - 1 ) sourcechannel encoder Yi sourcechannel decoder U p ( y| x ) U Fig. 7.17. Joint source-channel coding. This technique is called joint source-channel coding. It is then natural to ask whether it is possible to convey information through the channel reliably at a higher rate by using joint source-channel coding. In the rest of the section, we show that the answer to this question is no to the extent that for asymptotic reliability, we must have H C. However, whether asymptotical reliability can be achieved for H = C depends on the specific information source and channel. We base our discussion on the general assumption that complete feedback is available at the encoder as shown in Figure 7.17. Let fisc , 1 i n, be the encoding functions and g sc be the decoding function of the source-channel code. Then Xi = fisc (U, Yi-1 ) (7.244) for 1 i n, where Yi-1 = (Y1 , Y2 , , Yi-1 ), and ^ U = g sc (Y), (7.245) ^ ^ ^ ^ where U = (U1 , U2 , , Un ). In exactly the same way as we proved (7.226) in the last section, we can prove that I(U; Y) nC. ^ Since U is a function of Y, ^ ^ I(U; U) I(U; U, Y) = I(U; Y) nC. For any > 0, H(U) n(H - ) for sufficiently large n. Then ^ ^ ^ n(H - ) H(U) = H(U|U) + I(U; U) H(U|U) + nC. Applying Fano's inequality (Corollary 2.48), we obtain (7.251) (7.250) (7.247) (7.248) (7.249) (7.246) Chapter Summary 175 n(H - ) 1 + nPe log |U| + nC, or (7.252) 1 + Pe log |U| + C. (7.253) n For asymptotic reliability, Pe 0 as n . Therefore, by letting n and then 0, we conclude that H- H C. (7.254) This result, sometimes called the separation theorem for source and channel coding, says that asymptotic optimality can be achieved by separating source coding and channel coding. This theorem has significant engineering implication because the source code and the channel code can be designed separately without losing asymptotic optimality. Specifically, we only need to design the best source code for the information source and design the best channel code for the channel. Moreover, separation of source coding and channel coding facilitates the transmission of different information sources on the same channel because we need only change the source code for different information sources. Likewise, separation of source coding and channel coding also facilitates the transmission of an information source on different channels because we need only change the channel code for different channels. We remark that although asymptotic optimality can be achieved by separating source coding and channel coding, for finite block length, the probability of error generally can be reduced by using joint source-channel coding. Chapter Summary Capacity of Discrete Memoryless Channel: C = max I(X; Y ), p(x) where p(x) is the input distribution of the channel. 1. C min(log |X |, log |Y|). 2. For a binary symmetric channel with crossover probability , C = 1-hb ( ). 3. For a binary erasure channel with erasure probability , C = 1 - . Lemma: Let X and Y be a pair of random variables and (X , Y ) be n i.i.d. copies of a pair of generic random variables (X , Y ), where X and Y are independent and have the same marginal distributions as X and Y , respectively. Then n Pr{(X , Y ) T[XY ] } 2-n(I(X;Y )- ) , where 0 as 0. 176 7 Discrete Memoryless Channels Channel Coding Theorem: A message drawn uniformly from the set {1, 2, , 2n(R- ) } can be transmitted through a discrete memoryless channel with negligible probability of error as n if and only if R C. Feedback: The capacity of a discrete memoryless channel is not increased by feedback. Separation of Source and Channel Coding: An information source with entropy rate H can be transmitted through a discrete memoryless channel with capacity C reliably if H < C (only if H C), and asymptotic optimality can be achieved by separating source coding and channel coding. Problems In the following, X = (X1 , X2 , , Xn ), x = (x1 , x2 , , xn ), and so on. 1. Refer to the discussion in Section 7.4. a) Construct the dependency graph for the random variables involved in the random coding scheme. b) By considering the joint distribution of these random variables, prove the Markov chain in (7.160). 2. Show that the capacity of a DMC with complete feedback cannot be increased by using probabilistic encoding and/or decoding schemes. 3. Memory increases capacity Consider a BSC with crossover probability 0 < < 1 represented by Xi = Yi + Zi mod 2, where Xi , Yi , and Zi are respectively the input, the output, and the noise variable at time i. Then Pr{Zi = 0} = 1 - and Pr{Zi = 1} = for all i. We assume that {Xi } and {Zi } are independent, but we make no assumption that Zi are i.i.d. so that the channel may have memory. a) Prove that I(X; Y) n - hb ( ). b) Show that the upper bound in a) can be achieved by letting Xi be i.i.d. bits taking the values 0 and 1 with equal probability and Z1 = Z2 = = Zn . c) Show that with the assumptions in b), I(X; Y) > nC, where C = 1 - hb ( ) is the capacity of the BSC if it is memoryless. 4. Consider the channel in Problem 3, Part b). a) Show that the channel capacity is not increased by feedback. b) Devise a coding scheme without feedback that achieves the channel capacity. Problems 177 5. In Remark 1 toward the end of Section 7.6, it was mentioned that in the ^ presence of feedback, both the Markov chain W X Y W and Lemma 7.16 do not hold in general. Give examples to substantiate this remark. 6. Prove that when a DMC is used with complete feedback, Pr{Yi = yi |Xi = xi , Yi-1 = yi-1 } = Pr{Yi = yi |Xi = xi } for all i 1. This relation, which is a consequence of the causality of the code, says that given the current input, the current output does not depend on all the past inputs and outputs of the DMC. 7. Let 1- P( ) = 1- be the transition matrix for a BSC with crossover probability . Define a b = (1 - a)b + a(1 - b) for 0 a, b 1. a) Prove that a DMC with transition matrix P ( 1 )P ( 2 ) is equivalent to a BSC with crossover probability 1 2 . Such a channel is the cascade of two BSC's with crossover probabilities 1 and 2 , respectively. b) Repeat a) for a DMC with transition matrix P ( 2 )P ( 1 ). c) Prove that 1 - hb ( 1 2) min(1 - hb ( 1 ), 1 - hb ( 2 )). This means that the capacity of the cascade of two BSC's is upper bounded by the capacity of either of the two BSC's. d) Prove that a DMC with transition matrix P ( )n is equivalent to a BSC with crossover probabilities 1 (1 - (1 - 2 )n ). 2 8. Symmetric channel A DMC is symmetric if the rows of the transition matrix p(y|x) are permutations of each other and so are the columns. Determine the capacity of such a channel. See Section 4.5 in Gallager [129] for a more general discussion. 9. Let C1 and C2 be the capacities of two DMC's with transition matrices P1 and P2 , respectively, and let C be the capacity of the DMC with transition matrix P1 P2 . Prove that C min(C1 , C2 ). 10. Two parallel channels Let C1 and C2 be the capacities of two DMC's p1 (y1 |x1 ) and p2 (y2 |x2 ), respectively. Determine the capacity of the DMC p(y1 , y2 |x1 , x2 ) = p1 (y1 |x1 )p2 (y2 |x2 ). Hint: Prove that I(X1 , X2 ; Y1 , Y2 ) I(X1 ; Y1 ) + I(X2 ; Y2 ) if p(y1 , y2 |x1 , x2 ) = p1 (y1 |x1 )p2 (y2 |x2 ). 178 7 Discrete Memoryless Channels 11. In the system below, there are two channels with transition matrices p1 (y1 |x) and p2 (y2 |x). These two channels have a common input alphabet X and output alphabets Y1 and Y2 , repespectively, where Y1 and Y2 are disjoint. The position of the switch is determined by a random variable Z which is independent of X, where Pr{Z = 1} = . p1(y1 |x) Y1 Z=1 X p2(y2 |x) Z=2 Y Y2 a) Show that I(X; Y ) = I(X; Y1 ) + (1 - )I(X; Y2 ). b) The capacity of the system is given by C = maxp(x) I(X; Y ). Show that C C1 +(1-)C2 , where Ci = maxp(x) I(X; Yi ) is the capacity of the channel with transition matrix pi (yi |x), i = 1, 2. c) If both C1 and C2 can be achieved by a common input distribution, show that C = C1 + (1 - )C2 . 12. Feedback increases capacity Consider a ternary channel with memory with input/output alphabet {0, 1, 2} as follows. At time 1, the output of the channel Y1 has a uniform distribution on {0, 1, 2} and is independent of the input X1 (i.e., the channel outputs each of the values 0, 1, and 2 with 1 probability 3 regardless of the input). At time 2, the transition from X2 to Y2 which depends on the value of Y1 is depicted below: Y1 = 0 0 1 2 0 1 2 0 1 2 Y1 = 1 0 1 2 0 1 2 Y1 = 2 0 1 2 For every two subsequent transmissions, the channel replicates itself independently. So we only need to consider the first two transmissions. In the sequel, we regard this channel as described by a generic discrete channel (with transmission duration equals 2) with two input symbols X1 and X2 and two output symbols Y1 and Y2 , and we will refer to this channel as the block channel. Problems 179 a) Determine the capacity this block channel when it is used without feedback. Hint: Use the results in Problems 8 and 11. b) Consider the following coding scheme when the block channel is used with feedback. Let the message W = (W1 , W2 ) with W1 = {0, 1, 2} and W2 = {0, 1}. Let W1 and W2 be independent, and each of them is distributed uniformly on its alphabet. First, Let X1 = W1 and transmit X1 through the channel to obtain Y1 , which is independent of X1 . Then based on the value of Y1 , we determine X2 as follows: i) If Y1 = 0, let X2 = 0 if W2 = 0, and let X2 = 1 if W2 = 1. ii) If Y1 = 1, let X2 = 1 if W2 = 0, and let X2 = 2 if W2 = 1. iii) If Y1 = 2, let X2 = 0 if W2 = 0, and let X2 = 2 if W2 = 1. Then transmit X2 through the channel to obtain Y2 . Based on this coding scheme, show that for the capacity of this block channel can be increased by feedback. 13. Channel with memory and directed information The memorylessness of a DMC is characterized by the Markov chain Ti- Xi Yi according to the discussion following Definition 7.4. In general, a channel with memory satisfies the Markov chain Ti- (Xi , Yi-1 ) Yi , where Ti- denotes all the random variables generated in the system before Xi (i.e., the random variables denoted by Ti- ) except for Xi-1 and Yi-1 . Consider the use of such a channel in the presence of complete feedback. a) Give the dependency graph for all the random variables involved in the coding scheme. Note that the memory of the channel is manifested by the dependence of Yi on Xi-1 and Yi-1 (in addition to its dependence on Xi ) for 1 i n. b) Verify the correctness of the following derivation: I(W ; Y) = H(Y) - H(Y|W ) n = i=1 n [H(Yi |Yi-1 ) - H(Yi |W, Yi-1 )] [H(Yi |Yi-1 ) - H(Yi |W, Xi , Yi-1 )] i=1 n = i=1 n [H(Yi |Yi-1 ) - H(Yi |Xi , Yi-1 )] I(Yi ; Xi |Yi-1 ). i=1 = The above upper bound on I(W ; Y), denoted by I(X Y), is called the directed information from X to Y. c) Show that the inequality in the derivation in b) is in fact an equality. Hint: Use Definition 7.18. d) In the spirit of the informal discussion in Section 7.3, we impose the constraint H(W |Y) = 0. Show that 180 7 Discrete Memoryless Channels H(W ) = I(X Y). This is the generalization of (7.105) for a channel with memory in the presence of complete feedback. e) Show that I(X Y) = I(X; Y) if the channel code does not make use of the feedback. Hint: First show that H(Yi |Xi , Yi-1 ) = H(Yi |W, Xi , Yi-1 ) = H(Yi |W, X, Yi-1 ). (Marko [245] and Massey [250].) 14. Maximum likelihood decoding In maximum likelihood decoding for a given channel and a given codebook, if a received sequence y is decoded to a codeword x, then x maximizes Pr{y|x } among all codewords x in the codebook. a) Prove that maximum likelihood decoding minimizes the average probability of error. b) Does maximum likelihood decoding also minimize the maximal probability of error? Give an example if your answer is no. 15. Minimum distance decoding The Hamming distance between two binary sequences x and y, denoted by d(x, y), is the number of places where x and y differ. In minimum distance decoding for a memoryless BSC, if a received sequence y is decoded to a codeword x, then x minimizes d(x , y) over all codewords x in the codebook. Prove that minimum distance decoding is equivalent to maximum likelihood decoding if the crossover probability of the BSC is less than 0.5. 16. The following figure shows a communication system with two DMC's with complete feedback. The capacities of the two channels are respectively C1 and C2 . W W Encoder 1 Channel 1 Encoder 2 Channel 2 Decoder 2 a) Give the dependency graph for all the random variables involved in the coding scheme. b) Prove that the capacity of the system is min(C1 , C2 ). 17. Binary arbitrarily varying channel Consider a memoryless BSC whose crossover probability is time-varying. Specifically, the crossover probability (i) at time i is an arbitrary value in [ 1 , 2 ], where 0 1 < 2 < 0.5. Prove that the capacity of this channel is 1 - hb ( 2 ). (Ahlswede and Wolfowitz [12].) 18. Consider a BSC with crossover probability [ 1 , 2 ], where 0 < 1 < is unknown. Prove that the capacity of 2 < 0.5, but the exact value of this channel is 1 - hb ( 2 ). Historical Notes 181 Historical Notes The concept of channel capacity was introduced in Shannon's original paper [322], where he stated the channel coding theorem and outlined a proof. The first rigorous proof was due to Feinstein [110]. The random coding error exponent was developed by Gallager [128] in a simplified proof. The converse of the channel coding theorem was proved by Fano [107], where he used an inequality now bearing his name. The strong converse was first proved by Wolfowitz [384]. An iterative algorithm for calculating the channel capacity developed independently by Arimoto [19] and Blahut [37] will be discussed in Chapter 9. Shannon [326] proved that the capacity of a discrete memoryless channel cannot be increased by feedback. The definition of a discrete memoryless channel in this chapter is new. With this definition, coding over such a channel with or without feedback can be rigorously formulated. 8 Rate-Distortion Theory Consider an information source with entropy rate H. By the source coding theorem, it is possible to design a source code with rate R which reconstructs the source sequence X = (X1 , X2 , , Xn ) with an arbitrarily small probability of error provided R > H and the block length n is sufficiently large. However, there are situations in which we want to convey an information source by a source code with rate less than H. Then we are motivated to ask: what is the best we can do when R < H? A natural approach is to design a source code such that for part of the time the source sequence is reconstructed correctly, while for the other part of the time the source sequence is reconstructed incorrectly, i.e., an error occurs. In designing such a code, we try to minimize the probability of error. However, this approach is not viable asymptotically because the converse of the source coding theorem says that if R < H, then the probability of error inevitably tends to 1 as n . Therefore, if R < H, no matter how the source code is designed, the source sequence is almost always reconstructed incorrectly when n is large. An alternative approach is to design a source code called a rate-distortion code which reproduces the source sequence with distortion. In order to formulate the problem properly, we need a distortion measure between each source sequence and each reproduction sequence. Then we try to design a rate-distortion code which with high probability reproduces the source sequence with a distortion within a tolerance level. Clearly, a smaller distortion can potentially be achieved if we are allowed to use a higher coding rate. Rate-distortion theory, the subject matter of this chapter, gives a characterization of the asymptotic optimal tradeoff between the coding rate of a rate-distortion code for a given information source and the allowed distortion in the reproduction sequence with respect to a distortion measure. 184 8 Rate-Distortion Theory 8.1 Single-Letter Distortion Measures Let {Xk , k 1} be an i.i.d. information source with generic random variable X. We assume that the source alphabet X is finite. Let p(x) be the probability distribution of X, and we assume without loss of generality that the support of X is equal to X . Consider a source sequence x = (x1 , x2 , , xn ) and a reproduction sequence ^ x = (^1 , x2 , , xn ). x ^ ^ (8.2) (8.1) ^ The components of x can take values in X , but more generally, they can take ^ ^ values in any finite set X which may be different from X . The set X , which is also assumed to be finite, is called the reproduction alphabet. To measure the ^ distortion between x and x, we introduce the single-letter distortion measure and the average distortion measure. Definition 8.1. A single-letter distortion measure is a mapping ^ d:X X + , (8.3) where + is the set of nonnegative real numbers1 . The value d(x, x) denotes ^ the distortion incurred when a source symbol x is reproduced as x. ^ Definition 8.2. The average distortion between a source sequence x X n ^ ^ and a reproduction sequence x X n induced by a single-letter distortion measure d is defined by n 1 ^ d(xk , xk ). ^ (8.4) d(x, x) = n k=1 In Definition 8.2, we have used d to denote both the single-letter distortion measure and the average distortion measure, but this abuse of notation should cause no ambiguity. Henceforth, we will refer to a single-letter distortion measure simply as a distortion measure. Very often, the source sequence x represents quantized samples of a continuous signal, and the user attempts to recognize certain objects and derive ^ meaning from the reproduction sequence x. For example, x may represent a video signal, an audio signal, or an image. The ultimate purpose of a distor^ tion measure is to reflect the distortion between x and x as perceived by the user. This goal is difficult to achieve in general because measurements of the ^ distortion between x and x must be made within context unless the symbols in X carry no physical meaning. Specifically, when the user derives meaning 1 ^ Note that d(x, x) is finite for all (x, x) X X . ^ ^ 8.1 Single-Letter Distortion Measures 185 ^ ^ from x, the distortion in x as perceived by the user depends on the context. For example, the perceived distortion is small for a portrait contaminated by a fairly large noise, while the perceived distortion is large for the image of a book page contaminated by the same noise. Hence, a good distortion measure should be context dependent. Although the average distortion is not necessarily the best way to measure the distortion between a source sequence and a reproduction sequence, it has the merit of being simple and easy to use. Moreover, rate-distortion theory, which is based on the average distortion measure, provides a framework for data compression when distortion is inevitable. ^ Example 8.3. When the symbols in X and X represent real values, a popular distortion measure is the square-error distortion measure which is defined by d(x, x) = (x - x)2 . ^ ^ (8.5) The average distortion measure so induced is often referred to as the meansquare error. ^ Example 8.4. When X and X are identical and the symbols in X do not carry any particular meaning, a frequently used distortion measure is the Hamming distortion measure, which is defined by d(x, x) = ^ 0 if x = x ^ 1 if x = x. ^ (8.6) The Hamming distortion measure indicates the occurrence of an error. In ^ particular, for an estimate X of X, we have ^ ^ ^ ^ Ed(X, X) = Pr{X = X} 0 + Pr{X = X} 1 = Pr{X = X}, (8.7) ^ i.e., the expectation of the Hamming distortion measure between X and X is the probability of error. ^ ^ ^ For x X n and x X n , the average distortion d(x, x) induced by the Hamming distortion measure gives the frequency of error in the reproduction ^ sequence x. ^ Definition 8.5. For a distortion measure d, for each x X , let x (x) X ^ ^ . A distortion measure d is said to be normal minimize d(x, x) over all x X ^ ^ if def cx = d(x, x (x)) = 0 ^ (8.8) for all x X . The square-error distortion measure and the Hamming distortion measure are examples of normal distortion measures. Basically, a normal distortion measure is one which allows X to be reproduced with zero distortion. Although 186 8 Rate-Distortion Theory a distortion measure d is not normal in general, a normalization of d can always be obtained by defining the distortion measure ~ ^ d(x, x) = d(x, x) - cx ^ (8.9) ~ ^ for all (x, x) X X . Evidently, d is a normal distortion measure, and it is ^ referred to as the normalization of d. Example 8.6. Let d be a distortion measure defined by d(x, x) a b c ^ 1 275 2 438 ~ Then d, the normalization of d, is given by ~ ^ d(x, x) a b c 1 053 2 105 ~ ^ ^ Note that for every x X , there exists an x X such that d(x, x) = 0. ^ ^ ^ Let X be any estimate of X which takes values in X , and denote the joint ^ by p(x, x). Then distribution for X and X ^ ^ Ed(X, X) = x x ^ p(x, x)d(x, x) ^ ^ p(x, x) d(x, x) + cx ^ ~ ^ x x ^ (8.10) (8.11) (8.12) = ~ ^ = E d(X, X) + x p(x) x ^ p(^|x)cx x p(^|x) x x ^ ~ ^ = E d(X, X) + x p(x)cx p(x)cx x (8.13) (8.14) (8.15) ~ ^ = E d(X, X) + ~ ^ = E d(X, X) + , where = x p(x)cx (8.16) is a constant which depends only on p(x) and d but not on the conditional distribution p(^|x). In other words, for a given X and a distortion measure d, x ^ the expected distortion between X and an estimate X of X is always reduced ~ instead of d as the distortion measure. For reasons by a constant upon using d which will be explained in Section 8.3, it is sufficient for us to assume that a distortion measure is normal. 8.2 The Rate-Distortion Function R(D) 187 ^ Definition 8.7. Let x minimizes Ed(X, x) over all x X , and define ^ ^ ^ Dmax = Ed(X, x ). ^ (8.17) x is the best estimate of X if we know nothing about X, and Dmax is the ^ minimum expected distortion between X and a constant estimate of X. The ^ significance of Dmax can be seen by taking the reproduction sequence X to be (^ , x , , x ). Since d(Xk , x ) are i.i.d., by the weak law of large numbers x ^ ^ ^ ^ d(X, X) = 1 n n d(Xk , x ) Ed(X, x ) = Dmax ^ ^ k=1 (8.18) in probability, i.e., for any > 0, (8.19) ^ Pr{d(X, X) > Dmax + } ^ for sufficiently large n. Note that X is a constant sequence which does not depend on X. In other words, even when no description of X is available, we can still achieve an average distortion no more than Dmax + with probability arbitrarily close to 1 when n is sufficiently large. The notation Dmax may seem confusing because the quantity stands for the minimum rather than the maximum expected distortion between X and a constant estimate of X. But we see from the above discussion that this notation is in fact appropriate because Dmax is the maximum distortion we have to be concerned about. Specifically, it is not meanful to impose a constraint D Dmax on the reproduction sequence because it can be achieved even without receiving any information about the sequence produced by the source. 8.2 The Rate-Distortion Function R(D) Throughout this chapter, all the discussions are with respect to an i.i.d. information source {Xk , k 1} with generic random variable X and a distortion measure d. All logarithms are in the base 2 unless otherwise specified. Definition 8.8. An (n, M ) rate-distortion code is defined by an encoding function f : X n {1, 2, , M } (8.20) and a decoding function ^ g : {1, 2, , M } X n . (8.21) The set {1, 2, , M }, denoted by I, is called the index set. The reproduction ^ sequences g(f (1)), g(f (2)), , g(f (M )) in X n are called codewords, and the set of codewords is called the codebook. 188 8 Rate-Distortion Theory X source sequence f (X) Encoder Decoder X reproduction sequence Fig. 8.1. A rate-distortion code with block length n. Figure 8.1 is an illustration of a rate-distortion code. Definition 8.9. The rate of an (n, M ) rate-distortion code is n-1 log M in bits per symbol. Definition 8.10. A rate-distortion pair (R, D) is asymptotically achievable if for any > 0, there exists for sufficiently large n an (n, M ) rate-distortion code such that 1 log M R + (8.22) n and ^ Pr{d(X, X) > D + } , (8.23) ^ where X = g(f (X)). For brevity, an asymptotically achievable pair will be referred to as an achievable pair. Remark It is clear from the definition that if (R, D) is achievable, then (R , D) and (R, D ) are also achievable for all R R and D D. Definition 8.11. The rate-distortion region is the subset of achievable pairs (R, D). 2 containing all Theorem 8.12. The rate-distortion region is closed and convex. Proof. We first show that the rate-distortion region is closed. Consider achievable rate-distortion pairs (R(k) , D(k) ) such that k lim (R(k) , D(k) ) = (R, D) (8.24) componentwise. Then for any > 0, for all k, there exists for sufficiently large n an (n, M (k) ) code such that 1 log M (k) R(k) + n and ^ Pr{d(X(k) , X(k) ) > D(k) + } , (k) (k) (8.25) (8.26) where f and g are respectively the encoding function and the decoding ^ function of the (n, M (k) ) code, and X(k) = g (k) (f (k) (X)). By virtue of (8.24), let k( ) be an integer such that for all k > k( ), 8.2 The Rate-Distortion Function R(D) 189 |R - R(k) | < and |D - D(k) | < , which imply R(k) < R + and D(k) < D + , respectively. Then for all k > k( ), 1 log M (k) R(k) + < R + 2 n and ^ ^ Pr{d(X(k) , X(k) ) > D + 2 } Pr{d(X(k) , X(k) ) > D(k) + } . Note that (8.32) follows because D + 2 > D(k) + (8.27) (8.28) (8.29) (8.30) (8.31) (8.32) (8.33) (8.34) by (8.30). From (8.31) and (8.33), we see that (R, D) is also achievable. Thus we have proved that the rate-distortion region is closed. We will prove the convexity of the rate-distortion region by a time-sharing argument whose idea is the following. Roughly speaking, if we can use a code C1 to achieve (R(1) , D(1) ) and a code C2 to achieve (R(2) , D(2) ), then for any rational number between 0 and 1, we can use C1 for a fraction of the time and use C2 for a fraction of the time to achieve (R() , D() ), where R() = R(1) + R(2) () (1) D = D + D(2) , (8.35) (8.36) and = 1 - . Since the rate-distortion region is closed as we have proved, can be taken as any real number between 0 and 1, and the convexity of the region follows. We now give a formal proof for the convexity of the rate-distortion region. Let r , (8.37) = r+s where r and s are positive integers. Then is a rational number between 0 and 1. We now prove that if (R(1) , D(1) ) and (R(2) , D(2) ) are achievable, then (R() , D() ) is also achievable. Assume (R(1) , D(1) ) and (R(2) , D(2) ) are achievable. Then for any > 0 and sufficiently large n, there exist an (n, M (1) ) code and an (n, M (2) ) code such that 190 8 Rate-Distortion Theory 1 log M (i) R(i) + n and ^ Pr{d(X, X(i) ) > D(i) + } , i = 1, 2. Let M () = (M (1) )r (M (2) )s and n() = (r + s)n. (8.38) (8.39) (8.40) (8.41) We now construct an (n(), M ()) code by concatenating r copies of the (n, M (1) ) code followed by s copies of the (n, M (2) ) code. We call these r + s codes subcodes of the (n(), M ()) code. For this code, let Y = (X(1), X(2), , X(r + s)) and ^ ^ ^ ^ Y = (X(1), X(2), , X(r + s)), (8.43) ^ where X(j) and X(j) are the source sequence and the reproduction sequence of the jth subcode, respectively. Then for this (n(), M ()) code, 1 1 log M () = log[(M (1) )r (M (2) )s ] n() (r + s)n 1 = (r log M (1) + s log M (2) ) (r + s)n 1 1 log M (2) = log M (1) + n n (1) (2) + ) (R + ) + (R = (R (1) (8.42) (8.44) (8.45) (8.46) (8.47) (8.48) (8.49) + R(2) ) + = R() + , where (8.47) follows from (8.38), and ^ Pr{d(Y, Y) > D() + } 1 r+s ^ = Pr d(X(j), X(j)) > D() + r + s j=1 ^ Pr d(X(j), X(j)) > D(1) + for some 1 j r or ^ d(X(j), X(j)) > D(2) + for some r + 1 j r + s r (8.50) (8.51) j=1 ^ Pr{d(X(j), X(j)) > D(1) + } 8.2 The Rate-Distortion Function R(D) r+s 191 + j=r+1 ^ Pr{d(X(j), X(j)) > D(2) + } (8.52) (8.53) (r + s) , where (8.52) follows from the union bound and (8.53) follows from (8.39). Hence, we conclude that the rate-distortion pair (R() , D() ) is achievable. This completes the proof of the theorem. Definition 8.13. The rate-distortion function R(D) is the minimum of all rates R for a given distortion D such that (R, D) is achievable. Definition 8.14. The distortion-rate function D(R) is the minimum of all distortions D for a given rate R such that (R, D) is achievable. Both the functions R(D) and D(R) are equivalent descriptions of the boundary of the rate-distortion region. They are sufficient to describe the rate-distortion region because the region is closed. Note that in defining R(D), the minimum instead of the infimum is taken because for a fixed D, the set of all R such that (R, D) is achievable is closed and lower bounded by zero. Similarly, the minimum instead of the infimum is taken in defining D(R). In the subsequent discussions, only R(D) will be used. Theorem 8.15. The following properties hold for the rate-distortion function R(D): 1. R(D) is non-increasing in D. 2. R(D) is convex. 3. R(D) = 0 for D Dmax . 4. R(0) H(X). Proof. From the remark following Definition 8.10, since (R(D), D) is achievable, (R(D), D ) is also achievable for all D D. Therefore, R(D) R(D ) because R(D ) is the minimum of all R such that (R, D ) is achievable. This proves Property 1. Property 2 follows immediately from the convexity of the rate-distortion region which was proved in Theorem 8.12. From the discussion toward the end of the last section, we see for any > 0, it is possible to achieve ^ Pr{d(X, X) > Dmax + } (8.54) for sufficiently large n with no description of X available. Therefore, (0, D) is achievable for all D Dmax , proving Property 3. Property 4 is a consequence of the assumption that the distortion measure d is normalized, which can be seen as follows. By the source coding theorem, for any > 0, by using a rate no more than H(X) + , we can describe the 192 8 Rate-Distortion Theory R(D) H(X) R(0) The rate distortion region Dmax D Fig. 8.2. A rate-distortion function R(D). source sequence X of length n with probability of error less than sufficiently large. Since d is normalized, for each k 1, let ^ Xk = x (Xk ) ^ (cf. Definition 8.5), so that whenever an error does not occur, ^ d(Xk , Xk ) = d(Xk , x (Xk )) = 0 ^ by (8.8) for each k, and ^ d(X, X) = Therefore, ^ Pr{d(X, X) > } , 1 n n when n is (8.55) (8.56) ^ d(Xk , Xk ) = k=1 1 n n d(Xk , x (Xk )) = 0. ^ k=1 (8.57) (8.58) which shows that the pair (H(X), 0) is achievable. This in turn implies that R(0) H(X) because R(0) is the minimum of all R such that (R, 0) is achievable. Figure 8.2 is an illustration of a rate-distortion function R(D). The reader should note the four properties of R(D) in Theorem 8.15. The rate-distortion theorem, which will be stated in the next section, gives a characterization of R(D). 8.3 The Rate-Distortion Theorem Definition 8.16. For D 0, the information rate-distortion function is defined by ^ RI (D) = min I(X; X). (8.59) ^ ^ X:Ed(X,X)D 8.3 The Rate-Distortion Theorem 193 ^ In defining RI (D), the minimization is taken over all random variables X jointly distributed with X such that ^ Ed(X, X) D. (8.60) Since p(x) is given, the minimization is taken over the set of all p(^|x) such x that (8.60) is satisfied, namely the set p(^|x) : x p(x)p(^|x)d(x, x) D . x ^ (8.61) x,^ x ^ ^ Since this set is compact in |X ||X | and I(X; X) is a continuous functional of ^ can be attained2 . This justifies taking p(^|x), the minimum value of I(X; X) x the minimum instead of the infimum in the definition of RI (D). ~ We have seen in Section 8.1 that we can obtain a normalization d for any distortion measure d with ~ ^ ^ E d(X, X) = Ed(X, X) - (8.62) ^ for any X, where is a constant which depends only on p(x) and d. Thus if d ~ is not normal, we can always replace d by d and D by D - in the definition of RI (D) without changing the minimization problem. Therefore, we do not lose any generality by assuming that a distortion measure d is normal. Theorem 8.17 (The Rate-Distortion Theorem). R(D) = RI (D). The rate-distortion theorem, which is the main result in rate-distortion theory, says that the minimum coding rate for achieving a distortion D is RI (D). This theorem will be proved in the next two sections. In the next section, we will prove the converse of this theorem, i.e., R(D) RI (D), and in Section 8.5, we will prove the achievability of RI (D), i.e., R(D) RI (D). In order for RI (D) to be a characterization of R(D), it has to satisfy the same properties as R(D). In particular, the four properties of R(D) in Theorem 8.15 should also be satisfied by RI (D). Theorem 8.18. The following properties hold for the information rate-distortion function RI (D): 1. RI (D) is non-increasing in D. 2. RI (D) is convex. 3. RI (D) = 0 for D Dmax . 4. RI (0) H(X). 2 ^ The assumption that both X and X are finite is essential in this argument. 194 8 Rate-Distortion Theory Proof. Referring to the definition of RI (D) in (8.59), for a larger D, the minimization is taken over a larger set. Therefore, RI (D) is non-increasing in D, proving Property 1. To prove Property 2, consider any D(1) , D(2) 0 and let be any number ^ between 0 and 1. Let X (i) achieves RI (D(i) ) for i = 1, 2, i.e., ^ RI (D(i) ) = I(X; X (i) ), where ^ Ed(X, X (i) ) D(i) , and let X be defined by the transition matrix pi (^|x). Let X x distributed with X which is defined by x p (^|x) = p1 (^|x) + p2 (^|x), x x where = 1 - . Then ^ Ed(X, X () ) = x,^ x (8.63) (8.64) ^ () be jointly ^ (i) (8.65) p(x)p (^|x)d(x, x) x ^ x p(x)(p1 (^|x) + p2 (^|x))d(x, x) x ^ x,^ x (8.66) (8.67) p(x)p2 (^|x)d(x, x) x ^ x,^ x = = x,^ x p(x)p1 (^|x)d(x, x) + x ^ (8.68) (8.69) (8.70) (8.71) ^ ^ = Ed(X, X (1) ) + Ed(X, X (2) ) D(1) + D(2) =D where D() = D(1) + D(2) , and (8.70) follows from (8.64). Now consider ^ ^ RI (D(1) ) + RI (D(2) ) = I(X; X (1) ) + I(X; X (2) ) ^ I(X; X () ) RI (D() ), () , (8.72) (8.73) (8.74) (8.75) where the inequality in (8.74) follows from the convexity of mutual information with respect to the transition matrix p(^|x) (see Example 3.13), and the x inequality in (8.75) follows from (8.71) and the definition of RI (D). Therefore, we have proved Property 2. ^ To prove Property 3, let X take the value x as defined in Definition 8.7 ^ with probability 1. Then 8.3 The Rate-Distortion Theorem 195 ^ I(X; X) = 0 and ^ Ed(X; X) = Ed(X; x ) = Dmax . ^ Then for D Dmax , ^ RI (D) I(X; X) = 0. (8.76) (8.77) (8.78) On the other hand, since RI (D) is nonnegative, we conclude that RI (D) = 0. This proves Property 3. Finally, to prove Property 4, we let ^ X = x (X), ^ where x (x) is defined in Definition 8.5. Then ^ ^ Ed(X, X) = Ed(X, x (X)) ^ = x (8.79) (8.80) (8.81) (8.82) (8.83) p(x)d(x, x (x)) ^ =0 by (8.8) since we assume that d is a normal distortion measure. Moreover, ^ RI (0) I(X; X) H(X). Then Property 4 and hence the theorem is proved. Corollary 8.19. If RI (0) > 0, then RI (D) is strictly decreasing for 0 D Dmax , and the inequality constraint in Definition 8.16 for RI (D) can be replaced by an equality constraint. Proof. Assume that RI (0) > 0. We first show that RI (D) > 0 for 0 D < Dmax by contradiction. Suppose RI (D ) = 0 for some 0 D < Dmax , and ^ let RI (D ) be achieved by some X. Then ^ RI (D ) = I(X; X) = 0 ^ implies that X and X are independent, or p(x, x) = p(x)p(^) ^ x for all x and x. It follows that ^ (8.86) (8.85) (8.84) 196 8 Rate-Distortion Theory ^ D Ed(X, X) = x x ^ (8.87) (8.88) (8.89) (8.90) (8.91) (8.92) (8.93) (8.94) p(x, x)d(x, x) ^ ^ p(x)p(^)d(x, x) x ^ x x ^ = = x ^ p(^) x x p(x)d(x, x) ^ = x ^ p(^)Ed(X, x) x ^ p(^)Ed(X, x ) x ^ x ^ = x ^ p(^)Dmax x = Dmax , where x and Dmax are defined in Definition 8.7. This leads to a contradiction ^ because we have assumed that 0 D < Dmax . Therefore, we conclude that RI (D) > 0 for 0 D < Dmax . Since RI (0) > 0 and RI (Dmax ) = 0, and RI (D) is non-increasing and convex from the above theorem, RI (D) must be strictly decreasing for 0 D Dmax . We now prove by contradiction that the inequality constraint in Definition 8.16 for RI (D) can be replaced by an equality constraint. Assume ^ that RI (D) is achieved by some X such that ^ Ed(X, X ) = D < D. Then RI (D ) = ^ ^ X:Ed(X,X)D (8.95) min ^ ^ I(X; X) I(X; X ) = RI (D). (8.96) This is a contradiction because RI (D) is strictly decreasing for 0 D Dmax . Hence, ^ Ed(X, X ) = D. (8.97) This implies that the inequality constraint in Definition 8.16 for RI (D) can be replaced by an equality constraint. Remark In all problems of interest, R(0) = RI (0) > 0. Otherwise, R(D) = 0 for all D 0 because R(D) is nonnegative and non-increasing. Example 8.20 (Binary Source). Let X be a binary random variable with Pr{X = 0} = 1 - and Pr{X = 1} = . (8.98) ^ Let X = {0, 1} be the reproduction alphabet for X, and let d be the Hamming 1 distortion measure. We first consider the case that 0 2 . Then if we make 8.3 The Rate-Distortion Theorem 197 a guess on the value of X, we should guess 0 in order to minimize the expected distortion. Therefore, x = 0 and ^ Dmax = Ed(X, 0) = Pr{X = 1} = . We will show that for 0 1 , 2 RI (D) = hb () - hb (D) if 0 D < 0 if D . (8.102) (8.99) (8.100) (8.101) ^ ^ Let X be an estimate of X taking values in X , and let Y be the Hamming ^ i.e., distortion measure between X and X, ^ Y = d(X, X). (8.103) ^ Observe that conditioning on X, X and Y determine each other. Therefore, ^ ^ H(X|X) = H(Y |X). ^ Then for D < = Dmax and any X such that ^ Ed(X, X) D, we have ^ ^ I(X; X) = H(X) - H(X|X) ^ = hb () - H(Y |X) hb () - H(Y ) ^ = hb () - hb (Pr{X = X}) hb () - hb (D), where the last inequality is justified because ^ ^ Pr{X = X} = Ed(X, X) D (8.111) (8.106) (8.107) (8.108) (8.109) (8.110) (8.105) (8.104) 1 ^ and hb (a) is increasing for 0 a 2 . Minimizing over all X satisfying (8.105) in (8.110), we obtain the lower bound RI (D) hb () - hb (D). (8.112) ^ To show that this lower bound is achievable, we need to construct an X such that the inequalities in both (8.108) and (8.110) are tight. The tightness of the inequality in (8.110) simply says that ^ Pr{X = X} = D, (8.113) 198 8 Rate-Distortion Theory 1 D 1 2D X D D 1 2D 1 1 D 1 0 1 D D X 0 1 Fig. 8.3. Achieving RI (D) for a binary source via a reverse binary symmetric channel. while the tightness of the inequality in (8.108) says that Y should be inde^ pendent of X. ^ ^ It would be more difficult to make Y independent of X if we specify X ^ by means by p(^|x). Instead, we specify the joint distribution of X and X x of a reverse binary symmetric channel (BSC) with crossover probability D ^ as the shown in Figure 8.3. Here, we regard X as the input and X as the ^ output of the BSC. Then Y is independent of the input X because the error event is independent of the input for a BSC, and (8.113) is satisfied by setting the crossover probability to D. However, we need to ensure that the marginal distribution of X so specified is equal to p(x). Toward this end, we let ^ Pr{X = 1} = , and consider ^ ^ Pr{X = 1} = Pr{X = 0}Pr{X = 1|X = 0} ^ = 1}Pr{X = 1|X = 1}, ^ +Pr{X or = (1 - )D + (1 - D), which gives = Since D < Dmax = we have 0. On the other hand, , D gives + D 1. This implies (8.120) 1 2 (8.119) -D . 1 - 2D 1 , 2 (8.117) (8.116) (8.114) (8.115) (8.118) 8.3 The Rate-Distortion Theorem RI (D) 1 199 0 0.5 D Fig. 8.4. The function RI (D) for the uniform binary source with the Hamming distortion measure. - D 1 - 2D, or 1. Therefore, ^ 0 = Pr{X = 1} 1 and ^ 0 1 - = Pr{X = 0} 1. (8.121) (8.122) (8.123) Hence, we have shown that the lower bound on RI (D) in (8.110) can be achieved, and RI (D) is as given in (8.102). 1 For 2 1, by exchanging the roles of the symbols 0 and 1 in the above argument, we obtain RI (D) as in (8.102) except that is replaced by 1 - . Combining the two cases, we have RI (D) = hb () - hb (D) if 0 D < min(, 1 - ) 0 if D min(, 1 - ). 1 2 (8.124) for 0 1. The function RI (D) for = is illustrated in Figure 8.4. Remark In the above example, we see that RI (0) = hb () = H(X). Then by the rate-distortion theorem, H(X) is the minimum rate of a rate-distortion code which achieves an arbitrarily small average Hamming distortion. It is tempting to regarding this special case of the rate-distortion theorem as a version of the source coding theorem and conclude that the rate-distortion theorem is a generalization of the source coding theorem. However, this is incorrect because the rate-distortion theorem only guarantees that the average ^ Hamming distortion between X and X is small with probability arbitrarily ^ close to 1, but the source coding theorem guarantees that X = X with probability arbitrarily close to 1, which is much stronger. It is in general not possible to obtain the rate-distortion function in closed form, and we have to resort to numerical computation. In Chapter 9, we 200 8 Rate-Distortion Theory will discuss the Blahut-Arimoto algorithm for computing the rate-distortion function. 8.4 The Converse In this section, we prove that the rate-distortion function R(D) is lower bounded by the information rate-distortion function RI (D), i.e., R(D) RI (D). Specifically, we will prove that for any achievable rate-distortion pair (R, D), R RI (D). Then by fixing D and minimizing R over all achievable pairs (R, D), we conclude that R(D) RI (D). Let (R, D) be any achievable rate-distortion pair. Then for any > 0, there exists for sufficiently large n an (n, M ) code such that 1 log M R + n and ^ Pr{d(X, X) > D + } , ^ where X = g(f (X)). Then a) (8.125) (8.126) n(R + ) log M H(f (X)) H(g(f (X))) ^ = H(X) ^ ^ = H(X) - H(X|X) ^ = I(X; X) ^ = H(X) - H(X|X) n n (8.127) (8.128) (8.129) (8.130) (8.131) (8.132) (8.133) (8.134) (8.135) (8.136) (8.137) (8.138) = k=1 b) n H(Xk ) - k=1 n ^ H(Xk |X, X1 , X2 , , Xk-1 ) ^ H(Xk |Xk ) k=1 k=1 n H(Xk ) - = = ^ [H(Xk ) - H(Xk |Xk )] k=1 n ^ I(Xk ; Xk ) ^ RI (Ed(Xk , Xk )) k=1 c) n k=1 8.4 The Converse 201 =n d) 1 n n ^ RI (Ed(Xk , Xk )) k=1 (8.139) nRI 1 n n ^ Ed(Xk , Xk ) k=1 (8.140) (8.141) ^ = nRI (Ed(X, X)). In the above, a) follows from (8.125); b) follows because conditioning does not increase entropy; c) follows from the definition of RI (D) in Definition 8.16; d) follows from the convexity of RI (D) proved in Theorem 8.18 and Jensen's inequality. Now let dmax = max d(x, x) ^ x,^ x (8.142) be the maximum value which can be taken by the distortion measure d. The reader should not confuse dmax with Dmax in Definition 8.7. Then from (8.126), we have ^ Ed(X, X) ^ ^ ^ = E[d(X, X)|d(X, X) > D + ]Pr{d(X, X) > D + } ^ ^ ^ +E[d(X, X)|d(X, X) D + ]Pr{d(X, X) D + } dmax + (D + ) 1 = D + (dmax + 1) . (8.143) (8.144) (8.145) This shows that if the probability that the average distortion between X and ^ X exceeds D + is small, then the expected average distortion between X and ^ X can exceed D only by a small amount3 . Following (8.141), we have ^ R + RI (Ed(X, X)) RI (D + (dmax + 1) ), (8.146) (8.147) where the last inequality follows from (8.145) because RI (D) is non-increasing in D. We note that the convexity of RI (D) implies that it is a continuous function of D. Then taking the limit as 0, we obtain R lim RI (D + (dmax + 1) ) 0 (8.148) (8.149) (8.150) = RI D + (dmax + 1) lim = RI (D), 3 0 The converse is not true. 202 8 Rate-Distortion Theory where we have invoked the continuity of RI (D) in obtaining (8.149). Upon minimizing R over all achievable pairs (R, D) for a fixed D in (8.150), we have proved that R(D) RI (D). (8.151) This completes the proof for the converse of the rate-distortion theorem. 8.5 Achievability of RI (D) In this section, we prove that the rate-distortion function R(D) is upper bounded by the information rate-distortion function RI (D), i.e., R(D) RI (D). Then by combining with the result that R(D) RI (D) from the last section, we conclude that R(D) = RI (D), and the rate-distortion theorem is proved. ^ For any 0 D Dmax , we will prove that for every random variable X ^ such that taking values in X ^ Ed(X, X) D, (8.152) ^ the rate-distortion pair (I(X; X), D) is achievable. This will be proved by showing for sufficiently large n the existence of a rate-distortion code such that ^ 1. the rate of the code is not more than I(X; X) + ; ^ 2. d(X, X) D + with probability almost 1. ^ ^ Then by minimizing I(X; X) over all X satisfying (8.152), we conclude that the rate-distortion pair (RI (D), D) is achievable, which implies RI (D) R(D) because R(D) is the minimum of all R such that (R, D) is achievable. Fix any 0 D Dmax and any > 0, and let be a small positive quantity to be specified later. Toward proving the existence of a desired code, ^ we fix a random variable X which satisfies (8.152) and let M be an integer satisfying 1 ^ ^ (8.153) I(X; X) + log M I(X; X) + , 2 n where n is sufficiently large. We now describe a random coding scheme in the following steps: 1. Construct a codebook C of an (n, M ) code by randomly generating M ^ codewords in X n independently and identically according to p(^)n . Denote x ^ ^ ^ these codewords by X(1), X(2), , X(M ). 2. Reveal the codebook C to both the encoder and the decoder. 3. The source sequence X is generated according to p(x)n . 4. The encoder encodes the source sequence X into an index K in the set I = {1, 2, , M }. The index K takes the value i if n ^ a) (X, X(i)) T[X X] , ^ 8.5 Achievability of RI (D) n ^ b) for all i I, if (X, X(i )) T[X X] , then i i; ^ otherwise, K takes the constant value 1. 5. The index K is delivered to the decoder. ^ ^ 6. The decoder outputs X(K) as the reproduction sequence X. 203 Remark Strong typicality is used in defining the encoding function in Step 4. This is made possible by the assumption that both the source alphabet X ^ and the reproduction alphabet X are finite. Let us further explain the encoding scheme described in Step 4. After the source sequence X is generated, we search through all the codewords in the codebook C for those which are jointly typical with X with respect to p(x, x). ^ If there is at least one such codeword, we let i be the largest index of such codewords and let K = i. If such a codeword does not exist, we let K = 1. The event {K = 1} occurs in one of the following two scenarios: ^ 1. X(1) is the only codeword in C which is jointly typical with X. 2. No codeword in C is jointly typical with X. ^ ^ In either scenario, X is not jointly typical with the codewords X(2), X(3), , ^ X(M ). In other words, if K = 1, then X is jointly typical with none of the ^ ^ ^ codewords X(2), X(3), , X(M ). Define n ^ Ei = (X, X(i)) T[X X] (8.154) ^ ^ to be the event that X is jointly typical with the codeword X(i). We see from the above discussion that c c c {K = 1} E2 E3 EM . (8.155) Since the codewords are generated i.i.d., conditioning on {X = x} for any x X n , the events Ei are mutually independent4 , and they all have the same probability. Then for any x X n , c c c Pr{K = 1|X = x} Pr{E2 E3 EM |X = x} M (8.156) (8.157) (8.158) (8.159) = i=2 c Pr{Ei |X = x} c = (Pr{E1 |X = x})M -1 = (1 - Pr{E1 |X = x})M -1 . n We now obtain a lower bound on Pr{E1 |X = x} for x S[X] , where n n S[X] = {x T[X] : |T[n^ (x)| 1} X|X] 4 (8.160) Without conditioning on {X = x}, the events Ei are not mutually independent because they depend on each other through X. 204 8 Rate-Distortion Theory (cf. Section 6.3). Consider n ^ Pr{E1 |X = x} = Pr (x, X(1)) T[X X] ^ (8.161) (8.162) = ^ x:(x,^ )T n x ^ [X X] p(^ ). x n ^ ^ The summation above is over all x such that (x, x) T[X X] . From the con^ n ^ ^ sistency of strong typicality (Theorem 6.7), if (x, x) T[X X] , then x T[n^ . ^ X] By the strong AEP (Theorem 6.2), all p(^ ) in the above summation satisfy x p(^ ) 2-n(H(X)+) , x ^ (8.163) where 0 as 0. By the conditional strong AEP (Theorem 6.10), |T[n^ (x)| 2n(H(X|X)-) , X|X] where 0 as 0. Then from (8.162), we have Pr{E1 |X = x} 2n(H(X|X)-) 2-n(H(X)+) =2 =2 where =+ 0 as 0. Following (8.159), we have Pr{K = 1|X = x} 1 - 2-n(I(X;X)+) The lower bound in (8.153) implies M 2n(I(X;X)+ 2 ) . Then upon taking natural logarithm in (8.169), we obtain ln Pr{K = 1|X = x} (M - 1) ln 1 - 2-n(I(X;X)+) 2n(I(X;X)+ 2 ) - 1 ln 1 - 2-n(I(X;X)+) - 2n(I(X;X)+ 2 ) - 1 2-n(I(X;X)+) = - 2n( 2 -) - 2-n(I(X;X)+) . ^ b) ^ ^ a) ^ ^ ^ ^ ^ M -1 ^ ^ -n(H(X)-H(X|X)++) ^ -n(I(X;X)+) ^ ^ ^ (8.164) (8.165) (8.166) (8.167) (8.168) , . (8.169) (8.170) (8.171) (8.172) (8.173) (8.174) 8.5 Achievability of RI (D) 205 In the above, a) follows from (8.170) by noting that the logarithm in (8.171) is negative, and b) follows from the fundamental inequality ln a a - 1. By letting be sufficiently small so that 2 - > 0, (8.175) the above upper bound on ln Pr{K = 1|X = x} tends to - as n , i.e., Pr{K = 1|X = x} 0 as n . This implies Pr{K = 1|X = x} for sufficiently large n. It then follows that Pr{K = 1} = n xS[X] 2 (8.176) (8.177) Pr{K = 1|X = x}Pr{X = x} + n xS[X] Pr{K = 1|X = x}Pr{X = x} Pr{X = x} + n xS[X] (8.178) (8.179) (8.180) (8.181) (8.182) n xS[X] 2 1 Pr{X = x} = < 2 2 2 n n Pr{X S[X] } + Pr{X S[X] } n 1 + (1 - Pr{X S[X] }) + , where we have invoked Proposition 6.13 in the last step. By letting be sufficiently small so that < (8.183) 2 and (8.175) is satisfied, we obtain Pr{K = 1} < . (8.184) The main idea of the above upper bound on Pr{K = 1} for sufficiently large n is the following. In constructing the codebook, we randomly generate ^ M codewords in X n according to p(^)n . If M grows with n at a rate higher x ^ then the probability that there exists at least one codeword than I(X; X), which is jointly typical with the source sequence X with respect to p(x, x) ^ is very high when n is large. Further, the average distortion between X and ^ such a codeword is close to Ed(X, X) because the empirical joint distribution of the symbol pairs in X and such a codeword is close to p(x, x). Then by let^ ^ ting the reproduction sequence X be such a codeword, the average distortion 206 8 Rate-Distortion Theory ^ between X and X is less than D + with probability arbitrarily close to 1 ^ D. These will be formally shown in the rest of the proof. since Ed(X, X) Now for sufficiently large n, consider ^ Pr{d(X, X) > D + } ^ = Pr{d(X, X) > D + |K = 1}Pr{K = 1} ^ +Pr{d(X, X) > D + |K = 1}Pr{K = 1} ^ 1 + Pr{d(X, X) > D + |K = 1} 1 ^ = + Pr{d(X, X) > D + |K = 1}. (8.185) (8.186) (8.187) We will show that by choosing the value of carefully, it is possible to make ^ ^ d(X, X) always less than or equal to D + provided K = 1. Since (X, X) n T[X X] conditioning on {K = 1}, we have ^ ^ d(X, X) = = = 1 n 1 n 1 n n ^ d(Xk , Xk ) k=1 (8.188) (8.189) (8.190) ^ d(x, x)N (x, x|X, X) ^ ^ x,^ x ^ d(x, x)(np(x, x) + N (x, x|X, X) - np(x, x)) ^ ^ ^ ^ x,^ x x,^ x d(x, x) ^ x,^ x = p(x, x)d(x, x) + ^ ^ 1 ^ N (x, x|X, X) - p(x, x) ^ ^ n (8.191) ^ = Ed(X, X) + x,^ x d(x, x) ^ d(x, x) ^ x,^ x 1 ^ N (x, x|X, X) - p(x, x) ^ ^ n 1 ^ N (x, x|X, X) - p(x, x) ^ ^ n (8.192) (8.193) (8.194) (8.195) (8.196) ^ Ed(X, X) + a) ^ Ed(X, X) + dmax x,^ x 1 ^ N (x, x|X, X) - p(x, x) ^ ^ n ^ Ed(X, X) + dmax c) b) D + dmax , where a) follows from the definition of dmax in (8.142); n ^ b) follows because (X, X) T[X X] ; ^ Chapter Summary 207 c) follows from (8.152). By taking we obtain ^ d(X, X) D + dmax if K = 1. Therefore, ^ Pr{d(X, X) > D + |K = 1} = 0, and it follows that from (8.187) that ^ Pr{d(X, X) > D + } . (8.200) (8.199) dmax =D+ (8.198) dmax , (8.197) Thus we have shown that for sufficiently large n, there exists an (n, M ) random code which satisfies 1 ^ log M I(X; X) + n (8.201) (this follows from the upper bound in (8.153)) and (8.200). This implies the existence of an (n, M ) rate-distortion code which satisfies (8.201) and (8.200). ^ Therefore, the rate-distortion pair (I(X; X), D) is achievable. Then upon min^ which satisfy (8.152), we conclude that the rate-distortion imizing over all X pair (RI (D), D) is achievable, which implies RI (D) R(D). The proof is completed. Chapter Summary Rate-Distortion Function: For an information source X and a single-letter ^ distortion measure d : X X , the rate-distortion function is defined as R(D) = ^ ^ X:Ed(X,X)D min ^ I(X; X). Rate-Distortion Theorem: An i.i.d. random sequence X1 , X2 , , Xn with generic random variable X can be compressed at rate R + such that ^ Pr{d(X, X) > D + } 0 as n if and only if R R(D). Binary Source: Let X be binary with distribution {, 1 - } and let d be the Hamming distortion measure. Then R(D) = hb () - hb (D) if 0 D < min(, 1 - ) 0 if D min(, 1 - ). 208 8 Rate-Distortion Theory Problems 1. Obtain the forward channel description of R(D) for the binary source with the Hamming distortion measure. 2. Binary covering radius The Hamming ball with center c = (c1 , c2 , , cn ) {0, 1}n and radius r is the set n Sr (c) = x {0, 1}n : i=1 |xi - ci | r . Let Mr,n be the minimum number M such that there exists Hamming balls Sr (cj ), j = 1, 2, , M such that for all x {0, 1}n , x Sr (cj ) for some j. a) Show that 2n Mr,n r n . k=0 k b) What is the relation between Mr,n and the rate-distortion function for the binary source with the Hamming distortion measure? 3. Consider a source random variable X with the Hamming distortion measure. a) Prove that R(D) H(X) - D log(|X | - 1) - hb (D) for 0 D Dmax . b) Show that the above lower bound on R(D) is tight if X is distributed uniformly on X . See Jerohin [190] (also see [84], p.133) for the tightness of this lower bound for a general source. This bound is a special case of the Shannon lower bound for the rate-distortion function [327] (also see [80], p.369). 4. Product source Let X and Y be two independent source random variables ^ ^ with reproduction alphabets X and Y and distortion measures dx and dy , and the rate-distortion functions for X and Y are denoted by Rx (Dx ) and Ry (Dy ), respectively. Now for the product source (X, Y ), define a ^ ^ distortion measure d : X Y X Y by d((x, y), (^, y )) = dx (x, x) + dy (y, y ). x ^ ^ ^ Prove that the rate-distortion function R(D) for (X, Y ) with distortion measure d is given by R(D) = Dx +Dy =D min (Rx (Dx ) + Ry (Dy )). ^ ^ ^ ^ Hint: Prove that I(X, Y ; X, Y ) I(X; X) + I(Y ; Y ) if X and Y are independent. (Shannon [327].) Historical Notes 209 5. Compound source Let be an index set and Z = {X : } be a collection of source random variables. The random variables in Z have ^ a common source alphabet X , a common reproduction alphabet X , and a common distortion measure d. A compound source is an i.i.d. information source whose generic random variable is X , where is equal to some but we do not know which one it is. The rate-distortion function R (D) for X has the same definition as the rate-distortion function defined in this chapter except that (8.23) is replaced by ^ Pr{d(X , X) > D + } Show that R (D) = sup R (D), for all . where R (D) is the rate-distortion function for X . 6. Show that asymptotic optimality can always be achieved by separating rate-distortion coding and channel coding when the information source is i.i.d. (with a single-letter distortion measure) and the channel is memoryless. 7. Slepian-Wolf coding Let , , and be small positive quantities. For 1 i 2n(H(Y |X)+ ) , randomly and independently select with replacement n 2n(I(X;Y )-) sequences from T[Y ] according to the uniform distribution n to form a bin Bi . Let (x, y) be a fixed pair of sequences in T[XY ] . Prove the following by choosing , , and appropriately: a) the probability that y is in some Bi tends to 1 as n ; b) given that y Bi , the probability that there exists another y Bi n such that (x, y ) T[XY ] tends to 0 as n . Let (X, Y) pn (x, y). The results in a) and b) say that if (X, Y) is jointly typical, which happens with probability close to 1 for large n, then it is very likely that Y is in some bin Bi , and that Y is the unique vector in Bi which is jointly typical with X. If X is available as side-information, then by specifying the index of the bin containing Y, which takes about 2nH(Y |X) bits, Y can be uniquely specified. Note that no knowledge about X is involved in specifying the index of the bin containing Y. This is the basis of the Slepian-Wolf coding [339] which launched the whole area of multiterminal source coding (see Berger [28]). Historical Notes Transmission of an information source with distortion was first conceived by Shannon in his 1948 paper [322]. He returned to the problem in 1959 and proved the rate-distortion theorem [327]. The normalization of the ratedistortion function is due to Pinkston [290]. The rate-distortion theorem proved here is a stronger version of the original theorem. Extensions of the 210 8 Rate-Distortion Theory theorem to more general sources were proved in the book by Berger [27]. An iterative algorithm for computing the rate-distortion function developed by Blahut [37] will be discussed in Chapter 9. Rose [312] has developed an algorithm for the same purpose based on a mapping approach. 9 The Blahut-Arimoto Algorithms For a discrete memoryless channel p(y|x), the capacity C = max I(X; Y ), r(x) (9.1) where X and Y are respectively the input and the output of the generic channel and r(x) is the input distribution, characterizes the maximum asymptotically achievable rate at which information can be transmitted through the channel reliably. The expression for C in (9.1) is called a single-letter characterization in the sense that it depends only on the transition matrix of the generic channel but not on the block length n of a code for the channel. When both the input alphabet X and the output alphabet Y are finite, the computation of C becomes a finite-dimensional maximization problem. For an i.i.d. information source {Xk , k 1} with generic random variable X, the rate-distortion function R(D) = ^ Q(^|x):Ed(X,X)D x min ^ I(X; X) (9.2) characterizes the minimum asymptotically achievable rate of a rate-distortion code which reproduces the information source with an average distortion no more than D with respect to a single-letter distortion measure d. Again, the expression for R(D) in (9.2) is a single-letter characterization because it depends only on the generic random variable X but not on the block length n of a rate-distortion code. When both the source alphabet X and the reproduction ^ alphabet X are finite, the computation of R(D) becomes a finite-dimensional minimization problem. Unless for very special cases, it is not possible to obtain an expression for C or R(D) in closed form, and we have to resort to numerical computation. However, computing these quantities is not straightforward because the associated optimization problem is nonlinear. In this chapter, we discuss the Blahut-Arimoto algorithms (henceforth the BA algorithms), which is an iterative algorithm devised for this purpose. 212 9 The Blahut-Arimoto Algorithms In order to better understand how and why the BA algorithm works, we will first describe the algorithm in a general setting in the next section. Specializations of the algorithm for the computation of C and R(D) will be discussed in Section 9.2, and convergence of the algorithm will be proved in Section 9.3. 9.1 Alternating Optimization In this section, we describe an alternating optimization algorithm. This algorithm will be specialized in the next section for computing the channel capacity and the rate-distortion function. Consider the double supremum sup sup f (u1 , u2 ), (9.3) u1 A1 u2 A2 where Ai is a convex subset of ni for i = 1, 2, and f is a real function defined on A1 A2 . The function f is bounded from above, and is continuous and has continuous partial derivatives on A1 A2 . Further assume that for all u2 A2 , there exists a unique c1 (u2 ) A1 such that f (c1 (u2 ), u2 ) = max f (u1 , u2 ), u1 A1 (9.4) and for all u1 A1 , there exists a unique c2 (u1 ) A2 such that f (u1 , c2 (u1 )) = max f (u1 , u2 ). u2 A2 (9.5) Let u = (u1 , u2 ) and A = A1 A2 . Then (9.3) can be written as sup f (u). uA (9.6) In other words, the supremum of f is taken over a subset of n1 +n2 which is equal to the Cartesian product of two convex subsets of n1 and n2 , respectively. We now describe an alternating optimization algorithm for computing f , (k) (k) the value of the double supremum in (9.3). Let u(k) = (u1 , u2 ) for k 0 (0) which are defined as follows. Let u1 be an arbitrarily chosen vector in A1 , (0) (0) and let u2 = c2 (u1 ). For k 1, u(k) is defined by u1 = c1 (u2 and u2 = c2 (u1 ). (k) (k) (k) (k-1) ) (9.7) (9.8) 9.1 Alternating Optimization 213 Fig. 9.1. Alternating optimization. (k) (k) (0) (0) (1) (1) In other words, u1 and u2 are generated in the order u1 , u2 , u1 , u2 , (2) (2) u1 , u2 , , where each vector in the sequence is a function of the previous (0) vector except that u1 is arbitrarily chosen in A1 . Let f (k) = f (u(k) ). Then from (9.4) and (9.5), f (k) = f (u1 , u2 ) (k) (k-1) f (u1 , u2 ) (k-1) (k-1) f (u1 , u2 ) (k-1) (k) (k) (9.9) (9.10) (9.11) (9.12) (9.13) =f for k 1. Since the sequence f (k) is non-decreasing, it must converge because f is bounded from above. We will show in Section 9.3 that f (k) f if f is concave. Figure 9.1 is an illustration of the alternating maximization algorithm, where in this case both n1 and n2 are equal to 1, and f (k) f . The alternating optimization algorithm can be explained by the following analogy. Suppose a hiker wants to reach the summit of a mountain. Starting from a certain point in the mountain, the hiker moves north-south and eastwest alternately. (In our problem, the north-south and east-west directions can be multi-dimensional.) In each move, the hiker moves to the highest possible point. The question is whether the hiker can eventually approach the summit starting from any point in the mountain. Replacing f by -f in (9.3), the double supremum becomes the double infimum inf inf f (u1 , u2 ). (9.14) u1 A1 u2 A2 214 9 The Blahut-Arimoto Algorithms All the previous assumptions on A1 , A2 , and f remain valid except that f is now assumed to be bounded from below instead of bounded from above. The double infimum in (9.14) can be computed by the same alternating optimization algorithm. Note that with f replaced by -f , the maximums in (9.4) and (9.5) become minimums, and the inequalities in (9.11) and (9.12) are reversed. 9.2 The Algorithms In this section, we specialize the alternating optimization algorithm described in the last section to compute the channel capacity and the rate-distortion function. The corresponding algorithms are known as the BA algorithms. 9.2.1 Channel Capacity We will use r to denote an input distribution r(x), and we write r > 0 if r is strictly positive, i.e., r(x) > 0 for all x X . If r is not strictly positive, we write r 0. Similar notations will be introduced as appropriate. Lemma 9.1. Let r(x)p(y|x) be a given joint distribution on X Y such that r > 0, and let q be a transition matrix from Y to X . Then max q x y r(x)p(y|x) log q(x|y) = r(x) r(x)p(y|x) log x y q (x|y) , r(x) (9.15) where the maximization is taken over all q such that q(x|y) = 0 and q (x|y) = if and only if p(y|x) = 0, (9.16) r(x)p(y|x) , x r(x )p(y|x ) (9.17) i.e., the maximizing q is the one which corresponds to the input distribution r and the transition matrix p(y|x). In (9.15) and the sequel, we adopt the convention that the summation is taken over all x and y such that r(x) > 0 and p(y|x) > 0. Note that the right hand side of (9.15) gives the mutual information I(X; Y ) when r is the input distribution for the generic channel p(y|x). Proof. Let w(y) = x r(x )p(y|x ) (9.18) in (9.17). We assume without loss of generality that for all y Y, p(y|x) > 0 for some x X . Since r > 0, w(y) > 0 for all y, and hence q (x|y) is welldefined. Rearranging (9.17), we have 9.2 The Algorithms 215 r(x)p(y|x) = w(y)q (x|y). Consider r(x)p(y|x) log x y (9.19) q (x|y) - r(x) r(x)p(y|x) log x y q(x|y) r(x) (9.20) (9.21) (9.22) (9.23) (9.24) = x y q (x|y) r(x)p(y|x) log q(x|y) w(y)q (x|y) log q (x|y) q(x|y) q (x|y) q(x|y) = y x = y w(y) x q (x|y) log = y w(y)D(q (x|y) q(x|y)) 0, where (9.21) follows from (9.19), and the last step is an application of the divergence inequality. Then the proof is completed by noting in (9.17) that q satisfies (9.16) because r > 0. Theorem 9.2. For a discrete memoryless channel p(y|x), C = sup max r>0 q x y r(x)p(y|x) log q(x|y) , r(x) (9.25) where the maximization is taken over all q that satisfies (9.16). Proof. Let I(r, p) denote the mutual information I(X; Y ) when r is the input distribution for the generic channel p(y|x). Then we can write C = max I(r, p). r0 (9.26) Let r achieves C. If r > 0, then C = max I(r, p) r0 (9.27) (9.28) r(x)p(y|x) log q(x|y) r(x) (9.29) (9.30) = max I(r, p) r>0 = max max r>0 q x y = sup max r>0 q x y r(x)p(y|x) log q(x|y) , r(x) where (9.29) follows from Lemma 9.1 (and the maximization is over all q that satisfies (9.16)). 216 9 The Blahut-Arimoto Algorithms Next, we consider the case when r 0. Since I(r, p) is continuous in r, for any > 0, there exists > 0 such that if r - r < , then C - I(r, p) < , (9.31) (9.32) where r-r denotes the Euclidean distance between r and r . In particular, there exists ~ > 0 which satisfies (9.31) and (9.32). Then r C = max I(r, p) r0 (9.33) (9.34) (9.35) (9.36) sup I(r, p) r>0 I(~, p) r >C- , where the last step follows because ~ satisfies (9.32). Thus we have r C - < sup I(r, p) C. r>0 (9.37) Finally, by letting 0, we conclude that r(x)p(y|x) log x y C = sup I(r, p) = sup max r>0 r>0 q q(x|y) . r(x) (9.38) This accomplishes the proof. Now for the double supremum in (9.3), let f (r, q) = x y r(x)p(y|x) log q(x|y) , r(x) (9.39) with r and q playing the roles of u1 and u2 , respectively. Let A1 = {(r(x), x X ) : r(x) > 0 and and A2 = {(q(x|y), (x, y) X Y) : q(x|y) > 0 if p(x|y) > 0, q(x|y) = 0 if p(y|x) = 0, and x x r(x) = 1} , (9.40) q(x|y) = 1 for all y Y}. (9.41) Then A1 is a subset of |X | and A2 is a subset of |X ||Y| , and it can readily be checked that both A1 and A2 are convex. For all r A1 and q A2 , by Lemma 9.1, 9.2 The Algorithms 217 f (r, q) = x y r(x)p(y|x) log r(x)p(y|x) log x y q(x|y) r(x) q (x|y) r(x) (9.42) (9.43) (9.44) (9.45) (9.46) = I(X; Y ) H(X) log |X |. Thus f is bounded from above. Since for all q A2 , q(x|y) = 0 for all x and y such that p(x|y) = 0, these components of q are degenerated. In fact, these components of q do not appear in the definition of f (r, q) in (9.39), which can be seen as follows. Recall the convention that the double summation in (9.39) is over all x and y such that r(x) > 0 and p(y|x) > 0. If q(x|y) = 0, then p(y|x) = 0, and hence the corresponding term is not included in the double summation. Therefore, it is readily seen that f is continuous and has continuous partial derivatives on A because all the probabilities involved in the double summation in (9.39) are strictly positive. Moreover, for any given r A1 , by Lemma 9.1, there exists a unique q A2 that maximizes f . It will be shown shortly that for any given q A2 , there also exists a unique r A1 that maximizes f . The double supremum in (9.3) now becomes sup sup rA1 qA2 x y r(x)p(y|x) log q(x|y) , r(x) (9.47) which by Theorem 9.2 is equal to C, where the supremum over all q A2 is in fact a maximum. We then apply the alternating optimization algorithm in the last section to compute C. First, we arbitrarily choose a strictly positive input distribution in A1 and let it be r(0) . Then we define q(0) and in general q(k) for k 0 by r(k) (x)p(y|x) q (k) (x|y) = (9.48) (k) (x )p(y|x ) x r in view of Lemma 9.1. In order to define r(1) and in general r(k) for k 1, we need to find the r A1 that maximizes f for a given q A2 , where the constraints on r are r(x) = 1 (9.49) x and r(x) > 0 for all x X . (9.50) We now use the method of Lagrange multipliers to find the best r by ignoring temporarily the positivity constraints in (9.50). Let 218 9 The Blahut-Arimoto Algorithms J= x y r(x)p(y|x) log q(x|y) - r(x) r(x). x (9.51) For convenience sake, we assume that the logarithm is the natural logarithm. Differentiating with respect to r(x) gives J = r(x) Upon setting J r(x) p(y|x) log q(x|y) - log r(x) - 1 - . y (9.52) = 0, we have log r(x) = y p(y|x) log q(x|y) - 1 - , (9.53) or r(x) = e-(+1) y q(x|y)p(y|x) . (9.54) By considering the normalization constraint in (9.49), we can eliminate and obtain p(y|x) y q(x|y) r(x) = . (9.55) p(y|x ) x y q(x |y) The above product is over all y such that p(y|x) > 0, and q(x|y) > 0 for all such y. This implies that both the numerator and the denominator on the right hand side above are positive, and therefore r(x) > 0. In other words, the r thus obtained happen to satisfy the positivity constraints in (9.50) although these constraints were ignored when we set up the Lagrange multipliers. We will show in Section 9.3.2 that f is concave. Then r as given in (9.55), which is unique, indeed achieves the maximum of f for a given q A2 because r is in the interior of A1 . In view of (9.55), we define r(k) for k 1 by r(k) (x) = x y q (k-1) (x|y)p(y|x) y q (k-1) (x |y)p(y|x ) . (9.56) The vectors r(k) and q(k) are defined in the order r(0) , q(0) , r(1) , q(1) , r(2) , q , , where each vector in the sequence is a function of the previous vector except that r(0) is arbitrarily chosen in A1 . It remains to show by induction that r(k) A1 for k 1 and q(k) A2 for k 0. If r(k) A1 , i.e., r(k) > 0, then we see from (9.48) that q (k) (x|y) = 0 if and only if p(x|y) = 0, i.e., q(k) A2 . On the other hand, if q(k) A2 , then we see from (9.56) that r(k+1) > 0, i.e., r(k+1) A2 . Therefore, r(k) A1 and q(k) A2 for all k 0. Upon determining (r(k) , q(k) ), we can compute f (k) = f (r(k) , q(k) ) for all k. It will be shown in Section 9.3 that f (k) C. (2) 9.2 The Algorithms 219 9.2.2 The Rate-Distortion Function The discussion in this section is analogous to the discussion in Section 9.2.1. Some of the details will be omitted for brevity. For all problems of interest, R(0) > 0. Otherwise, R(D) = 0 for all D 0 since R(D) is nonnegative and non-increasing. Therefore, we assume without loss of generality that R(0) > 0. We have shown in Corollary 8.19 that if R(0) > 0, then R(D) is strictly decreasing for 0 D Dmax . Since R(D) is convex, for any s 0, there exists a point on the R(D) curve for 0 D Dmax such that the slope of a tangent1 to the R(D) curve at that point is equal to s. Denote such a point on the R(D) curve by (Ds , R(Ds )), which is not necessarily unique. Then this tangent intersects with the ordinate at R(Ds ) - sDs . This is illustrated in Figure 9.2. ^ Let I(p, Q) denote the mutual information I(X, X) and D(p, Q) denote ^ the expected distortion Ed(X, X) when p is the distribution for X and Q is the ^ ^ transition matrix from X to X defining X. Then for any Q, (I(p, Q), D(p, Q)) is a point in the rate-distortion region, and the line with slope s passing through (I(p, Q), D(p, Q)) intersects the ordinate at I(p, Q) - sD(p, Q). Since the R(D) curve defines the boundary of the rate-distortion region and it is above the tangent in Figure 9.2, we see that R(Ds ) - sDs = min[I(p, Q) - sD(p, Q)]. Q (9.57) For each s 0, if we can find a Qs that achieves the above minimum, then the line passing through (0, I(p, Qs ) - sD(p, Qs )), i.e., the tangent in Figure 9.2, R(D) R ( D s) - s D s R ( Ds ) (D s ,R ( D s )) D Ds D max Fig. 9.2. A tangent to the R(D) curve with slope equal to s. 1 We say that a line is a tangent to the R(D) curve if it touches the R(D) curve from below. 220 9 The Blahut-Arimoto Algorithms gives a tight lower bound on the R(D) curve. In particular, if (R(Ds ), Ds ) is unique, Ds = D(p, Qs ) (9.58) and R(Ds ) = I(p, Qs ). (9.59) By varying over all s 0, we can then trace out the whole R(D) curve. In the rest of the section, we will devise an iterative algorithm for the minimization problem in (9.57). ^ Lemma 9.3. Let p(x)Q(^|x) be a given joint distribution on X X such that x ^ Q > 0, and let t be any distribution on X such that t > 0. Then min t>0 x x ^ p(x)Q(^|x) log x Q(^|x) x = t(^) x p(x)Q(^|x) log x x x ^ Q(^|x) x , (9.60) (^) t x where t (^) = x x p(x)Q(^|x), x (9.61) i.e., the minimizing t is the one which corresponds to the input distribution p and the transition matrix Q. Proof. It suffices to prove that p(x)Q(^|x) log x x x ^ Q(^|x) x t(^) x p(x)Q(^|x) log x x x ^ Q(^|x) x (^) t x (9.62) for all t > 0. The details are left as an exercise. Note in (9.61) that t > 0 because Q > 0. Since I(p, Q) and D(p, Q) are continuous in Q, via an argument similar to the one we used in the proof of Theorem 9.2, we can replace the minimum over all Q in (9.57) by the infimum over all Q > 0. By noting that the right hand side of (9.60) is equal to I(p, Q) and D(p, Q) = x x ^ p(x)Q(^|x)d(x, x), x ^ (9.63) we can apply Lemma 9.3 to obtain R(Ds ) - sDs = = inf Q>0 min t>0 x,^ x p(x)Q(^|x) log x Q(^|x) x -s t(^) x x,^ x Q(^|x) x -s t(^) x x,^ x p(x)Q(^|x)d(x,^) x x (9.64) . (9.65) inf min Q>0 t>0 x,^ x p(x)Q(^|x) log x p(x)Q(^|x)d(x,^) x x Now in the double infimum in (9.14), let 9.2 The Algorithms 221 f (Q, t) = x x ^ p(x)Q(^|x) log x -s x x ^ Q(^|x) x t(^) x (9.66) p(x)Q(^|x)d(x, x), x ^ A1 = ^ (Q(^|x), (x, x) X X ) : Q(^|x) > 0, x ^ x Q(^|x) = 1 for all x X , x x ^ (9.67) and ^ A2 = {(t(^), x X ) : t(^) > 0 and x ^ x x ^ t(^) = 1}, x (9.68) with Q and t playing the roles of u1 and u2 , respectively. Then A1 is a subset ^ ^ of |X ||X | and A2 is a subset of |X | , and it can readily be checked that both A1 and A2 are convex. Since s 0, f (Q, t) = x x ^ p(x)Q(^|x) log x Q(^|x) x -s t(^) x p(x)Q(^|x)d(x, x) x ^ x x ^ (9.69) x x ^ Q(^|x) x p(x)Q(^|x) log x +0 t (^) x (9.70) (9.71) (9.72) ^ = I(X; X) 0. Therefore, f is bounded from below. The double infimum in (9.14) now becomes inf inf p(x)Q(^|x) log x x x ^ QA1 tA2 Q(^|x) x -s t(^) x p(x)Q(^|x)d(x, x) , x ^ x x ^ (9.73) where the infimum over all t A2 is in fact a minimum. We then apply the alternating optimization algorithm described in Section 9.2 to compute f , the value of (9.73). First, we arbitrarily choose a strictly positive transition matrix in A1 and let it be Q(0) . Then we define t(0) and in general t(k) for k 1 by t(k) (^) = x p(x)Q(k) (^|x) x (9.74) x in view of Lemma 9.3. In order to define Q(1) and in general Q(k) for k 1, we need to find the Q A1 that minimizes f for a given t A2 , where the constraints on Q are 222 9 The Blahut-Arimoto Algorithms Q(^|x) > 0 x and ^ for all (x, x) X X , ^ for all x X . (9.75) (9.76) Q(^|x) = 1 x x ^ As we did for the computation of the channel capacity, we first ignore the positivity constraints in (9.75) when setting up the Lagrange multipliers. Then we obtain x t(^)esd(x,^) x Q(^|x) = x > 0. (9.77) x x sd(x,^ ) x t(^ )e ^ The details are left as an exercise. We then define Q(k) for k 1 by Q(k) (^|x) = x x t(k-1) (^)esd(x,^) x . (k-1) (^ )esd(x,^ ) x x x t ^ (9.78) It will be shown in the next section that f (k) = f (Q(k) , t(k) ) f as k . If there exists a unique point (R(Ds ), Ds ) on the R(D) curve such that the slope of a tangent at that point is equal to s, then (I(p, Q(k) ), D(p, Q(k) )) (R(Ds ), Ds ). (9.79) Otherwise, (I(p, Q(k) ), D(p, Q(k) )) is arbitrarily close to the segment of the R(D) curve at which the slope is equal to s when k is sufficiently large. These facts are easily shown to be true. 9.3 Convergence In this section, we first prove that if f is concave, then f (k) f . We then apply this sufficient condition to prove the convergence of the BA algorithm for computing the channel capacity. The convergence of the BA algorithm for computing the rate-distortion function can be proved likewise. The details are omitted. 9.3.1 A Sufficient Condition In the alternating optimization algorithm in Section 9.1, we see from (9.7) and (9.8) that u(k+1) = (u1 for k 0. Define f (u) = f (c1 (u2 ), c2 (c1 (u2 ))) - f (u1 , u2 ). Then (9.81) (k+1) , u2 (k+1) ) = (c1 (u2 ), c2 (c1 (u2 ))) (k) (k) (9.80) 9.3 Convergence 223 f (k+1) - f (k) = f (u(k+1) ) - f (u(k) ) = = f (u (k) (k) f (c1 (u2 ), c2 (c1 (u2 ))) (k) (9.82) - (k) (k) f (u1 , u2 ) (9.83) (9.84) ). We will prove that f being concave is sufficient for f (k) f . To this end, we first prove that if f is concave, then the algorithm cannot be trapped at u if f (u) < f . Lemma 9.4. Let f be concave. If f (k) < f , then f (k+1) > f (k) . Proof. We will prove that f (u) > 0 for any u A such that f (u) < f . Then if f (k) = f (u(k) ) < f , we see from (9.84) that f (k+1) - f (k) = f (u(k) ) > 0, (9.85) and the lemma is proved. Consider any u A such that f (u) < f . We will prove by contradiction that f (u) > 0. Assume f (u) = 0. Then it follows from (9.81) that f (c1 (u2 ), c2 (c1 (u2 ))) = f (u1 , u2 ). Now we see from (9.5) that f (c1 (u2 ), c2 (c1 (u2 ))) f (c1 (u2 ), u2 ). If c1 (u2 ) = u1 , then f (c1 (u2 ), u2 ) > f (u1 , u2 ) because c1 (u2 ) is unique. Combining (9.87) and (9.88), we have f (c1 (u2 ), c2 (c1 (u2 ))) > f (u1 , u2 ), which is a contradiction to (9.86). Therefore, u1 = c1 (u2 ). Using this, we see from (9.86) that f (u1 , c2 (u1 )) = f (u1 , u2 ), which implies u2 = c2 (u1 ). because c2 (c1 (u2 )) is unique. Since f (u) < f , there exists v A such that f (u) < f (v). Consider (9.93) (9.92) (9.91) (9.90) (9.89) (9.88) (9.87) (9.86) 224 9 The Blahut-Arimoto Algorithms (u 1 , v 2 ) (v 1 , v 2 ) z2 z (u 1 , u 2 ) z1 (v 1 , u 2) ~ Fig. 9.3. The vectors u, v, z, z1 , and z2 . v - u = (v1 - u1 , 0) + (0, v2 - u2 ). (9.94) ~ Let z be the unit vector in the direction of v - u, z1 be the unit vector in the direction of (v1 - u1 , 0), and z2 be the unit vector in the direction of (v2 - u2 , 0). Then ~ v - u z = v1 - u1 z1 + v2 - u2 z2 , or ~ z = 1 z1 + 2 z2 , where i = vi - ui , v-u (9.96) (9.97) (9.95) ~ i = 1, 2. Figure 9.3 is an illustration of the vectors u, v, z, z1 , and z2 . We see from (9.90) that f attains its maximum value at u = (u1 , u2 ) when u2 is fixed. In particular, f attains its maximum value at u along the line passing through (u1 , u2 ) and (v1 , u2 ). Let f denotes the gradient of f . Since f is continuous and has continuous partial derivatives, the directional derivative of f at u in the direction of z1 exists and is given by f z1 . It follows from the concavity of f that f is concave along the line passing through (u1 , u2 ) and (v1 , u2 ). Since f attains its maximum value at u, the derivative of f along the line passing through (u1 , u2 ) and (v1 , u2 ) vanishes. Then we see that f z1 = 0. (9.98) Similarly, we see from (9.92) that f z2 = 0. (9.99) ~ Then from (9.96), the directional derivative of f at u in the direction of z is given by 9.3 Convergence 225 ~ f z = 1 ( f z1 ) + 2 ( f z2 ) = 0. (9.100) Since f is concave along the line passing through u and v, this implies f (u) f (v), (9.101) which is a contradiction to (9.93). Hence, we conclude that f (u) > 0. Although we have proved that the algorithm cannot be trapped at u if f (u) < f , f (k) does not necessarily converge to f because the increment in f (k) in each step may be arbitrarily small. In order to prove the desired convergence, we will show in next theorem that this cannot be the case. Theorem 9.5. If f is concave, then f (k) f . Proof. We have already shown in Section 9.1 that f (k) necessarily converges, say to f . Hence, for any > 0 and all sufficiently large k, f - f (k) f . Let = min f (u), uA (9.102) (9.103) (9.104) where A = {u A : f - f (u) f }. Since f has continuous partial derivatives, f (u) is a continuous function of u. Then the minimum in (9.103) exists because A is compact2 . We now show that f < f will lead to a contradiction if f is concave. If f < f , then from Lemma 9.4, we see that f (u) > 0 for all u A and hence > 0. Since f (k) = f (u(k) ) satisfies (9.102), u(k) A , and f (k+1) - f (k) = f (u(k) ) (9.105) for all sufficiently large k. Therefore, no matter how smaller is, f (k) will eventually be greater than f , which is a contradiction to f (k) f . Hence, we conclude that f (k) f . 9.3.2 Convergence to the Channel Capacity In order to show that the BA algorithm for computing the channel capacity converges as intended, i.e., f (k) C, we only need to show that the function f defined in (9.39) is concave. Toward this end, for f (r, q) = x 2 r(x)p(y|x) log y q(x|y) r(x) (9.106) A is compact because it is the inverse image of a closed interval under a continuous function and A is bounded. 226 9 The Blahut-Arimoto Algorithms defined in (9.39), we consider two ordered pairs (r1 , q1 ) and (r2 , q2 ) in A, where A1 and A2 are defined in (9.40) and (9.41), respectively. For any 0 1 and = 1 - , an application of the log-sum inequality (Theorem 2.32) gives (r1 (x) + r2 (x)) log r1 (x) + r2 (x) 2 (x|y) q1 (x|y) + q r1 (x) r2 (x) r1 (x) log + r2 (x) log . q1 (x|y) q2 (x|y) (9.107) Taking reciprocal in the logarithms yields q1 (x|y) + q2 (x|y) (r1 (x) + r2 (x)) log r1 (x) + r2 (x) q1 (x|y) q2 (x|y) r1 (x) log + r2 (x) log , r1 (x) r2 (x) (9.108) and upon multiplying by p(y|x) and summing over all x and y, we obtain f (r1 + r2 , q1 + q2 ) f (r1 , q1 ) + f (r2 , q2 ). Therefore, f is concave. Hence, we have shown that f (k) C. (9.109) Chapter Summary Channel Capacity: For a discrete memoryless channel p(y|x), C = sup max r>0 q x y r(x)p(y|x) log q(x|y) , r(x) where the maximization is taken over all q that satisfies q(x|y) = 0 if and only if p(y|x) = 0. Computation of Channel Capacity: Start with any strictly positive input distribution r(0) . Compute q(0) , r(1) , q(1) , r(2) , alternately by q (k) (x|y) = and r(k) (x) = x y r(k) (x)p(y|x) (k) (x )p(y|x ) x r q (k-1) (x|y)p(y|x) y q (k-1) (x |y)p(y|x ) . Then r(k) tends to the capacity-achieving input distribution as k . Problems and Historical Notes 227 Rate-Distortion Function: For s 0, the tangent to the rate-distortion function R(D) at (Ds , R(Ds )) has slope s and intersects with the ordinate at R(Ds ) - sDs , which is given by Q(^|x) x -s p(x)Q(^|x)d(x, x) . x ^ inf min p(x)Q(^|x) log x Q>0 t>0 t(^) x x,^ x x,^ x The curve R(D), 0 D Dmax is traced out by the collection of all such tangents. Computation of Rate-Distortion Function: Start with any strictly positive transition matrix Q(0) . Compute t(0) , Q(1) , t(1) , Q(2) , alternately by t(k) (^) = x x p(x)Q(k) (^|x) x and Q(k) (^|x) = x Let f (Q, t) = x x ^ x t(k-1) (^)esd(x,^) x . (k-1) (^ )esd(x,^ ) x x x t ^ p(x)Q(^|x) log x Q(^|x) x -s t(^) x p(x)Q(^|x)d(x, x). x ^ x x ^ Then f (Q(k) , t(k) ) R(Ds ) - sDs as k . Problems 1. Implement the BA algorithm for computing channel capacity. 2. Implement the BA algorithm for computing the rate-distortion function. 3. Explain why in the BA Algorithm for computing channel capacity, we should not choose an initial input distribution which contains zero probability masses. 4. Prove Lemma 9.3. 5. Consider f (Q, t) in the BA algorithm for computing the rate-distortion function. a) Show that for fixed s and t, f (Q, t) is minimized by Q(^|x) = x b) Show that f (Q, t) is convex. x t(^)esd(x,^) x . sd(x,^ ) x x x t(^ )e ^ 228 9 The Blahut-Arimoto Algorithms Historical Notes An iterative algorithm for computing the channel capacity was developed by Arimoto [19], where the convergence of the algorithm was proved. Blahut [37] independently developed two similar algorithms, the first for computing the channel capacity and the second for computing the rate-distortion function. The convergence of Blahut's second algorithm was proved by Csiszr [82]. a These two algorithms are now commonly referred to as the Blahut-Arimoto algorithms. The simplified proof of convergence in this chapter is based on Yeung and Berger [404]. The Blahut-Arimoto algorithms are special cases of a general iterative algorithm due to Csiszr and Tusndy [89] which also include the expectationa a maximization (EM) algorithm for fitting models from incomplete data [94] and the algorithm for finding the log-optimal portfolio for a stock market due to Cover [75]. 10 Differential Entropy Our discussion in the previous chapters involved only discrete random variables. The actual values taken by these random variables did not play any role in establishing the results. In this chapter and the next, our discussion will involve random variables taking real values. The values taken by these random variables do play a crucial role in the discussion. Let X be a real random variable with cumulative distribution function (CDF) FX (x) = Pr{X x}, which by definition is right-continuous. The random variable X is said to be discrete if FX (x) increases only at a countable number of values of x; continuous if FX (x) is continuous, or equivalently, Pr{X = x} = 0 for every value of x; mixed if it is neither discrete nor continuous. The support of X, denoted by SX , is the set of all x such that FX (x) > FX (x - ) for all > 0. For a function g defined on SX , we write Eg(X) = SX g(x)dFX (x), (10.1) where the right hand side is a Lebesgue-Stieltjes integration which covers all cases (i.e., discrete, continuous, and mixed) for the CDF FX (x). It may be regarded as a notation for the expectation of g(X) with respect to FX (x) if the reader is not familiar with measure theory. A nonnegative function fX (x) is called a probability density function (pdf) of X if x FX (x) = - fX (u)du (10.2) for all x. Since fX (x)dx = FX () = 1 < , (10.3) 230 10 Differential Entropy a pdf fX (x) can possibly take infinite value only on a set with zero Lebesgue measure. Therefore, we can assume without loss of generality that fX (x) instead takes any finite values on this set. If X has a pdf, then X is continuous, but not vice versa. Let X and Y be two real random variables with joint CDF FXY (x, y) = Pr{X x, Y y}. The marginal CDF of X is given by FX (x) = FXY (x, ) (likewise for Y ). A nonnegative function fXY (x, y) is called a joint pdf of X and Y if y x FXY (x, y) = - - fXY (u, v) dvdu (10.4) for all x and y. As for the case of a single random variable, we can assume without loss of generality that a joint pdf fXY (x, y) is finite for all x and y. For x SX , the conditional CDF of Y given {X = x} is defined as y FY |X (y|x) = - fY |X (v|x)dv, (10.5) where fY |X (y|x) = fXY (x, y) fX (x) (10.6) is the conditional pdf of Y given {X = x}. All the above definitions and notations naturally extend to more than two real random variables. When there is no ambiguity, the subscripts specifying the random variables will be omitted. All the random variables in this chapter are assumed to be real1 . The variance of a random variable X is defined as varX = E(X - EX)2 = EX 2 - (EX)2 . The covariance between two random variables X and Y is defined as cov(X, Y ) = E(X - EX)(Y - EY ) = E(XY ) - (EX)(EY ). (10.8) (10.7) For a random vector X = [X1 X2 Xn ] , the covariance matrix is defined as KX = E(X - EX)(X - EX) = [cov(Xi , Xj )], (10.9) and the correlation matrix is defined as ~ KX = EXX = [EXi Xj ]. Then 1 (10.10) For a discrete random variable X with a countable alphabet X , by replacing X by any countable subset of , all information measures involving X (and possibly other random variables) are unchanged. Therefore, we assume without loss of generality that a discrete random variable is real. 10.1 Preliminaries 231 KX = E(X - EX)(X - EX) = E[XX - X(EX ) - (EX)X + (EX)(EX )] = EXX - (EX)(EX ) - (EX)(EX ) + (EX)(EX ) = EXX - (EX)(EX ) ~ = KX - (EX)(EX) . (10.11) (10.12) (10.13) (10.14) (10.15) ~ This implies that if EX = 0, then KX = KX . On the other hand, it can readily be verified that ~ KX = KX-EX . (10.16) Therefore, a covariance matrix is a correlation matrix. When there is no am~ biguity, the subscripts in KX and KX will be omitted. Let N (, 2 ) denote the Gaussian distribution with mean and variance 2 , i.e., the pdf of the distribution is given by f (x) = 1 2 2 e- (x-)2 2 2 (10.17) for - < x < . More generally, let N (, K) denote the multivariate Gaussian distribution with mean and covariance matrix K, i.e., the joint pdf of the distribution is given by f (x) = 1 2 n |K|1/2 e- 2 (x-) 1 K -1 (x-) (10.18) for all x n , where K is a symmetric positive definite matrix2 and |K| is the determinant of K. In the rest of the chapter, we will define various information measures under suitable conditions. Whenever these information measures are subsequently used, they are assumed to be defined. 10.1 Preliminaries In this section, we present some preliminary results on matrices and linear transformation of random variables. All vectors and matrices are assumed to be real. Definition 10.1. A square matrix K is symmetric if K = K. Definition 10.2. An n n matrix K is positive definite if x Kx > 0 for all nonzero column n-vector x, and is positive semidefinite if x Kx 0 for all column n-vector x. 2 (10.19) (10.20) See Definitions 10.1 and 10.2. 232 10 Differential Entropy Proposition 10.3. A covariance matrix is both symmetric and positive semidefinite. Proof. Omitted. If a matrix K is symmetric, it can be diagonalized as K = QQ , (10.21) where is a diagonal matrix and Q (also Q ) is an orthogonal matrix, i.e., Q-1 = Q , or QQ = Q Q = I. (10.23) The latter says that the rows (columns) of Q form an orthonormal system. Since |Q|2 = |Q||Q | = |QQ | = |I| = 1, (10.24) we have |Q| = |Q | = 1. If (10.21) holds, we also say that QQ From (10.21) and (10.23), we have is a diagonalization of K. (10.25) (10.22) KQ = (QQ )Q = Q(Q Q) = Q. (10.26) Let i and qi = 0 denote the ith diagonal element of and the ith column of Q, respectively. Then (10.26) can be written as Kqi = i qi (10.27) for all i, i.e., qi is an eigenvector of K with eigenvalue i . The next proposition further shows that these eigenvalues are nonnegative if K is positive semidefinite. Proposition 10.4. The eigenvalues of a positive semidefinite matrix are nonnegative. Proof. Let K be a positive semidefinite matrix, and let q = 0 be an eigenvector of K with eigenvalue , i.e., Kq = q. (10.28) Since K is positive semidefinite, 0 q Kq = q (q) = (q q). Then we conclude that 0 because q q > 0. (10.29) 10.1 Preliminaries 233 The above discussions on diagonalization apply to a covariance matrix because a covariance matrix is both symmetric and positive semidefinite. As we will see, by diagonalizing the covariance matrix, a set of correlated random variables can be decorrelated by an orthogonal transformation. On the other hand, a set of correlated random variables can be regarded as an orthogonal transformation of a set of uncorrelated random variables. This is particularly important in the context of Gaussian random variables because a set of jointly distributed Gaussian random variables are mutually independent if and only if they are uncorrelated. Proposition 10.5. Let Y = AX, where X and Y are column vectors of n random variables and A is an n n matrix. Then KY = AKX A and ~ ~ KY = AKX A . (10.31) (10.30) Proof. To prove (10.30), consider KY = E(Y - EY)(Y - EY) = E[A(X - EX)][A(X - EX)] = E[A(X - EX)(X - EX) A ] = A[E(X - EX)(X - EX) ]A = AKX A . The proof of (10.31) is similar. Proposition 10.6. Let X and Y be column vectors of n random variables such that Y = Q X, (10.37) where QQ is a diagonalization of KX . Then KY = , i.e., the random variables in Y are uncorrelated and var Yi = i , the ith diagonal element of . Remark The matrix KX is positive semidefinite, so that i , being an eigenvalue of KX , is nonnegative by Proposition 10.4, as required for being the variance of a random variable. Proof of Propostion 10.6. By Proposition 10.5, KY = Q KX Q = Q (QQ )Q = (Q Q)(Q Q) = . (10.38) (10.39) (10.40) (10.41) (10.32) (10.33) (10.34) (10.35) (10.36) 234 10 Differential Entropy Since KY = is a diagonal matrix, the random variables in Y are uncorrelated. Furthermore, the variance of Yi is given by the ith diagonal element of KY = , i.e., i . The proposition is proved. Corollary 10.7. Let X be a column vector of n random variables such that QQ is a diagonalization of KX . Then X = QY, (10.42) where Y is the column vector of n uncorrelated random variables prescribed in Proposition 10.6. Proposition 10.8. Let X, Y, and Z be vectors of n random variables such that X and Z are independent and Y = X + Z. Then KY = KX + KZ . (10.43) Proof. Omitted. In communication engineering, the second moment of a random variable X is very often referred to as the energy of X. The total energy of a random 2 vector X is then equal to E i Xi . The following proposition shows that the total energy of a random vector is preserved by an orthogonal transformation. Proposition 10.9. Let Y = QX, where X and Y are column vectors of n random variables and Q is an orthogonal matrix. Then n n E i=1 Yi2 = E i=1 2 Xi . (10.44) Proof. Consider n Yi2 = Y Y i=1 (10.45) (10.46) (10.47) (10.48) (10.49) = (QX) (QX) = X (Q Q)X = X X n = i=1 2 Xi . The proposition is proved upon taking expectation on both sides. 10.2 Definition 235 10.2 Definition We now introduce the differential entropy for continuous random variables as the analog of the entropy for discrete random variables. Definition 10.10. The differential entropy h(X) of a continuous random variable X with pdf f (x) is defined as h(X) = - S f (x) log f (x)dx = -E log f (X). (10.50) The entropy of a discrete random variable X is a measure of the average amount of information contained in X, or equivalently, the average amount of uncertainty removed upon revealing the outcome of X. This was justified by the asymptotic achievability of the entropy bound for zero-error data compression discussed in Chapter 4 as well as the source coding theorem discussed in Chapter 5. However, although entropy and differential entropy have similar mathematical forms, the latter does not serve as a measure of the average amount of information contained in a continuous random variable. In fact, a continuous random variable generally contains an infinite amount of information, as explained in the following example. Example 10.11. Let X be uniformly distributed on [0, 1). Then we can write X = .X1 X2 X3 , (10.51) the dyadic expansion of X, where X1 , X2 , X3 , is a sequence of fair bits3 . Then H(X) = H(X1 , X2 , X3 , ) (10.52) (10.53) (10.54) (10.55) = i=1 H(Xi ) 1 i=1 = = . In the following, we give two examples in which the differential entropy can be evaluated explicitly. Example 10.12 (Uniform Distribution). Let X be uniformly distributed on [0, a). Then a 1 1 h(X) = - log dx = log a. (10.56) a 0 a 3 Fair bits refer to i.i.d. bits, each distributed uniformly on {0, 1}. 236 10 Differential Entropy From this example, we see immediately that h(X) < 0 if a < 1. This poses no contradiction because as we have mentioned, the differential entropy does not serve as a measure of the average amount of information contained in X. The physical meaning of differential entropy will be understood through the AEP for continuous random variables to be discussed in Section 10.4. Example 10.13 (Gaussian Distribution). Let X N (0, 2 ) and let e be the base of the logarithm. Then h(X) = - =- = f (x) ln f (x)dx f (x) - x2 - ln 2 2 dx 2 2 x2 f (x)dx + ln 2 2 f (x)dx (10.57) (10.58) (10.59) (10.60) (10.61) (10.62) (10.63) (10.64) (10.65) 1 2 2 EX 2 1 = + ln(2 2 ) 2 2 2 1 varX + (EX)2 = + ln(2 2 ) 2 2 2 2 + 0 1 = + ln(2 2 ) 2 2 2 1 1 = + ln(2 2 ) 2 2 1 1 = ln e + ln(2 2 ) 2 2 1 = ln(2e 2 ) 2 in nats. Changing the base of the logarithm to any chosen positive value, we obtain 1 h(X) = log(2e 2 ). (10.66) 2 The following two basic properties of differential entropy can readily be proved from the definition. Theorem 10.14 (Translation). h(X + c) = h(X). (10.67) Proof. Let Y = X + c. Then fY (y) = fX (y - c) and SY = {x + c : x SX }. Letting x = y - c in (10.50), we have 10.2 Definition 237 h(X) = - SX fX (x) log fX (x)dx fX (y - c) log fX (y - c)dy SY (10.68) (10.69) (10.70) (10.71) (10.72) =- =- SY fY (y) log fY (y)dy = h(Y ) = h(X + c), accomplishing the proof. Theorem 10.15 (Scaling). For a = 0, h(aX) = h(X) + log |a|. (10.73) Proof. Let Y = aX. Then fY (y) = y Letting x = a in (10.50), we have h(X) = - SX y 1 |a| fX ( a ) and SY = {ax : x SX }. fX (x) log fX (x)dx fX SY (10.74) (10.75) + log |a| dy fY (y)dy SY =- =- SY y y dy log fX a a |a| 1 1 y y fX log fX |a| a |a| a (10.76) (10.77) (10.78) (10.79) (10.80) =- SY fY (y) log fY (y)dy - log |a| = h(Y ) - log |a| = h(aX) - log |a|. Hence, h(aX) = h(X) + log |a|, accomplishing the proof. Example 10.16. We illustrate Theorem 10.14 and Theorem 10.15 by means 2 of the Gaussian distribution. Let X N (X , X ). By Theorem 10.14 (and Example 10.13), 1 2 h(X) = log(2eX ). (10.81) 2 2 2 2 Let Y = aX. Then Y N (Y , Y ), where Y = aX and Y = a2 X . By (10.81), 1 1 1 2 2 2 log(2eY ) = log(2ea2 X ) = log(2eX ) + log |a|, (10.82) 2 2 2 verifying Theorem 10.15. h(Y ) = 238 10 Differential Entropy Theorem 10.14 says that the differential entropy of a random variable is unchanged by translation. Theorem 10.15 says that the differential entropy of a random variable is generally changed by scaling. Specifically, if |a| > 1, the differential entropy is increased by log |a|. If |a| < 1, the differential entropy is decreased by - log |a| (note that - log |a| > 0). If a = -1, the differential entropy is unchanged. These properties suggest that the differential entropy of a random variable depends only on the "spread" of the pdf. More specifically, the differential entropy increases with the "spread" of the pdf. This point will be further elaborated in Section 10.6. 10.3 Joint Differential Entropy, Conditional (Differential) Entropy, and Mutual Information The definition for differential entropy is readily extended to multiple continuous random variables. In the rest of the chapter, we let X = [X1 X2 Xn ]. Definition 10.17. The joint differential entropy h(X) of a random vector X with joint pdf f (x) is defined as h(X) = - S f (x) log f (x)dx = -E log f (X). (10.83) It follows immediately from the above definition that if X1 , X2 , , Xn are mutually independent, then n h(X) = i=1 h(Xi ). (10.84) The following two theorems are straightforward generalizations of Theorems 10.14 and 10.15, respectively. The proofs are omitted. Theorem 10.18 (Translation). Let c be a column vector in h(X + c) = h(X). n . Then (10.85) Theorem 10.19 (Scaling). Let A be a nonsingular n n matrix. Then h(AX) = h(X) + log |det(A)|. (10.86) 10.3 Joint and Conditional Differential Entropy 239 Theorem 10.20 (Multivariate Gaussian Distribution). Let X N (, K). Then 1 (10.87) h(X) = log [(2e)n |K|] . 2 Proof. Let K be diagonalizable as QQ . Write X = QY as in Corollary 10.7, where the random variables in Y are uncorrelated with var Yi = i , the ith diagonal element of . Since X is Gaussian, so is Y. Then the random variables in Y are mutually independent because they are uncorrelated. Now consider h(X) = h(QY) = h(Y) + log |det(Q)| = h(Y) + 0 = = i=1 d) i=1 n c) n b) a) (10.88) (10.89) (10.90) (10.91) (10.92) h(Yi ) 1 log(2ei ) 2 n = = e) 1 log (2e)n i 2 i=1 (10.93) (10.94) (10.95) 1 log[(2e)n ||] 2 f) 1 = log[(2e)n |K|]. 2 In the above a) follows from Theorem 10.19; b) follows from (10.25); c) follows from (10.84) since Y1 , Y2 , , Yn are mutually independent; d) follows from Example 10.16; e) follows because is a diagonal matrix; f) follows because || = |Q||||Q | = |QQ | = |K|. The theorem is proved. In describing a communication system, we very often specify the relation between two random variables X and Y through a conditional distribution p(y|x) (if Y is discrete) or a conditional pdf f (y|x) (if Y is continuous) defined for all x, even though certain x may not be in SX . This is made precise by the following two definitions. (10.96) 240 10 Differential Entropy Definition 10.21. Let X and Y be two jointly distributed random variables with Y being discrete. The random variable Y is related to the random variable X through a conditional distribution p(y|x) defined for all x means that for all x and y, x Pr{X x, Y = y} = - pY |X (y|u)dFX (u). (10.97) Definition 10.22. Let X and Y be two jointly distributed random variables with Y being continuous. The random variable Y is related to the random variable X through a conditional pdf f (y|x) defined for all x means that for all x and y, x FXY (x, y) = - FY |X (y|u)dFX (u), y (10.98) where FY |X (y|x) = fY |X (v|x)dv. - (10.99) Definition 10.23. Let X and Y be jointly distributed random variables where Y is continuous and is related to X through a conditional pdf f (y|x) defined for all x. The conditional differential entropy of Y given {X = x} is defined as h(Y |X = x) = - f (y|x) log f (y|x)dy (10.100) SY (x) where SY (x) = {y : f (y|x) > 0}, and the conditional differential entropy of Y given X is defined as h(Y |X) = - SX h(Y |X = x)dF (x) = -E log f (Y |X). (10.101) Proposition 10.24. Let X and Y be jointly distributed random variables where Y is continuous and is related to X through a conditional pdf f (y|x) defined for all x. Then f (y) exists and is given by f (y) = f (y|x)dF (x). (10.102) Proof. From (10.98) and (10.99), we have y FY (y) = FXY (, y) = - fY |X (v|x) dv dF (x). (10.103) Since fY |X (v|x) is nonnegative and 10.3 Joint and Conditional Differential Entropy y 241 fY |X (v|x) dv dF (x) - fY |X (v|x) dv dF (x) dF (x) (10.104) (10.105) (10.106) = = 1, fY |X (v|x) is absolutely integrable. By Fubini's theorem4 , the order of integration in the iterated integral in (10.103) can be exchanged. Therefore, y FY (y) = - fY |X (v|x)dF (x) dv, (10.107) implying (10.102) (cf. (10.2)). The proposition is proved. The above proposition says that if Y is related to X through a conditional pdf f (y|x), then the pdf of Y exists regardless of the distribution of X. The next proposition is a generalization to random vectors, and the proof is omitted. The theory in the rest of this chapter and in the next chapter will be developed around this important fact. Proposition 10.25. Let X and Y be jointly distributed random vectors where Y is continuous and is related to X through a conditional pdf f (y|x) defined for all x. Then f (y) exists and is given by f (y) = f (y|x)dF (x). (10.108) Definition 10.26. Let X and Y be jointly distributed random variables where Y is continuous and is related to X through a conditional pdf f (y|x) defined for all x. The mutual information between X and Y is defined as I(X; Y ) = SX SY (x) f (y|x) log f (Y |X) , f (Y ) f (y|x) dy dF (x) f (y) (10.109) (10.110) = E log where f (y) exists and is given in (10.102) by Proposition 10.24. When both X and Y are continuous and f (x, y) exists, I(X; Y ) = E log f (Y |X) f (X, Y ) = E log . f (Y ) f (X)f (Y ) (10.111) 4 See for example [314]. 242 10 Differential Entropy Together with our discussion on discrete random variables in Chapter 2, the mutual information I(X; Y ) is defined when each of the random variables involved can be either discrete or continuous. In the same way, we can define the conditional mutual information I(X; Y |T ). Definition 10.27. Let X, Y , and T be jointly distributed random variables where Y is continuous and is related to (X, T ) through a conditional pdf f (y|x, t) defined for all x and t. The mutual information between X and Y given T is defined as I(X; Y |T ) = ST I(X; Y |T = t)dF (t) = E log f (Y |X, T ) , f (Y |T ) (10.112) where I(X; Y |T = t) = SX (t) SY (x,t) f (y|x, t) log f (y|x, t) dy dF (x|t). f (y|t) (10.113) We now give a physical interpretation of I(X; Y ) when X and Y have a joint pdf f (x, y). For simplicity, we assume that f (x, y) > 0 for all x and y. Let be a small positive quantity. For all integer i, define the interval Ai = [ i, (i + 1)) x in , and for all integer j, define the interval Aj = [ j, (j + 1)). y For all integers i and j, define the set Ai,j = Ai Aj , xy x y (10.116) (10.115) (10.114) which corresponds to a rectangle in 2 . ^ ^ We now introduce a pair of discrete random variables X and Y defined by ^ X = i if X Ai x (10.117) ^ = j if Y Aj . Y y ^ ^ The random variables X and Y are quantizations of the continuous random variables X and Y , respectively. For all i and j, let (xi , yj ) Ai,j . Then xy ^ ^ I(X ; Y ) = i j ^ ^ Pr{(X , Y ) = (i, j)} log f (xi , yj )2 log i j ^ ^ Pr{(X , Y ) = (i, j)} ^ ^ Pr{X = i}Pr{Y = j} (10.118) (10.119) f (xi , yj )2 (f (xi ))(f (yj )) 10.3 Joint and Conditional Differential Entropy 243 = i j f (xi , yj )2 log f (x, y) log f (xi , yj ) f (xi )f (yj ) (10.120) (10.121) (10.122) f (x, y) dxdy f (x)f (y) = I(X; Y ). ^ ^ Therefore, I(X; Y ) it can be interpreted as the limit of I(X ; Y ) as 0. This interpretation carries over to the case when X and Y have a general joint ^ ^ distribution5 (see Dobrushin [96]). As I(X ; Y ) is always nonnegative, this suggests that I(X; Y ) is also always nonnegative, which will be established in Theorem 10.31. Definition 10.28. Let Y be a continuous random variable and X be a discrete random variable, where Y is related to X through a conditional pdf f (y|x). The conditional entropy of X given Y is defined as H(X|Y ) = H(X) - I(X; Y ), where I(X; Y ) is defined as in Definition 10.26. Proposition 10.29. For two random variables X and Y , h(Y ) = h(Y |X) + I(X; Y ) if Y is continuous, and H(Y ) = H(Y |X) + I(X; Y ) if Y is discrete. Proposition 10.30 (Chain Rule for Differential Entropy). n (10.123) (10.124) (10.125) h(X1 , X2 , , Xn ) = i=1 h(Xi |X1 , , Xi-1 ). (10.126) The proofs of these propositions are left as an exercise. Theorem 10.31. I(X; Y ) 0, with equality if and only if X is independent of Y . 5 (10.127) In the general setting, the mutual information between X and Y is defined as I(X; Y ) = SXY log dPXY d(PX PY ) dPXY , where PXY , PX , and PY are the probability measures of (X, Y ), X, and Y , dP respectively, and d(PXXY Y ) denotes the Radon-Nikodym derivative of PXY with P respect to the product measure PX PY . 244 10 Differential Entropy Proof. Consider I(X; Y ) = SX SY (x) f (y|x) log f (y|x) dy dFX (x) f (y) f (y) f (y|x) dy dFX (x) f (y)dy dFX (x) SY (x) (10.128) (10.129) (10.130) (10.131) (log e) SX SY (x) f (y|x) 1 - f (y|x)dy - SX SY (x) = (log e) 0, where (10.129) results from an application of the fundamental inequality (Corollary 2.30), and (10.131) follows from f (y)dy 1 = SY (x) SY (x) f (y|x)dy. (10.132) This proves (10.127). For equality to hold in (10.127), equality must hold in (10.129) for all x SX and all y SY (x), and equality must hold in (10.131) for all x SX . For the former, this is the case if and only if f (y|x) = f (y) which implies f (y)dy = SY (x) SY (x) for all x SX and y SY (x), (10.133) f (y|x)dy = 1, (10.134) i.e., equality holds in (10.131). Thus (10.133) is a necessary and sufficient condition for equality to hold in (10.127). It is immediate that if X and Y are independent, then (10.133) holds. It remains to prove the converse. To this end, observe that (10.134), implied by (10.133), is equivalent to that f (y) = 0 on SY \SY (x) a.e. (almost everywhere). c By the definition of SY , this means that SY \SY (x) SY , or SY = SY (x). Since this holds for all x SX , we conclude that f (y|x) = f (y) for all (x, y) SX SY , i.e., X and Y are independent. The theorem is proved. Corollary 10.32. I(X; Y |T ) 0, (10.135) with equality if and only if X is independent of Y conditioning on T . Proof. This follows directly from (10.112). Corollary 10.33 (Conditioning Does Not Increase Differential Entropy). h(X|Y ) h(X) (10.136) with equality if and only if X and Y are independent. 10.4 The AEP for Continuous Random Variables 245 Corollary 10.34 (Independence Bound for Differential Entropy). n h(X1 , X2 , , Xn ) i=1 h(Xi ) (10.137) with equality if and only if i = 1, 2, , n are mutually independent. 10.4 The AEP for Continuous Random Variables The Weak AEP for discrete random variables discussed in Chapter 5 states that for n i.i.d. random variables X1 , X2 , , Xn with generic discrete random variable X, p(X1 , X2 , , Xn ) is close to 2-nH(X) with high probability when n is large (Theorem 5.1, Weak AEP I). This fundamental property of entropy leads to the definition of weak typicality, and as a consequence, the total number of weakly typical sequences is approximately equal to 2nH(X) (Theorem 5.3, Weak AEP II). In the following, we develop the AEP for continuous random variables in the same way we developed the Weak AEP for discrete random variables. Some of the proofs are exactly the same as their discrete analogs, and they are omitted. We note that for continuous random variables, the notion of strong typicality does not apply because the probability that a continuous random variable takes a particular value is equal to zero. Theorem 10.35 (AEP I for Continuous Random Variables). - 1 log f (X) h(X) n > 0, for n sufficiently large, >1- . (10.139) (10.138) in probability as n , i.e., for any Pr - 1 log f (X) - h(X) < n n Definition 10.36. The typical set W[X] with respect to f (x) is the set of n sequences x = (x1 , x2 , , xn ) X such that - or equivalently, 1 log f (x) - h(X) < , n (10.140) 1 log f (x) < h(X) + , (10.141) n n where is an arbitrarily small positive real number. The sequences in W[X] are called -typical sequences. h(X) - < - 246 10 Differential Entropy The quantity - 1 1 log f (x) = - n n n log f (xk ) k=1 (10.142) is called the empirical differential entropy of the sequence x. The empirical differential entropy of a typical sequence is close to the true differential entropy h(X). If the pdf f (x) is continuous, we see from (10.142) that the empirical differential entropy is continuous in x. Therefore, if x is -typical, then all the sequences in the neighborhood of x are also -typical. As a consequence, the number of -typical sequences is uncountable, and it is not meaningful to discuss the cardinality of a typical set as in the discrete case. Instead, the "size" of a typical set is measured by its volume. Definition 10.37. The volume of a set A in Vol(A) = A n is defined as (10.143) dx. Theorem 10.38 (AEP II for Continuous Random Variables). The following hold for any > 0: n 1) If x W[X] , then 2-n(h(X)+ ) < f (x) < 2-n(h(X)- ) . 2) For n sufficiently large, n Pr{X W[X] } > 1 - . (10.144) (10.145) 3) For n sufficiently large, n (1 - )2n(h(X)- ) < Vol(W[X] ) < 2n(h(X)+ ) . (10.146) n Proof. Property 1 follows immediately from the definition of W[X] in (10.141). Property 2 is equivalent to Theorem 10.35. To prove Property 3, we use the lower bound in (10.144) and consider n 1 Pr{W[X] } (10.147) (10.148) (10.149) (10.150) (10.151) = n W[X] f (x) dx 2-n(h(X)+ ) dx n W[X] > > 2-n(h(X)+ ) n W[X] dx n = 2-n(h(X)+ ) Vol(W[X] ), 10.5 Informational Divergence 247 which implies n Vol(W[X] ) < 2n(h(X)+ ) . (10.152) Note that this upper bound holds for any n 1. On the other hand, using the upper bound in (10.144) and Theorem 10.35, for n sufficiently large, we have n 1 - < Pr{W[X] } (10.153) (10.154) (10.155) (10.156) (10.157) = n W[X] f (x) dx 2-n(h(X)- ) dx n W[X] < n = 2-n(h(X)- ) Vol(W[X] ). Then n Vol(W[X] ) > (1 - )2n(h(X)- ) . Combining (10.152) and (10.157) gives Property 3. The theorem is proved. From the AEP for continuous random variables, we see that the volume of the typical set is approximately equal to 2nh(X) when n is large. This gives the following physical interpretations of differential entropy. First, the fact that h(X) can be negative does not incur any difficulty because 2nh(X) is always positive. Second, if the differential entropy is large, then the volume of the typical set is large; if the differential entropy is small (not in magnitude but in value), then the volume of the typical set is small. 10.5 Informational Divergence We first extend the definition of informational divergence introduced in Section 2.5 to pdf's. Definition 10.39. Let f and g be two pdf 's defined on n with supports Sf and Sg , respectively. The informational divergence between f and g is defined as f (X) f (x) dx = Ef log , (10.158) D(f g) = f (x) log g(x) g(X) Sf where Ef denotes expectation with respect to f . Remark In the above definition, we adopt the convention c log c > 0. Therefore, if D(f g) < , then Sf \ Sg = {x : f (x) > 0 and g(x) = 0} has zero Lebesgue measure, i.e., Sf is essentially a subset of Sg . c 0 = for (10.159) 248 10 Differential Entropy Theorem 10.40 (Divergence Inequality). Let f and g be two pdf 's defined on n . Then D(f g) 0, (10.160) with equality if and only if f = g a.e. Proof. Consider D(f g) = Sf f (x) log f (x) dx g(x) f (x) dx g(x) g(x) f (x) dx (10.161) (10.162) (10.163) (10.164) (10.165) = (log e) Sf f (x) ln (log e) Sf f (x) 1 - f (x)dx - Sf = (log e) 0, g(x)dx Sf where (10.163) follows from the fundamental inequality (Corollary 2.30) and (10.165) follows from g(x)dx 1 = Sf Sf f (x)dx. (10.166) Equality holds in (10.163) if and only if f (x) = g(x) on Sf a.e., which implies g(x)dx = Sf Sf f (x)dx = 1, (10.167) i.e., equality holds in (10.165). Then we see from (10.167) that g(x) = 0 on c Sf a.e. Hence, we conclude that equality holds in (10.160) if and only if f = g a.e. The theorem is proved. 10.6 Maximum Differential Entropy Distributions In Section 2.9, we have discussed maximum entropy distributions for a discrete random variable. We now extend this theme to multiple continuous random variables. Specifically, we are interested in the following problem: Maximize h(f ) over all pdf f defined on a subset S of ri (x)f (x)dx = ai Sf n , subject to (10.168) for 1 i m, where Sf S and ri (x) is defined for all x S. 10.6 Maximum Differential Entropy Distributions 249 Theorem 10.41. Let f (x) = e-0 - m i=1 i ri (x) (10.169) for all x S, where 0 , 1 , , m are chosen such that the constraints in (10.168) are satisfied. Then f maximizes h(f ) over all pdf f defined on S, subject to the constraints in (10.168). Proof. The proof is analogous to that of Theorem 2.50. The details are omitted. Corollary 10.42. Let f be a pdf defined on S with f (x) = e-0 - m i=1 i ri (x) (10.170) for all x S. Then f maximizes h(f ) over all pdf f defined on S, subject to the constraints ri (x)f (x)dx = Sf S ri (x)f (x)dx for 1 i m. (10.171) Theorem 10.43. Let X be a continuous random variable with EX 2 = . Then 1 (10.172) h(X) log(2e), 2 with equality if and only if X N (0, ). Proof. The problem here is to maximize h(f ) subject to the constraint x2 f (x)dx = . An application of Theorem 10.41 yields f (x) = ae-bx 2 (10.173) (10.174) which is identified as a Gaussian distribution with zero mean. In order that the constraint (10.173) is satisfied, we must have a= 1 2 and b = 1 . 2 (10.175) Hence, in light of (10.66) in Example 10.13, we have proved (10.172) with equality if and only if X N (0, ). Theorem 10.44. Let X be a continuous random variable with mean and variance 2 . Then 1 (10.176) h(X) log(2e 2 ), 2 with equality if and only if X N (, 2 ). 250 10 Differential Entropy Proof. Let X = X - . Then EX = E(X - ) = EX - = 0 and E(X )2 = E(X - )2 = varX = 2 . Applying Theorem 10.14 and Theorem 10.43, we have h(X) = h(X ) 1 log(2e 2 ), 2 (10.179) (10.178) (10.177) and equality holds if and only if X N (0, 2 ), or X N (, 2 ). The theorem is proved. Remark Theorem 10.43 says that with the constraint EX 2 = , the differential entropy is maximized by the distribution N (0, ). If we impose the additional constraint that EX = 0, then varX = EX 2 = . By Theorem 10.44, the differential entropy is still maximized by N (0, ). We have mentioned at the end of Section 10.2 that the differential entropy of a random variable increases with the "spread" of the pdf. Though a simple consequence of Theorem 10.43, the above theorem makes this important interpretation precise. By rewriting the upper bound in (10.179), we obtain h(X) log + 1 log(2e). 2 (10.180) That is, the differential entropy is at most equal to the logarithm of the standard deviation plus a constant. In particular, the differential entropy tends to - as the standard deviation tends to 0. The next two theorems are the vector generalizations of Theorems 10.43 and 10.44. Theorem 10.45. Let X be a vector of n continuous random variables with ~ correlation matrix K. Then h(X) 1 ~ log (2e)n |K| , 2 (10.181) ~ with equality if and only if X N (0, K). Proof. By Theorem 10.41, the joint pdf that maximizes h(X) has the form f (x) = e -0 - i,j ij xi xj = e-0 -x Lx , (10.182) where L = [ij ]. Thus f is a multivariate Gaussian distribution with zero mean. Therefore, Chapter Summary 251 cov(Xi , Xj ) = EXi Xj - (EXi )(EXj ) = EXi Xj (10.183) ~ for all i and j. Since f is constrained by K, 0 and L have the unique solution given by 1 (10.184) e-0 = n ~ 2 |K|1/2 and L= so that f (x) = 2 1 n 1 ~ -1 K , 2 ~ |K|1/2 e- 2 x 1 (10.185) ~ K -1 x , (10.186) ~ the joint pdf of X N (0, K). Hence, by Theorem 10.20, we have proved ~ (10.181) with equality if and only if X N (0, K). Theorem 10.46. Let X be a vector of n continuous random variables with mean and covariance matrix K. Then h(X) 1 log [(2e)n |K|] , 2 (10.187) with equality if and only if X N (, K). Proof. Similar to the proof of Theorem 10.44. Chapter Summary In the following, X = [X1 X2 Xn ] . Covariance Matrix: KX = E(X - EX)(X - EX) = [cov(Xi , Xj )]. ~ Correlation Matrix: KX = EXX = [EXi Xj ]. Diagonalization of a Covariance Matrix: A covariance matrix can be diagonalized as QQ . The diagonal elements of , which are nonnegative, are the eigenvalues of the covariance matrix. Linear Transformation of a Random Vector: Let Y = AX. Then KY = ~ ~ AKX A and KY = AKX A . Decorrelation of a Random Vector: Let Y = Q X, where QQ is a diagonalization of KX . Then KY = , i.e., the random variables in Y are uncorrelated and var Yi = i , the ith diagonal element of . Differential Entropy: h(X) = - S f (x) log f (x)dx = -E log f (X) = -E log f (X). 252 10 Differential Entropy 1. 2. 3. 4. Translation: h(X + c) = h(X). Scaling: h(aX) = h(X) + log |a|. Uniform Distribution on [0, a): h(X) = log a. 1 Gaussian Distribution N (, 2 ): h(X) = 2 log(2e 2 ). Joint Differential Entropy: h(X) = - S f (x) log f (x)dx = -E log f (X). 1. Translation: h(X + c) = h(X). 2. Scaling: h(AX) = h(X) + log |det(A)|. 3. Multivariate Gaussian Distribution N (, K): h(X) = 1 2 log [(2e)n |K|]. Proposition: For fixed f (y|x), f (y) exists for any F (x) and is given by f (y) = f (y|x)dF (x). Conditional (Differential) Entropy and Mutual Information: 1. If Y is continuous, h(Y |X = x) = - SY (x) f (y|x) log f (y|x)dy h(Y |X) = - SX h(Y |X = x)dF (x) = -E log f (Y |X) f (y|x) log f (Y |X) f (y|x) dy dF (x) = E log f (y) f (Y ) f (Y |X, T ) f (Y |T ) I(X; Y ) = SX SY (x) I(X; Y |T ) = ST I(X; Y |T = t)dF (t) = E log h(Y ) = h(Y |X) + I(X; Y ). 2. If Y is discrete, H(Y |X) = H(Y ) - I(X; Y ) H(Y ) = H(Y |X) + I(X; Y ). Chain Rule for Differential Entropy: n h(X1 , X2 , , Xn ) = i=1 h(Xi |X1 , , Xi-1 ). Some Useful Inequalities: Chapter Summary 253 I(X; Y ) 0 I(X; Y |T ) 0 h(Y |X) h(Y ) n h(X1 , X2 , , Xn ) i=1 h(Xi ). AEP I for Continuous Random Variables: - Typical Set: n W[X] = {x X n : -n-1 log f (x) - h(X) < }. 1 log f (X) h(X) in probability. n AEP II for Continuous Random Variables: n 1. 2-n(h(X)+ ) < f (x) < 2-n(h(X)- ) for x W[X] n 2. Pr{X W[X] } > 1 - for sufficiently large n n 3. (1 - )2n(h(X)- ) < Vol(W[X] ) < 2n(h(X)+ ) for sufficiently large n. Informational Divergence: For two probability density functions f and g defined on n , D(f g) = Sf f (x) log f (x) f (X) dx = Ef log . g(x) g(X) Divergence Inequality: D(f g) 0, with equality if and only if f = g a.e. Maximum Differential Entropy Distributions: Let f (x) = e-0 - m i=1 i ri (x) for all x S, where 0 , 1 , , m are chosen such that the constraints ri (x)f (x)dx = ai Sf for 1 i m are satisfied. Then f maximizes h(f ) over all pdf f defined on S subject to the above constraints. Maximum Differential Entropy for a Given Correlation Matrix: h(X) 1 ~ log (2e)n |K| , 2 ~ ~ with equality if and only if X N (0, K), where K is the correlation matrix of X. 254 10 Differential Entropy Maximum Differential Entropy for a Given Covariance Matrix: h(X) 1 log [(2e)n |K|] , 2 with equality if and only if X N (, K), where and K are the mean and the covariance matrix of X, respectively. Problems 1. Prove Propositions 10.3 and 10.8. 2. Show that the joint pdf of a multivariate Gaussian distribution integrates to 1. 3. Show that |K| > 0 in (10.18), the formula for the joint pdf of a multivariate Gaussian distribution. 4. Show that a symmetric positive definite matrix is a covariance matrix. 5. Let 2/4 -3/4 7/4 K = 2/4 5/2 - 2/4 . -3/4 - 2/4 7/4 a) Find the eigenvalues and eigenvectors of K. b) Show that K is positive definite. c) Suppose K is the covariance matrix of a random vector X = [X1 X2 X3 ] . i) Find the coefficient of correlation between Xi and Xj for 1 i < j 3. ii) Find an uncorrelated random vector Y = [Y1 Y2 Y3 ] such that X is a linear transformation of Y. iii) Determine the covariance matrix of Y. 6. Prove Theorem 10.19. 7. For continuous random variables X and Y , discuss why I(X; X) is not equal to h(X). 8. Each of the following continuous distributions can be obtained as the distribution that maximizes the differential entropy subject to a suitable set of constraints: a) the exponential distribution, f (x) = e-x for x 0, where > 0; b) the Laplace distribution, f (x) = for - < x < , where > 0; 1 -|x| e 2 Historical Notes 255 c) the gamma distribution, f (x) = (x)-1 e-x () z-1 -t t e dt; 0 for x 0, where , > 0 and (z) = d) the beta distribution, f (x) = (p + q) p-1 x (1 - x)q-1 (p) (q) for 0 x 1 , where p, q > 0; e) the Cauchy distribution, f (x) = 1 (1 + x2 ) 9. 10. 11. 12. 13. for - < x < . Identify the corresponding set of constraints for each of these distributions. Let be the mean of a continuous random variable X defined on + . Obtain an upper bound on h(X) in terms of . The inequality in (10.180) gives an upper bound on the differential entropy in terms of the variance. Can you give an upper bound on the variance in terms of the differential entropy? For i = 1, 2, suppose fi maximizes h(f ) over all the pdf's defined on Si n subject to the constraints in (10.168), where S1 S2 . Show that h(f1 ) h(f2 ). Hadamard's inequality Show that for a positive semidefinite matrix K, n |K| i=1 Kii , with equality if and only if K is diagonal. Hint: Consider the differential entropy of a multivariate Gaussian distribution. ~ Let KX and KX be the covariance matrix and the correlation matrix ~ of a random vector X, respectively. Show that |KX | |KX |. This is a generalization of varX EX 2 for a random variable X. Hint: Consider a multivariate Gaussian distribution with another multivariate Gaussian distribution with zero mean and the same correlation matrix. Historical Notes The concept of differential entropy was introduced by Shannon [322]. Informational divergence and mutual information were subsequently defined in Kolmogorov [204] and Pinsker [292] in the general setting of measure theory. A measure-theoretic treatment of information theory for continuous systems can be found in the book by Ihara [180]. The treatment in this chapter and the next chapter aims to keep the generality of the results without resorting to heavy use of measure theory. The bounds in Section 10.6 for differential entropy subject to constraints are developed in the spirit of maximum entropy expounded in Jayes [186]. 11 Continuous-Valued Channels In Chapter 7, we have studied the discrete memoryless channel. For such a channel, transmission is in discrete time, and the input and output are discrete. In a physical communication system, the input and output of a channel often take continuous real values. If transmission is in continuous time, the channel is called a waveform channel. In this chapter, we first discuss discrete-time channels with real input and output. We will then extend our discussion to waveform channels. All the logarithms in this chapter are in the base 2. 11.1 Discrete-Time Channels Definition 11.1. Let f (y|x) be a conditional pdf defined for all x, where - SY (x) f (y|x) log f (y|x)dy < (11.1) for all x. A discrete-time continuous channel f (y|x) is a system with input random variable X and output random variable Y such that Y is related to X through f (y|x) (cf. Definition 10.22). Remark The integral in (11.1) is precisely the conditional differential entropy h(Y |X = x) defined in (10.100), which is required to be finite in this definition of a discrete-time continuous channel. Definition 11.2. Let : , and Z be a real random variable, called the noise variable. A discrete-time continuous channel (, Z) is a system with a real input and a real output. For any input random variable X, the noise random variable Z is independent of X, and the output random variable Y is given by Y = (X, Z). (11.2) 258 11 Continuous-Valued Channels For brevity, a discrete-time continuous channel will be referred to as a continuous channel. Definition 11.3. Two continuous channels f (y|x) and (, Z) are equivalent if for every input distribution F (x), x y Pr{(X, Z) y, X x} = - - fY |X (v|u)dv dFX (u) (11.3) for all x and y. Remark In the above definitions, the input random variable X is not necessarily continuous. Definitions 11.1 and 11.2 are two definitions for a continuous channel which are analogous to Definitions 7.1 and 7.2 for a discrete channel. While Definitions 7.1 and 7.2 are equivalent, Definition 11.2 is more general than Definition 11.1. For a continuous channel defined in Definition 11.2, the noise random variable Z may not have a pdf, and the function (x, ) may be manyto-one. As a result, the corresponding conditional pdf f (y|x) as required in Definition 11.1 may not exist. In this chapter, we confine our discussion to continuous channels that can be defined by Definition 11.1 (and hence also by Definition 11.2). Definition 11.4. A continuous memoryless channel (CMC) f (y|x) is a sequence of replicates of a generic continuous channel f (y|x). These continuous channels are indexed by a discrete-time index i, where i 1, with the ith channel being available for transmission at time i. Transmission through a channel is assumed to be instantaneous. Let Xi and Yi be respectively the input and the output of the CMC at time i, and let Ti- denote all the random variables that are generated in the system before Xi . The Markov chain Ti- Xi Yi holds, and x y Pr{Yi y, Xi x} = - - fY |X (v|u)dv dFXi (u). (11.4) Definition 11.5. A continuous memoryless channel (, Z) is a sequence of replicates of a generic continuous channel (, Z). These continuous channels are indexed by a discrete-time index i, where i 1, with the ith channel being available for transmission at time i. Transmission through a channel is assumed to be instantaneous. Let Xi and Yi be respectively the input and the output of the CMC at time i, and let Ti- denote all the random variables that are generated in the system before Xi . The noise variable Zi for the transmission at time i is a copy of the generic noise variable Z, and is independent of (Xi , Ti- ). The output of the CMC at time i is given by Yi = (Xi , Zi ). (11.5) 11.1 Discrete-Time Channels 259 Definition 11.6. Let be a real function. An average input constraint (, P ) for a CMC is the requirement that for any codeword (x1 , x2 , , xn ) transmitted over the channel, n 1 (xi ) P. (11.6) n i=1 For brevity, an average input constraint is referred to as an input constraint. Definition 11.7. The capacity of a continuous memoryless channel f (y|x) with input constraint (, P ) is defined as C(P ) = sup F (x):E(X)P I(X; Y ), (11.7) where X and Y are respectively the input and output of the generic continuous channel, and F (x) is the distribution of X. Theorem 11.8. C(P ) is non-decreasing, concave, and left-continuous. Proof. In the definition of C(P ), the supremum is taken over a larger set for a larger P . Therefore, C(P ) is non-decreasing in P . We now show that C(P ) is concave. Let j = 1, 2. For an input distribution Fj (x), denote the corresponding input and output random variables by Xj and Yj , respectively. Then for any Pj , for all > 0, there exists Fj (x) such that E(Xj ) Pj (11.8) and I(Xj ; Yj ) C(Pj ) - . For 0 1, let = 1 - and define the random variable X () F1 (x) + F2 (x). Then E(X () ) = E(X1 ) + E(X2 ) P1 + P2 , (11.11) By the concavity of mutual information with respect to the input distribution1 , we have I(X () ; Y () ) I(X1 ; Y1 ) + I(X2 ; Y2 ) (C(P1 ) - ) + (C(P2 ) - ) = C(P1 ) + C(P2 ) - . Then 1 (11.9) (11.10) (11.12) (11.13) (11.14) Specifically, we refer to the inequality (3.124) in Example 3.14 with X and Y being real random variables related by a conditional pdf f (y|x). The proof of this inequality is left as an exercise. 260 11 Continuous-Valued Channels C(P1 + P2 ) I(X () ; Y () ) C(P1 ) + C(P2 ) - . Letting 0, we have C(P1 + P2 ) C(P1 ) + C(P2 ), (11.15) (11.16) proving that C(P ) is concave. Finally, we prove that C(P ) is left-continuous. Let P1 < P2 in (11.16). Since C(P ) is non-decreasing, we have C(P2 ) C(P1 + P2 ) C(P1 ) + C(P2 ). Letting 0, we have C(P2 ) lim C(P1 + P2 ) C(P2 ), 0 (11.17) (11.18) which implies 0 lim C(P1 + P2 ) = C(P2 ). (11.19) Hence, we conclude that P P2 lim C(P ) = C(P2 ), (11.20) i.e., C(P ) is left-continuous. The theorem is proved. 11.2 The Channel Coding Theorem Definition 11.9. An (n, M ) code for a continuous memoryless channel with input constraint (, P ) is defined by an encoding function e : {1, 2, , M } and a decoding function g: n n (11.21) {1, 2, , M }. (11.22) The set {1, 2, , M }, denoted by W, is called the message set. The sequences e(1), e(2), , e(M ) in X n are called codewords, and the set of codewords is called the codebook. Moreover, 1 n n (xi (w)) P i=1 for 1 w M , (11.23) where e(w) = (x1 (w), x2 (w), , xn (w)). 11.2 The Channel Coding Theorem 261 We assume that a message W is randomly chosen from the message set W according to the uniform distribution. Therefore, H(W ) = log M. With respect to a channel code for a given CMC, we let X = (X1 , X2 , , Xn ) and Y = (Y1 , Y2 , , Yn ) (11.26) be the input sequence and the output sequence of the channel, respectively. Evidently, X = e(W ). (11.27) We also let ^ W = g(Y) be the estimate of the message W by the decoder. Definition 11.10. For all 1 w M , let ^ w = Pr{W = w|W = w} = yY n :g(y)=w (11.24) (11.25) (11.28) Pr{Y = y|X = e(w)} (11.29) be the conditional probability of error given that the message is w. We now define two performance measures for a channel code. Definition 11.11. The maximal probability of error of an (n, M ) code is defined as max = max w . (11.30) w Definition 11.12. The average probability of error of an (n, M ) code is defined as ^ Pe = Pr{W = W }. (11.31) Evidently, Pe max . Definition 11.13. A rate R is asymptotically achievable for a continuous memoryless channel if for any > 0, there exists for sufficiently large n an (n, M ) code such that 1 log M > R - (11.32) n and max < . (11.33) For brevity, an asymptotically achievable rate will be referred to as an achievable rate. Theorem 11.14 (Channel Coding Theorem). A rate R is achievable for a continuous memoryless channel if and only if R C, the capacity of the channel. 262 11 Continuous-Valued Channels 11.3 Proof of the Channel Coding Theorem 11.3.1 The Converse We can establish the Markov chain ^ W XYW (11.34) very much like the discrete case as discussed in Section 7.3. Here, although X is a real random vector, it takes only discrete values as it is a function of the message W which is discrete. The only continuous random variable in the above Markov chain is the random vector Y, which needs to be handled with caution. The following lemma is essentially the data processing theorem we proved in Theorem 2.42 except that Y is continuous. The reader may skip the proof at the first reading. Lemma 11.15. ^ I(W ; W ) I(X; Y). (11.35) Proof. We first consider ^ ^ I(W ; W ) I(W, X; W ) ^ ^ = I(X; W ) + I(W ; W |X) ^ = I(X; W ). (11.36) (11.37) (11.38) Note that all the random variables above are discrete. Continuing from the above, we have ^ ^ I(W ; W ) I(X; W ) ^ ^ I(X; W ) + I(X; Y|W ) ^ ^ f (Y|X, W ) p(X, W ) + E log = E log ^ ^ p(X)p(W ) f (Y|W ) ^ ^ p(X, W )f (Y|X, W ) = E log ^ ^ p(X)[p(W )f (Y|W )] ^ f (Y)p(X, W |Y) = E log ^ p(X)[f (Y)p(W |Y)] ^ p(X, W |Y) = E log ^ p(X)p(W |Y) ^ p(X|Y)p(W |X, Y) = E log ^ p(X)p(W |Y) ^ p(X|Y) p(W |X, Y) = E log + E log ^ p(X) p(W |Y) (11.39) (11.40) (11.41) (11.42) (11.43) (11.44) (11.45) (11.46) 11.3 Proof of the Channel Coding Theorem 263 ^ f (Y|X) p(X|Y, W ) + E log f (Y) p(X|Y) p(X|Y) = I(X; Y) + E log p(X|Y) = I(X; Y) + E log 1 = E log = I(X; Y) + 0 = I(X; Y). The above steps are justified as follows: The relation f (y|x) = i=1 n (11.47) (11.48) (11.49) (11.50) (11.51) f (yi |xi ) (11.52) can be established in exactly the same way as we established (7.101) for the discrete case (when the channel is used without feedback). Then f (y|x, w) = ^ p(x)f (y|x)p(w|y) ^ , p(x, w) ^ (11.53) ^ and f (y|w) exists by Proposition 10.24. Therefore, I(X; Y|W ) in (11.40) ^ can be defined according to Definition 10.27. (11.40) follows from Corollary 10.32. In (11.43), given f (y|x) as in (11.52), it follows from Proposition 10.24 that f (y) exists. (11.47) follows from p(x)f (y|x) = f (y)p(x|y) and p(x|y)p(w|x, y) = p(w|y)p(x|y, w). ^ ^ ^ (11.55) (11.54) ^ (11.48) follows from the Markov chain X Y W . The proof is accomplished. We now proceed to prove the converse. Let R be an achievable rate, i.e., for any > 0, there exists for sufficiently large n and (n, M ) code such that 1 log M > R - n and max < . Consider (11.57) (11.56) 264 11 Continuous-Valued Channels log M = H(W ) ^ ^ = H(W |W ) + I(W ; W ) ^ H(W |W ) + I(X; Y) ^ = H(W |W ) + h(Y) - h(Y|X) n (11.58) (11.59) (11.60) (11.61) (11.62) (11.63) (11.64) ^ H(W |W ) + i=1 n h(Yi ) - h(Y|X) n ^ = H(W |W ) + i=1 n h(Yi ) - i=1 h(Yi |Xi ) ^ = H(W |W ) + i=1 I(Xi ; Yi ). The above steps are justified as follows: (11.60) follows from Lemma 11.15. It follows from (11.52) that n h(Y|X) = i=1 h(Yi |Xi ). (11.65) Then (11.1) in Definition 11.1 implies that h(Yi |Xi ) is finite for all i, and hence h(Y|X) is also finite. From the foregoing, f (y) exists. Therefore, h(Y) can be defined according to Definition 10.10 (but h(Y) may be infinite), and (11.61) follows from Proposition 10.29 because h(Y|X) is finite. Note that it is necessary to require h(Y|X) to be finite because otherwise h(Y) is also infinite and Proposition 10.29 cannot be applied. (11.62) follows from Corollary 10.34, the independence bound for differential entropy. (11.63) follows from (11.65) above. (11.64) follows from Proposition 10.29. Let V be a mixing random variable distributed uniformly on {1, 2, , n} which is independent of Xi , 1 i n. Let X = XV and Y be the output of the channel with X being the input. Then E(X) = EE[(X)|V ] n (11.66) (11.67) (11.68) (11.69) = i=1 n Pr{V = i}E[(X)|V = i] Pr{V = i}E[(Xi )|V = i] i=1 n = = i=1 1 E(Xi ) n 11.3 Proof of the Channel Coding Theorem 265 =E P, 1 n n (Xi ) i=1 (11.70) (11.71) where the above inequality follows from (11.23) in the definition of the code. By the concavity of mutual information with respect to the input distribution, we have n 1 I(Xi ; Yi ) I(X; Y ) C, (11.72) n i=1 where the last inequality holds in light of the definition of C and (11.71). Then it follows from (11.64) that ^ log M H(W |W ) + nC, (11.73) which is precisely (7.126) in the proof of the converse of the channel coding theorem for the DMC. Following exactly the same steps therein, we conclude that R C. (11.74) 11.3.2 Achievability The proof of the achievability of the channel capacity, which involves the construction of a random code, is somewhat different from the construction for the discrete case in Section 7.4. On the one hand, we need to take into account the input constraint. On the other hand, since the input distribution F (x) we use for constructing the random code may not have a pdf, it is difficult to formulate the notion of joint typicality as in the discrete case. Instead, we will introduce a different notion of typicality based on mutual information. Consider a bivariate information source {(Xk , Yk ), k 1}, where (Xk , Yk ) are i.i.d. with (X, Y ) being the pair of generic real random variables. The conditional pdf f (y|x) exists in the sense prescribed in Definition 10.22. By Proposition 10.24, f (y) exists so that the mutual information I(X; Y ) can be defined according to Definition 10.26. n Definition 11.16. The mutually typical set [XY ] with respect to F (x, y) is the set of (x, y) X n Y n such that 1 f (y|x) log - I(X; Y ) , n f (y) where f (y|x) = i=1 n (11.75) f (yi |xi ) (11.76) and 266 11 Continuous-Valued Channels n f (y) = i=1 f (yi ), (11.77) and is an arbitrarily small positive number. A pair of sequences (x, y) is n called mutually -typical if it is in [XY ] . Lemma 11.17. For any > 0, for sufficiently large n, n Pr{(X, Y) [XY ] )} 1 - . (11.78) Proof. By (11.76) and (11.77), we write f (Y|X) 1 1 1 f (Yi |Xi ) log = log = n f (Y) n f (Yi ) n i=1 n n log i=1 f (Yi |Xi ) . f (Yi ) f (Yi |Xi ) f (Yi ) . (11.79) Thus we Since (Xi , Yi ) are i.i.d., so are the random variables log conclude by the weak law of large numbers that 1 n n log i=1 f (Yi |Xi ) f (Y |X) E log = I(X; Y ) f (Yi ) f (Y ) (11.80) in probability, i.e., (11.78) holds for all sufficiently large n, proving the lemma. The following lemma is analogous to Lemma 7.17 for the discrete case. Lemma 11.18. Let (X , Y ) be n i.i.d. copies of a pair of generic random variables (X , Y ), where X and Y are independent and have the same marginal distributions as X and Y , respectively. Then n Pr{(X , Y ) [XY ] } 2-n(I(X;Y )-) . (11.81) n Proof. For (x, y) [XY ] , from (11.75), we obtain 1 f (y|x) log I(X; Y ) - , n f (y) or f (y|x) f (y)2n(I(X;Y )-) Then n 1 Pr{(X, Y) [XY ] )} (11.82) (11.83) (11.84) (11.85) (11.86) (11.87) = n [XY ] f (y|x)dF (x) dy f (y)dF (x) dy n [XY ] 2n(I(X;Y )-) n = 2n(I(X;Y )-) Pr{(X , Y ) [XY ] }, 11.3 Proof of the Channel Coding Theorem 267 where the last inequality follows from (11.83). Therefore, n Pr{(X , Y ) [XY ] } 2-n(I(X;Y )-) , (11.88) proving the lemma. Fix any > 0 and let be a small quantity to be specified later. Since C(P ) is left-continuous, there exists a sufficiently small > 0 such that C(P - ) > C(P ) - . 6 (11.89) By the definition of C(P - ), there exists an input random variable X such that E(X) P - (11.90) and I(X; Y ) C(P - ) - . 6 Then choose for a sufficiently large n an even integer M satisfying I(X; Y ) - from which we obtain 1 log M > I(X; Y ) - n 6 C(P - ) - > C(P ) - . 2 We now describe a random coding scheme: 1. Construct the codebook C of an (n, M ) code randomly by generating M codewords in n independently and identically according to F (x)n . ~ ~ ~ Denote these codewords by X(1), X(2), , X(M ). 2. Reveal the codebook C to both the encoder and the decoder. 3. A message W is chosen from W according to the uniform distribution. ~ 4. The sequence X = X(W ), namely the W th codeword in the codebook C, is transmitted through the channel. 5. The channel outputs a sequence Y according to n yi (11.91) 6 < 1 log M < I(X; Y ) - , n 8 (11.92) (11.93) 3 (11.94) (11.95) Pr{Yi yi , 1 i n|X(W ) = x} = i=1 - f (y|xi )dy. (11.96) This is the continuous analog of (7.101) and can be established similarly. 268 11 Continuous-Valued Channels n 6. The sequence Y is decoded to the message w if (X(w), Y) [XY ] and n there does not exist w = w such that (X(w ), Y) [XY ] . Otherwise, ^ Y is decoded to a constant message in W. Denote by W the message to which Y is decoded. We now analyze the performance of this random coding scheme. Let ~ ~ ~ ~ X(w) = (X1 (w), X2 (w), , Xn (w)) and define the error event Err = Ee Ed , where Ee = 1 n n (11.97) (11.98) ~ (Xi (W )) > P i=1 (11.99) is the event that the input constraint is violated, and ^ Ed = {W = W } (11.100) is the event that a decoding error occurs. By symmetry in the code construction, Pr{Err} = Pr{Err|W = 1} Pr{Ee |W = 1} + Pr{Ed |W = 1}. (11.101) (11.102) With Lemma 11.18 in place of Lemma 7.17, the analysis of Pr{Ed |W = 1} is exactly the same as the analysis of the decoding error in the discrete case. The details are omitted, and we conclude that by choosing to be a sufficiently small positive quantity, Pr{Ed |W = 1} (11.103) 4 for sufficiently large n. We now analyze Pr{Ee |W = 1}. By (11.90) and the weak law of large numbers, Pr{Ee |W = 1} = Pr = Pr 1 n 1 n 1 n 1 n n ~ (Xi (1)) > P W = 1 i=1 n (11.104) ~ (Xi (1)) > P i=1 n (11.105) = Pr Pr ~ (Xi (1)) > (P - ) + i=1 n (11.106) ~ (Xi (1)) > E(X) + i=1 (11.107) (11.108) 4 11.3 Proof of the Channel Coding Theorem 269 for sufficiently large n. It then follows from (11.102) and (11.103) that Pr{Err} 2 (11.109) for sufficiently large n. It remains to show the existence of a codebook such that max < the input constraint (11.23) is satisfied by every codeword. Consider Pr{Err} = C and Pr{C}Pr{Err|C}, (11.110) where Pr{C} is the probability of choosing a codebook C from the ensemble of all possible codebooks in Step 1 of the random coding scheme. In light of (11.109), there exists at least one codebook C such that Pr{Err|C } Furthermore, M 2 . (11.111) Pr{Err|C } = w=1 M Pr{W = w|C }Pr{Err|C , W = w} Pr{W = w}Pr{Err|C , W = w} w=1 M (11.112) = 1 = M (11.113) Pr{Err|C , W = w}. w=1 (11.114) ~ By discarding the worst half of the codewords in C , if a codeword X(w) remains in C , then Pr{Err|C , W = w} . (11.115) Since Err = Ee Ed , this implies Pr{Ee |C , W = w} and Pr{Ed |C , W = w} , (11.117) where the latter implies max for the codebook C . Finally, observe that ~ conditioning on {C , W = w}, the codeword X(w) is deterministic. Therefore, ~ Pr{Ee |C , W = w} is equal to 1 if the codeword X(w) violates the input constraint (11.23), otherwise it is equal to 0. Then (11.116) implies that for ~ every codeword X(w) that remains in C , Pr{Ee |C , W = w} = 0, i.e., the input constraint is satisfied. This completes the proof. (11.116) 270 11 Continuous-Valued Channels 11.4 Memoryless Gaussian Channels In communication engineering, the Gaussian channel is the most commonly used model for a noisy channel with real input and output. The reasons are two-fold. First, the Gaussian channel is highly analytically tractable. Second, the Gaussian noise can be regarded as the worst kind of additive noise subject to a constraint on the noise power. This will be discussed in Section 11.9. We first give two equivalent definitions of a Gaussian channel. Definition 11.19 (Gaussian Channel). A Gaussian channel with noise energy N is a continuous channel with the following two equivalent specifications: 1. f (y|x) = (y-x) 1 e- 2N 2N 2 . 2. Z N (0, N ) and (X, Z) = X + Z. Definition 11.20 (Memoryless Gaussian Channel). A memoryless Gaussian channel with noise power N and input power constraint P is a memoryless continuous channel with the generic continuous channel being the Gaussian channel with noise energy N . The input power constraint P refers to the input constraint (, P ) with (x) = x2 . Using the formula in Definition 11.7 for the capacity of a CMC, the capacity of a Gaussian channel can be evaluated. Theorem 11.21 (Capacity of a Memoryless Gaussian Channel). The capacity of a memoryless Gaussian channel with noise power N and input power constraint P is 1 P log 1 + . (11.118) 2 N The capacity is achieved by the input distribution N (0, P ). We first prove the following lemma. Lemma 11.22. Let Y = X + Z. Then h(Y |X) = h(Z|X) provided that fZ|X (z|x) exists for all x SX . Proof. For all x SX , since fZ|X (z|x) exists, fY |X (y|x) also exists and is given by fY |X (y|x) = fZ|X (y - x|x). (11.119) Then h(Y |X = x) is defined as in (10.100), and 11.4 Memoryless Gaussian Channels 271 h(Y |X) = = = = h(Y |X = x)dFX (x) h(X + Z|X = x)dFX (x) h(x + Z|X = x)dFX (x) h(Z|X = x)dFX (x) (11.120) (11.121) (11.122) (11.123) (11.124) = h(Z|X). In the above, (11.120) and (11.124) follow from (10.101), while (11.123) follows from the translation property of differential entropy (Theorem 10.14). Remark Since Y and Z uniquely determine each other given X, it is tempting to write h(Y |X) = h(Z|X) immediately. However, this interpretation is incorrect because differential entropy is not the same as entropy. Proof of Theorem 11.21. Let F (x) be the CDF of the input random variable X such that EX 2 P , where X is not necessarily continuous. Since Z N (0, N ), f (y|x) is given by (11.119). Then by Proposition 10.24, f (y) exists and hence h(Y ) is defined. Since Z is independent of X, by Lemma 11.22 and Corollary 10.33, h(Y |X) = h(Z|X) = h(Z). (11.125) Then I(X; Y ) = h(Y ) - h(Y |X) = h(Y ) - h(Z), (11.126) (11.127) where (11.126) follows from Proposition 10.29 and (11.127) follows from (11.125). Since Y = X + Z and Z is independent of X, we have EY 2 = E(X + Z)2 = EX + 2(EXZ) + EZ 2 2 2 (11.128) (11.129) (11.130) (11.131) (11.132) (11.133) 2 = EX 2 + 2(EX)(EZ) + EZ 2 = EX + 2(EX)(0) + EZ = EX 2 + EZ 2 P + N. Given the above constraint on Y , by Theorem 10.43, we have h(Y ) 1 log[2e(P + N )], 2 (11.134) with equality if Y N (0, P + N ). 272 11 Continuous-Valued Channels Recall from Example 10.13 that h(Z) = 1 log(2eN ). 2 (11.135) It then follows from (11.127), (11.134), and (11.135) that I(X; Y ) = h(Y ) - h(Z) 1 1 log[2e(P + N )] - log(2eN ) 2 2 1 P = log 1 + . 2 N Evidently, this upper bound is tight if X N (0, P ), because then Y = X + Z N (0, P + N ). Therefore, C= = = The theorem is proved. Theorem 11.21 says that the capacity of a memoryless Gaussian channel depends only on the ratio of the input power constraint P to the noise power N . This important quantity is called the signal-to-noise ratio. Note that no matter how small the signal-to-noise ratio is, the capacity is still strictly positive. In other words, reliable communication can still be achieved, though at a low rate, when the noise power is much higher than the signal power. We also see that the capacity is infinite if there is no constraint on the input power. sup F (x):EX 2 P F (x):EX 2 P (11.136) (11.137) (11.138) (11.139) I(X; Y ) I(X; Y ) . (11.140) (11.141) (11.142) max 1 P log 1 + 2 N 11.5 Parallel Gaussian Channels In Section 11.4, we have discussed the capacity of a memoryless Gaussian channel. Now suppose k such channels are available for communication, where k 1. This is illustrated in Figure 11.1, with Xi , Yi , and Zi being the input, the output, and the noise variable of the ith channel, respectively, where Zi N (0, Ni ) and Zi , 1 i k are independent. We are interested in the capacity of such a system of parallel Gaussian channels, with the total input power constraint 11.5 Parallel Gaussian Channels Z1 X1 273 + Z2 Y1 X2 + Y2 Xk + Fig. 11.1. A system of parallel Gaussian channels. k E i=1 2 Xi P. Let X = [X1 X2 Xk ], Y = [Y1 Y2 Yk ], and Z = [Z1 Z2 Zk ]. Then k fY|X (y|x) = i=1 k fYi |Xi (yi |xi ) fZi |Xi (yi - xi |xi ) i=1 k ... Zk Yk (11.143) (11.144) = (11.145) = i=1 fZi (yi - xi ). (11.146) With the existence of f (y|x), by extending Definition 10.23, we have h(Y|X) = - SX SY (x) f (y|x) log f (y|x)dy dF (x). (11.147) Then by Proposition 10.25, f (y) exists and therefore h(Y) is defined. By extending Definition 10.26, we have I(X; Y) = SX SY (x) f (y|x) log f (y|x) dy dF (x). f (y) (11.148) It then follows from Definition 11.7 that the capacity of the system is given by 274 11 Continuous-Valued Channels C(P ) = sup F (x):E i 2 Xi P I(X; Y), (11.149) where F (x) is the joint CDF of the input vector X. As we will see, the supremum above is indeed a maximum. When we calculated the capacity of the memoryless Gaussian channel in Theorem 11.21, we obtained in (11.132) that EY 2 = EX 2 + EZ 2 , (11.150) i.e., the output power is equal to the sum of the input power and the noise power, provided that the noise has zero mean. By exactly the same argument, we see that 2 2 EYi2 = EXi + EZi (11.151) for all i. Toward calculating C(P ), consider I(X; Y) = h(Y) - h(Y|X) = h(Y) - h(Z|X) = h(Y) - h(Z) k (11.152) (11.153) (11.154) (11.155) = h(Y) - i=1 h(Zi ) 1 2 k = h(Y) - k log(2eNi ) i=1 (11.156) i=1 h(Yi ) - 1 2 1 2 1 2 1 2 1 2 k 1 2 k log(2eNi ) i=1 (11.157) log[2e(EYi2 )] - i=1 k 1 2 k log(2eNi ) i=1 (11.158) = log(EYi2 ) - i=1 k 1 2 k log Ni i=1 (11.159) = 2 2 log(EXi + EZi ) - i=1 k k 1 2 k log Ni i=1 (11.160) = log(Pi + Ni ) - i=1 k 1 2 log Ni i=1 (11.161) = log 1 + i=1 Pi Ni , (11.162) 2 where Pi = EXi is the input power of the ith channel. In the above, (11.153) is the vector generalization of Lemma 11.22, (11.155) follows because Zi are independent, and (11.160) follows from (11.151). 11.5 Parallel Gaussian Channels 275 Equality holds in (11.157) and (11.158) if Yi , 1 i k are independent and Yi N (0, Pi + Ni ). This happens when Xi are independent of each other and Xi N (0, Pi ). Therefore, maximizing I(X; Y) becomes maximizing i log(Pi + Ni ) in (11.161) with the constraint i Pi P and Pi 0 for all i. In other words, we are to find the optimal input power allocation among the channels. Comparing (11.162) with (11.142), we see that the capacity of the system of parallel Gaussian channels is equal to the sum of the capacities of the individual Gaussian channels with the input power optimally allocated. Toward this end, we first apply the method of Lagrange multipliers by temporarily ignoring the nonnegativity constraints on Pi . Observe that in order for the summation i log(Pi +Ni ) in (11.161) to be maximized, i Pi = P must hold because log (Pi + Ni ) is increasing in Pi . Therefore, the inequality constraint i Pi P can be replaced by the equality constraint i Pi = P . Let k k J= i=1 log (Pi + Ni ) - i=1 Pi . (11.163) Differentiating with respect to Pi gives 0= which implies Pi = Upon letting = log e , log e J = - , Pi P i + Ni log e - Ni . (11.164) (11.165) we have Pi = - Ni , (11.166) where is chosen such that k k Pi = i=1 i=1 ( - Ni ) = P. (11.167) However, Pi as given in (11.166) is not guaranteed to be nonnegative, so it may not be a valid solution. Nevertheless, (11.166) suggests the general solution to be proved in the next proposition. Proposition 11.23. The problem For given i 0, maximize and ai 0 has the solution a = ( - i )+ , i where 1 i k, (11.168) k i=1 log(ai + i ) subject to i ai P 276 11 Continuous-Valued Channels (x)+ = and satisfies k x if x 0 0 if x < 0 (11.169) ( - i )+ = P. i=1 (11.170) Proof. Rewrite the maximization problem as For given i 0, maximize k i log (ai + i ) subject to (11.171) 1 i k. (11.172) ai P i=1 -ai 0, We will prove the proposition by verifying that the proposed solution in (11.168) satisfies the Karush-Kuhn-Tucker (KKT) condition. This is done by finding nonnegative and i satisfying the equations (a i log e - + i = 0 + i ) k (11.173) (11.174) 1 i k, (11.175) P- i=1 a i =0 i a = 0, i where and i are the multipliers associated with the constraints in (11.171) and (11.172), respectively. To avoid triviality, assume P > 0 so that > 0, and observe that there exists at least one i such that a > 0. For those i, (11.175) implies i i = 0. On the other hand, a = ( - i )+ = - i . i Substituting (11.176) and (11.177) into (11.173), we obtain = log e > 0. 2 (11.178) (11.177) (11.176) For those i such that a = 0, it follows from (11.168) that i . From i (11.178) and (11.173), we obtain i = (log e) 1 1 - i 0. (11.179) Thus we have found nonnegative and i that satisfy (11.173) to (11.175), verifying the KKT condition. The proposition is proved. 11.6 Correlated Gaussian Channels 277 Hence, following (11.162) and applying the above proposition with ai = Pi and i = Ni , we see that the capacity of the system of parallel Gaussian channels is equal to k 1 P log 1 + i , (11.180) 2 i=1 Ni where {Pi , 1 i k} is the optimal input power allocation among the channels given by Pi = ( - Ni )+ , 1 i k (11.181) with satisfying k ( - Ni )+ = P. i=1 (11.182) The process for obtaining {Pi }, called water-filling, is illustrated in Figure 11.2. One can image that an amount P of water is poured into a reservoir with an uneven bottom, and is the level the water rises to. Under this scheme, high input power is allocated to a channel with low noise power. For a channel with noise power higher than , no input power is allocated, i.e., the channel is not used. 11.6 Correlated Gaussian Channels In this section, we generalize the results in the last section to the case when the noise variables Zi , 1 i k are correlated with covariance matrix KZ . We continue to assume that Zi has zero mean for all i, i.e., Z N (0, KZ ), and the total input power constraint P3* P* 1 Power P2* N1 N2 Channel 1 Channel 2 Channel 3 Channel 4 N4 N3 Fig. 11.2. Water-filling for parallel Gaussian channels. 278 11 Continuous-Valued Channels Z X Y X' Q + Q! Y' Fig. 11.3. An equivalent system of parallel Gaussian channels. k E i=1 2 Xi P (11.183) prevails. We will derive the capacity of such a system of correlated Gaussian channels by decorrelating the noise vector Z. Let KZ be diagonalizable as QQ and consider Y = X + Z. (11.184) Then Q Y = Q X + Q Z. Upon letting X =Q X Y =Q Y and Z = Q Z, we obtain Y =X +Z. Note that EZ = E(QZ) = Q(EZ) = Q 0 = 0, (11.190) and Z is jointly Gaussian because it is a linear transformation of Z. By Proposition 10.6, the random variables in Z are uncorrelated, and KZ = . (11.191) (11.189) (11.188) (11.186) (11.187) (11.185) Hence, Zi N (0, i ), where i is the ith diagonal element of , and Zi , 1 i k are mutually independent. We are then motivated to convert the given system of correlated Gaussian channels into the system shown in Figure 11.3, with X and Y being the input and output, respectively. Note that in this system, X and Y are related to X and Y as prescribed in (11.186) and (11.187), respectively. We then see from (11.189) that Z is the equivalent noise vector of the system with Zi being the noise variable of the ith channel. Hence, the system in Figure 11.3 11.6 Correlated Gaussian Channels Z Q! X' X Y 279 X'' Q + Q! Y' Q Y'' Fig. 11.4. A system identical to the system of correlated Gaussian channels. is a system of parallel Gaussian channels. By Proposition 10.9, the total input power constraint in (11.183) for the original system translates to the total input power constraint k E i=1 (Xi )2 P (11.192) for the system in Figure 11.3. The question is whether the capacity of the system in Figure 11.3 is the same as the capacity of the original system. Let us called these two capacities C and C, respectively. Intuitively, C and C should be the same because the matrix Q is invertible. A formal proof goes as follows. We remind the reader that the capacity of a channel is the highest possible asymptotic rate at which information can be transmitted reliably through the channel by means of any encoding/decoding process. In Figure 11.3, by regarding the transformation Q on X as part of the encoding process and the transformation Q on Y as part of the decoding process, we see that C C. Now further convert the system in Figure 11.3 into the system in Figure 11.4 with input X and output Y , and call the capacity of this system C . By repeating the same argument, we see that C C . Thus C C C. However, the system in Figure 11.4 is equivalent to the original system because Q Q = I. Therefore, C = C, which implies C = C. Upon converting the given system of correlated Gaussian channels into an equivalent system of parallel Gaussian channels, we see that the capacity of the system is equal to k a 1 log 1 + i (11.193) 2 i=1 i where a is the optimal power allocated to the ith channel in the equivalent i system, and its value can be obtained by water-filling as prescribed in Proposition 11.23. The reader should compare (11.193) with the formula in (11.180) for the capacity of parallel Gaussian channels. Let A be the kk diagonal matrix with a being the ith diagonal element. i From the discussion in the last section, the optimal distribution for the input X in the equivalent system of parallel channels is N (0, A ). Accordingly, the distribution of X is N (0, QA Q ). We leave it as an exercise for the reader to verify that this indeed gives the optimal input distribution for the original system of correlated Gaussian channels. 280 11 Continuous-Valued Channels 11.7 The Bandlimited White Gaussian Channel In this section, we discuss a bandlimited waveform channel with zero-mean additive white Gaussian noise (AWGN). In the rest of this chapter, the letters j and f are reserved for -1 and "frequency," respectively. We begin with a few definitions from signal analysis. All the signals are assumed to be real. Definition 11.24. The Fourier transform of a signal g(t) is defined as G(f ) = - g(t)e-j2f t dt. (11.194) The signal g(t) can be recovered from G(f ) as g(t) = - G(f )ej2f t df, (11.195) and g(t) is called the inverse Fourier transform of G(f ). The functions g(t) and G(f ) are said to form a transform pair, denoted by g(t) G(f ). (11.196) The variables t and f are referred to as time and frequency, respectively. In general, the Fourier transform of a signal g(t) may not exist. A sufficient condition for the Fourier transform of g(t) to exist is that g(t) has finite energy, i.e., |g(t)|2 dt < . - (11.197) A signal with finite energy is called an energy signal. Definition 11.25. Let g1 (t) and g2 (t) be a pair of energy signals. The crosscorrelation function for g1 (t) and g2 (t) is defined as R12 ( ) = - g1 (t)g2 (t - )dt. (11.198) Proposition 11.26. For a pair of energy signals g1 (t) and g2 (t), R12 ( ) G1 (f )G (f ), 2 (11.199) where G (f ) denotes the complex conjugate of G2 (f ). 2 Definition 11.27. For a wide-sense stationary process {X(t), - < t < }, the autocorrelation function is defined as RX ( ) = E[X(t + )X(t)], (11.200) 11.7 The Bandlimited White Gaussian Channel 281 which does not depend on t, and the power spectral density is defined as SX (f ) = - RX ( )e-j2f d, (11.201) i.e., RX ( ) SX (f ). (11.202) Definition 11.28. Let {(X(t), Y (t)), - < t < } be a bivariate wide-sense stationary process. Their cross-correlation functions are defined as RXY ( ) = E[X(t + )Y (t)] and RY X ( ) = E[Y (t + )X(t)], which do not depend on t. The cross-spectral densities are defined as (11.203) (11.204) SXY (f ) = - RXY ( )e-j2f d (11.205) and SY X (f ) = RY X ( )e-j2f d, - (11.206) i.e., RXY ( ) and RY X ( ) SY X (f ). (11.208) We now describe the simplest nontrivial model for a waveform channel. In wired-line and wireless communication, the frequency spectrum of the medium is often partitioned into a number of communication channels, where each channel occupies a certain frequency band. Consider such a channel that occupies the frequency band [fl , fh ] with 0 fl < fh , where W = fh - fl (11.209) SXY (f ) (11.207) is called the bandwidth. The input process X(t) is contaminated by a zeromean additive white Gaussian noise process Z(t) with power N0 , i.e., 2 SZ (f ) = N0 , 2 - < f < . (11.210) In reality, such a noise process cannot exist because its total power is infinite. For practical purposes, one can regard the power spectral density to be constant within the range of interest of the problem. Let h(t) be the impulse response of an ideal bandpass filter for the frequency band [fl , fh ], i.e., 282 11 Continuous-Valued Channels H(f ) = 1 if fl |f | fh 0 otherwise. (11.211) At the receiver for this channel, the ideal bandpass filter h(t) is applied to the received signal in order to filter out the frequency components due to other channels. Effectively, we can regard this filtered version of the received signal given by Y (t) = [X(t) + Z(t)] h(t) = X(t) h(t) + Z(t) h(t) (11.212) as the output of the channel, where denotes convolution in the time domain. Letting X (t) = X(t) h(t) (11.213) and Z (t) = Z(t) h(t), (11.212) can be written as Y (t) = X (t) + Z (t). (11.215) (11.214) The only difference between X(t) and X (t) is that all the frequency components of X (t) are in [fl , fh ], while X(t) can have frequency components outside this range. However, even if such frequency components exist in X(t), they are filtered out by the ideal bandpass filter h(t) and do not appear in the output process Y (t). Therefore, we can regard X (t) instead of X(t) as the input process of the channel. By the same token, we regard Z (t) instead of Z(t) as the noise process of the channel. As for the memoryless Gaussian channel discussed in the last section, we impose an average power constraint P on the input process X (t). For simplicity, we consider in this section the case that the channel we have described occupies the frequency band [0, W ]. This channel, called the bandlimited white Gaussian channel, is the special case of the general model with fl = 0. While a rigorous formulation of the bandlimited white Gaussian channel involves mathematical tools beyond the scope of this book, we will nevertheless give a heuristic argument that suggests the formula for the channel capacity. The sampling theorem in signal analysis will allow us to "convert" this waveform channel into a memoryless Gaussian channel discussed in the last section. Theorem 11.29 (Sampling Theorem). Let g(t) be a signal with Fourier transform G(f ) that vanishes for f [-W, W ]. Then g(t) = i=- g i 2W sinc(2W t - i) (11.216) for - < t < , where 11.7 The Bandlimited White Gaussian Channel 283 sin(t) , t called the sinc function, is defined to be 1 at t = 0 by continuity. sinc(t) = Letting gi = and i (t) = 1 g 2W i 2W (11.217) (11.218) (11.219) 2W sinc(2W t - i), the formula in (11.216) can be rewritten as g(t) = i=- gi i (t). (11.220) Proposition 11.30. i (t), - < i < form an orthonormal basis for signals which are bandlimited to [0, W ]. Proof. Since i (t) = 0 t - i 2W , (11.221) i (t) and 0 (t) have the same energy. We first show that sinc2 (2W t)dt = - 1 . 2W (11.222) This integral is difficult to evaluate directly. Instead we consider sinc(2W t) where rect(f ) = 1 rect 2W f 2W 1 2 , (11.223) 1 1 -2 f 0 otherwise. (11.224) Then by Rayleigh's energy theorem, we have sinc2 (2W t)dt = - - 1 2W 2 rect2 f 2W df = 1 . 2W (11.225) It then follows that 2 i (t)dt = - - 2 0 (t)dt (11.226) sinc2 (2W t)dt (11.227) (11.228) (11.229) = ( 2W )2 = 2W = 1. 1 2W - 284 11 Continuous-Valued Channels Next, we show that sinc(2W t - i) sinc(2W t - i )dt - (11.230) vanishes whenever i = i . Again, this integral is difficult to evaluate directly. Since (11.225) implies that both sinc(2W t - i) and sinc(2W t - i ) have finite energy, we can consider their cross-correlation function, denoted by Rii ( ). Now sinc(2W t - i) and sinc(2W t - i ) Then we have Rii ( ) Gi (f )G (f ), i (11.233) and the integral in (11.230) is given by e-j2f ( 2W ) i 1 2W rect f 2W := Gi (f ) (11.231) e -j2f i 2W 1 2W rect f 2W := Gi (f ). (11.232) Rii (0) = - Gi (f )G (f )df i (11.234) (cf. (11.195)), which vanishes whenever i = i . Therefore, i (t)i (t)dt = 2W - - sinc(2W t - i) sinc(2W t - i )dt (11.235) (11.236) = 0. Together with (11.229), this shows that i (t), - < i < form an orthonormal set. Finally, since g(t) in (11.220) is any signal bandlimited to [0, W ], we conclude that i (t), - < i < form an orthonormal basis for such signals. The theorem is proved. Let us return to our discussion of the waveform channel. The sampling theorem implies that the input process X (t), assuming the existence of the Fourier transform, can be written as X (t) = i=- Xi i (t), (11.237) where Xi = 1 X 2W i 2W , (11.238) 11.7 The Bandlimited White Gaussian Channel 285 and there is a one-to-one correspondence between X (t) and {Xi , - < i < }. The same applies to (a realization of) the output process Y (t), which we assume can be written as Y (t) = i=- Yi i (t), (11.239) where Yi = 1 Y 2W i 2W . (11.240) With these assumptions on X (t) and Y (t), the waveform channel is equivalent i to a discrete-time channel defined at t = 2W , with the ith input and output of the channel being Xi and Yi , respectively. Toward determining the capacity of this equivalent discrete-time channel, we prove in the following a characterization of the effect of the noise process Z (t) at the sampling times. i Proposition 11.31. Z 2W , - < i < are i.i.d. Gaussian random variables with zero mean and variance N0 W . Proof. First of all, Z(t) is a zero-mean Gaussian process and Z (t) is a filtered version of Z(t), so Z (t) is also a zero-mean Gaussian process. Consequently, i Z 2W , - < i < are zero-mean Gaussian random variables. The power spectral density of Z (t) is given by SZ (f ) = N0 2 0 -W f W otherwise. (11.241) Then the autocorrelation function of Z (t), which is the inverse Fourier transform of SZ (f ), is given by RZ ( ) = N0 W sinc(2W ). (11.242) i It is seen that the value of RZ ( ) is equal to 0 when = 2W for all i = 0, because the sinc function in (11.217) vanishes at all nonzero integer values i of t. This shows that Z 2W , - < i < are uncorrelated and hence i independent because they are jointly Gaussian. Finally, since Z 2W has zero mean, in light of (11.200), its variance is given by RZ (0) = N0 W . Recall from (11.215) that Y (t) = X (t) + Z (t). Then Y i 2W i 2W i 2W (11.243) =X +Z . (11.244) 286 11 Continuous-Valued Channels Upon dividing by 2W and letting Zi = 1 Z 2W i 2W , (11.245) it follows from (11.238) and (11.240) that Yi = Xi + Zi . (11.246) i Since Z 2W , - < i < are i.i.d. with distribution N (0, N0 W ), Zi , - < i < are i.i.d. with distribution N (0, N0 ). 2 Thus we have shown that the bandlimited white Gaussian channel is equivalent to a memoryless Gaussian channel with noise power equal to N0 . As we 2 are converting the waveform channel into a discrete-time channel, we need to relate the input power constraint of the waveform channel to the input power constraint of the discrete-time channel. Let P be the average energy (i.e., the second moment) of the Xi 's. We now calculate the average power of X (t) in terms of P . Since i (t) has unit energy, the average contribution to the energy of X (t) by each sample is P . As there are 2W samples per unit time and i (t), - < i < are orthonormal, X (t) accumulates energy from the samples at a rate equal to 2W P . Upon considering 2W P P, (11.247) where P is the average power constraint on the input process X (t), we obtain P P , 2W (11.248) i.e., an input power constraint P for the bandlimited Gaussian channel transP lates to an input power constraint 2W for the discrete-time channel. By Theorem 11.21, the capacity of the memoryless Gaussian channel is 1 P/2W log 1 + 2 N0 /2 = P 1 log 1 + 2 N0 W bits per sample. (11.249) Since there are 2W samples per unit time, we conclude that the capacity of the bandlimited Gaussian channel is W log 1 + P N0 W bits per unit time. (11.250) The argument we have given above is evidently not rigorous because if there is no additional constraint on the Xi 's other than their average energy P not exceeding 2W , then X (t) may not have finite energy. This induces a gap in the argument because the Fourier transform of X (t) may not exist and hence the sampling theorem cannot be applied. A rigorous formulation of the bandlimited white Gaussian channel involves the consideration of an input signal of finite duration, which is analogous to a 11.8 The Bandlimited Colored Gaussian Channel 287 code for the DMC with a finite block length. Since a signal with finite duration cannot be bandlimited, this immediate leads to a contradiction. Overcoming this technical difficulty requires the use of prolate spheroidal wave functions [338, 218, 219] which are bandlimited functions with most of the energy on a finite interval. The main idea is that there are approximately 2W T orthonormal basis functions for the set of signals which are bandlimited to W and have most of the energy on [0, T ) in time. We refer the reader to Gallager[129] for a rigorous treatment of the bandlimited white Gaussian channel. 11.8 The Bandlimited Colored Gaussian Channel In the last section, we have discussed the bandlimited white Gaussian channel occupying the frequency band [0, W ]. We presented a heuristic argument that led to the formula in (11.250) for the channel capacity. Suppose the channel instead occupies the frequency band [fl , fh ], with fl being a multiple of W = fh - fl . Then the noise process Z (t) has power spectral density SZ (f ) = N0 2 0 if fl |f | fh otherwise. (11.251) We refer to such a channel as the bandpass white Gaussian channel. By an extenstion of the heuristic argument for the bandlimited white Gaussian channel, which would involve the bandpass version of the sampling theorem, the same formula for the channel capacity can be obtained. The details are omitted here. We now consider a waveform channel occupying the frequency band [0, W ] with input power constraint P and zero-mean additive colored Gaussian noise Z(t). We refer to this channel as the bandlimited colored Gaussian channel. To analyze the capacity of this channel, divide the interval [0, W ] into subintervals i [fli , fh ] for 1 i k, where fli = (i - 1)k i fh (11.252) (11.253) = ik , k = and W (11.254) k is the width of each subinterval. As an approximation, assume SZ (f ) is equal i to a constant SZ,i over the subinterval [fli , fh ]. Then the channel consists of k sub-channels, with the ith sub-channel being a bandpass (bandlimited if i i = 1) white Gaussian channel occupying the frequency band [fli , fh ]. Thus by letting N0 = 2SZ,i in (11.251), we obtain from (11.250) that the capacity of the ith sub-channel is equal to k log 1 + Pi 2SZ,i k (11.255) 288 11 Continuous-Valued Channels if Pi is the input power allocated to that sub-channel. The noise process of the ith sub-channel, denoted by Zi (t), is obtained by passing Z(t) through the ideal bandpass filter with frequency response Hi (f ) = i 1 if fli |f | fh 0 otherwise. (11.256) It can be shown (see Problem 9) that the noise processes Zi (t), 1 i k are independent. By converting each sub-channel into an equivalent memoryless Gaussian channel as discussed in the last section, we see that the k subchannels can be regarded as a system of parallel Gaussian channels. Thus the channel capacity is equal to the sum of the capacities of the individual sub-channels when the power allocation among the k sub-channels is optimal. Let Pi be the optimal power allocation for the ith sub-channel. Then it follows from (11.255) that the channel capacity is equal to Pi k log 1 + 2SZ,i k i=1 where by Proposition 11.23, Pi = ( - SZ,i )+ , 2k or Pi = 2k ( - SZ,i )+ , with k k k = i=1 k log 1 + Pi 2k SZ,i , (11.257) (11.258) (11.259) Pi = P. i=1 (11.260) Then from (11.259) and (11.260), we obtain k ( - SZ,i )+ k = i=1 P . 2 (11.261) As k , following (11.257) and (11.258), k k log 1 + i=1 Pi 2k k SZ,i = 0 k log 1 + i=1 W ( - SZ,i )+ SZ,i df df, (11.262) (11.263) (11.264) log 1 + 1 2 W ( - SZ (f ))+ SZ (f ) = and following (11.261), log 1 + -W ( - SZ (f ))+ SZ (f ) 11.9 Zero-Mean Gaussian Noise is the Worst Additive Noise 289 SZ ( f ) ! -W 0 W f Fig. 11.5. Water-filling for the bandlimited colored Gaussian channel. k W ( - SZ,i )+ k i=1 0 ( - SZ (f ))+ df 1 2 W (11.265) (11.266) = ( - SZ (f ))+ df, -W where (11.264) and (11.266) are obtained by noting that SZ (-f ) = SZ (f ) (11.267) for - < f < (see Problem 8). Hence, we conclude that the capacity of the bandlimited colored Gaussian channel is equal to 1 2 W log 1 + -W ( - SZ (f ))+ SZ (f ) W df bits per unit time, (11.268) where satisfies ( - SZ (f ))+ df = P -W (11.269) in view of (11.261) and (11.266). Figure 11.5 is an illustration of the waterfilling process for determining , where the amount of water to be poured into the reservoir is equal to P . 11.9 Zero-Mean Gaussian Noise is the Worst Additive Noise In the last section, we derived the capacity for a system of correlated Gaussian channels, where the noise vector is a zero-mean Gaussian random vector. In this section, we show that in terms of the capacity of the system, the zeromean Gaussian noise is the worst additive noise given that the noise vector has a fixed correlation matrix. Note that the diagonal elements of this correlation 290 11 Continuous-Valued Channels matrix specify the power of the individual noise variables, while the other elements in the matrix give a characterization of the correlation between the noise variables. Theorem 11.32. For a fixed zero-mean Gaussian random vector X , let Y = X + Z, (11.270) where the joint pdf of Z exists and Z is independent of X . Under the constraint that the correlation matrix of Z is equal to K, where K is any symmetric positive definite matrix, I(X ; Y) is minimized if and only if Z N (0, K). Before proving this theorem, we first prove the following two lemmas. Lemma 11.33. Let X be a zero-mean random vector and Y = X + Z, where Z is independent of X. Then ~ ~ ~ KY = KX + KZ . (11.272) (11.271) Proof. For any i and j, consider EYi Yj = E(Xi + Zi )(Xj + Zj ) = E(Xi Xj + Xi Zj + Zi Xj + Zi Zj ) = EXi Xj + EXi Zj + EZi Xj + EZi Zj = EXi Xj + (EXi )(EZj ) + (EZi )(EXj ) + EZi Zj = EXi Xj + (0)(EZj ) + (EZi )(0) + EZi Zj = EXi Xj + EZi Zj , (11.273) (11.274) (11.275) (11.276) (11.277) (11.278) where (11.277) follows from the assumption that Xi has zero mean for all i. The proposition is proved. The reader should compare this lemma with Lemma 10.8 in Chapter 10. Note that in Lemma 10.8, it is not necessary to assume that either X or Z has zero mean. Lemma 11.34. Let Y N (0, K) and Y be any random vector with correlation matrix K. Then fY (y) log fY (y)dy = SY fY (y) log fY (y)dy. (11.279) 11.9 Zero-Mean Gaussian Noise is the Worst Additive Noise 291 Proof. The random vector Y has joint pdf fY (y) = 1 2 k e- 2 (y |K|1/2 1 K -1 y) (11.280) ~ for all y k . Since EY = 0, KY = KY = K. Therefore, Y and Y have the same correlation matrix. Consider [ln fY (y)] fY (y)dy 1 - (y K -1 y) - ln ( 2)k |K|1/2 fY (y)dy 2 1 =- (y K -1 y)fY (y)dy - ln ( 2)k |K|1/2 2 1 =- (K -1 )ij yi yj fY (y)dy - ln ( 2)k |K|1/2 2 i,j = =- =- = SY (11.281) (11.282) (11.283) (11.284) (11.285) (11.286) (11.287) 1 2 1 2 (K -1 )ij i,j (yi yj )fY (y)dy - ln ( 2)k |K|1/2 (yi yj )fY (y)dy - ln ( 2)k |K|1/2 SY (K -1 )ij i,j 1 - y K -1 y - ln ( 2)k |K|1/2 2 [ln fY (y)] fY (y)dy. fY (y)dy = SY In the above, (11.285) is justified because Y and Y have the same correlation matrix, and (11.286) is obtained by backtracking the manipulations from (11.281) to (11.284) with fY (y) in place of fY (y). The lemma is proved upon changing the base of the logarithm. Proof of Theorem 11.32. Let Z N (0, K) such that Z is independent of X , and let Y = X + Z . (11.288) Obviously, the support of Y is k because Y has a multivariate Gaussian distribution. Note that the support of Y is also k regardless of the distribution of Z because the support of X is k . We need to prove that for any random vector Z with correlation matrix K, where Z is independent of X and the joint pdf of Z exists, I(X ; Y ) I(X ; Y). (11.289) ~ Since EZ = 0, KZ = KZ = K. Therefore, Z and Z have the same correlation matrix. By noting that X has zero mean, we apply Lemma 11.33 to see that Y and Y have the same correlation matrix. 292 11 Continuous-Valued Channels The inequality in (11.289) can be proved by considering I(X ; Y ) - I(X ; Y) = h(Y ) - h(Z ) - h(Y) + h(Z) =- + =- + = = SZ d) c) b) a) (11.290) fZ (z) log fZ (z)dz fY (y) log fY (y)dy + fY (y) log fY (y)dy - SZ fZ (z) log fZ (z)dz fZ (z) log fZ (z)dz SZ (11.291) fY (y) log fY (y)dy + fY (y) log fY (y)dy - SZ fZ (z) log fZ (z)dz log SZ (11.292) (11.293) (11.294) (11.295) (11.296) (11.297) (11.298) log fY (y) fY (y) log fY (y)dy + fZ (z) fZ (z) fZ (z)dz fY (y)fZ (z) fY (y)fZ (z) fYZ (y, z)dydz log SZ fY (y)fZ (z) fYZ (y, z)dydz fY (y)fZ (z) 1 fY (y) fX (y - z)fZ (z)dz fY (y)dy SZ = log f) e) log = 0. fY (y) fY (y)dy fY (y) The above steps are explained as follows: We assume that the pdf of Z exists so that h(Z) is defined. Moreover, fY|X (y|x) = fZ (y - x). (11.299) Then by Proposition 10.24, fY (y) exists and hence h(Y) is defined. In b), we have replaced fY (y) by fY (y) in the first integral and replaced fZ (z) by fZ (z) in the second integral. The former is justified by an application of Lemma 11.34 to Y and Y by noting that Y is a zero-mean Gaussian random vector and Y and Y have the same correlation matrix. The latter is justified similarly. To justify c), we need SYZ = k SZ , which can be seen by noting that fYZ (y, z) = fY|Z (y|z)fZ (z) = fX (y - z)fZ (z) > 0 (11.300) for all y k and all z SZ . d) follows from Jensen's inequality and the concavity of the logarithmic function. 11.9 Zero-Mean Gaussian Noise is the Worst Additive Noise 293 e) follows from (11.300). f) follows because fX (y - z)fZ (z)dz = Sz Sz fY |Z (y|z)fZ (z)dz fY |Z (y|z)fZ (z)dz (11.301) (11.302) (11.303) = fY (y). It remains to show that Z N (0, K) is a necessary and sufficient condition for I(X ; Y) to be minimized. Toward this end, we need to show that equality holds in both (11.295) and (11.297) if and only if Z N (0, K). Assume that Z N (0, K). Then fZ (z) = fZ (z) for all z SZ = k . This implies fY (y) = fY (y) for all y k in view of (11.270) and (11.288), so that fY (y)fZ (z) = fY (y)fZ (z) for all y k , z SZ . (11.304) Then the iterated integral in (11.294) is evaluated to zero. Therefore, it follows from (11.294) to (11.298) that equality holds in both (11.295) and (11.297). Conversely, assume that equality holds in both (11.295) and (11.297). In view of (11.302), equality holds in (11.297) if and only if k \SZ has zero Lebesgue measure. On the other hand, equality holds in (11.295) if and only if fY (y)fZ (z) = cfY (y)fZ (z) (11.305) a.e. on k SZ for some constant c. Integrating with respect to dy over k , we obtain fZ (z) fY (y)dy = cfZ (z) fY (y)dy, (11.306) which implies fZ (z) = cfZ (z) (11.307) a.e. on SZ . Further integrating with respect to dz over SZ , we obtain fZ (z)dz = c SZ SZ fZ (z)dz = c. (11.308) Since k \SZ has zero Lebesgue measure, fZ (z)dz = SZ fZ (z)dz = 1, (11.309) which together with (11.308) implies c = 1. It then follows from (11.307) that fZ (z) = fZ (z) a.e. on SZ and hence on k , i.e., Z N (0, K). Therefore, we conclude that Z N (0, K) is a necessary and sufficient condition for I(X ; Y) to be minimized. The theorem is proved. 294 11 Continuous-Valued Channels Consider the system of correlated Gaussian channels discussed in the last section. Denote the noise vector by Z and its correlation matrix by K. Note that K is also the covariance matrix of Z because Z has zero mean. In other words, Z N (0, K). Refer to this system as the zero-mean Gaussian system and let C be its capacity. Then consider another system with exactly the same specification except that the noise vector, denoted by Z, may neither be zero-mean nor Gaussian. We, however, require that the joint pdf of Z exists. Refer to this system as the alternative system and let C be its capacity. We now apply Theorem 11.32 to show that C C . Let X be the input random vector that achieves the capacity of the zero-mean Gaussian system. We have mentioned at the end of Section 11.6 that X is a zero-mean Gaussian random vector. Let Y and Y be defined in (11.288) and (11.270), which correspond to the outputs of the zero-mean Gaussian system and the alternative system, respectively when X is the input of both systems. Then C I(X ; Y) I(X ; Y ) = C , (11.310) where the second inequality follows from (11.289) in the proof of Theorem 11.32. Hence, we conclude that the zero-mean Gaussian noise is indeed the worst additive noise subject to a constraint on the correlation matrix. Chapter Summary Capacity of Continuous Memoryless Channel: For a continuous memoryless channel f (y|x) with average input constraint (, P ), namely the requirement that n 1 (xi ) P n i=1 for any codeword (x1 , x2 , , xn ) transmitted over the channel, C(P ) = sup F (x):E(X)P I(X; Y ), where F (x) is the input distribution of the channel. C(P ) is non-decreasing, concave, and left-continuous. Mutually Typical Set: For a joint distribution F (x, y), n [XY ] = (x, y) X n Y n : 1 f (y|x) log - I(X; Y ) . n f (y) Lemma: For any > 0 and sufficiently large n, n Pr{(X, Y) [XY ] )} 1 - . Chapter Summary 295 Lemma: Let X and Y be a pair of random variables, and (X , Y ) be n i.i.d. copies of a pair of generic random variables (X , Y ) where X and Y are independent and have the same marginal distributions as X and Y , respectively. Then n Pr{(X , Y ) [XY ] } 2-n(I(X;Y )-) . Channel Coding Theorem: A message drawn uniformly from the set {1, 2, , 2n(R- ) } can be transmitted through a continuous memoryless channel with negligible probability of error as n if and only if R C. Capacity of Memoryless Gaussian Channel: C= P 1 log 1 + 2 N . Capacity of Parallel Gaussian Channels: For a system of parallel Gaussian channels with noise variable Zi N (0, Ni ) for the ith channel and total input power constraint P , 1 C= 2 where satisfies k i=1 ( k log 1 + i=1 ( - Ni )+ Ni , - Ni )+ = P. Capacity of Correlated Gaussian Channels: For a system of correlated Gaussian channels with noise vector Z N (0, KZ ) and total input power constraint P , k 1 ( - i )+ C= log 1 + , 2 i=1 i where KZ is diagonalizable as QQ with i being the ith diagonal element k of , and satisfies i=1 ( - i )+ = P. Capacity of the Bandlimited White Gaussian Channel: C = W log 1 + P N0 W bits per unit time. Capacity of the Bandlimited Colored Gaussian Channel: 1 2 W log 1 + -W W ( -W ( - SZ (f ))+ SZ (f ) df bits per unit time, where satisfies - SZ (f ))+ df = P. Zero-Mean Gaussian is the Worst Additive Noise: For a system of channels with additive noise, if the correlation matrix of the noise vector is given, the capacity is minimized when the noise vector is zero-mean and Gaussian. 296 11 Continuous-Valued Channels Problems In the following, X, Y, Z, etc denote vectors of random variables. 1. Verify the two properties in Theorem 11.8 for the capacity of the memoryless Gaussian channel. 2. Let X and Y be two jointly distributed random variables with Y being continuous. The random variable Y is related to the random variable X through a conditional pdf f (y|x) defined for all x (cf. Definition 10.22). Prove that I(X; Y ) is concave in F (x). 3. Refer to Lemma 11.18 and prove that n Pr{(X , Y ) [XY ] } (1 - )2-n(I(X;Y )-) for n sufficiently large. 4. Show that the capacity of a continuous memoryless channel is not changed if (11.23) is replaced by 1 E n n (xi (W )) P, i=1 i.e., the average input constraint is satisfied on the average by a randomly selected codeword instead of by every codeword in the codebook. 5. Show that Rii (0) in (11.234) vanishes if and only if i = i . 6. Consider a system of Gaussian channels with noise vector Z (0, KZ ) and input power constraint equal to 3. Determine the capacity of the system for the following two cases: a) 400 KZ = 0 5 0 ; 002 b) 7/4 2/4 -3/4 KZ = 2/4 5/2 - 2/4 . -3/4 - 2/4 7/4 For b), you may use the results in Problem 5 in Chapter 10. 7. In the system of correlated Gaussian channels, let KZ be diagonalizable as QQ . Let A be the k k diagonal matrix with a being the ith diagonal i element, where a is prescribed in the discussion following (11.193). Show i that N (0, QA Q ) is the optimal input distribution. 8. Show that for a wide-sense stationary process X(t), SX (f ) = SX (-f ) for all f . Historical Notes 297 9. Consider a zero-mean white Gaussian noise process Z(t). Let h1 (t) and h2 (t) be two impulse responses such that the supports of H1 (f ) and H2 (f ) do not overlap. a) Show that for any t and t , the two random variables Z(t) h1 (t) and Z(t) h2 (t ) are independent. b) Show that the two processes Z(t) h1 (t) and Z(t) h2 (t) are independent. c) Repeat a) and b) if Z(t) is a zero-mean colored Gaussian noise process. Hint: Regard Z(t) as obtained by passing a zero-mean white Gaussian noise process through a coloring filter. 10. Interpret the bandpass white Gaussian channel as a special case of the bandlimited colored Gaussian channel in terms of the channel capacity. 11. Independent Gaussian noise is the worst Let C be the capacity of a system of k Gaussian channels with Zi N (0, Ni ). By ignoring the possible correlation among the noise variables, we can use the channels in the system independently as parallel Gaussian channels. Thus C is lower bounded by the expression in (11.180). In this sense, a Gaussian noise vector is the worst if its components are uncorrelated. Justify this claim analytically. Hint: Show that I(X; Y) i I(Xi ; Yi ) if Xi are independent. Historical Notes Channels with additive Gaussian noise were first analyzed by Shannon in [322], where the formula for the capacity of the bandlimited white Gaussian channel was given. The form of the channel coding theorem for the continuous memoryless channel presented in this chapter was first proved in the book by Gallager [129]. A rigorous proof of the capacity formula for the bandlimited white Gaussian channel was obtained by Wyner [387]. The water-filling solution to the capacity of the bandlimited colored Gaussian channel was developed by Shannon in [324] and was proved rigorously by Pinsker [292]. The discussion in this chapter on the continuous memoryless channel with an average input constraint is adapted from the discussions in the book by Gallager [129] and the book by Ihara [180], where in the former a comprehensive treatment of waveform channels can also be found. The Gaussian noise being the worst additive noise was proved by Ihara [179]. The proof presented here is due to Diggavi and Cover [95]. 12 Markov Structures We have proved in Section 3.5 that if X1 X2 X3 X4 forms a Markov chain, the I-Measure always vanishes on the five atoms ~ ~c ~ ~c X1 X2 X3 X4 ~ ~c ~ ~ X1 X2 X3 X4 ~ ~c ~c ~ X1 X2 X3 X4 ~ ~ ~c ~ X1 X2 X3 X4 ~c ~ ~c ~ X1 X2 X3 X4 . Consequently, the I-Measure is completely specified by the values of on the other ten nonempty atoms of F4 , and the information diagram for four random variables forming a Markov chain can be displayed in two dimensions as in Figure 3.11. Figure 12.1 is a graph which represents the Markov chain X1 X2 X3 X4 . The observant reader would notice that always vanishes on a nonempty atom A of F4 if and only if the graph in Figure 12.1 becomes disconnected upon removing all the vertices corresponding to the complemented set ~ ~c ~ ~c variables in A. For example, always vanishes on the atom X1 X2 X3 X4 , and the graph in Figure 12.1 becomes disconnected upon removing vertices 2 and 4. On the other hand, does not necessarily vanish on the atom ~c ~ ~ ~c X1 X2 X3 X4 , and the graph in Figure 12.1 remains connected upon removing vertices 1 and 4. This observation will be explained in a more general setting in the subsequent sections. (12.1) 1 2 3 4 Fig. 12.1. The graph representing the Markov chain X1 X2 X3 X4 . 300 12 Markov Structures The theory of I-Measure establishes a one-to-one correspondence between Shannon's information measures and set theory. Based on this theory, we develop in this chapter a set-theoretic characterization of a Markov structure called full conditional mutual independence. A Markov chain, and more generally a Markov random field, is a collection of full conditional mutual independencies. We will show that if a collection of random variables forms a Markov random field, then the structure of can be simplified. In particular, when the random variables form a Markov chain, exhibits a very simple structure so that the information diagram can be displayed in two dimensions regardless of the length of the Markov chain, and is always nonnegative. (See also Sections 3.5 and 3.6.) The topics to be covered in this chapter are fundamental. Unfortunately, the proofs of the results are very heavy. At the first reading, the reader should understand the theorems through the examples instead of getting into the details of the proofs. 12.1 Conditional Mutual Independence In this section, we explore the effect of conditional mutual independence on the structure of the I-Measure . We begin with a simple example. Example 12.1. Let X, Y , and Z be mutually independent random variables. Then I(X; Y ) = I(X; Y ; Z) + I(X; Y |Z) = 0. (12.2) Since I(X; Y |Z) 0, we let I(X; Y |Z) = a 0, so that I(X; Y ; Z) = -a. Similarly, I(Y ; Z) = I(X; Y ; Z) + I(Y ; Z|X) = 0 and I(X; Z) = I(X; Y ; Z) + I(X; Z|Y ) = 0. Then from (12.4), we obtain I(Y ; Z|X) = I(X; Z|Y ) = a. (12.7) (12.6) (12.5) (12.4) (12.3) The relations (12.3), (12.4), and (12.7) are shown in the information diagram in Figure 12.2, which indicates that X, Y , and Z are pairwise independent. We have proved in Theorem 2.39 that X, Y , and Z are mutually independent if and only if H(X, Y, Z) = H(X) + H(Y ) + H(Z). (12.8) 12.1 Conditional Mutual Independence 301 Y a -a a a X Z Fig. 12.2. X, Y , and Z are pairwise independent. By counting atoms in the information diagram, we see that 0 = H(X) + H(Y ) + H(Z) - H(X, Y, Z) = I(X; Y |Z) + I(Y ; Z|X) + I(X; Z|Y ) + 2I(X; Y ; Z) = a. Thus a = 0, which implies I(X; Y |Z), I(Y ; Z|X), I(X; Z|Y ), I(X; Y ; Z) are all equal to 0. Equivalently, vanishes on ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ X Y - Z, Y Z - X, X Z - Y , X Y Z, (12.13) (12.12) (12.9) (12.10) (12.11) which are precisely the atoms in the intersection of any two of the set variables ~ ~ ~ X, Y , and Z. Conversely, if vanishes on the sets in (12.13), then we see from (12.10) that (12.8) holds, i.e., X, Y , and Z are mutually independent. Therefore, X, Y , and Z are mutually independent if and only if vanishes on the sets in (12.13). This is shown in the information diagram in Figure 12.3. The theme of this example will be extended to conditional mutual independence among collections of random variables in Theorem 12.9, which is Y 0 0 0 0 X Z Fig. 12.3. X, Y , and Z are mutually independent. 302 12 Markov Structures the main result in this section. In the rest of the section, we will develop the necessary tools for proving this theorem. At first reading, the reader should try to understand the results by studying the examples without getting into the details of the proofs. In Theorem 2.39, we have proved that X1 , X2 , , Xn are mutually independent if and only if n H(X1 , X2 , , Xn ) = i=1 H(Xi ). (12.14) By conditioning on a random variable Y , one can readily prove the following. Theorem 12.2. X1 , X2 , , Xn are mutually independent conditioning on Y if and only if n H(X1 , X2 , , Xn |Y ) = i=1 H(Xi |Y ). (12.15) We now prove two alternative characterizations of conditional mutual independence. Theorem 12.3. X1 , X2 , , Xn are mutually independent conditioning on Y if and only if for all 1 i n, I(Xi ; Xj , j = i|Y ) = 0, i.e., Xi and (Xj , j = i) are independent conditioning on Y . Remark A conditional independency is a special case of a conditional mutual independency. However, this theorem says that a conditional mutual independency is equivalent to a set of conditional independencies. Proof of Theorem 12.3. It suffices to prove that (12.15) and (12.16) are equivalent. Assume (12.15) is true, i.e., X1 , X2 , , Xn are mutually independent conditioning on Y . Then for all i, Xi is independent of (Xj , j = i) conditioning on Y . This proves (12.16). Now assume that (12.16) is true for all 1 i n. Consider 0 = I(Xi ; Xj , j = i|Y ) = I(Xi ; X1 , X2 , , Xi-1 |Y ) +I(Xi ; Xi+1 , , Xn |Y, X1 , X2 , , Xi-1 ). Since mutual information is always nonnegative, this implies I(Xi ; X1 , , Xi-1 |Y ) = 0, (12.19) (12.18) (12.17) (12.16) or Xi and (X1 , X2 , , Xi-1 ) are independent conditioning on Y . Therefore, X1 , X2 , , Xn are mutually independent conditioning on Y (see the proof of Theorem 2.39), proving (12.15). Hence, the theorem is proved. 12.1 Conditional Mutual Independence 303 Theorem 12.4. X1 , X2 , , Xn are mutually independent conditioning on Y if and only if n H(X1 , X2 , , Xn |Y ) = i=1 H(Xi |Y, Xj , j = i). (12.20) Proof. It suffices to prove that (12.15) and (12.20) are equivalent. Assume (12.15) is true, i.e., X1 , X2 , , Xn are mutually independent conditioning on Y . Since for all i, Xi is independent of Xj , j = i conditioning on Y , H(Xi |Y ) = H(Xi |Y, Xj , j = i) Therefore, (12.15) implies (12.20). Now assume that (12.20) is true. Consider H(X1 , X2 , , Xn |Y ) n (12.21) = i=1 n H(Xi |Y, X1 , , Xi-1 ) [H(Xi |Y, Xj , j = i) + I(Xi ; Xi+1 , , Xn |Y, X1 , , Xi-1 )] i=1 (12.22) = (12.23) n n = i=1 H(Xi |Y, Xj , j = i) + i=1 I(Xi ; Xi+1 , , Xn |Y, X1 , , Xi-1 ). (12.24) Then (12.20) implies n I(Xi ; Xi+1 , , Xn |Y, X1 , , Xi-1 ) = 0. i=1 (12.25) Since all the terms in the above summation are nonnegative, they must all be equal to 0. In particular, for i = 1, we have I(X1 ; X2 , , Xn |Y ) = 0. By symmetry, it can be shown that I(Xi ; Xj , j = i|Y ) = 0 (12.27) (12.26) for all 1 i n. Then this implies (12.15) by the last theorem, completing the proof. 304 12 Markov Structures Theorem 12.5. Let C and Qi be disjoint index sets and Wi be a subset of Qi for 1 i k, where k 2. Assume that there exist at least two i such that Wi = . Let XQi = (Xl , l Qi ), 1 i k, and XC = (Xl , l C) be collections of random variables. If XQi , 1 i k, are mutually independent conditioning on XC , then XWi such that Wi = are mutually independent conditioning on (XC , XQi -Wi , 1 i k). We first give an example before we prove the theorem. Example 12.6. Suppose X1 , (X2 , X3 , X4 ), and (X5 , X6 ) are mutually independent conditioning on X7 . By Theorem 12.5, X1 , X2 , and (X5 , X6 ) are mutually independent conditioning on (X3 , X4 , X7 ). Proof of Theorem 12.5. Assume XQi , 1 i k, are mutually independent conditioning on XC , i.e., k H(XQi , 1 i k|XC ) = i=1 H(XQi |XC ). (12.28) Consider H(XWi , 1 i k|XC , XQi -Wi , 1 i k) = H(XQi , 1 i k|XC ) - H(XQi -Wi , 1 i k|XC ) k (12.29) = i=1 H(XQi |XC ) k - i=1 k H(XQi -Wi |XC , XQj -Wj , 1 j i - 1) H(XQi |XC , XQj -Wj , 1 j i - 1) (12.30) i=1 k - i=1 k H(XQi -Wi |XC , XQj -Wj , 1 j i - 1) H(XWi |XC , XQj -Wj , 1 j i) (12.31) = i=1 k (12.32) i=1 H(XWi |XC , XQj -Wj , 1 j k). (12.33) In the second step we have used (12.28), and the two inequalities follow because conditioning does not increase entropy. On the other hand, by the chain rule for entropy, we have 12.1 Conditional Mutual Independence 305 H(XWi , 1 i k|XC , XQi -Wi , 1 i k) k = i=1 H(XWi |XC , (XQj -Wj , 1 j k), (XWl , 1 l i - 1)). (12.34) Therefore, it follows from (12.33) that k H(XWi |XC , XQj -Wj , 1 j k) i=1 (12.35) (12.36) H(XWi , 1 i k|XC , XQi -Wi , 1 i k) k = i=1 H(XWi |XC , (XQj -Wj , 1 j k), (XWl , 1 l i - 1)). (12.37) However, since conditioning does not increase entropy, the ith term in the summation in (12.35) is lower bounded by the ith term in the summation in (12.37). Thus we conclude that the inequality in (12.36) is an equality. Hence, the conditional entropy in (12.36) is equal to the summation in (12.35), i.e., H(XWi , 1 i k|XC , XQi -Wi , 1 i k) k (12.38) (12.39) = i=1 H(XWi |XC , XQj -Wj , 1 j k). The theorem is proved. Theorem 12.5 specifies a set of conditional mutual independencies (CMI's) which is implied by a CMI. This theorem is crucial for understanding the effect of a CMI on the structure of the I-Measure , which we discuss next. Lemma 12.7. Let (Zi1 , , Ziti ), 1 i r be r collections of random variables, where r 2, and let Y be a random variable such that (Zi1 , , Ziti ), 1 i r are mutually independent conditioning on Y . Then r ti i=1 j=1 ~ ~ Zij - Y = 0. (12.40) We first prove the following set identity which will be used for proving the above lemma. Lemma 12.8. Let S and T be disjoint index sets, and Ai and B be sets. Let be a set-additive function. Then 306 12 Markov Structures iS Ai jT Aj - B - B)) , (12.41) = S S T T (-1)|S |+|T | ((AS - B) + (AT - B) - (AS T where AS denotes iS Ai . Proof. The right hand side of (12.41) is equal to (-1)|S S S T T |+|T | (AS - B) + S S T T (-1)|S |+|T | (AT - B) (12.42) - S S T T (-1)|S |+|T | (AS T - B). Now (-1)|S S S T T |+|T | (AS - B) = S S (-1)|S | (AS - B) T T (-1)|T | . (12.43) Since |T | (-1)|T | = T T k=0 |T | k (-1)k = 0 (12.44) by the binomial formula1 , we conclude that (-1)|S S S T T |+|T | (AS - B) = 0. (12.45) Similarly, (-1)|S S S T T |+|T | (AT - B) = 0. (12.46) Therefore, (12.41) is equivalent to iS Ai jT Aj - B = S S T T (-1)|S |+|T |+1 (AS T - B) (12.47) 1 This can be obtained by letting a = 1 and b = -1 in the binomial formula |T | (a + b) |T | = k=0 |T | k ak b|T |-k . 12.1 Conditional Mutual Independence 307 which can readily be obtained from Theorem 3.19. Hence, the lemma is proved. Proof of Lemma 12.7. We first prove the lemma for r = 2. By Lemma 12.8, 2 ti i=1 j=1 ~ ~ Zij - Y = (-1)|S |+|T | jS ~ ~ Z1j - Y ~ Z2k ~ - Y . (12.48) S {1,,t1 } T {1,,t2 } + kT ~ Z1j kT ~ ~ Z2k - Y - jS The expression in the square bracket is equal to H(Z1j , j S |Y ) + H(Z2k , k T |Y ) -H((Z1j , j S ), (Z2k , k T )|Y ), (12.49) which vanishes because (Z1j , j S ) and (Z2k , k T ) are independent conditioning on Y . Therefore the lemma is proved for r = 2. For r > 2, we write r ti r-1 ti tr i=1 j=1 ~ ~ Zij - Y = i=1 j=1 ~ Zij j=1 ~ ~ Zrj - Y . (12.50) Since ((Zi1 , , Ziti ), 1 i r - 1) and (Zr1 , , Zrtr ) are independent conditioning on Y , upon applying the lemma for r = 2, we see that r ti i=1 j=1 ~ ~ Zij - Y = 0. (12.51) The lemma is proved. Theorem 12.9. Let T and Qi , 1 i k, be disjoint index sets, where k 2, and let XQi = (Xl , l Qi ), 1 i k, and XT = (Xl , l T ) be collections of random variables. Then XQi , 1 i k, are mutually independent conditioning on XT if and only if for any W1 , W2 , , Wk , where Wi Qi , 1 i k, if there exist at least two i such that Wi = , then k i=1 jWi ~ ~ Xj - XT (k (Qi -Wi )) = 0. i=1 (12.52) 308 12 Markov Structures We first give an example before proving this fundamental result. The reader should compare this example with Example 12.6. Example 12.10. Suppose X1 , (X2 , X3 , X4 ), and (X5 , X6 ) are mutually independent conditioning on X7 . By Theorem 12.9, ~ ~ ~ ~ ~ ~ ~ (X1 X2 X5 X6 - (X3 X4 X7 )) = 0. However, the theorem does not say, for instance, that ~ ~ ~ ~ ~ ~ ~ (X2 X4 - (X1 X3 X5 X6 X7 )) is equal to 0. Proof of Theorem 12.9. We first prove the `if' part. Assume that for any W1 , W2 , , Wk , where Wi Qi , 1 i k, if there exist at least two i such that Wi = , then (12.52) holds. Consider ~ ~ H(XQi , 1 i k|XT ) = Xk Qi - XT i=1 = BS (12.53) (12.54) (12.55) (12.56) (B), where S consists of sets of the form k i=1 jWi ~ ~ Xj - XT (k (Qi -Wi )) i=1 (12.57) with Wi Qi for 1 i k and there exists at least one i such that Wi = . By our assumption, if B S is such that there exist at least two i for which Wi = , then (B) = 0. Therefore, if (B) is possibly nonzero, then B must be such that there exists a unique i for which Wi = . Now for 1 i k, let Sl be the set consisting of sets of the form in (12.57) with Wi Qi , Wi = , and Wl = for l = i. In other words, Si consists of atoms of the form jWi ~ ~ Xj - XT (l=i Ql )(Qi -Wi ) (12.58) with Wi Qi and Wi = . Then k (B) = BS i=1 BSi (B). (12.59) Now 12.2 Full Conditional Mutual Independence 309 ~ ~ XQi - XT (l=i Ql ) = Wi Qi Wi = (12.60) jWi ~ ~ Xj - XT (l=i Ql )(Qi -Wi ) = BSi B. (12.61) Since is set-additive, we have ~ ~ XQi - XT (l=i Ql ) = BSi (B). (12.62) Hence, from (12.56) and (12.59), we have H(XQi , 1 i k|XT ) k = i=1 BSi k (B) ~ ~ XQi - XT (l=i Ql ) i=1 k (12.63) = (12.64) = i=1 H(XQi |XT , XQl , l = i), (12.65) where (12.64) follows from (12.62). By Theorem 12.4, XQi , 1 i k, are mutually independent conditioning on XT . We now prove the `only if' part. Assume XQi , 1 i k, are mutually independent conditioning on XT . For any collection of sets W1 , W2 , , Wk , where Wi Qi , 1 i k, if there exist at least two i such that Wi = , by Theorem 12.5, XWi , 1 i k, are mutually independent conditioning on (XT , XQi -Wi , 1 i k). By Lemma 12.7, we obtain (12.52). The theorem is proved. 12.2 Full Conditional Mutual Independence Definition 12.11. A conditional mutual independency on X1 , X2 , , Xn is full if all X1 , X2 , , Xn are involved. Such a conditional mutual independency is called a full conditional mutual independency (FCMI). Example 12.12. For n = 5, X1 , X2 , X4 , and X5 are mutually independent conditioning on X3 is an FCMI. However, 310 12 Markov Structures X1 , X2 , and X5 are mutually independent conditioning on X3 is not an FCMI because X4 is not involved. As in the previous chapters, we let Nn = {1, 2, , n}. In Theorem 12.9, if k (12.66) T i=1 Qi = Nn , (12.67) then the tuple (T, Qi , 1 i k) defines the following FCMI on X1 , X2 , , Xn : K : XQ1 , XQ2 , , XQk are mutually independent conditioning on XT . We will denote K by (T, Qi , 1 i k). Definition 12.13. Let K = (T, Qi , 1 i k) be an FCMI on X1 , X2 , , Xn . The image of K, denoted by Im(K), is the set of all atoms of Fn which has the form of the set in (12.57), where Wi Qi , 1 i k, and there exist at least two i such that Wi = . Recall from Chapter 3 that A is the set of all nonempty atoms of Fn . Proposition 12.14. Let K = (T, Q1 , Q2 ) be an FCI (full conditional independency) on X1 , X2 , , Xn . Then ~ ~ ~ Im(K) = {A A : A (XQ1 XQ2 - XT )}. (12.68) Proposition 12.15. Let K = (T, Qi , 1 i k) be an FCMI on X1 , X2 , , Xn . Then ~ ~ ~ Im(K) = A A : A (XQi XQj - XT ) . (12.69) 1i<jk These two propositions greatly simplify the description of Im(K). Their proofs are elementary and they are left as an exercise. We first illustrate these two propositions in the following example. 12.2 Full Conditional Mutual Independence 311 Example 12.16. Consider n = 4 and FCMI's K1 = ({3}, {1}, {2, 4}) and K2 = (, {1}, {2, 3}, {4}). Then ~ ~ ~ Im(K1 ) = {A A : A (X1 X{2,4} - X3 )} and ~ ~ ~ ~ ~ ~ Im(K2 ) = {A A : A (X1 X{2,3} ) (X{2,3} X4 ) (X1 X4 )}. (12.71) Theorem 12.17. Let K be an FCMI on X1 , X2 , , Xn . Then K holds if and only if (A) = 0 for all A Im(K). Proof. First, (12.67) is true if K is an FCMI. Then the set in (12.57) can be written as jk Wi i=1 (12.70) ~ ~ Xj - XNn -k Wi , i=1 (12.72) which is seen to be an atom of Fn . The theorem can then be proved by a direct application of Theorem 12.9 to the FCMI K. ~ Let A = n Yi be a nonempty atom of Fn . Define the set i=1 ~ ~c UA = {i Nn : Yi = Xi }. Note that A is uniquely specified by UA because A= iNn -UA (12.73) ~ Xi iUA ~c Xi = iNn -UA ~ Xi ~ - XUA . (12.74) ~ Define w(A) = n - |UA | as the weight of the atom A, the number of Xi in A which are not complemented. We now show that an FCMI K = (T, Qi , 1 i k) is uniquely specified by Im(K). First, by letting Wi = Qi for 1 i k in Definition 12.13, we see that the atom jk Qi i=1 ~ ~ Xj - XT (12.75) is in Im(K), and it is the unique atom in Im(K) with the largest weight. From this atom, T can be determined. To determine Qi , 1 i k, we define a relation q on T c = Nn \T as follows. For l, l T c , (l, l ) is in q if and only if i) l = l ; or ii) there exists an atom of the form ~ ~ Xl Xl 1jn j=l,l ~ Yj (12.76) ~ ~ ~c in A - Im(K), where Yj = Xj or Xj . 312 12 Markov Structures Recall that A is the set of nonempty atoms of Fn . The idea of ii) is that (l, l ) is in q if and only if l, l Qi for some 1 i k. Then q is reflexive and symmetric by construction, and is transitive by virtue of the structure of Im(K). In other words, q is an equivalence relation which partitions T c into {Qi , 1 i k}. Therefore, K and Im(K) uniquely specify each other. The image of an FCMI K completely characterizes the effect of K on the I-Measure for X1 , X2 , , Xn . The joint effect of more than one FCMI can easily be described in terms of the images of the individual FCMI's. Let = {Kl , 1 l m} (12.77) be a set of FCMI's. By Theorem 12.9, Kl holds if and only if vanishes on the atoms in Im(Kl ). Then Kl , 1 l m hold simultaneously if and only if vanishes on the atoms in k Im(Kl ). This is summarized as follows. l=1 Definition 12.18. The image of a set of FCMI's = {Kl , 1 l m} is defined as k Im() = l=1 Im(Kl ). (12.78) Theorem 12.19. Let be a set of FCMI's on X1 , X2 , , Xn . Then holds if and only if (A) = 0 for all A Im(). In probability problems, we are often given a set of conditional independencies and we need to see whether another given conditional independency is logically implied. This is called the implication problem which will be discussed in detail in Section 13.5. The next theorem renders a solution to this problem if only FCMI's are involved. Theorem 12.20. Let 1 and 2 be two sets of FCMI's. Then 1 implies 2 if and only if Im(2 ) Im(1 ). Proof. We first prove that if Im(2 ) Im(1 ), then 1 implies 2 . Assume Im(2 ) Im(1 ) and 1 holds. Then by Theorem 12.19, (A) = 0 for all A Im(1 ). Since Im(2 ) Im(1 ), this implies that (A) = 0 for all A Im(2 ). Again by Theorem 12.19, this implies 2 also holds. Therefore, if Im(2 ) Im(1 ), then 1 implies 2 . We now prove that if 1 implies 2 , then Im(2 ) Im(1 ). To prove this, we assume that 1 implies 2 but Im(2 ) Im(1 ), and we will show that this leads to a contradiction. Fix a nonempty atom A Im(2 )-Im(1 ). By Theorem 3.11, we can construct random variables X1 , X2 , , Xn such that vanishes on all the atoms of Fn except for A. Then vanishes on all the atoms in Im(1 ) but not on all the atoms in Im(2 ). By Theorem 12.19, this implies that for X1 , X2 , , Xn so constructed, 1 holds but 2 does not hold. Therefore, 1 does not imply 2 , which is a contradiction. The theorem is proved. 12.2 Full Conditional Mutual Independence 313 Remark In the course of proving this theorem and all its preliminaries, we have used nothing more than the basic inequalities. Therefore, we have shown that the basic inequalities are a sufficient set of tools to solve the implication problem if only FCMI's are involved. Corollary 12.21. Two sets of FCMI's are equivalent if and only if their images are identical. Proof. Two set of FCMI's 1 and 2 are equivalent if and only if 1 2 and 2 1 . (12.79) Then by the last theorem, this is equivalent to Im(2 ) Im(1 ) and Im(1 ) Im(2 ), i.e., Im(2 ) = Im(1 ). The corollary is proved. Thus a set of FCMI's is completely characterized by its image. A set of FCMI's is a set of probabilistic constraints, but the characterization by its image is purely set-theoretic! This characterization offers an intuitive settheoretic interpretation of the joint effect of FCMI's on the I-Measure for X1 , X2 , , Xn . For example, Im(K1 ) Im(K2 ) is interpreted as the effect commonly due to K1 and K2 , Im(K1 ) - Im(K2 ) is interpreted as the effect due to K1 but not K2 , etc. We end this section with an example. Example 12.22. Consider n = 4. Let K1 = (, {1, 2, 3}, {4}), K3 = (, {1, 2}, {3, 4}), K2 = (, {1, 2, 4}, {3}) K4 = (, {1, 3}, {2, 4}) (12.80) (12.81) and let 1 = {K1 , K2 } and 2 = {K3 , K4 }. Then Im(1 ) = Im(K1 ) Im(K2 ) and Im(2 ) = Im(K3 ) Im(K4 ), where ~ ~ Im(K1 ) = {A A : A (X{1,2,3} X4 )} ~ ~ Im(K2 ) = {A A : A (X{1,2,4} X3 )} ~ ~ Im(K3 ) = {A A : A (X{1,2} X{3,4} )} ~ ~ Im(K4 ) = {A A : A (X{1,3} X{2,4} )}. (12.84) (12.85) (12.86) (12.87) (12.83) (12.82) It can readily be seen by using an information diagram that Im(1 ) Im(2 ). Therefore, 2 implies 1 . Note that no probabilistic argument is involved in this proof. 314 12 Markov Structures 12.3 Markov Random Field A Markov random field is a generalization of a discrete time Markov chain in the sense that the time index for the latter, regarded as a chain, is replaced by a general graph for the former. Historically, the study of Markov random field stems from statistical physics. The classical Ising model, which is defined on a rectangular lattice, was used to explain certain empirically observed facts about ferromagnetic materials. In this section, we explore the structure of the I-Measure for a Markov random field. We refer the reader to textbooks on graph theory (e.g. [45]) for formal definitions of the graph-theoretic terminologies to be used in the rest of the chapter. Let G = (V, E) be an undirected graph, where V is the set of vertices and E is the set of edges. We assume that there is no loop in G, i.e., there is no edge in G which connects a vertex to itself. For any (possibly empty) subset U of V , denote by G\U the graph obtained from G by eliminating all the vertices in U and all the edges joining a vertex in U . The connectivity of a graph partitions the graph into subgraphs called components, i.e., two vertices are in the same component if and only if they are connected. Let s(U ) be the number of distinct components in G\U . Denote the sets of vertices of these components by V1 (U ), V2 (U ), , Vs(U ) (U ). If s(U ) > 1, we say that U is a cutset in G. Definition 12.23 (Markov Random Field). Let G = (V, E) be an undirected graph with V = Nn = {1, 2, , n}, and let Xi be a random variable corresponding to vertex i. The random variables X1 , X2 , , Xn form a Markov random field represented by G if for all cutsets U in G, the sets of random variables XV1 (U ) , XV2 (U ) , , XVs(U ) (U ) are mutually independent conditioning on XU . This definition of a Markov random field is referred to as the global Markov property in the literature. If X1 , X2 , , Xn form a Markov random field represented by a graph G, we also say that X1 , X2 , , Xn form a Markov graph G. When G is a chain, we say that X1 , X2 , , Xn form a Markov chain. In the definition of a Markov random field, each cutset U in G specifies an FCMI on X1 , X2 , , Xn , denoted by [U ]. Formally, [U ] : XV1 (U ) , , XVs(U ) (U ) are mutually independent conditioning on XU . For a collection of cutsets U1 , U2 , , Uk in G, we introduce the notation [U1 , U2 , , Uk ] = [U1 ] [U2 ] [Uk ] (12.88) where `' denotes `logical AND.' Using this notation, X1 , X2 , , Xn form a Markov graph G if and only if [U V : U = V and s(U ) > 1] (12.89) 12.3 Markov Random Field 315 2 3 1 7 4 5 6 Fig. 12.4. The graph G in Example 12.27. holds. Therefore, a Markov random field is simply a collection of FCMI's induced by a graph. We now define two types of nonempty atoms of Fn with respect to a graph G. Recall the definition of the set UA for a nonempty atom A of Fn in (12.73). Definition 12.24. For a nonempty atom A of Fn , if s(UA ) = 1, i.e., G\UA is connected, then A is a Type I atom, otherwise A is a Type II atom. The sets of all Type I and Type II atoms of Fn are denoted by T1 and T2 , respectively. Theorem 12.25. X1 , X2 , , Xn form a Markov graph G if and only if vanishes on all the Type II atoms. Before we prove this theorem, we first state the following proposition which is the graph-theoretic analog of Theorem 12.5. The proof is trivial and is omitted. This proposition and Theorem 12.5 together establish an analogy between the structure of conditional mutual independence and the connectivity of a graph. This analogy will play a key role in proving Theorem 12.25. Proposition 12.26. Let C and Qi be disjoint subsets of the vertex set V of a graph G and Wi be a subset of Qi for 1 i k, where k 2. Assume that there exist at least two i such that Wi = . If Qi , 1 i k, are disconnected in k G\C, then those Wi which are nonempty are disconnected in G\(C i=1 (Qi - Wi )). Example 12.27. In the graph G in Figure 12.4, {1}, {2, 3, 4}, and {5, 6} are disjoint in G\{7}. Then Proposition 12.26 says that {1}, {2}, and {5, 6} are disjoint in G\{3, 4, 7}. Proof of Theorem 12.25. We note that {UA , A A} contains precisely all the proper subsets of Nn . Thus the set of FCMI's specified by the graph G can be written as [UA : A A and s(UA ) > 1] (12.90) (cf. (12.89)). By Theorem 12.19, it suffices to prove that Im([UA : A A and s(UA ) > 1]) = T2 , (12.91) 316 12 Markov Structures where T2 is defined in Definition 12.24. We first prove that T2 Im([UA : A A and s(UA ) > 1]). (12.92) Consider an atom A T2 . Then s(UA ) > 1. In Definition 12.13, let T = UA , k = s(UA ), and Qi = Vi (UA ) for 1 i s(UA ). By considering Wi = Vi (UA ) for 1 i s(UA ), we see that A Im([UA ]). Therefore, T2 = {A A : s(UA ) > 1} AA:s(UA )>1 (12.93) (12.94) (12.95) Im([UA ]) = Im([UA : A A and s(UA ) > 1]). We now prove that Im([UA : A A and s(UA ) > 1]) T2 . (12.96) Consider A Im([UA : A A and s(UA ) > 1]). Then there exists A A with s(UA ) > 1 such that A Im([UA ]). From Definition 12.13, A= s(U ) ji=1A Wi ~ ~ Xj - XU A i=1A s(U ) (Vi (UA )-Wi ) , (12.97) where Wi Vi (UA ), 1 i s(UA ), and there exist at least two i such that Wi = . It follows from (12.97) and the definition of UA that s(UA ) UA = UA i=1 (Vi (UA ) - Wi ). (12.98) With UA playing the role of C and Vi (UA ) playing the role of Qi in Proposition 12.26, we see by applying the proposition that those (at least two) Wi which are nonempty are disjoint in s(UA ) G \ UA i=1 (Vi (UA ) - Wi ) = G\UA . (12.99) This implies s(UA ) > 1, i.e., A T2 . Therefore, we have proved (12.96), and hence the theorem is proved. Example 12.28. With respect to the graph G in Figure 12.5, the Type II atoms are ~ ~ ~c ~ ~c ~ ~c ~ ~ ~c ~c ~ X1 X2 X3 X4 , X1 X2 X3 X4 , X1 X2 X3 X4 , (12.100) while the other twelve nonempty atoms of F4 are Type I atoms. The random variables X1 , X2 , X3 , and X4 form a Markov graph G if and only if (A) = 0 for all Type II atoms A. 12.4 Markov Chain 317 1 3 2 Fig. 12.5. The graph G in Example 12.28. 4 12.4 Markov Chain When the graph G representing a Markov random field is a chain, the Markov random field becomes a Markov chain. In this section, we will show that the information diagram for a Markov chain can be displayed in two dimensions. We will also show that the I-Measure for a Markov chain is always nonnegative. This characteristic of facilitates the use of the information diagram because if B is seen to be a subset of B in the information diagram, then (B ) = (B) + (B - B) (B). (12.101) These two properties are not possessed by a general Markov random field. Without loss of generality, we assume that the Markov chain is represented by the graph G in Figure 12.6. This corresponds to the Markov chain X1 X2 Xn . We first prove the following characterization of a Type I atom for a Markov chain. Lemma 12.29. For the Markov chain represented by the graph G in Figure 12.6, a nonempty atom A of Fn is a Type I atom if and only if Nn \UA = {l, l + 1, , u}, (12.102) where 1 l u n, i.e., the indices of the set variables in A which are not complemented are consecutive. Proof. It is easy to see that for a nonempty atom A, if (12.102) is satisfied, then G\UA is connected, i.e., s(UA ) = 1. Therefore, A is a Type I atom of Fn . On the other hand, if (12.102) is not satisfied, then G\UA is not connected, i.e., s(UA ) > 1, or A is a Type II atom of Fn . The lemma is proved. We now show how the information diagram for a Markov chain with any length n 3 can be constructed in two dimensions. Since vanishes on 1 2 ... n-1 n Fig. 12.6. The graph G representing the Markov chain X1 X2 Xn . 318 12 Markov Structures all the Type II atoms of Fn , it is not necessary to display these atoms in the information diagram. In constructing the information diagram, the regions representing the random variables X1 , X2 , , Xn should overlap with each other such that the regions corresponding to all the Type II atoms are empty, while the regions corresponding to all the Type I atoms are nonempty. Figure 12.7 shows such a construction. We have already shown that is nonnegative for a Markov chain with length 3 or 4. Toward proving that this is true for any length n 3, it suffices to show that (A) 0 for all Type I atoms A of Fn because (A) = 0 for all Type II atoms A of Fn . We have seen in Lemma 12.29 that for a Type I atom A of Fn , UA has the form prescribed in (12.102). Consider any such atom A. Then an inspection of the information diagram in Figure 12.7 reveals that ~ ~ ~ ~ (A) = (Xl Xl+1 Xu - XUA ) = I(Xl ; Xu |XUA ) 0. (12.103) (12.104) (12.105) This shows that is always nonnegative. However, since Figure 12.7 involves an indefinite number of random variables, we give a formal proof of this result in the following theorem. Theorem 12.30. For a Markov chain X1 X2 Xn , is nonnegative. Proof. Since (A) = 0 for all Type II atoms A of Fn , it suffices to show that (A) 0 for all Type I atoms A of Fn . We have seen in Lemma 12.29 that for a Type I atom A of Fn , UA has the form prescribed in (12.102). Consider any such atom A and define the set W = {l + 1, , u - 1}. Then I(Xl ; Xu |XUA ) (12.106) X1 X2 ... Xn-1 Xn Fig. 12.7. The information diagram for the Markov chain X1 X2 Xn . Chapter Summary 319 ~ ~ ~ = (Xl Xu - XUA ) = SW (12.107) ~ Xt ~ ~ Xu - XUA(W \S) ~ ~ Xu - XUA(W \S) . (12.108) ~ Xl tS = SW ~ Xl tS ~ Xt (12.109) In the above summation, except for the atom corresponding to S = W , namely ~ ~ ~ ~ (Xl Xl+1 Xu - XU ), all the atoms are Type II atoms. Therefore, A ~ ~ ~ ~ I(Xl ; Xu |XUA ) = (Xl Xl+1 Xu - XUA ). Hence, ~ ~ ~ ~ (A) = (Xl Xl+1 Xu - XUA ) = I(Xl ; Xu |XUA ) 0. The theorem is proved. (12.110) (12.111) (12.112) (12.113) Chapter Summary In the following, Nn = {1, 2, , n} and A is the set of all nonempty atoms of Fn . Full Conditional Mutual Independency (FCMI): For a partition {T, Qi , 1 i k} of Nn , the tuple (T, Qi , 1 i k) specifies the following FCMI on X1 , X2 , , Xn : XQ1 , XQ2 , , XQk are mutually independent conditioning on XT . Image of an FCMI: For an FCMI K = (T, Qi , 1 i k) on X1 , X2 , , Xn , Im(K) = A A : A 1i<jk (XQi ~ ~ ~ XQj - XT ) . Characterization of an FCMI: An FCMI K on X1 , X2 , , Xn holds if and only if (A) = 0 for all A Im(K). Image of a Set of FCMI's: For a set of FCMI's = {Kl , 1 l m}, k Im() = l=1 Im(Kl ). Characterization of a Set of FCMI's: A set of FCMI's on X1 , X2 , , Xn holds if and only if (A) = 0 for all A Im(). 320 12 Markov Structures Set-Theoretic Characterization of FCMI: 1 implies 2 if and only if Im(2 ) Im(1 ). Markov Random Field (Markov Graph): Let G = (V, E) be an undirected graph with V = Nn , and Xi be a random variable corresponding to vertex i. X1 , X2 , , Xn form a Markov graph G if for all cutsets U in G, XV1 (U ) , XV2 (U ) , , XVs(U ) (U ) are mutually independent conditioning on XU , where V1 (U ), V2 (U ), , Vs(U ) (U ) are the components in G\U . ~ Type I and Type II Atoms: For an atom A = n Yi in A, UA = {i i=1 ~ ~c Nn : Yi = Xi }. For an undirected graph G = (V, E) with V = Nn , an atom A A is Type I if G\UA is connected, otherwise it is Type II. I-Measure Characterization of Markov Random Field: X1 , X2 , , Xn form a Markov graph G if and only if vanishes on all the Type II atoms. I-Measure for Markov Chain: 1. is always nonnegative. 2. The information diagram can be displayed in two dimensions. Problems 1. Prove Proposition 12.14 and Proposition 12.15. 2. In Example 12.22, it was shown that 2 implies 1 . Show that 1 does not imply 2 . Hint: Use an information diagram to determine Im(2 )\Im(1 ). 3. Alternative definition of the global Markov property: For any partition {U, V1 , V2 } of V such that the sets of vertices V1 and V2 are disconnected in G\U , the sets of random variables XV1 and XV2 are independent conditioning on XU . Show that this definition is equivalent to the global Markov property in Definition 12.23. 4. The local Markov property: For 1 i n, Xi and XV -Ni -i are independent conditioning on XNi , where Ni is the set of neighbors2 of vertex i in G. a) Show that the global Markov property implies the local Markov property. b) Show that the local Markov property does not imply the global Markov property by giving a counterexample. Hint: Consider a joint distribution which is not strictly positive. 2 Vertices i and j in an undirected graph are neighbors if i and j are connected by an edge. Historical Notes 321 5. Construct a Markov random field whose I-Measure can take negative values. Hint: Consider a Markov "star." 6. a) Show that X1 , X2 , X3 , and X4 are mutually independent if and only if X1 (X2 , X3 , X4 ), X2 (X3 , X4 )|X1 , X3 X4 |(X1 , X2 ). Hint: Use an information diagram. b) Generalize the result in a) to n random variables. 7. Determine the Markov random field with four random variables X1 , X2 , X3 , and X4 which is characterized by the following conditional independencies: (X1 , X2 , X5 ) X4 |X3 X2 (X4 , X5 )|(X1 , X3 ) X1 (X3 , X4 )|(X2 , X5 ). What are the other conditional independencies pertaining to this Markov random field? Historical Notes A Markov random field can be regarded as a generalization of a discrete-time Markov chain. Historically, the study of Markov random field stems from statistical physics. The classical Ising model, which is defined on a rectangular lattice, was used to explain certain empirically observed facts about ferromagnetic materials. The foundation of the theory of Markov random fields can be found in Preston [296] or Spitzer [343]. The structure of the I-Measure for a Markov chain was first investigated in the unpublished work of Kawabata [196]. Essentially the same result was independently obtained by R. W. Yeung eleven years later in the context of the I-Measure, and the result was eventually published in Kawabata and Yeung [197]. Full conditional independencies were shown to be axiomatizable by Malvestuto [243]. The results in this chapter are due to Yeung et al. [407], where they obtained a set-theoretic characterization of full conditional independencies and investigated the structure of the I-Measure for a Markov random field. In this paper, they also obtained a hypergraph characterization of a Markov random field based on the I-Measure characterization in Theorem 12.25. Ge and Ye [131] have applied these results to characterize a class of graphical models for conditional independence of random variables. 13 Information Inequalities An information expression f refers to a linear combination1 of Shannon's information measures involving a finite number of random variables. For example, H(X, Y ) + 2I(X; Z) (13.1) and I(X; Y ) - I(X; Y |Z) are information expressions. An information inequality has the form f c, (13.3) (13.2) where the constant c is usually equal to zero. We consider non-strict inequalities only because these are usually the form of inequalities in information theory. Likewise, an information identity has the form f = c. (13.4) We point out that an information identity f = c is equivalent to the pair of information inequalities f c and f c. An information inequality or identity is said to always hold if it holds for any joint distribution for the random variables involved. For example, we say that the information inequality I(X; Y ) 0 (13.5) always holds because it holds for any joint distribution p(x, y). On the other hand, we say that an information inequality does not always hold if there exists a joint distribution for which the inequality does not hold. Consider the information inequality 1 More generally, an information expression can be nonlinear, but they do not appear to be useful in information theory. 324 13 Information Inequalities I(X; Y ) 0. Since I(X; Y ) 0 always holds, (13.6) is equivalent to I(X; Y ) = 0, (13.6) (13.7) (13.8) which holds if and only if X and Y are independent. In other words, (13.6) does not hold if X and Y are not independent. Therefore, we say that (13.6) does not always hold. As we have seen in the previous chapters, information inequalities are the major tools for proving converse coding theorems. These inequalities govern the impossibilities in information theory. More precisely, information inequalities imply that certain things cannot happen. For this reason, they are sometimes referred to as the laws of information theory. The basic inequalities form the most important set of information inequalities. In fact, almost all the information inequalities known to date are implied by the basic inequalities. These are called Shannon-type inequalities. On the other hand, if an information inequality always holds but is not implied by the basic inequalities, then it is called a non-Shannon-type inequality. We have not yet explained what it means by that an inequality is or is not implied by the basic inequalities, but this will become clear later in the chapter. Let us now rederive the inequality obtained in Example 3.15 (Imperfect secrecy theorem) without using an information diagram. In this example, three random variables X, Y , and Z are involved, and the setup of the problem is equivalent to the constraint H(X|Y, Z) = 0. Then I(X; Y ) = H(X) + H(Y ) - H(X, Y ) = H(X) + H(Y ) - [H(X, Y, Z) - H(Z|X, Y )] H(X) + H(Y ) - H(X, Y, Z) = H(X) + H(Y ) - [H(Z) + H(Y |Z) + H(X|Y, Z)] = H(X) - H(Z) + I(Y ; Z) - H(X|Y, Z) H(X) - H(Z), where we have used H(Z|X, Y ) 0 in obtaining (13.12), and I(Y ; Z) 0 (13.17) (13.16) (13.10) (13.11) (13.12) (13.13) (13.14) (13.15) (13.9) 13.1 The Region n 325 and (13.9) in obtaining (13.15). This derivation is less transparent than the one we presented in Example 3.15, but the point here is that the final inequality we obtain in (13.15) can be proved by invoking the basic inequalities (13.16) and (13.17). In other words, (13.15) is implied by the basic inequalities. Therefore, it is a (constrained) Shannon-type inequality. We are motivated to ask the following two questions: 1. How can Shannon-type inequalities be characterized? That is, given an information inequality, how can we tell whether it is implied by the basic inequalities? 2. Are there any non-Shannon-type information inequalities? These two are very fundamental questions in information theory. We point out that the first question naturally comes before the second question because if we cannot characterize all Shannon-type inequalities, even if we are given a non-Shannon-type inequality, we cannot tell that it actually is one. In this chapter, we develop a geometric framework for information inequalities which enables them to be studied systematically. This framework naturally leads to an answer to the first question, which makes machine-proving of all Shannon-type inequalities possible. This will be discussed in the next chapter. The second question will be answered positively in Chapter 15. In other words, there do exist laws in information theory beyond those laid down by Shannon. 13.1 The Region n Let Nn = {1, 2, , n}, where n 2, and let = {Xi , i Nn } be any collection of n random variables. Associated with are k = 2n - 1 (13.20) (13.19) (13.18) joint entropies. For example, for n = 3, the 7 joint entropies associated with random variables X1 , X2 , and X3 are H(X1 ), H(X2 ), H(X3 ), H(X1 , X2 ), H(X2 , X3 ), H(X1 , X3 ), H(X1 , X2 , X3 ). Let let X = (Xi , i ) and (13.22) (13.21) denote the set of real numbers. For any nonempty subset of Nn , 326 13 Information Inequalities H () = H(X ). Nn (13.23) For a fixed , we can then view H as a set function from 2 to with H () = 0, i.e., we adopt the convention that the entropy of an empty set of random variable is equal to zero. For this reason, we call H the entropy function of . Let Hn be the k-dimensional Euclidean space with the coordinates labeled by h , 2Nn \{}, where h corresponds to the value of H () for any collection of n random variables. We will refer to Hn as the entropy space for n random variables. Then an entropy function H can be represented by a column vector in Hn . On the other hand, a column vector h Hn is called entropic if h is equal to the entropy function H of some collection of n random variables. We are motivated to define the following region in Hn : n = {h Hn : h is entropic}. (13.24) For convenience, the vectors in n will also be referred to as entropy functions. As an example, for n = 3, the coordinates of H3 are labeled by h1 , h2 , h3 , h12 , h13 , h23 , h123 , (13.25) where h123 denotes h{1,2,3} , etc, and 3 is the region in H3 of all entropy functions for 3 random variables. While further characterizations of n will be given later, we first point out a few basic properties of n : 1. n contains the origin. 2. n , the closure of n , is convex. 3. n is in the nonnegative orthant of the entropy space Hn 2 . The origin of the entropy space corresponds to the entropy function of n degenerate random variables taking constant values. Hence, Property 1 follows. Property 2 will be proved in Chapter 15. Properties 1 and 2 imply that n is a convex cone. Property 3 is true because the coordinates in the entropy space Hn correspond to joint entropies, which are always nonnegative. 13.2 Information Expressions in Canonical Form Any Shannon's information measure other than a joint entropy can be expressed as a linear combination of joint entropies by application of one of the following information identities: H(X|Y ) = H(X, Y ) - H(Y ) I(X; Y ) = H(X) + H(Y ) - H(X, Y ) I(X; Y |Z) = H(X, Z) + H(Y, Z) - H(X, Y, Z) - H(Z). 2 (13.26) (13.27) (13.28) The nonnegative orthant of H 2Nn \{}}. n is the region {h Hn : h 0 for all 13.2 Information Expressions in Canonical Form 327 The first and the second identity are special cases of the third identity, which has already been proved in Lemma 3.8. Thus any information expression which involves n random variables can be expressed as a linear combination of the k associated joint entropies. We call this the canonical form of an information expression. When we write an information expression f as f (h), it means that f is in canonical form. Since an information expression in canonical form is a linear combination of the joint entropies, it has the form b h (13.29) where b denotes the transpose of a constant column vector b in k . The identities in (13.26) to (13.28) provide a way to express every information expression in canonical form. However, it is not clear whether such a canonical form is unique. To illustrate the point, we consider obtaining the canonical form of H(X|Y ) in two ways. First, H(X|Y ) = H(X, Y ) - H(Y ). Second, H(X|Y ) = H(X) - I(X; Y ) = H(X) - (H(Y ) - H(Y |X)) = H(X) - (H(Y ) - H(X, Y ) + H(X)) = H(X, Y ) - H(Y ). (13.31) (13.32) (13.33) (13.34) (13.30) Thus it turns out that we can obtain the same canonical form for H(X|Y ) via two different expansions. This is not accidental, as it is implied by the uniqueness of the canonical form which we will prove shortly. Recall from the proof of Theorem 3.6 that the vector h represents the values of the I-Measure on the unions in Fn . Moreover, h is related to the values of on the atoms of Fn , represented as u, by h = Cn u (13.35) where Cn is a unique k k matrix (cf. (3.27)). We now state the following lemma which is a rephrase of Theorem 3.11. This lemma is essential for proving the next theorem which implies the uniqueness of the canonical form. Lemma 13.1. Let n = {u k : Cn u n }. (13.36) Then the nonnegative orthant of k is a subset of n . Theorem 13.2. Let f be an information expression. Then the unconstrained information identity f = 0 always holds if and only if f is the zero function. 328 13 Information Inequalities Proof. Without loss of generality, assume f is in canonical form and let f (h) = b h. (13.37) Assume f = 0 always holds and f is not the zero function, i.e., b = 0. We will show that this leads to a contradiction. First, f = 0, or more precisely the set {h Hn : b h = 0}, (13.38) is a hyperplane3 in the entropy space which has zero Lebesgue measure4 . We claim that n is contained in the hyperplane f = 0. If this is not true, then there exists h0 n which is not on f = 0, i.e., f (h0 ) = 0. Since h0 n , it corresponds to the entropy function of some joint distribution. This means that there exists a joint distribution such that f (h) = 0 does not hold, which is a contradiction to our assumption that f = 0 always holds. This proves our claim. If n has positive Lebesgue measure, it cannot be contained in the hyperplane f = 0 which has zero Lebesgue measure. Therefore, it suffices to show that n has positive Lebesgue measure. To this end, we see from Lemma 13.1 that the nonnegative orthant of Hn , which has positive Lebesgue measure, is a subset of n . Thus n has positive Lebesgue measure. Since n is an invertible linear transformation of n , its Lebesgue measure is also positive. Therefore, n is not contained in the hyperplane f = 0, which implies that there exists a joint distribution for which f = 0 does not hold. This leads to a contradiction because we have assumed that f = 0 always holds. Hence, we have proved that if f = 0 always holds, then f must be the zero function. Conversely, if f is the zero function, then it is trivial that f = 0 always holds. The theorem is proved. Corollary 13.3. The canonical form of an information expression is unique. Proof. Let f1 and f2 be canonical forms of an information expression g. Since g = f1 and g = f2 always hold, f1 - f2 = 0 (13.41) always holds. By the above theorem, f1 -f2 is the zero function, which implies that f1 and f2 are identical. The corollary is proved. 3 4 (13.39) (13.40) If b = 0, then {h Hn : b h = 0} is equal to Hn . The Lebesque measure can be thought of as "volume" in the Euclidean space if the reader is not familiar with measure theory. 13.3 A Geometrical Framework 329 Due to the uniqueness of the canonical form of an information expression, it is an easy matter to check whether for two information expressions f1 and f2 the unconstrained information identity f1 = f2 (13.42) always holds. All we need to do is to express f1 - f2 in canonical form. Then (13.42) always holds if and only if all the coefficients are zero. 13.3 A Geometrical Framework In the last section, we have seen the role of the region n in proving unconstrained information identities. In this section, we explain the geometrical meanings of unconstrained information inequalities, constrained information inequalities, and constrained information identities in terms of n . Without loss of generality, we assume that all information expressions are in canonical form. 13.3.1 Unconstrained Inequalities Consider an unconstrained information inequality f 0, where f (h) = b h. Then f 0 corresponds to the set {h Hn : b h 0} (13.43) which is a half-space in the entropy space Hn containing the origin. Specifically, for any h Hn , f (h) 0 if and only if h belongs to this set. For simplicity, we will refer to this set as the half-space f 0. As an example, for n = 2, the information inequality I(X1 ; X2 ) = H(X1 ) + H(X2 ) - H(X1 , X2 ) 0, written as h1 + h2 - h12 0, corresponds to the half-space {h Hn : h1 + h2 - h12 0}. (13.46) (13.45) (13.44) in the entropy space H2 . Since an information inequality always holds if and only if it is satisfied by the entropy function of any joint distribution for the random variables involved, we have the following geometrical interpretation of an information inequality: f 0 always holds if and only if n {h Hn : f (h) 0}. 330 13 Information Inequalities f 0 n Fig. 13.1. An illustration for f 0 always holds. This gives a complete characterization of all unconstrained inequalities in terms of n . If n is known, we in principle can determine whether any information inequality involving n random variables always holds. The two possible cases for f 0 are illustrated in Figure 13.1 and Fig ure 13.2. In Figure 13.1, n is completely included in the half-space f 0, so f 0 always holds. In Figure 13.2, there exists a vector h0 n such that f (h0 ) < 0. Thus the inequality f 0 does not always hold. 13.3.2 Constrained Inequalities In information theory, we very often deal with information inequalities (identities) with certain constraints on the joint distribution for the random variables involved. These are called constrained information inequalities (identities), and the constraints on the joint distribution can usually be expressed as linear constraints on the entropies. The following are such examples: f 0 n .h 0 Fig. 13.2. An illustration for f 0 not always holds. 13.3 A Geometrical Framework 331 f 0 Fig. 13.3. An illustration for f 0 always holds under the constraint . 1. X1 , X2 , and X3 are mutually independent if and only if H(X1 , X2 , X3 ) = H(X1 ) + H(X2 ) + H(X3 ). 2. X1 , X2 , and X3 are pairwise independent if and only if I(X1 ; X2 ) = I(X2 ; X3 ) = I(X1 ; X3 ) = 0. 3. X1 is a function of X2 if and only if H(X1 |X2 ) = 0. 4. X1 X2 X3 X4 forms a Markov chain if and only if I(X1 ; X3 |X2 ) = 0 and I(X1 , X2 ; X4 |X3 ) = 0. Suppose there are q linear constraints on the entropies given by Qh = 0, (13.47) where Q is a q k matrix. Here we do not assume that the q constraints are linearly independent, so Q is not necessarily full rank. Let = {h Hn : Qh = 0}. (13.48) In other words, the q constraints confine h to a linear subspace in the entropy space. Parallel to our discussion on unconstrained inequalities, we have the following geometrical interpretation of a constrained inequality: Under the constraint , f 0 always holds if and only if (n ) {h Hn : f (h) 0}. This gives a complete characterization of all constrained inequalities in terms of n . Note that = Hn when there is no constraint on the entropies. In this sense, an unconstrained inequality is a special case of a constrained inequality. The two cases of f 0 under the constraint are illustrated in Figure 13.3 and Figure 13.4. Figure 13.3 shows the case when f 0 always holds under the constraint . Note that f 0 may or may not always hold when there is no constraint. Figure 13.4 shows the case when f 0 does not always hold 332 13 Information Inequalities f 0 Fig. 13.4. An illustration for f 0 not always holds under the constraint . under the constraint . In this case, f 0 also does not always hold when there is no constraint, because (n ) {h Hn : f (h) 0} (13.49) implies n {h Hn : f (h) 0}. (13.50) 13.3.3 Constrained Identities As we have pointed out at the beginning of the chapter, an identity f =0 (13.51) always holds if and only if both the inequalities f 0 and f 0 always hold. Then following our discussion on constrained inequalities, we have Under the constraint , f = 0 always holds if and only if (n ) {h Hn : f (h) 0} {h Hn : f (h) 0}, or Under the constraint , f = 0 always holds if and only if (n ) {h Hn : f (h) = 0}. This condition says that the intersection of n and is contained in the hyperplane f = 0. 13.4 Equivalence of Constrained Inequalities 333 13.4 Equivalence of Constrained Inequalities When there is no constraint on the entropies, two information inequalities b h0 and c h0 (13.53) are equivalent if and only if c = ab, where a is a positive constant. However, this is not the case under a non-trivial constraint = Hn . This situation is illustrated in Figure 13.5. In this figure, although the inequalities in (13.52) and (13.53) correspond to different half-spaces in the entropy space, they actually impose the same constraint on h when h is confined to . In this section, we present a characterization of (13.52) and (13.53) being equivalent under a set of linear constraint . The reader may skip this section at the first reading. Let r be the rank of Q in (13.47). Since h is in the null space of Q, we can write ~ h = Qh , (13.54) ~ is a k (k - r) matrix such that the rows of Q form a basis of the ~ where Q orthogonal complement of the row space of Q, and h is a column (k - r)vector. Then using (13.54), (13.52) and (13.53) can be written as ~ b Qh 0 and ~ c Qh 0, (13.56) ~ respectively in terms of the set of basis given by the columns of Q. Then (13.55) and (13.56) are equivalent if and only if ~ ~ c Q = ab Q, where a is a positive constant, or (13.57) (13.55) (13.52) c h 0 b h 0 Fig. 13.5. Equivalence of b h 0 and c h 0 under the constraint . 334 13 Information Inequalities ~ (c - ab) Q = 0. (13.58) In other words, (c - ab) is in the orthogonal complement of the row space ~ of Q , i.e., (c - ab) is in the row space of Q. Let Q be any r k matrix such that Q and Q have the same row space. (Q can be taken as Q if Q is full rank.) Since the rank of Q is r and Q has r rows, the rows of Q form a basis for the row space of Q, and Q is full rank. Then from (13.58), (13.55) and (13.56) are equivalent under the constraint if and only if c = ab + (Q ) e (13.59) for some positive constant a and some column r-vector e. Suppose for given b and c, we want to see whether (13.55) and (13.56) are equivalent under the constraint . We first consider the case when either b or c is in the row space of Q. This is actually not an interesting case because if b , for example, is in the row space of Q, then ~ b Q=0 (13.60) in (13.55), which means that (13.55) imposes no additional constraint under the constraint . Theorem 13.4. If either b or c is in the row space of Q, then b h 0 and c h 0 are equivalent under the constraint if and only if both b and c are in the row space of Q. The proof of this theorem is left as an exercise. We now turn to the more interesting case when neither b nor c is in the row space of Q. The following theorem gives an explicit condition for (13.55) and (13.56) to be equivalent under the constraint . Theorem 13.5. If neither b nor c is in the row space of Q, then b h 0 and c h 0 are equivalent under the constraint if and only if (Q ) b e = c. a (13.61) has a unique solution with a > 0, where Q is any full-rank matrix such that Q and Q have the same row space. Proof. For b and c not in the row space of Q, we want to see when we can find unknowns a and e satisfying (13.59) with a > 0. To this end, we write (13.59) in matrix form as (13.61). Since b is not in the column space of (Q ) and (Q ) is full rank, (Q ) b is also full rank. Then (13.61) has either a unique solution or no solution. Therefore, the necessary and sufficient condition for (13.55) and (13.56) to be equivalent is that (13.61) has a unique solution and a > 0. The theorem is proved. 13.4 Equivalence of Constrained Inequalities 335 Example 13.6. Consider three random variables X1 , X2 , and X3 with the Markov constraint I(X1 ; X3 |X2 ) = 0, (13.62) which is equivalent to H(X1 , X2 ) + H(X2 , X3 ) - H(X1 , X2 , X3 ) - H(X2 ) = 0. (13.63) In terms of the coordinates in the entropy space H3 , this constraint is written as Qh = 0, (13.64) where Q = [ 0 -1 0 1 1 0 -1 ] and h = [ h1 h2 h3 h12 h23 h13 h123 ] . We now show that under the constraint in (13.64), the inequalities H(X1 |X3 ) - H(X1 |X2 ) 0 and I(X1 ; X2 |X3 ) 0 (13.68) are in fact equivalent. Toward this end, we write (13.67) and (13.68) as b h 0 and c h 0, respectively, where b = [ 0 1 -1 -1 0 1 0 ] and c = [ 0 0 -1 0 1 1 -1 ] . Since Q is full rank, we may take Q = Q. Upon solving Q b e = c, a (13.71) (13.70) (13.69) (13.67) (13.66) (13.65) we obtain the unique solution a = 1 > 0 and e = 1 (e is a 1 1 matrix). Therefore, (13.67) and (13.68) are equivalent under the constraint in (13.64). Under the constraint , if neither b can be shown that the identities nor c is in the row space of Q, it (13.72) (13.73) b h=0 and c h=0 are equivalent if and only if (13.61) has a unique solution. We leave the proof as an exercise. 336 13 Information Inequalities 13.5 The Implication Problem of Conditional Independence We use X X |X to denote the conditional independency (CI) X and X are conditionally independent given X . We have proved in Theorem 2.34 that X X |X is equivalent to I(X ; X |X ) = 0. (13.74) When = , X X |X becomes an unconditional independency which we regard as a special case of a conditional independency. When = , (13.74) becomes H(X |X ) = 0, (13.75) which we see from Proposition 2.36 that X is a function of X . For this reason, we also regard functional dependency as a special case of conditional independency. In probability problems, we are often given a set of CI's and we need to determine whether another given CI is logically implied. This is called the implication problem, which is one of the most basic problems in probability theory. We have seen in Section 12.2 that the implication problem has a solution if only full conditional mutual independencies are involved. However, the general problem is extremely difficult, and it has been solved only up to four random variables [256]. We end this section by explaining the relation between the implication problem and the region n . A CI involving random variables X1 , X2 , , Xn has the form X X |X , (13.76) where , , Nn . Since I(X ; X |X ) = 0 is equivalent to H(X ) + H(X ) - H(X ) - H(X ) = 0, X X |X corresponds to the hyperplane {h Hn : h + h - h - h = 0}. (13.78) (13.77) For a CI K, we denote the hyperplane in Hn corresponding to K by E(K). Let = {Kl } be a collection of CI's, and we want to determine whether implies a given CI K. This would be the case if and only if the following is true: For all h n , if h E(Kl ), then h E(K). l Equivalently, Chapter Summary 337 implies K if and only if l E(Kl ) n E(K). Therefore, the implication problem can be solved if n can be characterized. Hence, the region n is not only of fundamental importance in information theory, but is also of fundamental importance in probability theory. Chapter Summary Entropy Space: The entropy space Hn is the (2n - 1)-dimensional Euclidean space with the coordinates labeled by h , 2Nn \{}, where Nn = {1, 2, , n}. The Region n is the subset of Hn of all entropy functions for n discrete random variables. Basic Properties of n : 1. n contains the origin. 2. n , the closure of n , is convex. 3. n is in the nonnegative orthant of the entropy space Hn . Canonical Form of an Information Expression: Any information expression can be expressed as a linear combination of joint entropies, called the canonical form. The canonical form of an information expression is unique. Unconstrained Information Identities: b h = 0 always holds if and only if b = 0. Unconstrained Information Inequalities: b h 0 always holds if and only if n {h Hn : b h 0}. Constrained Information Inequalities: Under the constraint = {h Hn : Qh = 0}, b h 0 always holds if and only if (n ) {h Hn : b h 0}. Equivalence of Constrained Inequalities (Identities): Under the constraint = {h Hn : Qh = 0}, b h 0 and c h 0 (b h = 0 and c h = 0) are equivalent if and only if one of the following holds: 1. Both b and c are in the row space of Q. 2. Neither b nor c is in the row space of Q, and (Q ) b e =c a has a unique solution with a > 0 (has a unique solution), where Q is any full-rank matrix such that Q and Q have the same row space. 338 13 Information Inequalities Problems 1. Symmetrical information expressions An information expression is said to be symmetrical if it is identical under every permutation of the random variables involved. However, sometimes a symmetrical information expression cannot be readily recognized symbolically. For example, I(X1 ; X2 ) - I(X1 ; X2 |X3 ) is symmetrical in X1 , X2 , and X3 but it is not symmetrical symbolically. Devise a general method for recognizing symmetrical information expressions. 2. The canonical form of an information expression is unique when there is no constraint on the random variables involved. Show by an example that this does not hold when certain constraints are imposed on the random variables involved. ~ 3. Alternative canonical form Denote iG Xi by XG and let C = XG : G is a nonempty subset of Nn . a) Prove that a signed measure on Fn is completely specified by {(C), C C}, which can be any set of real numbers. b) Prove that an information expression involving X1 , X2 , , Xn can be expressed uniquely as a linear combination of (XG ), where G are nonempty subsets of Nn . 4. Uniqueness of the canonical form for nonlinear information expressions Consider a function f : k , where k = 2n - 1 such that {h k : f (h) = 0} has zero Lebesgue measure. a) Prove that f cannot be identically zero on n . b) Use the result in a) to show the uniqueness of the canonical form for the class of information expressions of the form g(h) where g is a polynomial. (Yeung [401].) 5. Prove that under the constraint Qh = 0, if neither b nor c is in the row space of Q, the identities b h = 0 and c h = 0 are equivalent if and only if (13.61) has a unique solution. Historical Notes The uniqueness of the canonical form for linear information expressions was first proved by Han [146]. The same result was independently obtained in the book by Csiszr and Krner [84]. The geometrical framework for infora o mation inequalities is due to Yeung [401]. The characterization of equivalent constrained inequalities in Section 13.4 first appeared in the book by Yeung [402]. 14 Shannon-Type Inequalities The basic inequalities form the most important set of information inequalities. In fact, almost all the information inequalities known to date are implied by the basic inequalities. These are called Shannon-type inequalities. In this chapter, we show that verification of Shannon-type inequalities can be formulated as a linear programming problem, thus enabling machine-proving of all such inequalities. 14.1 The Elemental Inequalities Consider the conditional mutual information I(X, Y ; X, Z, U |Z, T ), (14.1) in which the random variables X and Z appear more than once. It is readily seen that I(X, Y ; X, Z, U |Z, T ) can be written as H(X|Z, T ) + I(Y ; U |X, Z, T ), (14.2) where in both H(X|Z, T ) and I(Y ; U |X, Z, T ), each random variable appears only once. A Shannon's information measure is said to be reducible if there exists a random variable which appears more than once in the information measure, otherwise the information measure is said to be irreducible. Without loss of generality, we will consider irreducible Shannon's information measures only, because a reducible Shannon's information measure can always be written as the sum of irreducible Shannon's information measures. The nonnegativity of all Shannon's information measures form a set of inequalities called the basic inequalities. The set of basic inequalities, however, is not minimal in the sense that some basic inequalities are implied by the others. For example, H(X|Y ) 0 (14.3) 340 14 Shannon-Type Inequalities and I(X; Y ) 0, (14.4) which are both basic inequalities involving random variables X and Y , imply H(X) = H(X|Y ) + I(X; Y ) 0, (14.5) again a basic inequality involving X and Y . Let Nn = {1, 2, , n}, where n 2. Unless otherwise specified, all information expressions in this chapter involve some or all of the random variables X1 , X2 , , Xn . The value of n will be specified when necessary. Through application of the identities H(X) = H(X|Y ) + I(X; Y ) H(X, Y ) = H(X) + H(Y |X) I(X; Y, Z) = I(X; Y ) + I(X; Z|Y ) H(X|Z) = H(X|Y, Z) + I(X; Y |Z) H(X, Y |Z) = H(X|Z) + H(Y |X, Z) I(X; Y, Z|T ) = I(X; Y |T ) + I(X; Z|Y, T ), (14.6) (14.7) (14.8) (14.9) (14.10) (14.11) any Shannon's information measure can be expressed as the sum of Shannon's information measures of the following two elemental forms: i) H(Xi |XNn -{i} ), i Nn ii) I(Xi ; Xj |XK ), where i = j and K Nn - {i, j}. This will be illustrated in the next example. It is not difficult to check that the total number of the two elemental forms of Shannon's information measures for n random variables is equal to m=n+ n 2 2n-2 . (14.12) The proof of (14.12) is left as an exercise. Example 14.1. We can expand H(X1 , X2 ) into a sum of elemental forms of Shannon's information measures for n = 3 by applying the identities in (14.6) to (14.11) as follows: H(X1 , X2 ) = H(X1 ) + H(X2 |X1 ) = H(X1 |X2 , X3 ) + I(X1 ; X2 , X3 ) + H(X2 |X1 , X3 ) +I(X2 ; X3 |X1 ) = H(X1 |X2 , X3 ) + I(X1 ; X2 ) + I(X1 ; X3 |X2 ) +H(X2 |X1 , X3 ) + I(X2 ; X3 |X1 ). (14.15) (14.14) (14.13) 14.2 A Linear Programming Approach 341 The nonnegativity of the two elemental forms of Shannon's information measures form a proper subset of the set of basic inequalities. We call the m inequalities in this smaller set the elemental inequalities. They are equivalent to the basic inequalities because each basic inequality which is not an elemental inequality can be obtained as the sum of a set of elemental inequalities in view of (14.6) to (14.11). This will be illustrated in the next example. The proof for the minimality of the set of elemental inequalities is deferred to Section 14.6. Example 14.2. In the last example, we expressed H(X1 , X2 ) as H(X1 |X2 , X3 ) + I(X1 ; X2 ) + I(X1 ; X3 |X2 ) +H(X2 |X1 , X3 ) + I(X2 ; X3 |X1 ). (14.16) All the five Shannon's information measures in the above expression are in elemental form for n = 3. Then the basic inequality H(X1 , X2 ) 0 can be obtained as the sum of the following elemental inequalities: H(X1 |X2 , X3 ) 0 I(X1 ; X2 ) 0 I(X1 ; X3 |X2 ) 0 H(X2 |X1 , X3 ) 0 I(X2 ; X3 |X1 ) 0. (14.18) (14.19) (14.20) (14.21) (14.22) (14.17) 14.2 A Linear Programming Approach Recall from Section 13.2 that any information expression can be expressed uniquely in canonical form, i.e., a linear combination of the k = 2n - 1 joint entropies involving some or all of the random variables X1 , X2 , , Xn . If the elemental inequalities are expressed in canonical form, they become linear inequalities in the entropy space Hn . Denote this set of inequalities by Gh 0, where G is an m k matrix, and define n = {h Hn : Gh 0}. (14.23) We first show that n is a pyramid in the nonnegative orthant of the entropy space Hn . Evidently, n contains the origin. Let ej , 1 j k, be the column k-vector whose jth component is equal to 1 and all the other components are equal to 0. Then the inequality ej h 0 (14.24) 342 14 Shannon-Type Inequalities corresponds to the nonnegativity of a joint entropy, which is a basic inequality. Since the set of elemental inequalities is equivalent to the set of basic inequalities, if h n , i.e., h satisfies all the elemental inequalities, then h also satisfies the basic inequality in (14.24). In other words, n {h Hn : ej h 0} (14.25) for all 1 j k. This implies that n is in the nonnegative orthant of the entropy space. Since n contains the origin and the constraints Gh 0 are linear, we conclude that n is a pyramid in the nonnegative orthant of Hn . Since the elemental inequalities are satisfied by the entropy function of any n random variables X1 , X2 , , Xn , for any h in n , h is also in n , i.e., n n . (14.26) Therefore, for any unconstrained inequality f 0, if n {h Hn : f (h) 0}, then n {h Hn : f (h) 0}, (14.27) (14.28) i.e., f 0 always holds. In other words, (14.27) is a sufficient condition for f 0 to always hold. Moreover, an inequality f 0 such that (14.27) is satisfied is implied by the basic inequalities, because if h satisfies the basic inequalities, i.e., h n , then h satisfies f (h) 0. For constrained inequalities, following our discussion in Section 13.3, we impose the constraint Qh = 0 (14.29) and let = {h Hn : Qh = 0}. For an inequality f 0, if (n ) {h Hn : f (h) 0}, then by (14.26), (n ) {h Hn : f (h) 0}, (14.30) (14.31) (14.32) i.e., f 0 always holds under the constraint . In other words, (14.31) is a sufficient condition for f 0 to always hold under the constraint . Moreover, an inequality f 0 under the constraint such that (14.31) is satisfied is implied by the basic inequalities and the constraint , because if h and h satisfies the basic inequalities, i.e., h n , then h satisfies f (h) 0. 14.2 A Linear Programming Approach 343 14.2.1 Unconstrained Inequalities To check whether an unconstrained inequality b h 0 is a Shannon-type inequality, we need to check whether n is a subset of {h Hn : b h 0}. The following theorem induces a computational procedure for this purpose. Theorem 14.3. b h 0 is a Shannon-type inequality if and only if the minimum of the problem Minimize b h, subject to Gh 0 is zero. In this case, the minimum occurs at the origin. Remark The idea of this theorem is illustrated in Figure 14.1 and Figure 14.2. In Figure 14.1, n is contained in {h Hn : b h 0}. The minimum of b h subject to n occurs at the origin with the minimum equal to 0. In Figure 14.2, n is not contained in {h Hn : b h 0}. The minimum of b h subject to n is -. A formal proof of the theorem is given next. Proof of Theorem 14.3. We have to prove that n is a subset of {h Hn : b h 0} if and only if the minimum of the problem in (14.33) is zero. First of all, since 0 n and b 0 = 0 for any b, the minimum of the problem in (14.33) is at most 0. Assume n is a subset of {h Hn : b h 0} and the minimum of the problem in (14.33) is negative. Then there exists h n such that b h < 0, (14.34) which implies n {h Hn : b h 0}, (14.35) a contradiction. Therefore, if n is a subset of {h Hn : b h 0}, then the minimum of the problem in (14.33) is zero. (14.33) b h 0 n Fig. 14.1. n is contained in {h Hn : b h 0}. 344 14 Shannon-Type Inequalities b h 0 n Fig. 14.2. n is not contained in {h Hn : b h 0}. To prove the converse, assume n is not a subset of {h Hn : b h 0}, i.e., (14.35) is true. Then there exists h n such that b h < 0. (14.36) This implies that the minimum of the problem in (14.33) is negative, i.e., it is not equal to zero. Finally, if the minimum of the problem in (14.33) is zero, since the n contains the origin and b 0 = 0, the minimum occurs at the origin. By virtue of this theorem, to check whether b h 0 is an unconstrained Shannon-type inequality, all we need to do is to apply the optimality test of the simplex method [91] to check whether the point h = 0 is optimal for the minimization problem in (14.33). Then b h 0 is an unconstrained Shannon-type inequality if and only if h = 0 is optimal. 14.2.2 Constrained Inequalities and Identities To check whether an inequality b h 0 under the constraint is a Shannontype inequality, we need to check whether n is a subset of {h Hn : b h 0}. Theorem 14.4. b h 0 is a Shannon-type inequality under the constraint if and only if the minimum of the problem Minimize b h, subject to Gh 0 and Qh = 0 is zero. In this case, the minimum occurs at the origin. The proof of this theorem is similar to that for Theorem 14.3, so it is omitted. By taking advantage of the linear structure of the constraint , we (14.37) 14.3 A Duality 345 can reformulate the minimization problem in (14.37) as follows. Let r be the rank of Q. Since h is in the null space of Q, we can write ~ h = Qh , (14.38) ~ ~ where Q is a k (k - r) matrix such that the rows of Q form a basis of the orthogonal complement of the row space of Q, and h is a column (k - r)vector. Then the elemental inequalities can be expressed as ~ GQh 0, and in terms of h , n becomes n = {h k-r (14.39) ~ : GQh 0}, (14.40) which is a pyramid in k-r (but not necessarily in the nonnegative orthant). ~ Likewise, b h can be expressed as b Qh . With all the information expressions in terms of h , the problem in (14.37) becomes ~ ~ (14.41) Minimize b Qh , subject to GQh 0. Therefore, to check whether b h 0 is a Shannon-type inequality under the constraint , all we need to do is to apply the optimality test of the simplex method to check whether the point h = 0 is optimal for the problem in (14.41). Then b h 0 is a Shannon-type inequality under the constraint if and only if h = 0 is optimal. By imposing the constraint , the number of elemental inequalities remains the same, while the dimension of the problem decreases from k to k - r. Finally, to verify that b h = 0 is a Shannon-type identity under the constraint , i.e., b h = 0 is implied by the basic inequalities, all we need to do is to verify that both b h 0 and b h 0 are Shannon-type inequalities under the constraint . 14.3 A Duality A nonnegative linear combination is a linear combination whose coefficients are all nonnegative. It is clear that a nonnegative linear combination of basic inequalities is a Shannon-type inequality. However, it is not clear that all Shannon-type inequalities are of this form. By applying the duality theorem in linear programming [336], we will see that this is in fact the case. The dual of the primal linear programming problem in (14.33) is Maximize y 0 subject to y 0 and y G b , where y = [ y1 ym ] . (14.43) (14.42) 346 14 Shannon-Type Inequalities By the duality theorem, if the minimum of the primal problem is zero, which happens when b h 0 is a Shannon-type inequality, the maximum of the dual problem is also zero. Since the cost function in the dual problem is zero, the maximum of the dual problem is zero if and only if the feasible region = {y is nonempty. Theorem 14.5. b h 0 is a Shannon-type inequality if and only if b = x G for some x 0, where x is a column m-vector, i.e., b is a nonnegative linear combination of the rows of G. Proof. We have to prove that is nonempty if and only if b some x 0. The feasible region is nonempty if and only if b z G = x G for (14.45) m : y 0 and y G b } (14.44) for some z 0, where z is a column m-vector. Consider any z which satisfies (14.45), and let s = b - z G 0. (14.46) Denote by ej the column k-vector whose jth component is equal to 1 and all the other components are equal to 0, 1 j k. Then ej h is a joint entropy. Since every joint entropy can be expressed as the sum of elemental forms of Shannon's information measures, ej can be expressed as a nonnegative linear combination of the rows of G. Write s = [ s1 s2 sk ] , where sj 0 for all 1 j k. Then k (14.47) s = j=1 sj ej (14.48) can also be expressed as a nonnegative linear combinations of the rows of G, i.e., s =w G (14.49) for some w 0. From (14.46), we see that b = (w + z )G = x G, where x 0. The proof is accomplished. From this theorem, we see that all Shannon-type inequalities are actually trivially implied by the basic inequalities! However, the verification of a Shannontype inequality requires a computational procedure as described in the last section. (14.50) 14.4 Machine Proving ITIP 347 14.4 Machine Proving ITIP Theorems 14.3 and 14.4 transform the problem of verifying a Shannon-type inequality into a linear programming problem. This enables machine-proving of all Shannon-type inequalities. A software package called ITIP1 has been developed for this purpose. The most updated versions of ITIP can be downloaded from the World Wide Web [409]. Using ITIP is very simple and intuitive. The following examples illustrate the use of ITIP: 1. >> ITIP('H(XYZ) <= H(X) + H(Y) + H(Z)') True 2. >> ITIP('I(X;Z) = 0','I(X;Z|Y) = 0','I(X;Y) = 0') True 3. >> ITIP('I(Z;U) - I(Z;U|X) - I(Z;U|Y) <= 0.5 I(X;Y) + 0.25 I(X;ZU) + 0.25 I(Y;ZU)') Not provable by ITIP In the first example, we prove an unconstrained inequality. In the second example, we prove that X and Z are independent if X Y Z forms a Markov chain and X and Y are independent. The first identity is what we want to prove, while the second and the third expressions specify the Markov chain X Y Z and the independency of X and Y , respectively. In the third example, ITIP returns the clause "Not provable by ITIP," which means that the inequality is not a Shannon-type inequality. This, however, does not mean that the inequality to be proved cannot always hold. In fact, this inequality is one of the known non-Shannon-type inequalities which will be discussed in Chapter 15. We note that most of the results we have previously obtained by using information diagrams can also be proved by ITIP. However, the advantage of using information diagrams is that one can visualize the structure of the problem. Therefore, the use of information diagrams and ITIP very often complement each other. In the rest of the section, we give a few examples which demonstrate the use of ITIP. Example 14.6. By Proposition 2.10, the long Markov chain X Y Z T implies the two short Markov chains X Y Z and Y Z T . We want to see whether the two short Markov chains also imply the long Markov chain. If so, they are equivalent to each other. Using ITIP, we have >> ITIP('X/Y/Z/T', 'X/Y/Z', 'Y/Z/T') Not provable by ITIP 1 ITIP stands for Information-Theoretic Inequality Prover. 348 14 Shannon-Type Inequalities Y X Z T Fig. 14.3. The information diagram for X, Y , Z, and T in Example 14.6. In the above, we have used a macro in ITIP to specify the three Markov chains. The above result from ITIP says that the long Markov chain cannot be proved from the two short Markov chains by means of the basic inequalities. This strongly suggests that the two short Markov chains is weaker than the long Markov chain. However, in order to prove that this is in fact the case, we need an explicit construction of a joint distribution for X, Y , Z, and T which satisfies the two short Markov chains but not the long Markov chain. Toward this end, we resort to the information diagram in Figure 14.3. The Markov chain X Y Z is equivalent to I(X; Z|Y ) = 0, i.e., ~ ~ ~ ~ ~ ~ ~ ~ (X Y c Z T ) + (X Y c Z T c ) = 0. Similarly, the Markov chain Y Z T is equivalent to ~ ~ ~ ~ ~ ~ ~ ~ (X Y Z c T ) + (X c Y Z c T ) = 0. (14.52) (14.51) The four atoms involved in the constraints (14.51) and (14.52) are marked by a dagger in Figure 14.3. In Section 3.5, we have seen that the Markov chain X Y Z T holds if and only if takes zero value on the set of atoms in Figure 14.4 which are marked with an asterisk2 . Comparing Figure 14.3 and Figure 14.4, we see that the only atom marked in Figure 14.4 but not ~ ~ ~ ~ in Figure 14.3 is X Y c Z c T . Thus if we can construct a such that ~ ~ ~ ~ it takes zero value on all the atoms except for X Y c Z c T , then the corresponding joint distribution satisfies the two short Markov chains but not the long Markov chain. This would show that the two short Markov chains are in fact weaker than the long Markov chain. Following Theorem 3.11, such a can be constructed. In fact, the required joint distribution can be obtained by simply letting X = T = U , where U is any random variable such that H(U ) > 0, and letting 2 This information diagram is essentially a reproduction of Figure 3.8. 14.4 Machine Proving ITIP 349 Y * * X * * Z * T Fig. 14.4. The atoms of F4 on which vanishes when X Y Z T forms a Markov chain. Y and Z be degenerate random variables taking constant values. Then it is easy to see that X Y Z and Y Z T hold, while X Y Z T does not hold. Example 14.7. The data processing theorem says that if X Y Z T forms a Markov chain, then I(Y ; Z) I(X; T ). (14.53) We want to see whether this inequality holds under the weaker condition that X Y Z and Y Z T form two short Markov chains. By using ITIP, we can show that (14.53) is not a Shannon-type inequality under the Markov conditions I(X; Z|Y ) = 0 (14.54) and I(Y ; T |Z) = 0. (14.55) This strongly suggests that (14.53) does not always hold under the constraint of the two short Markov chains. However, this has to be proved by an explicit construction of a joint distribution for X, Y , Z, and T which satisfies (14.54) and (14.55) but not (14.53). The construction at the end of the last example serves this purpose. Example 14.8 (Secret Sharing [42][321]). Let S be a secret to be encoded into three pieces, X, Y , and Z. We need to design a scheme that satisfies the following two requirements: 1. No information about S can be obtained from any one of the three encoded pieces. 2. S can be recovered from any two of the three encoded pieces. 350 14 Shannon-Type Inequalities This is called a (1,2)-threshold secret sharing scheme. The first requirement of the scheme is equivalent to the constraints I(S; X) = I(S; Y ) = I(S; Z) = 0, while the second requirement is equivalent to the constraints H(S|X, Y ) = H(S|Y, Z) = H(S|X, Z) = 0. Since the secret S can be recovered if all X, Y , and Z are known, H(X) + H(Y ) + H(Z) H(S). We are naturally interested in the maximum constant c that satisfies H(X) + H(Y ) + H(Z) cH(S). (14.59) (14.58) (14.57) (14.56) We can explore the possible values of c by ITIP. After a few trials, we find that ITIP returns a "True" for all c 3, and returns the clause "Not provable by ITIP" for any c slightly larger than 3, say 3.0001. This means that the maximum value of c is lower bounded by 3. This lower bound is in fact tight, as we can see from the following construction. Let S and N be mutually independent ternary random variables uniformly distributed on {0, 1, 2}, and define X=N Y = S + N mod 3, and Z = S + 2N mod 3. Then it is easy to verify that S = Y - X mod 3 = 2Y - Z mod 3 = Z - 2X mod 3. (14.63) (14.64) (14.65) (14.62) (14.60) (14.61) Thus the requirements in (14.57) are satisfied. It is also readily verified that the requirements in (14.56) are satisfied. Finally, all S, X, Y , and Z are distributed uniformly on {0, 1, 2}. Therefore, H(X) + H(Y ) + H(Z) = 3H(S). (14.66) This proves that the maximum constant c which satisfies (14.59) is 3. Using the approach in this example, almost all information-theoretic bounds reported in the literature for this class of problems can be obtained when a definite number of random variables are involved. 14.5 Tackling the Implication Problem 351 14.5 Tackling the Implication Problem We have already mentioned in Section 13.5 that the implication problem of conditional independence is extremely difficult except for the special case that only full conditional mutual independencies are involved. In this section, we employ the tools we have developed in this chapter to tackle this problem. In Bayesian networks (see [287]), the following four axioms are often used for proving implications of conditional independencies: Symmetry: X Y |Z Y X|Z Decomposition: X (Y, T )|Z X Y |Z X T |Z Weak Union: X (Y, T )|Z X Y |(Z, T ) Contraction: X Y |Z X T |(Y, Z) X (Y, T )|Z. (14.70) (14.69) (14.68) (14.67) These axioms form a system called semi-graphoid and were first proposed in [93] as heuristic properties of conditional independence. The axiom of symmetry is trivial in the context of probability3 . The other three axioms can be summarized by X (Y, T )|Z X Y |Z X T |(Y, Z). This can easily be proved as follows. Consider the identity I(X; Y, T |Z) = I(X; Y |Z) + I(X; T |Y, Z). (14.72) (14.71) Since conditional mutual informations are always nonnegative by the basic inequalities, if I(X; Y, T |Z) vanishes, I(X; Y |Z) and I(X; T |Y, Z) also vanish, and vice versa. This proves (14.71). In other words, (14.71) is the result of a specific application of the basic inequalities. Therefore, any implication which can be proved by invoking these four axioms are provable by ITIP. In fact, ITIP is considerably more powerful than the above four axioms. This will be shown in the next example in which we give an implication which can be proved by ITIP but not by these four axioms4 . We will see some implications which cannot be proved by ITIP when we discuss non-Shannontype inequalities in the next chapter. 3 4 These four axioms may be applied beyond the context of probability. This example is due to Zhen Zhang, private communication. 352 14 Shannon-Type Inequalities Y + _ _ X + _ + Z T Fig. 14.5. The information diagram for X, Y , Z, and T . For a number of years, researchers in Bayesian networks generally believed that the semi-graphoidal axioms form a complete set of axioms for conditional independence until it was refuted by Studen [346]. See Problem 10 for a y discussion. Example 14.9. We will show that I(X; Y |Z) = 0 I(X; T |Z) = 0 I(X; T |Y ) = 0 I(X; Y |T ) = 0 I(X; Z|Y ) = 0 I(X; Z|T ) = 0 can be proved by invoking the basic inequalities. First, we write I(X; Y |Z) = I(X; Y |Z, T ) + I(X; Y ; T |Z). Since I(X; Y |Z) = 0 and I(X; Y |Z, T ) 0, we let I(X; Y |Z, T ) = a for some nonnegative real number a, so that I(X; Y ; T |Z) = -a (14.76) (14.75) (14.74) (14.73) from (14.74). In the information diagram in Figure 14.5, we mark the atom I(X; Y |Z, T ) by a "+" and the atom I(X; Y ; T |Z) by a "-." Then we write I(X; T |Z) = I(X; Y ; T |Z) + I(X; T |Y, Z). Since I(X; T |Z) = 0 and I(X; Y ; T |Z) = -a, we obtain I(X; T |Y, Z) = a. (14.78) (14.77) 14.6 Minimality of the Elemental Inequalities 353 In the information diagram, we mark the atom I(X; T |Y, Z) with a "+." Continue in this fashion, the five CI's on the left hand side of (14.73) imply that all the atoms marked with a "+" in the information diagram take the value a, while all the atoms marked with a "-" take the value -a. From the information diagram, we see that I(X; Y |T ) = I(X; Y ; Z|T ) + I(X; Y |Z, T ) = (-a) + a = 0, (14.79) which proves our claim. Since we base our proof on the basic inequalities, this implication can also be proved by ITIP. Due to the form of the five given CI's in (14.73), none of the axioms in (14.68) to (14.70) can be applied. Thus we conclude that the implication in (14.73) cannot be proved by invoking the four axioms in (14.67) to (14.70). 14.6 Minimality of the Elemental Inequalities We have already seen in Section 14.1 that the set of basic inequalities is not minimal in the sense that in the set, some inequalities are implied by the others. We then showed that the set of basic inequalities is equivalent to the smaller set of elemental inequalities. Again, we can ask whether the set of elemental inequalities is minimal. In this section, we prove that the set of elemental inequalities is minimal. This result is important for efficient implementation of ITIP because it says that we cannot consider a smaller set of inequalities. The proof, however, is rather technical. The reader may skip this proof without missing the essence of this chapter. The elemental inequalities in set-theoretic notations have one of the following two forms: ~ ~ 1. (Xi - XNn -{i} ) 0, ~ ~ ~ 2. (Xi Xj - XK ) 0, i = j and K Nn - {i, j}, where denotes a set-additive function defined on Fn . They will be referred to as -inequalities and -inequalities, respectively. We are to show that all the elemental inequalities are nonredundant, i.e., none of them is implied by the others. For an -inequality ~ ~ (Xi - XNn -{i} ) 0, (14.80) ~ ~ since it is the only elemental inequality which involves the atom Xi - XNn -{i} , it is clearly not implied by the other elemental inequalities. Therefore we only need to show that all -inequalities are nonredundant. To show that a inequality is nonredundant, it suffices to show that there exists a measure on ^ Fn which satisfies all other elemental inequalities except for that -inequality. We will show that the -inequality 354 14 Shannon-Type Inequalities ~ ~ ~ (Xi Xj - XK ) 0 (14.81) is nonredundant. To facilitate our discussion, we denote Nn - K - {i, j} by ~ ~ ~ L(i, j, K), and we let Cij|K (S), S L(i, j, K) be the atoms in Xi Xj - XK , where ~ ~ ~ ~c ~c (14.82) Cij|K (S) = Xi Xj XS XK XL(i,j,K)-S . We first consider the case when L(i, j, K) = , i.e., K = Nn - {i, j}. We construct a measure by ^ (A) = ^ ~ ~ ~ -1 if A = Xi Xj - XK 1 otherwise, (14.83) ~ ~ ~ where A A. In other words, Xi Xj - XK is the only atom with measure ~ ~ -1; all other atoms have measure 1. Then (Xi Xj - XK ) < 0 is trivially ^ ~ true. It is also trivial to check that for any i Nn , ~ (Xi - XNn -{i } ) = 1 0, ^ ~ (14.84) and for any (i , j , K ) = (i, j, K) such that i = j and K Nn - {i , j }, ~ ~ (Xi Xj - XK ) = 1 0 ^ ~ (14.85) if K = Nn -{i , j }. On the other hand, if K is a proper subset of Nn -{i , j }, ~ ~ ~ then Xi Xj - XK contains at least two atoms, and therefore ~ ~ (Xi Xj - XK ) 0. ^ ~ (14.86) This completes the proof for the -inequality in (14.81) to be nonredundant when L(i, j, K) = . We now consider the case when L(i, j, K) = , or |L(i, j, K)| 1. We ~ ~ ~ construct a measure as follows. For the atoms in Xi Xj - XK , let ^ (Cij|K (S)) = ^ (-1)|S| - 1 S = L(i, j, K) (-1)|S| S = L(i, j, K). (14.87) ~ ~ ~ For Cij|K (S), if |S| is odd, it is referred to as an odd atom of Xi Xj - XK , ~ i Xj - XK . For any ~ ~ and if |S| is even, it is referred to as an even atom of X ~ ~ atom A Xi Xj - XK , we let / ~ (A) = 1. ^ This completes the construction of . ^ We first prove that ~ ~ (Xi Xj - XK ) < 0. ^ ~ Consider (14.88) (14.89) 14.6 Minimality of the Elemental Inequalities 355 ~ ~ (Xi Xj - XK ) = ^ ~ SL(i,j,K) (Cij|K (S)) ^ |L(i,j,K)| |L(i, j, K)| r (-1)r - 1 = r=0 = -1, where the last equality follows from the binomial formula n r=0 n r (-1)r = 0 (14.90) for n 1. This proves (14.89). Next we prove that satisfies all -inequalities. We note that for any ^ ~ ~ ~ ~ ~ i Nn , the atom Xi - XNn -{i } is not in Xi Xj - XK . Thus ~ (Xi - XNn -{i} ) = 1 0. ^ ~ (14.91) It remains to prove that satisfies all -inequalities except for (14.81), ^ i.e., for any (i , j , K ) = (i, j, K) such that i = j and K Nn - {i , j }, ~ ~ (Xi Xj - XK ) 0. ^ ~ Consider ~ ~ (Xi Xj - XK ) ^ ~ ~ ~ ~ ~ ~ = ((Xi Xj - XK ) (Xi Xj - XK )) ^ ~ ~ ~ ~ ~ ~ +^((Xi Xj - XK ) - (Xi Xj - XK )). ~ (14.92) (14.93) The nonnegativity of the second term above follows from (14.88). For the first term, ~ ~ ~ ~ ~ ~ (Xi Xj - XK ) (Xi Xj - XK ) (14.94) is nonempty if and only if {i , j } K = and {i, j} K = . (14.95) If this condition is not satisfied, then the first term in (14.93) becomes () = ^ 0, and (14.92) follows immediately. Let us assume that the condition in (14.95) is satisfied. Then by simple counting, we see that the number atoms in ~ ~ ~ ~ ~ ~ (Xi Xj - XK ) (Xi Xj - XK ) is equal to 2 , where = n - |{i, j} {i , j } K K |. (14.97) (14.96) 356 14 Shannon-Type Inequalities For example, for n = 6, there are 4 = 22 atoms in ~ ~ ~ ~ ~ (X1 X2 ) (X1 X3 - X4 ), (14.98) ~ ~ ~ ~c ~ ~c namely X1 X2 X3 X4 Y5 Y6 , where Yi = Xi or Xi for i = 5, 6. We check that = 6 - |{1, 2} {1, 3} {4}| = 2. (14.99) We first consider the case when = 0, i.e., Nn = {i, j} {i , j } K K . Then ~ ~ ~ ~ ~ ~ (Xi Xj - XK ) (Xi Xj - XK ) (14.101) ~ ~ ~ contains exactly one atom. If this atom is an even atom of Xi Xj - XK , then the first term in (14.93) is either 0 or 1 (cf., (14.87)), and (14.92) follows ~ ~ ~ immediately. If this atom is an odd atom of Xi Xj - XK , then the first term in (14.93) is equal to -1. This happens if and only if {i, j} and {i , j } have ~ ~ ~ ~ ~ ~ one common element, which implies that (Xi Xj - XK ) - (Xi Xj - XK ) is nonempty. Therefore the second term in (14.93) is at least 1, and hence (14.92) follows. Finally, we consider the case when 1. Using the binomial formula in ~ ~ ~ (14.90), we see that the number of odd atoms and even atoms of Xi Xj - XK in ~ ~ ~ ~ ~ ~ (Xi Xj - XK ) (Xi Xj - XK ) (14.102) are the same. Therefore the first term in (14.93) is equal to -1 if ~ ~ ~ Cij|K (L(i, j, K)) Xi Xj - XK , (14.103) (14.100) and is equal to 0 otherwise. The former is true if and only if K K, which ~ ~ ~ ~ ~ ~ implies that (Xi Xj - XK ) - (Xi Xj - XK ) is nonempty, or that the second term is at least 1. Thus in either case (14.92) is true. This completes the proof that (14.81) is nonredundant. Appendix 14.A: The Basic Inequalities and the Polymatroidal Axioms In this appendix, we show that the basic inequalities for a collection of n random variables = {Xi , i Nn } is equivalent to the following polymatroidal axioms: For all , Nn , P1. H () = 0. P2. H () H () if . P3. H () + H () H ( ) + H ( ). Chapter Summary 357 We first show that the polymatroidal axioms imply the basic inequalities. From P1 and P2, since for any Nn , we have H () H () = 0, or H(X ) 0. This shows that entropy is nonnegative. In P2, letting = \, we have H () H ( ), or H(X |X ) 0. Here, and are disjoint subsets of Nn . In P3, letting = \, = , and = \, we have H ( ) + H ( ) H () + H ( ), or I(X ; X |X ) 0. (14.109) Again, , , and are disjoint subsets of Nn . When = , from P3, we have I(X ; X ) 0. (14.110) (14.108) (14.107) (14.106) (14.105) (14.104) Thus P1 to P3 imply that entropy is nonnegative, and that conditional entropy, mutual information, and conditional mutual information are nonnegative provided that they are irreducible. However, it has been shown in Section 14.1 that a reducible Shannon's information measure can always be written as the sum of irreducible Shannon's information measures. Therefore, we have shown that the polymatroidal axioms P1 to P3 imply the basic inequalities. The converse is trivial and the proof is omitted. Chapter Summary Shannon-Type Inequalities are information inequalities implied by the basic inequalities. Elemental Form of Shannon's Information Measures: Any Shannon's information measure involving random variables X1 , X2 , , Xn can be expressed as the sum of the following two element forms: i) H(Xi |XNn -{i} ), i Nn ii) I(Xi ; Xj |XK ), where i = j and K Nn - {i, j}. 358 14 Shannon-Type Inequalities Elemental Inequalities: For a set of random variables, the nonnegativity of the two elemental forms of Shannon's information measures are called the elemental inequalities. The elemental inequalities are equivalent to the basic inequalities for the same set of random variables, and they form the minimal such subset of the basic inequalities. The region n = {h Hn : Gh 0}, is the subset of Hn defined by the basic inequalities for n random variables, and n n . Unconstrained Shannon-Type Inequalities: b h 0 is a Shannon-type inequality if and only if one of the following is true: 1. n {h Hn : b h 0}. 2. The minimum of the problem "Minimize b h, subject to Gh 0" is zero. Constrained Shannon-Type Inequalities: Under the constraint = {h Hn : Qh = 0}, b h 0 is a Shannon-type inequality if and only if 1. (n ) {h Hn : b h 0}. 2. The minimum of the problem "Minimize b h, subject to Gh 0 and Qh = 0" is zero. Duality: An unconstrained Shannon-type inequality is a nonnegative linear combination of the elemental inequalities for the same set of random variables. ITIP is a software package running on MATLAB for proving Shannon-type inequalities. Problems 1. Prove (14.12) for the total number of elemental forms of Shannon's information measures for n random variables. 2. Shannon-type inequalities for n random variables X1 , X2 , , Xn refer to all information inequalities implied by the basic inequalities for these n random variables. Show that no new information inequality can be generated by considering the basic inequalities for more than n random variables. 3. Show by an example that the decomposition of an information expression into a sum of elemental forms of Shannon's information measures is not unique. 4. Elemental forms of conditional independencies Consider random variables X1 , X2 , , Xn . A conditional independency is said to be elemental if it corresponds to setting an elemental form of Shannon's information measure to zero. Show that any conditional independency involving X1 , X2 , , Xn is equivalent to a collection of elemental conditional independencies. Problems 359 5. Symmetrical information inequalities a) Show that every symmetrical information expression (cf. Problem 1 in Chapter 13) involving random variable X1 , X2 , , Xn can be written in the form n-1 E= k=0 ak ck , (n) where c0 and for 1 k n - 1, ck (n) (n) (n) n = i=1 H(Xi |XN -i ) = 1i<jn KN -{i,j},|K|=k-1 I(Xi ; Xj |XK ). Note that c0 is the sum of all Shannon's information measures of (n) the first elemental form, and for 1 k n - 1, ck is the sum of all Shannon's information measures of the second elemental form conditioning on k - 1 random variables. b) Show that E 0 always holds if ak 0 for all k. c) Show that if E 0 always holds, then ak 0 for all k. Hint: Construct random variables X1 , X2 , , Xn for each 0 k n - 1 such that (n) (n) ck > 0 and ck = 0 for all 0 k n - 1 and k = k. (Han [147].) 6. Strictly positive probability distributions It was shown in Proposition 2.12 that X1 X4 |(X2 , X3 ) X1 (X3 , X4 )|X2 X1 X3 |(X2 , X4 ) if p(x1 , x2 , x3 , x4 ) > 0 for all x1 , x2 , x3 , and x4 . Show by using ITIP that this implication is not implied by the basic inequalities. This strongly suggests that this implication does not hold in general, which was shown to be the case by the construction following Proposition 2.12. 7. a) Verify by ITIP that I(X1 , X2 ; Y1 , Y2 ) I(X1 ; Y1 ) + I(X2 ; Y2 ) under the constraint H(Y1 , Y2 |X1 , X2 ) = H(Y1 |X1 ) + H(Y2 |X2 ). This constrained inequality was used in Problem 10 in Chapter 7 to obtain the capacity of two parallel channels. b) Verify by ITIP that I(X1 , X2 ; Y1 , Y2 ) I(X1 ; Y1 ) + I(X2 ; Y2 ) under the constraint I(X1 ; X2 ) = 0. This constrained inequality was used in Problem 4 in Chapter 8 to obtain the rate-distortion function for a product source. 360 14 Shannon-Type Inequalities 8. Verify by ITIP the information identity in Example 3.18. 9. Repeat Problem 13 in Chapter 3 with the help of ITIP. 10. Prove the implications in Problem 15 in Chapter 3 by ITIP and show that they cannot be deduced from the semi-graphoidal axioms. (Studen y [346].) Historical Notes For almost half a century, all information inequalities known in the literature are consequences of the basic inequalities due to Shannon [322]. Fujishige [126] showed that the entropy function is a polymatroid (see Appendix 14.A). Yeung [401] showed that verification of all such inequalities, referred to Shannon-type inequalities, can be formulated as a linear programming problem if the number of random variables involved is fixed. ITIP, a software package for this purpose, was developed by Yeung and Yan [409]. Non-Shannon-type inequalities, which were first discovered in the late 1990's, will be discussed in the next chapter. 15 Beyond Shannon-Type Inequalities In Chapter 13, we introduced the regions n and n in the entropy space Hn for n random variables. From n , one in principle can determine whether any information inequality always holds. The region n , defined by the set of all basic inequalities (equivalently all elemental inequalities) involving n random variables, is an outer bound on n . From n , one can determine whether any information inequality is implied by the basic inequalities. If so, it is called a Shannon-type inequality. Since the basic inequalities always hold, so do all Shannon-type inequalities. In the last chapter, we have shown how machine-proving of all Shannon-type inequalities can be made possible by taking advantage of the linear structure of n . If the two regions n and n are identical, then all information inequalities which always hold are Shannon-type inequalities, and hence all information inequalities can be completely characterized. However, if n is a proper subset of n , then there exist constraints on an entropy function which are not implied by the basic inequalities. Such a constraint, if in the form of an inequality, is referred to as a non-Shannon-type inequality. There is a point here which needs further explanation. The fact that n = n does not necessarily imply the existence of a non-Shannon-type inequality. As an example, suppose n contains all but an isolated point in n . Then this does not lead to the existence of a non-Shannon-type inequality for n random variables. In this chapter, we present characterizations of n which are more refined than n . These characterizations lead to the existence of non-Shannon-type inequalities for n 4. 15.1 Characterizations of , , and 2 3 n Recall from the proof of Theorem 3.6 that the vector h represents the values of the I-Measure on the unions in Fn . Moreover, h is related to the values of on the atoms of Fn , represented as u, by 362 15 Beyond Shannon-Type Inequalities h = Cn u (15.1) where Cn is a unique k k matrix with k = 2n - 1 (cf. (3.27)). Let In be the k-dimensional Euclidean space with the coordinates labeled by the components of u. Note that each coordinate in In corresponds to the value of on a nonempty atom of Fn . Recall from Lemma 13.1 the definition of the region n = {u In : Cn u n }, (15.2) which is obtained from the region n via the linear transformation induced -1 by Cn . Analogously, we define the region n = {u In : Cn u n }. (15.3) The region n , as we will see, is extremely difficult to characterize for a general n. Therefore, we start our discussion with the simplest case, namely n = 2. Theorem 15.1. 2 = 2 . Proof. For n = 2, the elemental inequalities are ~ ~ H(X1 |X2 ) = (X1 - X2 ) 0 ~ ~ H(X2 |X1 ) = (X2 - X1 ) 0 ~ ~ I(X1 ; X2 ) = (X1 X2 ) 0. (15.4) (15.5) (15.6) Note that the quantities on the left hand sides above are precisely the values of on the atoms of F2 . Therefore, 2 = {u I2 : u 0}, (15.7) i.e., 2 is the nonnegative orthant of I2 . Since 2 2 , 2 2 . On the other hand, 2 2 by Lemma 13.1. Thus 2 = 2 , which implies 2 = 2 . The proof is accomplished. Next, we prove that Theorem 15.1 cannot even be generalized to n = 3. Theorem 15.2. 3 = 3 . Proof. For n = 3, the elemental inequalities are ~ ~ ~ H(Xi |Xj , Xk ) = (Xi - Xj - Xk ) 0 ~ ~ ~ I(Xi ; Xj |Xk ) = (Xi Xj - Xk ) 0, and ~ ~ I(Xi ; Xj ) = (Xi Xj ) ~ ~ ~ ~ ~ ~ = (Xi Xj Xk ) + (Xi Xj - Xk ) 0 (15.10) (15.11) (15.12) (15.8) (15.9) 15.1 Characterizations of , , and n 2 3 363 X2 0 a X1 0 a a a 0 X3 Fig. 15.1. The set-theoretic structure of the point (0, 0, 0, a, a, a, -a) in 3 . for 1 i < j < k 3. For u I3 , let u = (u1 , u2 , u3 , u4 , u5 , u6 , u7 ), where ui , 1 i 7 correspond to the values ~ ~ ~ ~ ~ ~ ~ ~ ~ (X1 - X2 - X3 ), (X2 - X1 - X3 ), (X3 - X1 - X2 ), ~ ~ ~ ~ ~ ~ ~ ~ ~ (X1 X2 - X3 ), (X1 X3 - X2 ), (X2 X3 - X1 ), ~ ~ ~ (X1 X2 X3 ), (15.13) (15.14) respectively. These are the values of on the nonempty atoms of F3 . Then from (15.8), (15.9), and (15.12), we see that 3 = {u I3 : ui 0, 1 i 6; uj + u7 0, 4 j 6}. (15.15) It is easy to check that the point (0, 0, 0, a, a, a, -a) for any a 0 is in 3 . This is illustrated in Figure 15.1, and it is readily seen that the relations H(Xi |Xj , Xk ) = 0 and I(Xi ; Xj ) = 0 (15.17) for 1 i < j < k 3 are satisfied, i.e., each random variable is a function of the other two, and the three random variables are pairwise independent. Let SXi be the support of Xi , i = 1, 2, 3. For any x1 SX1 and x2 SX2 , since X1 and X2 are independent, we have p(x1 , x2 ) = p(x1 )p(x2 ) > 0. (15.18) (15.16) Since X3 is a function of X1 and X2 , there is a unique x3 SX3 such that p(x1 , x2 , x3 ) = p(x1 , x2 ) = p(x1 )p(x2 ) > 0. (15.19) Since X2 is a function of X1 and X3 , and X1 and X3 are independent, we can write 364 15 Beyond Shannon-Type Inequalities p(x1 , x2 , x3 ) = p(x1 , x3 ) = p(x1 )p(x3 ). Equating (15.19) and (15.20), we have p(x2 ) = p(x3 ). (15.20) (15.21) Now consider any x2 SX2 such that x2 = x2 . Since X2 and X3 are independent, we have p(x2 , x3 ) = p(x2 )p(x3 ) > 0. (15.22) Since X1 is a function of X2 and X3 , there is a unique x1 SX1 such that p(x1 , x2 , x3 ) = p(x2 , x3 ) = p(x2 )p(x3 ) > 0. (15.23) Since X2 is a function of X1 and X3 , and X1 and X3 are independent, we can write p(x1 , x2 , x3 ) = p(x1 , x3 ) = p(x1 )p(x3 ). (15.24) Similarly, since X3 is a function of X1 and X2 , and X1 and X2 are independent, we can write p(x1 , x2 , x3 ) = p(x1 , x2 ) = p(x1 )p(x2 ). (15.25) Equating (15.24) and (15.25), we have p(x2 ) = p(x3 ), and from (15.21), we have p(x2 ) = p(x2 ). (15.27) Therefore, X2 must have a uniform distribution on its support. The same can be proved for X1 and X3 . Now from Figure 15.1, H(X1 ) = H(X1 |X2 , X3 ) + I(X1 ; X2 |X3 ) + I(X1 ; X3 |X2 ) +I(X1 ; X2 ; X3 ) = 0 + a + a + (-a) = a, and similarly H(X2 ) = H(X3 ) = a. (15.31) Then the only values that a can take are log M , where M (a positive integer) is the cardinality of the supports of X1 , X2 , and X3 . In other words, if a is not equal to log M for some positive integer M , then the point (0, 0, 0, a, a, a, -a) is not in 3 . This proves that 3 = 3 , which implies 3 = 3 . The theorem is proved. The proof above has the following interpretation. For h H3 , let h = (h1 , h2 , h3 , h12 , h13 , h23 , h123 ). (15.32) (15.28) (15.29) (15.30) (15.26) 15.1 Characterizations of , , and n 2 3 a=0 log 2 log 3 log 4 ( a , a , a , a , 2a, 2a, 2a) 365 Fig. 15.2. The values of a for which (a, a, a, 2a, 2a, 2a, 2a) is in 3 . From Figure 15.1, we see that the point (0, 0, 0, a, a, a, -a) in 3 corresponds to the point (a, a, a, 2a, 2a, 2a, 2a) in 3 . Evidently, the point (a, a, a, 2a, 2a, 2a, 2a) in 3 satisfies the 6 elemental inequalities given in (15.8) and (15.12) for 1 i < j < k 3 with equality. Since 3 is defined by all the elemental inequalities, the set {(a, a, a, 2a, 2a, 2a, 2a) 3 : a 0} (15.33) is in the intersection of 6 hyperplanes in H3 (i.e., 7 ) defining the boundary of 3 , and hence it defines an extreme direction of 3 . Then the proof says that along this extreme direction of 3 , only certain discrete points, namely those points with a equals log M for some positive integer M , are entropic. This is illustrated in Figure 15.2. As a consequence, the region 3 is not convex. Having proved that 3 = 3 , it is natural to conjecture that the gap between 3 and 3 has zero Lebesgue measure. In other words, 3 = 3 , where 3 is the closure of 3 . This is indeed the case and will be proved at the end of the section. More generally, we are interested in characterizing n , the closure of n . Although the region n is not sufficient for characterizing all information inequalities, it is actually sufficient for characterizing all unconstrained information inequalities. This can be seen as follows. Following the discussion in Section 13.3.1, an unconstrained information inequality f 0 involving n random variables always hold if and only if n {h : f (h) 0}. (15.34) Since {h : f (h) 0} is closed, upon taking closure on both sides, we have n {h : f (h) 0}. On the other hand, if f 0 satisfies (15.35), then n n {h : f (h) 0}. (15.35) (15.36) Therefore, (15.34) and (15.35) are equivalent, and hence n is sufficient for characterizing all unconstrained information inequalities. We will prove in the next theorem an important property of the region n for all n 2. This result will be used in the proof for 3 = 3 . Further, this result will be used in Chapter 16 when we establish a fundamental relation between information theory and group theory. We first prove a simple lemma. In the following, we use Nn to denote the set {1, 2, , n}. 366 15 Beyond Shannon-Type Inequalities Lemma 15.3. If h and h are in n , then h + h is in n . Proof. Consider h and h in n . Let h represents the entropy function for random variables X1 , X2 , , Xn , and let h represents the entropy function for random variables X1 , X2 , , Xn . Let (X1 , X2 , , Xn ) and (X1 , X2 , , Xn ) be independent, and define random variables Y1 , Y2 , , Yn by Yi = (Xi , Xi ) for all i Nn . Then for any subset of Nn , H(Y ) = H(X ) + H(X ) = h + h . (15.37) (15.38) Therefore, h + h , which represents the entropy function for Y1 , Y2 , , Yn , is in n . The lemma is proved. Corollary 15.4. If h n , then kh n for any positive integer k. Proof. It suffices to write kh = h + h + + h k (15.39) and apply Lemma 15.3. Theorem 15.5. n is a convex cone. Proof. Consider the entropy function for random variables X1 , X2 , , Xn all taking constant values with probability 1. Then for all subset of Nn , H(X ) = 0. (15.40) Therefore, n contains the origin in Hn . Let h and h in n be the entropy functions for any two sets of random variables Y1 , Y2 , , Yn and Z1 , Z2 , , Zn , respectively. In view of Corollary 15.4, in order to prove that n is a convex cone, we only need to show that if h bh and h are in n , then bh + is in n for all 0 < b < 1, where = 1 - b. b Let (Y1 , Y2 , , Yn ) be k independent copies of (Y1 , Y2 , , Yn ) and (Z1 , Z2 , , Zn ) be k independent copies of (Z1 , Z2 , , Zn ). Let U be a ternary random variable independent of all other random variables such that Pr{U = 0} = 1 - - , Pr{U = 1} = , Pr{U = 2} = . Now construct random variables X1 , X2 , , Xn by letting 0 if U = 0 Xi = Yi if U = 1 Zi if U = 2. 15.1 Characterizations of , , and n 2 3 367 Note that H(U ) 0 as , 0. Then for any nonempty subset of Nn , H(X ) H(X , U ) = H(U ) + H(X |U ) = H(U ) + kH(Y ) + kH(Z ). On the other hand, H(X ) H(X |U ) = kH(Y ) + kH(Z ). Combining the above, we have 0 H(X ) - (kH(Y ) + kH(Z )) H(U ). Now take = and = to obtain 0 H(X ) - (bH(Y ) + bH(Z )) H(U ). (15.48) By letting k be sufficiently large, the upper bound can be made arbitrarily small. This shows that bh + n . The theorem is proved. bh In the next theorem, we prove that 3 and 3 are almost identical. Anal ogous to n , we will use n to denote the closure of n . (15.41) (15.42) (15.43) (15.44) (15.45) b k b k (15.46) (15.47) Theorem 15.6. 3 = 3 . Proof. We first note that 3 = 3 if and only if 3 = 3 . Since 3 3 (15.49) (15.50) and 3 is closed, by taking closure on both sides in the above, we obtain 3 3 . This implies that 3 3 . Therefore, in order to prove the theorem, it suffices to show that 3 3 . We first show that the point (0, 0, 0, a, a, a, -a) is in 3 for all a > 0. Let random variables X1 , X2 , and X3 be defined as in Example 3.10, i.e., X1 and X2 are two independent binary random variables taking values in {0, 1} according to the uniform distribution, and X3 = X1 + X2 mod 2. (15.51) 368 15 Beyond Shannon-Type Inequalities X2 0 1 X1 0 1 1 1 0 X3 Fig. 15.3. The I-Measure for X1 , X2 , and X3 in the proof of Theorem 15.6. Let h 3 represents the entropy function for X1 , X2 , and X3 , and let -1 u = C3 h. (15.52) As in the proof of Theorem 15.2, we let ui , 1 i 7, be the coordinates of I3 which correspond to the values of the quantities in (15.14), respectively. From Example 3.10, we have 0 for i = 1, 2, 3 ui = 1 for i = 4, 5, 6 (15.53) -1 for i = 7. Thus the point (0, 0, 0, 1, 1, 1, -1) is in 3 , and the I-Measure for X1 , X2 , and X3 is shown in Figure 15.3. Then by Corollary 15.4, (0, 0, 0, k, k, k, -k) is in 3 and hence in 3 for all positive integer k. Since 3 contains the origin, 3 also contains the origin. By Theorem 15.5, 3 is convex. This implies 3 is also convex. Therefore, (0, 0, 0, a, a, a, -a) is in 3 for all a > 0. Consider any u 3 . Referring to (15.15), we have ui 0 (15.54) for 1 i 6. Thus u7 is the only component of u which can possibly be negative. We first consider the case when u7 0. Then u is in the nonnegative orthant of I3 , and by Lemma 13.1, u is in 3 . Next, consider the case when u7 < 0. Let t = (0, 0, 0, -u7 , -u7 , -u7 , u7 ). (15.55) Then u = w + t, where w = (u1 , u2 , u3 , u4 + u7 , u5 + u7 , u6 + u7 , 0). Since -u7 > 0, we see from the foregoing that t ui + u7 0 3. (15.56) (15.57) From (15.15), we have (15.58) 15.2 A Non-Shannon-Type Unconstrained Inequality 369 for i = 4, 5, 6. Thus w is in the nonnegative orthant in I3 and hence in 3 by Lemma 13.1. Now for any > 0, let t 3 such that t-t where t - t < , (15.59) denotes the Euclidean distance between t and t , and let u =w+t. (15.60) Since both w and t are in 3 , by Lemma 15.3, u is also in 3 , and u-u = t-t < . (15.61) Therefore, u 3 . Hence, 3 3 , and the theorem is proved. Remark 1 Han [148] has found that 3 is the smallest cone that contains 3 . This result together with Theorem 15.5 implies Theorem 15.6. Theorem 15.6 was also obtained by Goli [137], and it is a consequence of the theorem in c Mat [254]. us Remark 2 We have shown that the region n completely characterizes all unconstrained information inequalities involving n random variables. Since 3 = 3 , it follows that there exists no unconstrained information inequalities involving three random variables other than the Shannon-type inequalities. Mat [258] has obtained piecewise-linear constrained non-Shannon-type us inequalities for three random variables that generalize the construction in the proof of Theorem 15.2. 15.2 A Non-Shannon-Type Unconstrained Inequality We have proved in Theorem 15.6 at the end of the last section that 3 = 3 . It is natural to conjecture that this theorem can be generalized to n 4. If this conjecture is true, then it follows that all unconstrained information inequalities involving a finite number of random variables are Shannon-type inequalities, and they can all be proved by ITIP running on a sufficiently powerful computer. However, it turns out that this is not the case even for n = 4. We will prove in the next theorem an unconstrained information inequality involving four random variables. Then we will show that this inequality is a non-Shannon-type inequality, and that 4 = 4 . Theorem 15.7. For any four random variables X1 , X2 , X3 , and X4 , 2I(X3 ; X4 ) I(X1 ; X2 ) + I(X1 ; X3 , X4 ) +3I(X3 ; X4 |X1 ) + I(X3 ; X4 |X2 ). (15.62) 370 15 Beyond Shannon-Type Inequalities Toward proving this theorem, we introduce two auxiliary random variables ~ ~ ~ X1 and X2 jointly distributed with X1 , X2 , X3 , and X4 such that X1 = X1 ~2 = X2 . To simplify notation, we will use p and X ~ ~ 1234~~ (x1 , x2 , x3 , x4 , x1 , x2 ) 12 to denote pX1 X2 X3 X4 X1 X2 (x1 , x2 , x3 , x4 , x1 , x2 ), etc. The joint distribution for ~ ~ ~ ~ ~ ~ the six random variables X1 , X2 , X3 , X4 , X1 , and X2 is defined by p1234~~ (x1 , x2 , x3 , x4 , x1 , x2 ) ~ ~ 12 = p1234 (x1 ,x2 ,x3 ,x4 )p1234 (~1 ,~2 ,x3 ,x4 ) x x p34 (x3 ,x4 ) 0 if p34 (x3 , x4 ) > 0 if p34 (x3 , x4 ) = 0. (15.63) Lemma 15.8. ~ ~ (X1 , X2 ) (X3 , X4 ) (X1 , X2 ) (15.64) ~ ~ forms a Markov chain. Moreover, (X1 , X2 , X3 , X4 ) and (X1 , X2 , X3 , X4 ) have the same marginal distribution. Proof. The Markov chain in (15.64) is readily seen by invoking Proposition 2.5. The second part of the lemma is readily seen to be true by noting ~ ~ that in (15.63), p1234~~ is symmetrical in X1 and X1 and in X2 and X2 . 12 From the above lemma, we see that the pair of auxiliary random vari~ ~ ables (X1 , X2 ) corresponds to the pair of random variables (X1 , X2 ) in ~ ~ the sense that (X1 , X2 , X3 , X4 ) have the same marginal distribution as (X1 , X2 , X3 , X4 ). We need to prove two inequalities regarding these six random variables before we prove Theorem 15.7. Lemma 15.9. For any four random variables X1 , X2 , X3 , and X4 and auxil~ ~ iary random variables X1 and X2 as defined in (15.63), ~ I(X3 ; X4 ) - I(X3 ; X4 |X1 ) - I(X3 ; X4 |X2 ) I(X1 ; X2 ). (15.65) Proof. Consider I(X3 ; X4 ) - I(X3 ; X4 |X1 ) - I(X3 ; X4 |X2 ) a) ~ = [I(X3 ; X4 ) - I(X3 ; X4 |X1 )] - I(X3 ; X4 |X2 ) ~ = I(X1 ; X3 ; X4 ) - I(X3 ; X4 |X2 ) (15.66) (15.67) (15.68) (15.69) (15.70) (15.71) (15.72) ~ ~ ~ = [I(X1 ; X3 ; X4 ; X2 ) + I(X1 ; X3 ; X4 |X2 )] - I(X3 ; X4 |X2 ) ~ 2 ) - [I(X3 ; X4 |X2 ) - I(X1 ; X3 ; X4 |X2 )] ~ ~ = I(X1 ; X3 ; X4 ; X ~ ~ = I(X1 ; X3 ; X4 ; X2 ) - I(X3 ; X4 |X1 , X2 ) ~ ~ ~ = [I(X1 ; X4 ; X2 ) - I(X1 ; X4 ; X2 |X3 )] - I(X3 ; X4 |X1 , X2 ) ~ ~ ~ = [I(X1 ; X2 ) - I(X1 ; X2 |X4 )] - [I(X1 ; X2 |X3 ) ~ ~ -I(X1 ; X2 |X3 , X4 )] - I(X3 ; X4 |X1 , X2 ) 15.2 A Non-Shannon-Type Unconstrained Inequality b) ~ ~ ~ = I(X1 ; X2 ) - I(X1 ; X2 |X4 ) - I(X1 ; X2 |X3 ) ~ -I(X3 ; X4 |X1 , X2 ) 371 (15.73) (15.74) ~ I(X1 ; X2 ), where a) follows because we see from Lemma 15.8 that (X2 , X3 , X4 ) and ~ (X2 , X3 , X4 ) have the same marginal distribution, and b) follows because ~ I(X1 ; X2 |X3 , X4 ) = 0 from the Markov chain in (15.64). The lemma is proved. Lemma 15.10. For any four random variables X1 , X2 , X3 , and X4 and aux~ ~ iliary random variables X1 and X2 as defined in (15.63), ~ I(X3 ; X4 ) - 2I(X3 ; X4 |X1 ) I(X1 ; X1 ). (15.76) (15.75) Proof. Notice that (15.76) can be obtained from (15.65) by replacing X2 by ~ ~ X1 and X2 by X1 in (15.65). The inequality (15.76) can be proved by replacing ~ ~ X2 by X1 and X2 by X1 in (15.66) through (15.74) in the proof of the last lemma. The details are omitted. Proof of Theorem 15.7. By adding (15.65) and (15.76), we have 2I(X3 ; X4 ) - 3I(X3 ; X4 |X1 ) - I(X3 ; X4 |X2 ) ~ ~ I(X1 ; X2 ) + I(X1 ; X1 ) ~ ~ ~ ~ ~ = I(X1 ; X2 ) + [I(X1 ; X1 |X2 ) + I(X1 ; X1 ; X2 )] ~ ~ ~ ~ ~ = [I(X1 ; X2 ) + I(X1 ; X1 |X2 )] + I(X1 ; X1 ; X2 ) ~ ~ ~ ~ = I(X1 ; X1 , X2 ) + I(X1 ; X1 ; X2 ) ~ ~ ~ ~ ~ ~ = I(X1 ; X1 , X2 ) + [I(X1 ; X2 ) - I(X1 ; X2 |X1 )] ~ ~ ~ ~ I(X1 ; X1 , X2 ) + I(X1 ; X2 ) ~ ~ I(X1 ; X3 , X4 ) + I(X1 ; X2 ) = I(X1 ; X3 , X4 ) + I(X1 ; X2 ), b) a) (15.77) (15.78) (15.79) (15.80) (15.81) (15.82) (15.83) (15.84) where a) follows from the Markov chain in (15.64), and b) follows because ~ ~ we see from Lemma 15.8 that (X1 , X2 ) and (X1 , X2 ) have the same marginal ~ ~ distribution. Note that the auxiliary random variables X1 and X2 disappear in (15.84) after the sequence of manipulations. The theorem is proved. Theorem 15.11. The inequality (15.62) is a non-Shannon-type inequality, and 4 = 4 . 372 15 Beyond Shannon-Type Inequalities 0 a a a a 0 0 a a 0 0 0 a X2 X1 a 0 X3 X4 ~ Fig. 15.4. The set-theoretic structure of h(a). ~ Proof. Consider for any a > 0 the point h(a) H4 , where ~ ~ ~ ~ h1 (a) = h2 (a) = h3 (a) = h4 (a) = 2a, ~ ~ ~ h12 (a) = 4a, h13 (a) = h14 (a) = 3a, ~ ~ ~ h23 (a) = h24 (a) = h34 (a) = 3a, ~ 123 (a) = h124 (a) = h134 (a) = h234 (a) = h1234 (a) = 4a. ~ ~ ~ ~ h (15.85) ~ The set-theoretic structure of h(a) is illustrated by the information diagram in Figure 15.4. The reader should check that this information diagram correctly ~ ~ represents h(a) as defined. It is also easy to check from this diagram that h(a) satisfies all the elemental inequalities for four random variables, and therefore ~ h(a) 4 . However, upon substituting the corresponding values in (15.62) for ~ h(a) with the help of Figure 15.4, we have 2a 0 + a + 0 + 0 = a, (15.86) ~ which is a contradiction because a > 0. In other words, h(a) does not satisfy (15.62). Equivalently, ~ h(a) {h H4 : h satisfies (15.62)}. ~ Since h(a) 4 , we conclude that 4 {h H4 : h satisfies (15.62)}, (15.88) (15.87) i.e., (15.62) is not implied by the basic inequalities for four random variables. Hence, (15.62) is a non-Shannon-type inequality. Since (15.62) is satisfied by all entropy functions for four random variables, we have 4 {h H4 : h satisfies (15.62)}, (15.89) 15.2 A Non-Shannon-Type Unconstrained Inequality 373 and upon taking closure on both sides, we have 4 {h H4 : h satisfies (15.62)}. ~ ~ ~ Then (15.87) implies h(a) Since h(a) 4 and h(a) that 4 = 4 . The theorem is proved. 4. 4, (15.90) we conclude Remark We have shown in the proof of Theorem 15.11 that the inequality (15.62) cannot be proved by invoking the basic inequalities for four random variables. However, (15.62) can be proved by invoking the basic inequalities for ~ ~ the six random variables X1 , X2 , X3 , X4 , X1 , and X2 with the joint probability distribution p1234~~ as constructed in (15.63). 12 The inequality (15.62) remains valid when the indices 1, 2, 3, and 4 are permuted. Since (15.62) is symmetrical in X3 and X4 , 4!/2! = 12 distinct versions of (15.62) can be obtained by permuting the indices, and all these 12 inequalities are simultaneously satisfied by the entropy function of any set of random variables X1 , X2 , X3 , and X4 . We will denote these 12 inequalities collectively by 15.62 . Now define the region ~ 4 = {h 4 : h satisfies 15.62 }. Evidently, ~ 4 4 4 . (15.91) (15.92) ~ Since both 4 and 4 are closed, upon taking closure, we also have ~ 4 4 4 . (15.93) Since 15.62 are non-Shannon-type inequalities as we have proved in the last ~ theorem, 4 is a proper subset of 4 and hence a tighter outer bound on 4 and 4 than 4 . In the course of proving that (15.62) is of non-Shannon-type, it was shown ~ in the proof of Theorem 15.11 that there exists h(a) 4 as defined in (15.85) which does not satisfy (15.62). By investigating the geometrical relation be~ tween h(a) and 4 , we prove in the next theorem that (15.62) in fact induces a class of 214 - 1 non-Shannon-type constrained inequalities. Applications of some of these inequalities will be discussed in Section 15.4. Theorem 15.12. The inequality (15.62) is a non-Shannon-type inequality conditioning on setting any nonempty subset of the following 14 Shannon's information measures to zero: I(X1 ; X2 ), I(X1 ; X2 |X3 ), I(X1 ; X2 |X4 ), I(X1 ; X3 |X4 ), I(X1 ; X4 |X3 ), I(X2 ; X3 |X4 ), I(X2 ; X4 |X3 ), I(X3 ; X4 |X1 ), I(X3 ; X4 |X2 ), I(X3 ; X4 |X1 , X2 ), H(X1 |X2 , X3 , X4 ), H(X2 |X1 , X3 , X4 ), H(X3 |X1 , X2 , X4 ), H(X4 |X1 , X2 , X3 ). (15.94) 374 15 Beyond Shannon-Type Inequalities ~ Proof. It is easy to verify from Figure 15.4 that h(a) lies in exactly 14 hy15 perplanes in H4 (i.e., ) defining the boundary of 4 which correspond to ~ setting the 14 Shannon's measures in (15.94) to zero. Therefore, h(a) for a 0 define an extreme direction of 4 . ~ Now for any linear subspace of H4 containing h(a), where a > 0, we have ~ h(a) 4 (15.95) ~ and h(a) does not satisfy (15.62). Therefore, (4 ) {h H4 : h satisfies (15.62)}. (15.96) This means that (15.62) is a non-Shannon-type inequality under the constraint . From the above, we see that can be taken to be the intersection of any ~ nonempty subset of the 14 hyperplanes containing h(a). Thus (15.62) is a non-Shannon-type inequality conditioning on any nonempty subset of the 14 Shannon's measures in (15.94) being equal to zero. Hence, (15.62) induces a class of 214 - 1 non-Shannon-type constrained inequalities. The theorem is proved. Remark It is not true that the inequality (15.62) is of non-Shannon-type under any constraint. Suppose we impose the constraint I(X3 ; X4 ) = 0. (15.97) Then the left hand side of (15.62) becomes zero, and the inequality is trivially implied by the basic inequalities because only mutual informations with positive coefficients appear on the right hand side. Then (15.62) becomes a Shannon-type inequality under the constraint in (15.97). 15.3 A Non-Shannon-Type Constrained Inequality In the last section, we proved a non-Shannon-type unconstrained inequality for four random variables which implies 4 = 4 . This inequality induces a region ~ 4 which is a tighter outer bound on 4 and 4 then 4 . We further showed that this inequality induces a class of 214 - 1 non-Shannon-type constrained inequalities for four random variables. In this section, we prove a non-Shannon-type constrained inequality for four random variables. Unlike the non-Shannon-type unconstrained inequality we proved in the last section, this constrained inequality is not strong enough to imply that 4 = 4 . However, the latter is not implied by the former. Lemma 15.13. Let p(x1 , x2 , x3 , x4 ) be any probability distribution. Then p(x1 , x2 , x3 , x4 ) = ~ p(x1 ,x3 ,x4 )p(x2 ,x3 ,x4 ) p(x3 ,x4 ) 0 if p(x3 , x4 ) > 0 if p(x3 , x4 ) = 0 (15.98) 15.3 A Non-Shannon-Type Constrained Inequality 375 is also a probability distribution. Moreover, p(x1 , x3 , x4 ) = p(x1 , x3 , x4 ) ~ and p(x2 , x3 , x4 ) = p(x2 , x3 , x4 ) ~ for all x1 , x2 , x3 , and x4 . Proof. The proof for the first part of the lemma is straightforward (see Problem 4 in Chapter 2). The details are omitted here. To prove the second part of the lemma, it suffices to prove (15.99) for all x1 , x3 , and x4 because p(x1 , x2 , x3 , x4 ) is symmetrical in x1 and x2 . We first ~ consider x1 , x3 , and x4 such that p(x3 , x4 ) > 0. From (15.98), we have p(x1 , x3 , x4 ) = ~ x2 (15.99) (15.100) p(x1 , x2 , x3 , x4 ) ~ p(x1 , x3 , x4 )p(x2 , x3 , x4 ) p(x3 , x4 ) p(x2 , x3 , x4 ) x2 (15.101) (15.102) (15.103) (15.104) (15.105) = x2 = = p(x1 , x3 , x4 ) p(x3 , x4 ) p(x1 , x3 , x4 ) p(x3 , x4 ) p(x3 , x4 ) = p(x1 , x3 , x4 ). For x1 , x3 , and x4 such that p(x3 , x4 ) = 0, we have 0 p(x1 , x3 , x4 ) p(x3 , x4 ) = 0, which implies p(x1 , x3 , x4 ) = 0. Therefore, from (15.98), we have p(x1 , x3 , x4 ) = ~ x2 (15.106) (15.107) p(x1 , x2 , x3 , x4 ) ~ 0 x2 (15.108) (15.109) (15.110) (15.111) = =0 = p(x1 , x3 , x4 ). Thus we have proved (15.99) for all x1 , x3 , and x4 , and the lemma is proved. 376 15 Beyond Shannon-Type Inequalities Theorem 15.14. For any four random variables X1 , X2 , X3 , and X4 , if I(X1 ; X2 ) = I(X1 ; X2 |X3 ) = 0, then I(X3 ; X4 ) I(X3 ; X4 |X1 ) + I(X3 ; X4 |X2 ). (15.113) (15.112) Proof. Consider I(X3 ; X4 ) - I(X3 ; X4 |X1 ) - I(X3 ; X4 |X2 ) = x1 ,x2 ,x3 ,x4 : p(x1 ,x2 ,x3 ,x4 )>0 p(x1 ,x2 ,x3 ,x4 ) log p(x3 ,x4 )p(x1 ,x3 )p(x1 ,x4 )p(x2 ,x3 )p(x2 ,x4 ) p(x3 )p(x4 )p(x1 )p(x2 )p(x1 ,x3 ,x4 )p(x2 ,x3 ,x4 ) = Ep log p(X3 , X4 )p(X1 , X3 )p(X1 , X4 )p(X2 , X3 )p(X2 , X4 ) , p(X3 )p(X4 )p(X1 )p(X2 )p(X1 , X3 , X4 )p(X2 , X3 , X4 ) (15.114) where we have used Ep to denote expectation with respect to p(x1 , x2 , x3 , x4 ). We claim that the above expectation is equal to Ep log ~ p(X3 , X4 )p(X1 , X3 )p(X1 , X4 )p(X2 , X3 )p(X2 , X4 ) , p(X3 )p(X4 )p(X1 )p(X2 )p(X1 , X3 , X4 )p(X2 , X3 , X4 ) (15.115) where p(x1 , x2 , x3 , x4 ) is defined in (15.98). ~ Toward proving that the claim is correct, we note that (15.115) is the sum of a number of expectations with respect to p. Let us consider one of these ~ expectations, say Ep log p(X1 , X3 ) = ~ x1 ,x2 ,x3 ,x4 : p(x1 ,x2 ,x3 ,x4 )>0 ~ p(x1 , x2 , x3 , x4 ) log p(x1 , x3 ). ~ (15.116) Note that in the above summation, if p(x1 , x2 , x3 , x4 ) > 0, then from (15.98), ~ we see that p(x1 , x3 , x4 ) > 0, (15.117) and hence p(x1 , x3 ) > 0. (15.118) Therefore, the summation in (15.116) is always well-defined. Further, it can be written as log p(x1 , x3 ) x1 ,x3 ,x4 x2 :p(x1 ,x2 ,x3 ,x4 )>0 ~ p(x1 , x2 , x3 , x4 ) ~ (15.119) = x1 ,x3 ,x4 p(x1 , x3 , x4 ) log p(x1 , x3 ). ~ 15.3 A Non-Shannon-Type Constrained Inequality 377 Thus Ep log p(X1 , X3 ) depends on p(x1 , x2 , x3 , x4 ) only through p(x1 , x3 , x4 ), ~ ~ ~ which by Lemma 15.13 is equal to p(x1 , x3 , x4 ). It then follows that Ep log p(X1 , X3 ) ~ = x1 ,x3 ,x4 p(x1 , x3 , x4 ) log p(x1 , x3 ) ~ p(x1 , x3 , x4 ) log p(x1 , x3 ) x1 ,x3 ,x4 (15.120) (15.121) (15.122) = = Ep log p(X1 , X3 ). In other words, the expectation on log p(X1 , X3 ) can be taken with respect to either p(x1 , x2 , x3 , x4 ) or p(x1 , x2 , x3 , x4 ) without affecting its value. By ~ observing that all the marginals of p in the logarithm in (15.115) involve only subsets of either {X1 , X3 , X4 } or {X2 , X3 , X4 }, we see that similar conclusions can be drawn for all the other expectations in (15.115), and hence the claim is proved. Thus the claim implies that I(X3 ; X4 ) - I(X3 ; X4 |X1 ) - I(X3 ; X4 |X2 ) p(X3 , X4 )p(X1 , X3 )p(X1 , X4 )p(X2 , X3 )p(X2 , X4 ) = Ep log ~ p(X3 )p(X4 )p(X1 )p(X2 )p(X1 , X3 , X4 )p(X2 , X3 , X4 ) = x1 ,x2 ,x3 ,x4 : p(x1 ,x2 ,x3 ,x4 )>0 ~ p(x1 ,x2 ,x3 ,x4 ) log ~ p(x3 ,x4 )p(x1 ,x3 )p(x1 ,x4 )p(x2 ,x3 )p(x2 ,x4 ) p(x3 )p(x4 )p(x1 )p(x2 )p(x1 ,x3 ,x4 )p(x2 ,x3 ,x4 ) =- x1 ,x2 ,x3 ,x4 : p(x1 ,x2 ,x3 ,x4 )>0 ~ p(x1 , x2 , x3 , x4 ) log ~ p(x1 , x2 , x3 , x4 ) ~ , p(x1 , x2 , x3 , x4 ) ^ (15.123) where p(x1 , x2 , x3 , x4 ) = ^ p(x1 ,x3 )p(x1 ,x4 )p(x2 ,x3 )p(x2 ,x4 ) p(x1 )p(x2 )p(x3 )p(x4 ) 0 if p(x1 ), p(x2 ), p(x3 ), p(x4 ) > 0 otherwise. (15.124) The equality in (15.123) is justified by observing that if x1 , x2 , x3 , and x4 are such that p(x1 , x2 , x3 , x4 ) > 0, then ~ p(x1 , x3 ), p(x1 , x4 ), p(x2 , x3 ), p(x2 , x4 ), p(x1 ), p(x2 ), p(x3 ), p(x4 ) (15.125) are all strictly positive, and we see from (15.124) that p(x1 , x2 , x3 , x4 ) > 0. ^ To complete the proof, we only need to show that p(x1 , x2 , x3 , x4 ) is a ^ probability distribution. Once this is proven, the conclusion of the theorem follows immediately because the summation in (15.123), which is identified as 378 15 Beyond Shannon-Type Inequalities the divergence between p(x1 , x2 , x3 , x4 ) and p(x1 , x2 , x3 , x4 ), is always non~ ^ negative by the divergence inequality (Theorem 2.31). Toward this end, we notice that for x1 , x2 , and x3 such that p(x3 ) > 0, p(x1 , x2 , x3 ) = by the assumption I(X1 ; X2 |X3 ) = 0, and for all x1 and x2 , p(x1 , x2 ) = p(x1 )p(x2 ) by the assumption I(X1 ; X2 ) = 0. Then p(x1 , x2 , x3 , x4 ) ^ x1 ,x2 ,x3 ,x4 p(x1 , x3 )p(x2 , x3 ) p(x3 ) (15.126) (15.127) (15.128) (15.129) = x1 ,x2 ,x3 ,x4 : p(x1 ,x2 ,x3 ,x4 )>0 ^ p(x1 , x2 , x3 , x4 ) ^ p(x1 , x3 )p(x1 , x4 )p(x2 , x3 )p(x2 , x4 ) p(x1 )p(x2 )p(x3 )p(x4 ) p(x1 , x2 , x3 )p(x1 , x4 )p(x2 , x4 ) p(x1 )p(x2 )p(x4 ) p(x1 , x2 , x3 )p(x1 , x4 )p(x2 , x4 ) p(x1 , x2 )p(x4 ) p(x3 |x1 , x2 ) x3 :p(x3 )>0 (15.130) = x1 ,x2 ,x3 ,x4 : p(x1 ),p(x2 ),p(x3 ),p(x4 )>0 (15.131) = x1 ,x2 ,x3 ,x4 : p(x1 ),p(x2 ),p(x3 ),p(x4 )>0 a) (15.132) = x1 ,x2 ,x3 ,x4 : p(x1 ),p(x2 ),p(x3 ),p(x4 )>0 b) (15.133) = x1 ,x2 ,x4 : p(x1 ),p(x2 ),p(x4 )>0 p(x1 , x4 )p(x2 , x4 ) p(x4 ) p(x1 , x4 )p(x2 , x4 ) p(x4 ) (15.134) = x1 ,x2 ,x4 : p(x1 ),p(x2 ),p(x4 )>0 (15.135) (15.136) = x2 ,x4 : p(x2 ),p(x4 )>0 p(x2 , x4 ) x1 :p(x1 )>0 p(x1 |x4 ) = x2 ,x4 : p(x2 ),p(x4 )>0 c) p(x2 , x4 ) (15.137) = 1, d) (15.138) where a) and b) follows from (15.126) and (15.128), respectively. The equality in c) is justified as follows. For x1 such that p(x1 ) = 0, 15.3 A Non-Shannon-Type Constrained Inequality 379 p(x1 |x4 ) = Therefore p(x1 )p(x4 |x1 ) = 0. p(x4 ) p(x1 |x4 ) = 1. x1 (15.139) p(x1 |x4 ) = x1 :p(x1 )>0 (15.140) Finally, the equality in d) is justified as follows. For x2 and x4 such that p(x2 ) or p(x4 ) vanishes, p(x2 , x4 ) must vanish because 0 p(x2 , x4 ) p(x2 ) and 0 p(x2 , x4 ) p(x4 ). Therefore, p(x2 , x4 ) = x2 ,x4 : p(x2 ),p(x4 )>0 (15.141) (15.142) (15.143) p(x2 , x4 ) = 1. x2 ,x4 The theorem is proved. Theorem 15.15. The constrained inequality in Theorem 15.14 is a nonShannon-type inequality. ~ Proof. The theorem can be proved by considering the point h(a) H4 for a > 0 as in the proof of Theorem 15.11. The details are left as an exercise. The constrained inequality in Theorem 15.14 has the following geometrical interpretation. The constraints in (15.112) correspond to the intersection of two hyperplanes in H4 which define the boundary of 4 . Then the inequality (15.62) says that a certain region on the boundary of 4 is not in 4 . It 1 can further be proved by computation that the constrained inequality in Theorem 15.14 is not implied by the 12 distinct versions of the unconstrained inequality in Theorem 15.7 (i.e., 15.62 ) together with the basic inequalities. We have proved in the last section that the non-Shannon-type inequality (15.62) implies a class of 214 - 1 constrained non-Shannon-type inequalities. We end this section by proving a similar result for the non-Shannon-type constrained inequality in Theorem 15.14. Theorem 15.16. The inequality I(X3 ; X4 ) I(X3 ; X4 |X1 ) + I(X3 ; X4 |X2 ) (15.144) is a non-Shannon-type inequality conditioning on setting both I(X1 ; X2 ) and I(X1 ; X2 |X3 ) and any subset of the following 12 Shannon's information mea- 1 Ying-On Yan, private communication. 380 15 Beyond Shannon-Type Inequalities sures to zero: I(X1 ; X2 |X4 ), I(X1 ; X3 |X4 ), I(X1 ; X4 |X3 ), I(X2 ; X3 |X4 ), I(X2 ; X4 |X3 ), I(X3 ; X4 |X1 ), I(X3 ; X4 |X2 ), I(X3 ; X4 |X1 , X2 ), H(X1 |X2 , X3 , X4 ), H(X2 |X1 , X3 , X4 ), H(X3 |X1 , X2 , X4 ), H(X4 |X1 , X2 , X3 ). Proof. The proof of this theorem is very similar to the proof of Theorem 15.12. We first note that I(X1 ; X2 ) and I(X1 ; X2 |X3 ) together with the 12 Shannon's information measures in (15.145) are exactly the 14 Shannon's information measures in (15.94). We have already shown in the proof of Theorem 15.12 ~ that h(a) (cf. Figure 15.4) lies in exactly 14 hyperplanes defining the boundary of 4 which correspond to setting these 14 Shannon's information measures ~ to zero. We also have shown that h(a) for a 0 define an extreme direction of 4 . Denote by 0 the intersection of the two hyperplanes in H4 which cor~ respond to setting I(X1 ; X2 ) and I(X1 ; X2 |X3 ) to zero. Since h(a) for any a > 0 satisfies I(X1 ; X2 ) = I(X1 ; X2 |X3 ) = 0, (15.146) ~ ~ h(a) is in 0 . Now for any linear subspace of H4 containing h(a) such that 0 , we have ~ h(a) 4 . (15.147) ~ Upon substituting the corresponding values in (15.113) for h(a) with the help of Figure 15.4, we have a 0 + 0 = 0, (15.148) ~ which is a contradiction because a > 0. Therefore, h(a) does not satisfy (15.113). Therefore, (4 ) {h H4 : h satisfies (15.113)}. (15.149) (15.145) This means that (15.113) is a non-Shannon-type inequality under the constraint . From the above, we see that can be taken to be the intersection of 0 and any subset of the 12 hyperplanes which correspond to setting the 12 Shannon's information measures in (15.145) to zero. Hence, (15.113) is a non-Shannon-type inequality conditioning on I(X1 ; X2 ), I(X1 ; X2 |X3 ), and any subset of the 12 Shannon's information measures in (15.145) being equal to zero. In other words, the constrained inequality in Theorem 15.14 in fact induces a class of 212 constrained non-Shannon-type inequalities. The theorem is proved. 15.4 Applications As we have mentioned in Chapter 13, information inequalities govern the impossibilities in information theory. In this section, we give several appli- 15.4 Applications 381 cations of the non-Shannon-type inequalities we have proved in this chapter in probability theory and information theory. An application in group theory of the unconstrained inequality proved in Section 15.2 will be discussed in Chapter 16. Non-Shannon-type inequalities also find applications in network coding theory to be discussed in Part II of this book. Example 15.17. For the constrained inequality in Theorem 15.14, if we further impose the constraints I(X3 ; X4 |X1 ) = I(X3 ; X4 |X2 ) = 0, then the right hand side of (15.113) becomes zero. This implies I(X3 ; X4 ) = 0 because I(X3 ; X4 ) is nonnegative. This means that X1 X2 X1 X2 |X3 X3 X4 . X3 X4 |X1 X3 X4 |X2 (15.151) (15.150) (15.152) We leave it as an exercise for the reader to show that this implication cannot be deduced from the basic inequalities. Example 15.18. If we impose the constraints I(X1 ; X2 ) = I(X1 ; X3 , X4 ) = I(X3 ; X4 |X1 ) = I(X3 ; X4 |X2 ) = 0, (15.153) then the right hand side of (15.62) becomes zero, which implies I(X3 ; X4 ) = 0. This means that X1 X1 X3 X3 X2 (X3 , X4 ) X3 X4 . X4 |X1 X4 |X2 (15.154) (15.155) Note that (15.152) and (15.155) differ only in the second constraint. Again, we leave it as an exercise for the reader to show that this implication cannot be deduced from the basic inequalities. Example 15.19. Consider a fault-tolerant data storage system consisting of random variables X1 , X2 , X3 , X4 such that any three random variables can recover the remaining one, i.e., H(Xi |Xj , j = i) = 0, 1 i, j 4. (15.156) 382 15 Beyond Shannon-Type Inequalities We are interested in the set of all entropy functions subject to these constraints, denoted by , which characterizes the amount of joint information which can possibly be stored in such a data storage system. Let = {h H4 : h satisfies (15.156)}. (15.157) Then the set is equal to the intersection between 4 and , i.e., 4 . Since each constraint in (15.156) is one of the 14 constraints specified in Theorem 15.12, we see that (15.62) is a non-Shannon-type inequality under ~ the constraints in (15.156). Then 4 (cf. (15.91)) is a tighter outer bound on than 4 . Example 15.20. Consider four random variables X1 , X2 , X3 , and X4 such that X3 (X1 , X2 ) X4 forms a Markov chain. This Markov condition is equivalent to I(X3 ; X4 |X1 , X2 ) = 0. (15.158) It can be proved by invoking the basic inequalities (using ITIP) that I(X3 ; X4 ) I(X3 ; X4 |X1 ) + I(X3 ; X4 |X2 ) + 0.5I(X1 ; X2 ) +cI(X1 ; X3 , X4 ) + (1 - c)I(X2 ; X3 , X4 ), (15.159) where 0.25 c 0.75, and this is the best possible. Now observe that the Markov condition (15.158) is one of the 14 constraints specified in Theorem 15.12. Therefore, (15.62) is a non-Shannon-type inequality under this Markov condition. By replacing X1 and X2 by each other in (15.62), we obtain 2I(X3 ; X4 ) I(X1 ; X2 ) + I(X2 ; X3 , X4 ) +3I(X3 ; X4 |X2 ) + I(X3 ; X4 |X1 ). Upon adding (15.62) and (15.160) and dividing by 4, we obtain I(X3 ; X4 ) I(X3 ; X4 |X1 ) + I(X3 ; X4 |X2 ) + 0.5I(X1 ; X2 ) +0.25I(X1 ; X3 , X4 ) + 0.25I(X2 ; X3 , X4 ). (15.161) (15.160) Comparing the last two terms in (15.159) and the last two terms in (15.161), we see that (15.161) is a sharper upper bound than (15.159). The Markov chain X3 (X1 , X2 ) X4 arises in many communication situations. As an example, consider a person listening to an audio source. Then the situation can be modeled by this Markov chain with X3 being the sound wave generated at the source, X1 and X2 being the sound waves received at the two ear drums, and X4 being the nerve impulses which eventually arrive at the brain. The inequality (15.161) gives an upper bound on I(X3 ; X4 ) which is tighter than what can be implied by the basic inequalities. There is some resemblance between the constrained inequality (15.161) and the data processing theorem, but they do not appear to be directly related. Problems 383 Chapter Summary Characterizations of n and n : 1. 2. 3. 4. 2 = 2 . 3 = 3 , but 3 = 3 . 4 = 4 . n is a convex cone. An Unconstrained Non-Shannon-Type Inequality: 2I(X3 ; X4 ) I(X1 ; X2 ) + I(X1 ; X3 , X4 ) + 3I(X3 ; X4 |X1 ) + I(X3 ; X4 |X2 ). A Constrained Non-Shannon-Type Inequality: If I(X1 ; X2 ) = I(X1 ; X2 |X3 ) = 0, then I(X3 ; X4 ) I(X3 ; X4 |X1 ) + I(X3 ; X4 |X2 ). Problems 1. Verify by ITIP that the unconstrained information inequality in Theorem 15.7 is of non-Shannon-type. 2. Verify by ITIP and prove analytically that the constrained information inequality in Theorem 15.14 is of non-Shannon-type. 3. Use ITIP to verify the unconstrained information inequality in Theorem 15.7. Hint: Create two auxiliary random variables as in the proof of Theorem 15.7 and impose appropriate constraints on the random variables. 4. Verify by ITIP that the implications in Examples 15.17 and 15.18 cannot be deduced from the basic inequalities. 5. Can you show that the sets of constraints in Examples 15.17 and 15.18 are in fact different? 6. Consider an information inequality involving random variables X1 , X2 , , Xn , which can be written as c H(X ) 0, 2Nn \{} where Nn = {1, 2, , n}. For i Nn , let ri = 2Nn \{} c n (i), where n (i) is equal to 1 if i and is equal to 0 otherwise. 384 15 Beyond Shannon-Type Inequalities a) Show that ri is the coefficient of H(Xi |XNn -{i} ) when the information inequality is expressed in terms of the elemental forms of Shannon's information measures for n random variables. b) Show that if the information inequality always holds, then ri 0 for all i Nn . (Chan [61].) 7. Let Xi , i = 1, 2, , n, Z, and T be discrete random variables. a) Prove that n nI(Z; T ) - j=1 I(Z; T |Xj ) - nI(Z; T |Xi ) n I(Xi ; Z, T ) + j=1 H(Xj ) - H(X1 , X2 , , Xn ). Hint: When n = 2, this inequality reduces to the unconstrained nonShannon-type inequality in Theorem 15.7. b) Prove that n nI(Z; T ) - 2 j=1 I(Z; T |Xj ) n 1 n n I(Xi ; Z, T ) + i=1 j=1 H(Xj ) - H(X1 , X2 , , Xn ). ( Zhang and Yeung [416].) 8. Let p(x1 , x2 , x3 , x4 ) be the joint distribution for random variables X1 , X2 , X3 , and X4 such that I(X1 ; X2 |X3 ) = I(X2 ; X4 |X3 ) = 0, and let p be ~ defined in (15.98). a) Show that p(x1 , x2 , x3 , x4 ) = c 0 p(x1 ,x2 ,x3 )p(x1 ,x4 )p(x2 ,x4 ) p(x1 ,x2 )p(x4 ) if p(x1 , x2 ), p(x4 ) > 0 otherwise defines a probability distribution for an appropriate c 1. b) Prove that p(x1 , x2 , x3 ) = p(x1 , x2 , x3 ) for all x1 , x2 , and x3 . ~ c) By considering D(~ p) 0, prove that p H(X13 ) + H(X14 ) + H(X23 ) + H(X24 ) + H(X34 ) H(X3 ) + H(X4 ) + H(X12 ) + H(X134 ) + H(X234 ), where H(X134 ) denotes H(X1 , X3 , X4 ), etc. d) Prove that under the constraints in (15.112), the inequality in (15.113) is equivalent to the inequality in c). Historical Notes 385 The inequality in c) is referred to as the Ingleton inequality for entropy in the literature. For the origin of the Ingleton inequality, see Problem 9 in Chapter 16. (Mat [256].) us Historical Notes In 1986, Pippenger [294] asked whether there exist constraints on the entropy function other than the polymatroidal axioms, which are equivalent to the basic inequalities. He called the constraints on the entropy function the laws of information theory. The problem had been open until Zhang and Yeung discovered for four random variables first a constrained non-Shannon-type inequality [415] and then an unconstrained non-Shannon-type inequality [416] in the late 1990's. The inequality reported in [416] has been further generalized by Makarychev et al. [242] and Zhang [413]. The existence of these inequalities implies that there are laws in information theory beyond those laid down by Shannon [322]. The non-Shannon-type inequalities that have been discovered induce outer bounds on the region 4 which are tighter than 4 . Mat and Studen [261] us y showed that an entropy function in 4 is entropic if it satisfies the Ingleton inequality (see Problem 9 in Chapter 16). This gives an inner bound on 4 . A more explicit proof of this inner bound can be found in [416], where the bound was shown not to be tight. Mat [259] has obtained asymptotically us tight inner bounds on n by constructing entropy functions from matroids. Dougherty et al. [98] discovered a host of unconstrained non-Shannon-type inequalities by means of a computer search based on ITIP and the Markov chain construction in [416] (see Problem 3). Recently, Mat [260] proved an us infinite class of unconstrained non-Shannon-type inequalities, implying that n is not a pyramid. Chan [61] proved a characterization for an inequality for differential entropy in terms of its discrete version. Lnnika [237] proved that the tightness e c of the continuous version of the unconstrained non-Shannon-type inequality reported in [416] can be achieved by a multivariate Gaussian distribution. In the 1990's, Mat and Studen [254][261][255] studied the structure us y of conditional independence (which subsumes the implication problem) of random variables. Mat [256] finally settled the problem for four random us variables by means of a constrained non-Shannon-type inequality which is a variation of the inequality reported in [415]. The von Neumann entropy is an extension of classical entropy (as discussed in this book) to the field of quantum mechanics. The strong subadditivity of the von Neumann entropy proved by Lieb and Ruskai [233] plays the same role as the basic inequalities for classical entropy. Pippenger [295] proved that for a three-party system, there exists no inequality for the von Neumann entropy beyond strong subadditivity. Subsequently, Linden and Winter [235] discovered for a four-party system a constrained inequality for the von Neumann 386 15 Beyond Shannon-Type Inequalities entropy which is independent of strong subadditivity. We refer the reader to the book by Nielsen and Chuang [274] for an introduction to quantum information theory. Along a related direction, Hammer et al. [144] have shown that all linear inequalities that always hold for Kolmogorov complexity also always hold for entropy, and vice versa. This establishes a one-to-one correspondence between entropy and Kolmogorov complexity. 16 Entropy and Groups The group is the first major mathematical structure in abstract algebra, while entropy is the most basic measure of information. Group theory and information theory are two seemingly unrelated subjects which turn out to be intimately related to each other. This chapter explains this intriguing relation between these two fundamental subjects. Those readers who have no knowledge in group theory may skip this introduction and go directly to the next section. Let X1 and X2 be any two random variables. Then H(X1 ) + H(X2 ) H(X1 , X2 ), which is equivalent to the basic inequality I(X1 ; X2 ) 0. (16.2) (16.1) Let G be any finite group and G1 and G2 be subgroups of G. We will show in Section 16.4 that |G||G1 G2 | |G1 ||G2 |, (16.3) where |G| denotes the order of G and G1 G2 denotes the intersection of G1 and G2 (G1 G2 is also a subgroup of G, see Proposition 16.13). By rearranging the terms, the above inequality can be written as log |G| |G| |G| + log log . |G1 | |G2 | |G1 G2 | (16.4) By comparing (16.1) and (16.4), one can easily identify the one-to-one correspondence between these two inequalities, namely that Xi corresponds to Gi , i = 1, 2, and (X1 , X2 ) corresponds to G1 G2 . While (16.1) is true for any pair of random variables X1 and X2 , (16.4) is true for any finite group G and subgroups G1 and G2 . Recall from Chapter 13 that the region n characterizes all information inequalities (involving n random variables). In particular, we have shown in 388 16 Entropy and Groups Section 15.1 that the region n is sufficient for characterizing all unconstrained information inequalities, i.e., by knowing n , one can determine whether any unconstrained information inequality always holds. The main purpose of this chapter is to obtain a characterization of n in terms of finite groups. An important consequence of this result is a one-to-one correspondence between unconstrained information inequalities and group inequalities. Specifically, for every unconstrained information inequality, there is a corresponding group inequality, and vice versa. A special case of this correspondence has been given in (16.1) and (16.4). By means of this result, unconstrained information inequalities can be proved by techniques in group theory, and a certain form of inequalities in group theory can be proved by techniques in information theory. In particular, the unconstrained non-Shannon-type inequality in Theorem 15.7 corresponds to the group inequality |G1 G3 |3 |G1 G4 |3 |G3 G4 |3 |G2 G3 ||G2 G4 | |G1 ||G1 G2 ||G3 |2 |G4 |2 |G1 G3 G4 |4 |G2 G3 G4 |, (16.5) where Gi are subgroups of a finite group G, i = 1, 2, 3, 4. The meaning of this inequality and its implications in group theory are yet to be understood. 16.1 Group Preliminaries In this section, we present the definition and some basic properties of a group which are essential for subsequent discussions. Definition 16.1. A group is a set of objects G together with a binary operation on the elements of G, denoted by "" unless otherwise specified, which satisfy the following four axioms: 1. Closure For every a, b in G, a b is also in G. 2. Associativity For every a, b, c in G, a (b c) = (a b) c. 3. Existence of Identity There exists an element e in G such that a e = e a = a for every a in G. 4. Existence of Inverse For every a in G, there exists an element b in G such that a b = b a = e. Proposition 16.2. For any group G, the identity element is unique. Proof. Let both e and e be identity elements in a group G. Since e is an identity element, e e = e, (16.6) and since e is also an identity element, e e=e. (16.7) It follows by equating the right hand sides of (16.6) and (16.7) that e = e , which implies the uniqueness of the identity element of a group. 16.1 Group Preliminaries 389 Proposition 16.3. For every element a in a group G, its inverse is unique. Proof. Let b and b be inverses of an element a, so that ab=ba=e and a b = b a = e. Then b = be = b (a b ) = (b a) b = eb =b, (16.10) (16.11) (16.12) (16.13) (16.14) (16.9) (16.8) where (16.11) and (16.13) follow from (16.9) and (16.8), respectively, and (16.12) is by associativity. Therefore, the inverse of a is unique. Thus the inverse of a group element a is a function of a, and it will be denoted by a-1 . Definition 16.4. The number of elements of a group G is called the order of G, denoted by |G|. If |G| < , G is called a finite group, otherwise it is called an infinite group. There is an unlimited supply of examples of groups. Some familiar examples are: the integers under addition, the rationals excluding zero under multiplication, and the set of real-valued 2 2 matrices under addition, where addition and multiplication refer to the usual addition and multiplication for real numbers and matrices. In each of these examples, the operation (addition or multiplication) plays the role of the binary operation "" in Definition 16.2. All the above are examples of infinite groups. In this chapter, however, we are concerned with finite groups. In the following, we discuss two examples of finite groups in details. Example 16.5 (Modulo 2 Addition). The trivial group consists of only the identity element. The simplest nontrivial group is the group of modulo 2 addition. The order of this group is 2, and the elements are {0, 1}. The binary operation, denoted by "+", is defined by following table: +01 0 01 1 10 390 16 Entropy and Groups The four axioms of a group simply say that certain constraints must hold in the above table. We now check that all these axioms are satisfied. First, the closure axiom requires that all the entries in the table are elements in the group, which is easily seen to be the case. Second, it is required that associativity holds. To this end, it can be checked in the above table that for all a, b, and c, a + (b + c) = (a + b) + c. (16.15) For example, 0 + (1 + 1) = 0 + 0 = 0, while (0 + 1) + 1 = 1 + 1 = 0, (16.17) which is the same as 0 + (1 + 1). Third, the element 0 is readily identified as the unique identity. Fourth, it is readily seen that an inverse exists for each element in the group. For example, the inverse of 1 is 1, because 1 + 1 = 0. (16.18) (16.16) Thus the above table defines a group of order 2. It happens in this example that the inverse of each element is the element itself, which is not true for a group in general. We remark that in the context of a group, the elements in the group should be regarded strictly as symbols only. In particular, one should not associate group elements with magnitudes as we do for real numbers. For instance, in the above example, one should not think of 0 as being less than 1. The element 0, however, is a special symbol which plays the role of the identity of the group. We also notice that for the group in the above example, a + b is equal to b + a for all group elements a and b. A group with this property is called a commutative group or an Abelian group1 . Example 16.6 (Symmetric Group). Consider a permutation of the components of a vector x = (x1 , x2 , , xr ) (16.19) given by [x] = (x(1) , x(2) , , x(r) ), where : {1, 2, , r} {1, 2, , r} (16.21) is a one-to-one mapping. The one-to-one mapping is called a permutation on {1, 2, , r}, which is represented by = ((1), (2), , (r)). 1 (16.20) (16.22) The Abelian group is named after the Norwegian mathematician Niels Henrik Abel (1802-1829). 16.1 Group Preliminaries 391 For two permutations 1 and 2 , define 1 2 as the composite function of 1 and 2 . For example, for r = 4, suppose 1 = (2, 1, 4, 3) and 2 = (1, 4, 2, 3). Then 1 2 is given by 1 2 (1) = 1 2 (2) = 1 2 (3) = 1 2 (4) = or 1 2 = (2, 3, 1, 4). The reader can easily check that 2 1 = (4, 1, 2, 3), (16.27) (16.26) 1 (2 (1)) = 1 (1) = 1 (2 (2)) = 1 (4) = 1 (2 (3)) = 1 (2) = 1 (2 (4)) = 1 (3) = 2 3 1 4, (16.24) (16.23) (16.25) which is different from 1 2 . Therefore, the operation "" is not commutative. We now show that the set of all permutations on {1, 2, , r} and the operation "" form a group, called the symmetric group on {1, 2, , r}. First, for two permutations 1 and 2 , since both 1 and 2 are one-to-one mappings, so is 1 2 . Therefore, the closure axiom is satisfied. Second, for permutations 1 , 2 , and 3 , 1 (2 3 )(i) = 1 (2 3 (i)) = 1 (2 (3 (i))) = 1 2 (3 (i)) = (1 2 ) 3 (i) (16.28) (16.29) (16.30) (16.31) for 1 i r. Therefore, associativity is satisfied. Third, it is clear that the identity map is the identity element. Fourth, for a permutation , it is clear that its inverse is -1 , the inverse mapping of which is defined because is one-to-one. Therefore, the set of all permutations on {1, 2, , r} and the operation "" form a group. The order of this group is evidently equal to (r!). Definition 16.7. Let G be a group with operation "", and S be a subset of G. If S is a group with respect to the operation "", then S is called a subgroup of G. Definition 16.8. Let S be a subgroup of a group G and a be an element of G. The left coset of S with respect to a is the set a S = {a s : s S}. Similarly, the right coset of S with respect to a is the set S a = {s a : s S}. 392 16 Entropy and Groups In the sequel, only the left coset will be used. However, any result which applies to the left coset also applies to the right coset, and vice versa. For simplicity, a S will be denoted by aS. Proposition 16.9. For a1 and a2 in G, a1 S and a2 S are either identical or disjoint. Further, a1 S and a2 S are identical if and only if a1 and a2 belong to the same left coset of S. Proof. Suppose a1 S and a2 S are not disjoint. Then there exists an element b in a1 S a2 S such that b = a1 s1 = a2 s2 , (16.32) for some si in S, i = 1, 2. Then a1 = (a2 s2 ) s-1 = a2 (s2 s-1 ) = a2 t, 1 1 (16.33) where t = s2 s-1 is in S. We now show that a1 S a2 S. For an element a1 s 1 in a1 S, where s S, a1 s = (a2 t) s = a2 (t s) = a2 u, (16.34) where u = t s is in S. This implies that a1 s is in a2 S. Thus, a1 S a2 S. By symmetry, a2 S a1 S. Therefore, a1 S = a2 S. Hence, if a1 S and a2 S are not disjoint, then they are identical. Equivalently, a1 S and a2 S are either identical or disjoint. This proves the first part of the proposition. We now prove the second part of the proposition. Since S is a group, it contains e, the identity element. Then for any group element a, a = a e is in aS because e is in S. If a1 S and a2 S are identical, then a1 a1 S and a2 a2 S = a1 S. Therefore, a1 and a2 belong to the same left coset of S. To prove the converse, assume a1 and a2 belong to the same left coset of S. From the first part of the proposition, we see that a group element belongs to one and only one left coset of S. Since a1 is in a1 S and a2 is in a2 S, and a1 and a2 belong to the same left coset of S, we see that a1 S and a2 S are identical. The proposition is proved. Proposition 16.10. Let S be a subgroup of a group G and a be an element of G. Then |aS| = |S|, i.e., the numbers of elements in all the left cosets of S are the same, and they are equal to the order of S. Proof. Consider two elements a s1 and a s2 in a S, where s1 and s2 are in S such that a s1 = a s2 . (16.35) Then a-1 (a s1 ) = a-1 (a s2 ) (a -1 (16.36) (16.37) (16.38) (16.39) a) s1 = (a -1 a) s2 e s1 = e s2 s1 = s2 . 16.2 Group-Characterizable Entropy Functions 393 Thus each element in S corresponds to a unique element in aS. Therefore, |aS| = |S| for all a G. We are just one step away from obtaining the celebrated Lagrange's theorem stated below. Theorem 16.11 (Lagrange's Theorem). If S is a subgroup of a finite group G, then |S| divides |G|. Proof. Since a aS for every a G, every element of G belongs to a left coset of S. Then from Proposition 16.9, we see that the distinct left cosets of S partition G. Therefore |G|, the total number of elements in G, is equal to the number of distinct cosets of S multiplied by the number of elements in each left coset, which is equal to |S| by Proposition 16.10. This implies that |S| divides |G|, proving the theorem. The following corollary is immediate from the proof of Lagrange's Theorem. Corollary 16.12. Let S be a subgroup of a group G. The number of distinct left cosets of S is equal to |G| . |S| 16.2 Group-Characterizable Entropy Functions Recall from Chapter 13 that the region n consists of all the entropy functions in the entropy space Hn for n random variables. As a first step toward establishing the relation between entropy and groups, we discuss in this sec tion entropy functions in n which can be described by a finite group G and subgroups G1 , G2 , , Gn . Such entropy functions are said to be groupcharacterizable. The significance of this class of entropy functions will become clear in the next section. In the sequel, we will make use of the intersections of subgroups extensively. We first prove that the intersection of two subgroups is also a subgroup. Proposition 16.13. Let G1 and G2 be subgroups of a group G. Then G1 G2 is also a subgroup of G. Proof. It suffices to show that G1 G2 together with the operation "" satisfy all the axioms of a group. First, consider two elements a and b of G in G1 G2 . Since both a and b are in G1 , (a b) is in G1 . Likewise, (a b) is in G2 . Therefore, a b is in G1 G2 . Thus the closure axiom holds for G1 G2 . Second, associativity for G1 G2 inherits from G. Third, G1 and G2 both contain the identity element because they are groups. Therefore, the identity element is in G1 G2 . Fourth, for an element a Gi , since Gi is a group, a-1 is in Gi , i = 1, 2. Thus for an element a G1 G2 , a-1 is also in G1 G2 . Therefore, G1 G2 is a group and hence a subgroup of G. 394 16 Entropy and Groups Corollary 16.14. Let G1 , G2 , , Gn be subgroups of a group G. Then n Gi i=1 is also a subgroup of G. In the rest of the chapter, we let Nn = {1, 2, , n} and denote i Gi by G , where is a nonempty subset of Nn . Lemma 16.15. Let Gi be subgroups of a group G and ai be elements of G, i . Then |i ai Gi | = |G | 0 if i ai Gi = otherwise. (16.40) Proof. For the special case that is a singleton, i.e., = {i} for some i Nn , (16.40) reduces to |ai Gi | = |Gi |, (16.41) which has already been proved in Proposition 16.10. Let be any nonempty subset of Nn . If i ai Gi = , then (16.40) is obviously true. If i ai Gi = , then there exists x i ai Gi such that for all i , x = ai si , (16.42) where si Gi . For any i and for any y G , consider x y = (ai si ) y = ai (si y). (16.43) Since both si and y are in Gi , si y is in Gi . Thus x y is in ai Gi for all i , or x y is in i ai Gi . Moreover, for y, y G , if x y = x y , then y = y . Therefore, each element in G corresponds to a unique element in i ai Gi . Hence, |i ai Gi | = |G |, (16.44) proving the lemma. The relation between a finite group G and subgroups G1 and G2 is illustrated by the membership table in Figure 16.1. In this table, an element of G is represented by a dot. The first column represents the subgroup G1 , with the dots in the first column being the elements in G1 . The other columns represent the left cosets of G1 . By Proposition 16.10, all the columns have the same number of dots. Similarly, the first row represents the subgroup G2 and the other rows represent the left cosets of G2 . Again, all the rows have the same number of dots. The upper left entry in the table represents the subgroup G1 G2 . There are |G1 G2 | dots in this entry, with one of them representing the identity element. Any other entry represents the intersection between a left coset of G1 and a left coset of G2 , and by Lemma 16.15, the number of dots in each of these entries is either equal to |G1 G2 | or zero. 16.2 Group-Characterizable Entropy Functions 395 G2 G1,2 G1 Fig. 16.1. The membership table for a finite group G and subgroups G1 and G2 . Since all the column have the same numbers of dots and all the rows have the same number of dots, we say that the table in Figure 16.1 exhibits a quasi-uniform structure. We have already seen a similar structure in Figure 6.1 for the two-dimensional strong joint typicality array, which we reproduce in Figure 16.2. In this array, when n is large, all the columns have approximately the same number of dots and all the rows have approximately the same number of dots. For this reason, we say that the two-dimensional strong typicality array exhibits an asymptotic quasi-uniform structure. In a strong typicality array, however, each entry can contain only one dot, while in a membership table, each entry can contain multiple dots. One can make a similar comparison between a strong joint typicality array for any n 2 random variables and the membership table for a finite group with n subgroups. The details are omitted here. Theorem 16.16. Let Gi , i Nn be subgroups of a group G. Then h Hn defined by |G| h = log (16.45) |G | for all nonempty subsets of Nn is entropic, i.e., h n . 2nH(Y) y 2nH(X) x S[n] X S[n] Y . . . . . . . . . . . . . . . 2nH(X,Y) n (x,y) T[XY] Fig. 16.2. A two-dimensional strong typicality array. 396 16 Entropy and Groups Proof. It suffices to show that there exists a collection of random variables X1 , X2 , , Xn such that H(X ) = log |G| |G | (16.46) for all nonempty subsets of Nn . We first introduce a uniform random variable defined on the sample space G with probability mass function Pr{ = a} = 1 |G| (16.47) for all a G. For any i Nn , let random variable Xi be a function of such that Xi = aGi if = a. Let be a nonempty subset of Nn . Since Xi = ai Gi for all i if and only if is equal to some b i ai Gi , Pr{Xi = ai Gi : i } = = | i ai Gi | |G| if i ai Gi = otherwise (16.48) (16.49) |G | |G| 0 by Lemma 16.15. In other words, (Xi , i ) is distributed uniformly on its |G| support whose cardinality is |G | . Then (16.46) follows and the theorem is proved. Definition 16.17. Let G be a finite group and G1 , G2 , , Gn be subgroups |G| of G. Let h be a vector in Hn . If h = log |G | for all nonempty subsets of Nn , then (G, G1 , , Gn ) is a group characterization of h. Theorem 16.16 asserts that certain entropy functions in n have a group characterization. These are called group-characterizable entropy functions, which will be used in the next section to obtain a group characterization of the region n . We end this section by giving a few examples of such entropy functions. Example 16.18. Fix any subset of N3 = {1, 2, 3} and define a vector h H3 by h = log 2 0 if = otherwise. (16.50) We now show that h has a group characterization. Let G = {0, 1} be the group of modulo 2 addition in Example 16.5, and for i = 1, 2, 3, let Gi = {0} if i G otherwise. (16.51) 16.2 Group-Characterizable Entropy Functions 397 Then for a nonempty subset of N3 , if = , there exists an i in such that i is also in , and hence by definition Gi = {0}. Thus, G = i Gi = {0}. (16.52) Therefore, log |G| 2 |G| = log = log = log 2. |G | |{0}| 1 (16.53) If = , then Gi = G for all i , and G = i Gi = G. (16.54) Therefore, log |G| |G| = log = log 1 = 0. |G | |G| |G| |G | (16.55) Then we see from (16.50), (16.53), and (16.55) that h = log (16.56) for all nonempty subsets of N3 . Hence, (G, G1 , G2 , G3 ) is a group characterization of h. Example 16.19. This is a generalization of the last example. Fix any nonempty subset of Nn and define a vector h Hn by h = log 2 0 if = otherwise. (16.57) Then (G, G1 , G2 , , Gn ) is a group characterization of h, where G is the group of modulo 2 addition, and Gi = {0} if i G otherwise. (16.58) By letting = , we have h = 0. Thus we see that (G, G1 , G2 , , Gn ) is a group characterization of the origin of Hn , with G = G1 = G2 = = Gn . Example 16.20. Define a vector h H3 as follows: h = min(||, 2). Let F be the group of modulo 2 addition, G = F F , and G1 = {(0, 0), (1, 0)} G2 = {(0, 0), (0, 1)} G3 = {(0, 0), (1, 1)}. Then (G, G1 , G2 , G3 ) is a group characterization of h. (16.60) (16.61) (16.62) (16.59) 398 16 Entropy and Groups 16.3 A Group Characterization of n We have introduced in the last section the class of entropy functions in n which have a group characterization. However, an entropy function h n may not have a group characterization due to the following observation. Sup pose h n . Then there exists a collection of random variables X1 , X2 , , Xn such that h = H(X ) (16.63) for all nonempty subsets of Nn . If (G, G1 , , Gn ) is a group characterization of h, then |G| H(X ) = log (16.64) |G | for all nonempty subsets of Nn . Since both |G| and |G | are integers, H(X ) must be the logarithm of a rational number. However, the joint entropy of a set of random variables in general is not necessarily the logarithm of a rational number (see Corollary 2.44). Therefore, it is possible to construct an entropy function h n which has no group characterization. Although h n does not imply h has a group characterization, it turns out that the set of all h n which have a group characterization is almost good enough to characterize the region n , as we will see next. Definition 16.21. Define the following region in Hn : n = {h Hn : h has a group characterization}. (16.65) By Theorem 16.16, if h Hn has a group characterization, then h n . Therefore, n n . We will prove as a corollary of the next theorem that con(n ), the convex closure of n , is in fact equal to n , the closure of n . Theorem 16.22. For any h n , there exists a sequence {f (r) } in n such 1 (r) that limr r f = h. We need the following lemma to prove this theorem. The proof of this lemma resembles the proof of the strong conditional AEP (Theorem 6.10). Nevertheless, we give a sketch of the proof for the sake of completeness. Lemma 16.23. Let X be a random variable such that |X | < and the distribution {p(x)} is rational, i.e., p(x) is a rational number for all x X . Without loss of generality, assume p(x) is a rational number with denominator q for all x X . Then for r = q, 2q, 3q, , lim 1 log r r! = H(X). (rp(x))! x (16.66) r 16.3 A Group Characterization of n 399 Proof. Applying Lemma 6.11, we can obtain 1 ln r - x r! (rp(x))! x p(x) ln p(x) + r+1 ln(r + 1) - ln r r ln 1 + 1 r . (16.67) (16.68) = He (X) + 1 1 ln r + 1 + r r This upper bound tends to He (X) as r . On the other hand, we can obtain 1 ln r - x r! (rp(x))! x p(x) + 1 r ln p(x) + 1 r - ln r . r (16.69) This lower bound also tends to He (X) as r . Then the proof is completed by changing the base of the logarithm if necessary. Proof of Theorem 16.22. For any h n , there exists a collection of random variables X1 , X2 , , Xn such that h = H(X ) (16.70) for all nonempty subsets of Nn . We first consider the special case that |Xi | < for all i Nn and the joint distribution of X1 , X2 , , Xn is rational. We want to show that there exists a sequence {f (r) } in n such that limr 1 f (r) = h. r Denote i Xi by X . For any nonempty subset of Nn , let Q be the marginal distribution of X . Assume without loss of generality that for any nonempty subset of Nn and for all a X , Q (a) is a rational number with denominator q. For each r = q, 2q, 3q, , fix a sequence xNn = (xNn ,1 , xNn ,2 , xNn ,r ) where for all j = 1, 2, , r, xNn ,j = (xi,j : i Nn ) XNn , such that N (a; xNn ), the number of occurrences of a in sequence xNn , is equal to rQNn (a) for all a XNn . The existence of such a sequence is guaranteed by that all the values of the joint distribution of XNn are rational numbers with denominator q. Also, we denote the sequence of r elements of X , (x,1 , x,2 , x,r ), where x,j = (xi,j : i ), by x . Let a X . It is easy to check that N (a; x ), the number of occurrences of a in the sequence x , is equal to rQ (a) for all a X . 400 16 Entropy and Groups Let G be the group of all permutations on {1, 2, , r}, i.e., the symmetric group on {1, 2, , r} (cf. Example 16.6). The group G depends on r, but for simplicity, we do not state this dependency explicitly. For any i Nn , define Gi = { G : [xi ] = xi }, where [xi ] = (xi,(1) , xi,(2) , , xi,(r) ). It is easy to check that Gi is a subgroup of G. Let be a nonempty subset of Nn . Then G = i (16.71) Gi { G : [xi ] = xi } i (16.72) (16.73) (16.74) (16.75) = = { G : [xi ] = xi for all i } = { G : [x ] = x }, where [x ] = (x,(1) , x,(2) , , x,(r) ). For any a X , define the set Lx (a) = {j {1, 2, , r} : x,j = a}. (16.76) (16.77) Lx (a) contains the "locations" of a in x . Then [x ] = x if and only if for all a X , j Lx (a) implies (j) Lx (a). Since |Lx (a)| = N (a; x ) = rQ (a), |G | = aX (16.78) (16.79) (rQ (a))! and therefore |G| = |G | r! . aX (rQ (a))! (16.80) By Lemma 16.23, r lim 1 |G| log = H(X ) = h . r |G | (16.81) Recall that G and hence all its subgroups depend on r. Define f (r) by (r) f = log |G| |G | (16.82) for all nonempty subsets of Nn . Then f (r) n and 16.4 Information Inequalities and Group Inequalities 401 r lim 1 (r) f = h. r (16.83) We have already proved the theorem for the special case that h is the entropy function of a collection of random variables X1 , X2 , , Xn with finite alphabets and a rational joint distribution. To complete the proof, we only have to note that for any h n , it is always possible to construct a sequence (k) {h } in n such that limk h(k) = h, where h(k) is the entropy function (k) (k) (k) of a collection of random variables X1 , X2 , , Xn with finite alphabets and a rational joint distribution. This can be proved by techniques similar to those used in Appendix 2.A together with the continuity of the entropy function for a fixed finite support (Section 2.3). The details are omitted here. Corollary 16.24. con(n ) = n . Proof. First of all, n n . By taking convex closure, we have con(n ) con(n ). By Theorem 15.5, n is convex. Therefore, con(n ) = n , and we have con(n ) n . On the other hand, we have shown in Example 16.19 that the origin of Hn has a group characterization and therefore is in n . It then follows from Theorem 16.22 that n con(n ). Hence, we conclude that n = con(n ), completing the proof. 16.4 Information Inequalities and Group Inequalities We have proved in Section 15.1 that an unconstrained information inequality b h0 always holds if and only if n {h Hn : b h 0}. (16.84) (16.85) In other words, all unconstrained information inequalities are fully charac terized by n . We also have proved at the end of the last section that con(n ) = n . Since n n n , if (16.85) holds, then n {h Hn : b h 0}. (16.86) On the other hand, if (16.86) holds, since {h Hn : b h 0} is closed and convex, by taking convex closure in (16.86), we obtain n = con(n ) {h Hn : b h 0}. Therefore, (16.85) and (16.86) are equivalent. Now (16.86) is equivalent to (16.87) 402 16 Entropy and Groups b h 0 for all h n . Since h n if and only if h = log |G| |G | (16.88) (16.89) for all nonempty subsets of Nn for some finite group G and subgroups G1 , G2 , , Gn , we see that the inequality (16.84) holds for all random variables X1 , X2 , , Xn if and only if the inequality obtained from (16.84) by replacing |G| h by log |G | for all nonempty subsets of Nn holds for any finite group G and subgroups G1 , G2 , , Gn . In other words, for every unconstrained information inequality, there is a corresponding group inequality, and vice versa. Therefore, inequalities in information theory can be proved by methods in group theory, and inequalities in group theory can be proved by methods in information theory. In the rest of the section, we explore this one-to-one correspondence between information theory and group theory. We first give a group-theoretic proof of the basic inequalities in information theory. At the end of the section, we will give an information-theoretic proof for the group inequality in (16.5). Definition 16.25. Let G1 and G2 be subgroups of a finite group G. Define G1 G2 = {a b : a G1 and b G2 }. (16.90) G1 G2 is in general not a subgroup of G. However, it can be shown that G1 G2 is a subgroup of G if G is Abelian (see Problem 1). Proposition 16.26. Let G1 and G2 be subgroups of a finite group G. Then |G1 G2 | = |G1 ||G2 | . |G1 G2 | (16.91) Proof. Fix (a1 , a2 ) G1 G2 , Then a1 a2 is in G1 G2 . Consider any (b1 , b2 ) G1 G2 such that b1 b2 = a1 a2 . (16.92) We will determine the number of (b1 , b2 ) in G1 G2 which satisfies this relation. From (16.92), we have b-1 (b1 b2 ) = b-1 (a1 a2 ) 1 1 (b-1 1 b1 ) b2 = b2 = Then b2 a-1 = b-1 a1 (a2 a-1 ) = b-1 a1 . 2 1 2 1 (16.96) b-1 1 b-1 1 a1 a2 a1 a2 . (16.93) (16.94) (16.95) 16.4 Information Inequalities and Group Inequalities 403 Let k be this common element in G, i.e., k = b2 a-1 = b-1 a1 . 2 1 (16.97) Since b-1 a1 G1 and b2 a-1 G2 , k is in G1 G2 . In other words, for 1 2 given (a1 , a2 ) G1 G2 , if (b1 , b2 ) G1 G2 satisfies (16.92), then (b1 , b2 ) satisfies (16.97) for some k G1 G2 . On the other hand, if (b1 , b2 ) G1 G2 satisfies (16.97) for some k G1 G2 , then (16.96) is satisfied, which implies (16.92). Therefore, for given (a1 , a2 ) G1 G2 , (b1 , b2 ) G1 G2 satisfies (16.92) if and only if (b1 , b2 ) satisfies (16.97) for some k G1 G2 . Now from (16.97), we obtain b1 (k) = (k a-1 )-1 1 and b2 (k) = k a2 , (16.99) (16.98) where we have written b1 and b2 as b1 (k) and b2 (k) to emphasize their dependence on k. Now consider k, k G1 G2 such that (b1 (k), b2 (k)) = (b1 (k ), b2 (k )). Since b1 (k) = b1 (k ), from (16.98), we have (k a-1 )-1 = (k a-1 )-1 , 1 1 which implies k=k. (16.102) Therefore, each k G1 G2 corresponds to a unique pair (b1 , b2 ) G1 G2 which satisfies (16.92). Therefore, we see that the number of distinct elements in G1 G2 is given by |G1 G2 | = completing the proof. Theorem 16.27. Let G1 , G2 , and G3 be subgroups of a finite group G. Then |G3 ||G123 | |G13 ||G23 |. (16.104) |G1 ||G2 | |G1 G2 | = , |G1 G2 | |G1 G2 | (16.103) (16.101) (16.100) Proof. First of all, G13 G23 = (G1 G3 ) (G2 G3 ) = G1 G2 G3 = G123 . By Proposition 16.26, we have (16.105) 404 16 Entropy and Groups |G13 G23 | = |G13 ||G23 | . |G123 | (16.106) It is readily seen that G13 G23 is a subset of G3 , Therefore, |G13 G23 | = The theorem is proved. Corollary 16.28. For random variables X1 , X2 , and X3 , I(X1 ; X2 |X3 ) 0. Proof. Let G1 , G2 , and G3 be subgroups of a finite group G. Then |G3 ||G123 | |G13 ||G23 | by Theorem 16.27, or |G|2 |G|2 . |G13 ||G23 | |G3 ||G123 | This is equivalent to log |G| |G| |G| |G| + log log + log . |G13 | |G23 | |G3 | |G123 | (16.111) (16.110) (16.109) (16.108) |G13 ||G23 | |G3 |. |G123 | (16.107) This group inequality corresponds to the information inequality H(X1 , X3 ) + H(X2 , X3 ) H(X3 ) + H(X1 , X2 , X3 ), which is equivalent to I(X1 ; X2 |X3 ) 0. (16.113) (16.112) The above corollary shows that all the basic inequalities in information theory has a group-theoretic proof. Of course, Theorem 16.27 is also implied by the basic inequalities. As a remark, the inequality in (16.3) is seen to be a special case of Theorem 16.27 by letting G3 = G. We are now ready to prove the group inequality in (16.5). The unconstrained non-Shannon-type inequality we have proved in Theorem 15.7 can be expressed in canonical form as H(X1 ) + H(X1 , X2 ) + 2H(X3 ) + 2H(X4 ) +4H(X1 , X3 , X4 ) + H(X2 , X3 , X4 ) 3H(X1 , X3 ) + 3H(X1 , X4 ) + 3H(X3 , X4 ) +H(X2 , X3 ) + H(X2 , X4 ), (16.114) Chapter Summary 405 which corresponds to the group inequality log |G| |G| |G| |G| + log + 2 log + 2 log |G1 | |G12 | |G3 | |G4 | |G| |G| +4 log + log |G134 | |G234 | |G| |G| |G| |G| 3 log + 3 log + 3 log + log |G13 | |G14 | |G34 | |G23 | |G| + log . |G24 | (16.115) Upon rearranging the terms, we obtain |G1 G3 |3 |G1 G4 |3 |G3 G4 |3 |G2 G3 ||G2 G4 | |G1 ||G1 G2 ||G3 |2 |G4 |2 |G1 G3 G4 |4 |G2 G3 G4 |, (16.116) which is the group inequality in (16.5). The meaning of this inequality and its implications in group theory are yet to be understood. Chapter Summary In the following, Nn = {1, 2, , n}. Properties of Subgroups of a Finite Group: 1. Lagrange's Theorem: If S is a subgroup of a finite group G, then |S| divides |G|. 2. Let Gi be subgroups of a finite group G and ai be elements of G, i . Then |G | if i ai Gi = |i ai Gi | = 0 otherwise, where ai Gi is the left coset of Gi containing ai and G = i Gi . Group Characterization of an Entropy Function: Let G be a finite group and G1 , G2 , , Gn be subgroups of G. For a vector h Hn , if h = |G| log |G | for all nonempty subsets of Nn , then (G, G1 , , Gn ) is a group characterization of h. A vector h that has a group characterization is entropic. Group Characterization of n : con(n ) = n , where n = {h Hn : h has a group characterization}. Information Inequalities and Group Inequalities: An unconstrained inequality b h 0 involving random variables X1 , X2 , , Xn , where h Hn , always holds if and only if the inequality obtained by replacing h by 406 16 Entropy and Groups |G| log |G | for all nonempty subsets of Nn holds for any finite group G and subgroups G1 , G2 , , Gn . A "Non-Shannon-Type" Group Inequality: |G1 G3 |3 |G1 G4 |3 |G3 G4 |3 |G2 G3 ||G2 G4 | |G1 ||G1 G2 ||G3 |2 |G4 |2 |G1 G3 G4 |4 |G2 G3 G4 |. Problems 1. Let G1 and G2 be subgroups of a finite group G. Show that G1 G2 is a subgroup if G is Abelian. 2. Let g1 and g2 be group characterizable entropy functions. a) Prove that m1 g1 + m2 g2 is group characterizable, where m1 and m2 are any positive integers. b) For any positive real numbers a1 and a2 , construct a sequence of group characterizable entropy functions f (k) for k = 1, 2, , such that h f (k) , = (k) || k ||f ||h|| lim where h = a1 g1 + a2 g2 . 3. Let (G, G1 , G2 , , Gn ) be a group characterization of g n , where g is the entropy function for random variables X1 , X2 , , Xn . Fix any nonempty subset of Nn , and define h by h = g - g for all nonempty subsets of Nn . It can easily be checked that h = H(X |X ). Show that (K, K1 , K2 , , Kn ) is a group characterization of h, where K = G and Ki = Gi G . 4. Let (G, G1 , G2 , , Gn ) be a group characterization of g n , where g is the entropy function for random variables X1 , X2 , , Xn . Show that if Xi is a function of (Xj : j ), then G is a subgroup of Gi . 5. Let G1 , G2 , G3 be subgroups of a finite group G. Prove that |G||G1 G2 G3 |2 |G1 G2 ||G2 G3 ||G1 G3 |. Hint: Use the information-theoretic approach. 6. Let h 2 be the entropy function for random variables X1 and X2 such that h1 + h2 = h12 , i.e. X1 and X2 are independent. Let (G, G1 , G2 ) be a group characterization of h, and define a mapping L : G1 G2 G by L(a, b) = a b. Problems 407 a) Prove that the mapping L is onto, i.e., for any element c G, there exists (a, b) G1 G2 such that a b = c. b) Prove that G1 G2 is a group. 7. Denote an entropy function h 2 by (h1 , h2 , h12 ). Construct a group characterization for each of the following entropy functions: a) h1 = (log 2, 0, log 2) b) h2 = (0, log 2, log 2) c) h3 = (log 2, log 2, log 2). Verify that 2 is the minimal convex set containing the above three entropy functions. 8. Denote an entropy function h 3 by (h1 , h2 , h3 , h12 , h23 , h13 , h123 ). Construct a group characterization for each of the following entropy functions: a) h1 = (log 2, 0, 0, log 2, 0, log 2, log 2) b) h2 = (log 2, log 2, 0, log 2, log 2, log 2, log 2) c) h3 = (log 2, log 2, log 2, log 2, log 2, log 2, log 2) d) h4 = (log 2, log 2, log 2, log 4, log 4, log 4, log 4). 9. Ingleton inequality Let G be a finite Abelian group and G1 , G2 , G3 , and G4 be subgroups of G. Let (G, G1 , G2 , G3 , G4 ) be a group characterization of g, where g is the entropy function for random variables X1 , X2 , X3 , and X4 . Prove the following statements: a) |(G1 G3 ) (G1 G4 )| |G1 (G3 G4 )| Hint: Show that (G1 G3 ) (G1 G4 ) G1 (G3 G4 ). b) |G1 G3 G4 | c) |G1 G2 G3 G4 | d) |G1 G2 G3 G4 | |G1 ||G2 ||G3 ||G4 ||G1 G3 G4 ||G2 G3 G4 | . |G1 G3 ||G1 G4 ||G2 G3 ||G2 G4 ||G3 G4 | e) |G1 G3 ||G1 G4 ||G2 G3 ||G2 G4 ||G3 G4 | |G3 ||G4 ||G1 G2 ||G1 G3 G4 ||G2 G3 G4 |. |G1 G3 G4 ||G2 G3 G4 | . |G3 G4 | |G1 ||G3 G4 ||G1 G3 G4 | . |G1 G3 ||G1 G4 | 408 16 Entropy and Groups f) H(X13 ) + H(X14 ) + H(X23 ) + H(X24 ) + H(X34 ) H(X3 ) + H(X4 ) + H(X12 ) + H(X134 ) + H(X234 ), where H(X134 ) denotes H(X1 , X3 , X4 ), etc. g) Is the inequality in f) implied by the basic inequalities? And does it always hold? Explain. The Ingleton inequality [181] (see also [283]) was originally obtained as a constraint on the rank functions of vector spaces. The inequality in e) was obtained in the same spirit by Chan [58] for subgroups of a finite group. The inequality in f) is referred to as the Ingleton inequality for entropy in the literature. (See also Problem 8 in Chapter 15.) Historical Notes The results in this chapter are due to Chan and Yeung [64], whose work was inspired by a one-to-one correspondence between entropy and quasi-uniform arrays previously established by Chan [58] (also Chan [59]). Romashchenko et al. [311] have developed an interpretation of Kolmogorov complexity similar to the combinatorial interpretation of entropy in Chan [58]. The results in this chapter have been used by Chan [62] to construct codes for multi-source network coding to be discussed in Chapter 21. Part II Fundamentals of Network Coding 17 Introduction For a point-to-point communication system, we see from Section 7.7 and Problem 6 in Chapter 8 that asymptotic optimality can be achieved by separating source coding and channel coding. Recall from Section 5.3 that the goal of source coding is to represent the information source in (almost) fair bits1 . Then the role of channel coding is to enable the transmission of fair bits through the channel essentially free of error with no reference to the meaning of these fair bits. Thus a theme in classical information theory for point-topoint communication is that fair bits can be drawn equivalence to a commodity. It is intuitively appealing that this theme in classical information theory would continue to hold in network communication where the network consists of noiseless point-to-point communication channels. If so, in order to multicast2 information from a source node to possibly more than one sink node, we only need to compress the information at the source node into fair bits, organize them into data packets, and route the packets to the sink node through the intermediate nodes in the network. In the case when there are more than one sink node, the information needs to be replicated at certain intermediate nodes so that every sink node can receive a copy of the information. This method of transmitting information in a network is generally referred to as store-and-forward or routing. As a matter of fact, almost all computer networks built in the last few decades are based on this principle, where routers are deployed at the intermediate nodes to switch a data packet from an input channel to an output channel without processing the data content. The delivery of data packets in a computer network resembles mail delivery in a postal system. We refer the readers to textbooks on data communication [35][215] and switching theory [176][227]. 1 2 Fair bits refer to i.i.d. bits, each distributed uniformly on {0, 1}. Multicast means to transmit information from a source node to a specified set of sink nodes. 412 17 Introduction However, we will see very shortly that in network communication, it does not suffice to simply route and/or replicate information within the network. Specifically, coding generally needs to be employed at the intermediate nodes in order to achieve bandwidth optimality. This notion, called network coding, is the subject of discussion in Part II of this book. 17.1 The Butterfly Network In this section, the advantage of network coding over routing is explained by means of a few simple examples. The application of network coding in wireless and satellite communication will be discussed in the next section. We will use a finite directed graph to represent a point-to-point communication network. A node in the network corresponds to a vertex in the graph, while a communication channel in the network corresponds to an edge in the graph. We will not distinguish a node from a vertex, nor will we distinguish a channel from an edge. In the graph, a node is represented by a circle, with the exception that the unique source node, denoted by s (if exists), is represented by a square. Each edge is labeled by a positive integer called the capacity3 or the rate constraint, which gives the maximum number of information symbols taken from some finite alphabet that can be transmitted over the channel per unit time. In this section, we assume that the information symbol is binary. When there is only one edge from node a to node b, we denote the edge by (a, b). Example 17.1 (Butterfly Network I). Consider the network in Figure 17.1(a). In this network, two bits b1 and b2 are generated at source node s, and they are to be multicast to two sink nodes t1 and t2 . In Figure 17.1(b), we try to devise a routing scheme for this purpose. By symmetry, we send the two bits on different output channels at node s. Without loss of generality, b1 is sent on channel (s, 1) and b2 is sent on channel (s, 2). At nodes 1 and 2, the received bit is replicated and the copies are sent on the two output channels. At node 3, since both b1 and b2 are received but there is only one output channel, we have to choose one of the two bits to be sent on the output channel (3, 4). Suppose we send b1 as in Figure 17.1(b). Then the bit is replicated at node 4 and the two copies are sent to nodes t1 and t2 , respectively. At node t2 , both b1 and b2 are received. However, at node t1 , two copies of b1 are received and b2 cannot be recovered. Thus this routing scheme does not work. Similarly, if b2 instead of b1 is sent on channel (3, 4), then b1 cannot be recovered at node t2 . However, if network coding is allowed, it is actually possible to achieve our goal. Figure 17.1(c) shows a scheme which multicasts both b1 and b2 to nodes t1 and t2 , where `+' denotes modulo 2 addition. At node t1 , b1 is received, and b2 can be recovered by adding b1 and b1 + b2 , because 3 Here the term "capacity" is used in the sense of graph theory. 17.1 The Butterfly Network 1 413 s 1 b1 s b2 1 1 2 1 1 b1 2 b2 3 1 1 1 b1 b1 3 b1 b1 b2 1 4 1 4 t1 t2 t1 t2 (a) (b) b1 s b2 b2 b1 s b2 2 1 b1 2 1 b1 3 b2 3 b1 b1+ b2 b 1+ b 2 b 1+ b 2 b2 b1 b2 b1 b2 4 b1 b2 4 t1 t2 t1 t2 (c) Fig. 17.1. Butterfly Network I. (d) b1 + (b1 + b2 ) = (b1 + b1 ) + b2 = 0 + b2 = b2 . (17.1) Similarly, b2 is received at node t2 , and b1 can be recovered by adding b2 and b1 + b2 . In this scheme, b1 and b2 are encoded into the bit b1 + b2 which is then sent on channel (3, 4). If network coding is not allowed, in order to multicast both b1 and b2 to nodes t1 and t2 , at least one more bit has to be sent. Figure 17.1(d) shows such a scheme. In this scheme, however, the capacity of channel (3, 4) is exceeded by 1 bit. If the capacity of channel (3, 4) cannot be exceeded and network coding is not allowed, it can be shown that at most 1.5 bits can be multicast per unit time on the average (see Problem 3). 414 17 Introduction The above example shows the advantage of network coding over routing for a single multicast in a network. The next example shows the advantage of network coding over routing for multiple unicasts4 in a network. Example 17.2 (Butterfly Network II). In Figure 17.1, instead of both being generated at node s, suppose bit b1 is generated at node 1 and bit b2 is generated at node 2. Then we can remove node s and obtain the network in Figure 17.2(a). We again want to multicast b1 and b2 to both nodes t1 and t2 . Since this network is essentially the same as the previous one, Figure 17.2(b) shows the obvious network coding solution. 1 1 2 1 1 b1 2 b2 3 1 1 1 b1 b1+b2 3 b1+b2 b1+b2 b2 1 4 1 4 t1 t2 t1 t2 (a) (b) b1 b2 3 t1 ' b1+b2 b1+b2 b1+b2 t2 ' 4 (c) Fig. 17.2. Butterfly Network II. There are two multicasts in this network. However, if we merge node 1 and node t1 into a new node t1 and merge node 2 and node t2 into a new node t2 , then we obtain the network and the corresponding network coding solution in Figure 17.2(c). In this new network, bits b1 and b2 are generated at nodes t1 and t2 , respectively, and the communication goal is to exchange the two 4 Unicast is the special case of multicast with one sink node. 17.2 Wireless and Satellite Communications 415 bits through the network. In other words, the two multicasts in Figure 17.2(a) become two unicasts in Figure 17.2(c). If network coding is not allowed, we need to route b1 from node t1 to node t2 and to route b2 from node t2 to node t1 . Since each of these routes has to go through node 3 and node 4, if b1 and b2 are routed simultaneously, the capacity of channel (3, 4) is exceeded. Therefore, we see the advantage of network coding over routing when there are multiple unicasts in the network. For the network in Figure 17.2(b), the two sink nodes are required to recover both of the information sources, namely the bits b1 and b2 . Even though they are generated at two different source nodes 1 and 2, they can be regarded as being generated at a super source node s connecting to nodes t1 and t2 as in Figure 17.1(c). Precisely, the network (network code) in Figure 17.2(b) can be obtained from the network (network code) in Figure 17.1(c) by removing node s and all its output channels. This observation will be further discussed in Example 19.26 in the context of single-source linear network coding. 17.2 Wireless and Satellite Communications In wireless communication, when a node broadcasts, different noisy versions of the signal is received by the neighboring nodes. Under certain conditions, with suitable channel coding, we can assume the existence of an error-free channel between the broadcast node and the neighboring nodes such that each of the latter receives exactly the same information. Such an abstraction, though generally suboptimal, provides very useful tools for communication systems design. Our model for network communication can be used for modeling the above broadcast scenario by imposing the following constraints on the broadcast node: 1. all the output channels have the same capacity; 2. the same symbol is sent on each of the output channels. We will refer to these constraints as the broadcast constraint. Figure 17.3(a) is an illustration of a broadcast node b with two neighboring nodes n1 and n2 , where the two output channels of node b have the same capacity. In order to express the broadcast constraint in the usual graph-theoretic terminology, we need to establish the following simple fact about network coding. Proposition 17.3. Network coding is not necessary at a node if the node has only one input channel and the capacity of each output channel is the same as that of the input channel. Proof. Consider a node in the network as prescribed and denote the symbol(s) received on the input channel by x. (There is more than one symbol in x if 416 17 Introduction b b n1 n2 n1 n2 (a) (b) Fig. 17.3. A broadcast node b with two neighboring nodes n1 and n2 . the input channel has capacity larger than 1.) Let a coding scheme be given, and denote the symbol sent on the ith output channel by gi (x). We now show that one may assume without loss of generality that x is sent on all the output channels. If x instead of gi (x) is sent on the ith output channel, then the receiving node can mimic the effect of receiving gi (x) by applying the function gi on x upon receiving it. In other words, any coding scheme that does not send x on all the output channels can readily be converted into one which does. This proves the proposition. We now show that the broadcast constraint depicted in Figure 17.3(a) is logically equivalent to the usual graph representation in Figure 17.3(b). In this figure, the unlabeled node is a dummy node associated with the broadcast node which is inserted for the purpose of modeling the broadcast constraint, where the input channel and all the output channels of the dummy node have the same capacity as an output channel of the broadcast node b in Figure 17.3(a). Although no broadcast constraint is imposed on the dummy node in Figure 17.3(b), by Proposition 17.3, we may assume without loss of generality that the dummy node simply sends the symbol received on the input channel on each of the output channels. Then Figures 17.3(a) and (b) are logically equivalent to each other because a coding scheme for the former corresponds to a coding scheme for the latter, and vice versa. Example 17.4 (A Wireless/Satellite System). Consider a communication system with two wireless nodes t1 and t2 that generate two bits b1 and b2 , respectively, and the two bits are to be exchanged through a relay node. Such a system can also be the model of a satellite communication system, where the relay node corresponds to a satellite, and the two nodes t1 and t2 correspond to ground stations that communicate with each other through the satellite. We make the usual assumption that a wireless node cannot simultaneously 1. transmit and receive; 2. receive the transmission from more than one neighboring node. 17.3 Source Separation t 1' t 2' t 1' t 2' 417 k=1 k=2 k=3 k=4 b1 b2 b1 b2 b1 b2 b1 + b2 b1 + b2 (a) (b) Fig. 17.4. A network coding application in wireless communication. A straightforward routing scheme which takes a total of 4 time units to complete is shown in Figure 17.4(a), with k being the discrete time index. By taking into account the broadcast nature of the relay node, the system can be modeled by the network in Figure 17.2(c), where node 3 corresponds to the relay node and node 4 corresponds to the associated dummy node. Then the network coding solution is shown in Figure 17.4(b), which takes a total of 3 time units to complete. In other words, a very simple coding scheme at the relay node can save 50 percent of the downlink bandwidth. 17.3 Source Separation In an error-free point-to-point communication system, suppose we want to transmit two information sources X and Y . If we compress the two sources separately, we need to transmit approximately H(X) + H(Y ) bits. If we compress the two sources jointly, we need to transmit approximately H(X, Y ) bits. If X and Y are independent, we have H(X, Y ) = H(X) + H(Y ). (17.2) In other words, if the information sources are independent, asymptotically there is no difference between coding them separately or jointly. We will refer to coding independent information sources separately as source separation. Example 17.2 reveals the important fact that source separation is not necessary optimal in network communication, which is explained as follows. Let B1 and B2 be random bits generated at nodes t1 and t2 , respectively, where B1 and B2 are independent and each of them are distributed uniformly on {0, 1}. With B2 as side-information which is independent of B1 , 418 17 Introduction node t2 has to receive at least 1 bit in order to decode B1 . Since node t2 can receive information only from node 4 which in turn can receive information only from node 3, any coding scheme that transmits B1 from node t1 to node t2 must send at least 1 bit on channel (3, 4). Similarly, any coding scheme that transmits B2 from node t2 to node t1 must send at least 1 bit on channel (3, 4). Therefore, any source separation solution must send at least 2 bits on channel (3, 4). Since the network coding solution in Figure 17.2(c) sends only 1 bit on channel (3, 4), we see that source separation is not optimal. For a network coding problem with multiple information sources, since source separation does not guarantee optimality, the problem cannot always be decomposed into a number single-source problems. We will see that while single-source network coding has a relatively simple characterization, the characterization of multi-source network coding is much more involved. Chapter Summary Advantage of Network Coding: For communication on a point-to-point network, store-and-forward may not be bandwidth optimal when 1. there is one information source to be multicast; 2. there are two or more independent information sources to be unicast (more generally multicast). In general, network coding needs to be employed for bandwidth optimality. Source Separation: For communication on a point-to-point network, when there are two or more independent information sources to be unicast (more generally multicast), source separation coding may not be bandwidth optimal. Problems In the following problems, the rate constraint for an edge is in bits per unit time. 1. Consider the following network. We want to multicast information to the sink nodes at the maximum rate without using network coding. Let B = {b1 , b2 , , b } be the set of bits to be multicast. Let Bi be the set of bits sent in edge (s, i), where |Bi | = 2, i = 1, 2, 3. At node i, the received bits are duplicated and sent in the two out-going edges. Thus two bits are sent in each edge in the network. a) Show that B = Bi Bj for any 1 i < j 3. b) Show that B3 (B1 B2 ) = B. c) Show that |B3 (B1 B2 )| |B3 | + |B1 | + |B2 | - |B1 B2 |. Historical Notes 419 1 t1 s 2 t2 3 t3 d) Determine the maximum value of and devise a network code which achieves this maximum value. e) What is the percentage of improvement if network coding is used? (Ahlswede et al. [6].) 2. Consider the following butterfly network. 1 s 2 3 4 5 6 Devise a network coding scheme which multicasts two bits b1 and b2 from node s to all the other nodes such that nodes 3, 5, and 6 receive b1 and b2 after 1 unit time and nodes 1, 2, and 4 receive b1 and b2 after 2 units of time. In other words, node i receives information at a rate equal to maxflow(s, i) for all i = s. 3. Determine the maximum rate at which information can be multicast to nodes 5 and 6 only in the network in Problem 2 if network coding is not used. Devise a network coding scheme which achieves this maximum rate. Historical Notes The concept of network coding was first introduced for satellite communication networks in Yeung and Zhang [411] and then fully developed in Ahlswede et al. [6], where in the latter the term "network coding" was coined. In this work, the advantage of network coding over store-and-forward was first demonstrated by the butterfly network, thus refuting the folklore that information transmission in a point-to-point network is equivalent to a commodity flow. 420 17 Introduction Prior to [411] and [6], network coding problems for special networks had been studied in the context of distributed source coding. The suboptimality of source separation was first demonstrated by Yeung [400]. Source separation was proved to be optimal for special networks by Hau [160], Roche et al. [308], and Yeung and Zhang [410]. Some other special cases of single-source network coding had been studied by Roche et al. [307], Rabin [297], Ayanoglu et al. [22], and Roche [306]. For a tutorial on the theory, we refer the reader to the unifying work by Yeung et al. [408]. Tutorials on the subject have also been written by Fragouli and Soljanin [123] and Chou and Wu [69] from the algorithm and application perspectives. We also refer the reader to the book by Ho and Lun [164]. For an update of the literature, the reader may visit the Network Coding Homepage [273]. By regarding communication as a special case of computation, it can be seen that network coding is in the spirit of communication complexity in computer science studied by Yao [394]. However, the problem formulations of network coding and communication complexity are quite different. 18 The Max-Flow Bound In this chapter, we discuss an important bound for single-source network coding which has a strong connection with graph theory. This bound, called the max-flow min-cut bound, or simply the max-flow bound, gives a fundamental limit on the amount of information that can be multicast in a network. The max-flow bound is established in a general setting where information can be transmitted within the network in some arbitrary manner. Toward this end, we first formally define a point-to-point network and a class of codes on such a network. In Chapters 19 and 20, we will prove the achievability of the max-flow bound by linear network coding1 . 18.1 Point-to-Point Communication Networks A point-to-point communication network is represented by a directed graph G = (V, E), where V is the set of nodes in the network and E is the set of edges in G which represent the point-to-point channels. Parallel edges between a pair of nodes is allowed2 . We assume that G is finite, i.e., |E| < (and hence |V | < ). The unique source node in the network, where information is generated, is denoted by s. All the other nodes are referred to as non-source nodes. The sets of input channels and output channels of a node i are denoted by In(i) and Out(i), respectively. For a channel e, let Re be the rate constraint, i.e., the maximum number of information symbols taken from a finite alphabet that can be sent on the channel per unit time. As before, we also refer to Re as the capacity of channel e in the sense of graph theory. Let R = (Re : e E) 1 (18.1) 2 A more specific form of the max-flow bound will be proved in Theorem 19.10 for linear network coding. Such a graph is sometimes called a multigraph. 422 18 The Max-Flow Bound be the rate constraints for the graph G. To simplify our discussion, we assume that Re are positive integers for all e E. In the following, we introduce some notions in graph theory which will facilitate the characterization of a point-to-point network. Temporarily regard an edge in the graph G as a water pipe and G as a network of water pipes. Fix a node t = s and call it the sink node. Suppose water is generated at a constant rate at node s. We assume that the rate of water flow in each pipe does not exceed its capacity. We also assume that there is no leakage in the network, so that water is conserved at every node other than s and t in the sense that the total rate of water flowing into the node is equal to the total rate of water flowing out of the node. The water generated at node s is eventually drained at node t. A flow F = (Fe : e E) (18.2) in G from node s to node t with respect to rate constraints R is a valid assignment of a nonnegative integer Fe to every edge e E such that Fe is equal to the rate of water flow in edge e under all the assumptions in the last paragraph. The integer Fe is referred to as the value of F on edge e. Specifically, F is a flow in G from node s to node t if for all e E, 0 Fe Re , and for all i V except for s and t, F+ (i) = F- (i), where F+ (i) = eIn(i) (18.3) (18.4) (18.5) Fe and F- (i) = eOut(i) Fe . (18.6) In the above, F+ (i) is the total flow into node i and F- (i) is the total flow out of node i, and (18.4) is called the conservation conditions. Since the conservation conditions require that the resultant flow out of any node other than s and t is zero, it is intuitively clear and not difficult to show that the resultant flow out of node s is equal to the resultant flow into node t. This common value is called the value of F. F is a max-flow from node s to node t in G with respect to rate constraints R if F is a flow from node s to node t whose value is greater than or equal to the value of any other flow from node s to node t. A cut between node s and node t is a subset U of V such that s U and t U . Let EU = {e E : e Out(i) In(j) for some i U and j U } (18.7) 18.1 Point-to-Point Communication Networks 423 ! ! (a) (b) Fig. 18.1. Illustrations of the max-flow and the min-cut from the source node to (a) a collection of non-source node T and (b) a collection of edges . be the set of edges across the cut U . The capacity of the cut U with respect to rate constraints R is defined as the sum of the capacities of all the edges across the cut, i.e., Re . (18.8) eEU A cut U is a min-cut between node s and node t if it is a cut between node s and node t whose capacity is less than or equal to the capacity of any other cut between s and t. A min-cut between node s and node t can be thought of as a bottleneck between node s and node t. Therefore, it is intuitively clear that the value of a max-flow from node s to node t cannot exceed the capacity of a min-cut between the two nodes. The following theorem, known as the max-flow mincut theorem, states that the capacity of a min-cut is always achievable. This theorem will play a key role in the subsequent discussions. Theorem 18.1 (Max-Flow Min-Cut Theorem [116]). Let G be a graph with source node s, sink node t, and rate constraints R. Then the value of a max-flow from node s to node t is equal to the capacity of a min-cut between the two nodes. The notions of max-flow and min-cut can be generalized to a collection of non-source nodes T . To define the max-flow and the min-cut from s to T , we expand the graph G = (V, E) into G = (V , E ) by installing a new node which is connected from every node in T by an edge. The capacity of an edge (t, ), t T , is set to infinity. Intuitively, node acts as a single sink node that collects all the flows into T . Then the max-flow and the min-cut from node s to T in graph G are defined as the max-flow and the min-cut from node s to node in graph G , respectively. This is illustrated in Figure 18.1(a). The notions of max-flow and min-cut can be further generalized to a collection of edges . For an edge e , let the edge be from node ve to node we . ~ ~ ~ We modify the graph G = (V, E) to obtain the graph G = (V , E) by installing a new node te for each edge e and replacing edge e by two new edges e and e , where e is from node ve to node te and e is from node te to node we . Let T be the set of nodes te , e . Then the max-flow and the min-cut between 424 18 The Max-Flow Bound s 2 2 1 1 2 2 s 1 1 1 2 b1b2 s b3 b1 b2 1 2 1 2 1 2 b 1b 3 t1 (a) t1 (b) Fig. 18.2. A one-sink network. t1 (c) node s and the collection of edges in graph G are defined as the max-flow ~ and the min-cut between node s and the collection of nodes T in graph G, respectively. This is illustrated in Figure 18.1(b). 18.2 Examples Achieving the Max-Flow Bound Let be the rate at which information is multicast from source node s to sink nodes t1 , t2 , , tL in a network G with rate constraints R. We are naturally interested in the maximum possible value of . With a slight abuse of notation, we denote the value of a max-flow from source node s to a sink node tl by maxflow(tl ). It is intuitive that maxflow(tl ) for all l = 1, 2, , L, i.e., min maxflow(tl ). l (18.9) (18.10) This is called the max-flow bound, which will be formally established in the next two sections. In this section, we first show by a few examples that the max-flow bound can be achieved. In these examples, the unit of information is the bit. First, we consider the network in Figure 18.2 which has one sink node. Figure 18.2(a) shows the capacity of each edge. By identifying the min-cut to be {s, 1, 2} and applying the max-flow min-cut theorem, we see that maxflow(t1 ) = 3. (18.11) Therefore the flow in Figure 18.2(b) is a max-flow. In Figure 18.2(c), we show how we can send three bits b1 , b2 , and b3 from node s to node t1 based on the max-flow in Figure 18.2(b). Evidently, the max-flow bound is achieved. In fact, we can easily see that the max-flow bound can always be achieved when there is only one sink node in the network. In this case, we only need to 18.2 Examples Achieving the Max-Flow Bound 425 s 2 1 1 3 1 b4b5 b3 s b3 b1b2b3 1 4 2 2 3 3 3 b3b4b5 1 b1b2 2 b4b5 3 b1b2b3 t1 (a) t2 t1 (b) t2 Fig. 18.3. A two-sink network without coding. treat the information bits constituting the message as a commodity and route them through the network according to any fixed routing scheme. Eventually, all the bits will arrive at the sink node. Since the routing scheme is fixed, the sink node knows which bit is coming in from which edge, and the message can be recovered accordingly. Next, we consider the network in Figure 18.3 which has two sink nodes. Figure 18.3(a) shows the capacity of each edge. It is easy to see that maxflow(t1 ) = 5 and maxflow(t2 ) = 6. (18.13) So the max-flow bound asserts that we cannot send more than 5 bits to both t1 and t2 . Figure 18.3(b) shows a scheme which sends 5 bits b1 , b2 , b3 , b4 , and b5 to t1 and t2 simultaneously. Therefore, the max-flow bound is achieved. In this scheme, b1 and b2 are replicated at node 3, b3 is replicated at node s, while b4 and b5 are replicated at node 1. Note that each bit is replicated exactly once in the network because two copies of each bit are needed to be sent to the two sink nodes. We now revisit the butterfly network reproduced in Figure 18.4(a), which again has two sink nodes. It is easy to see that maxflow(tl ) = 2 (18.14) (18.12) for l = 1, 2. So the max-flow bound asserts that we cannot send more than 2 bits to both sink nodes t1 and t2 . We have already seen the network coding scheme in Figure 18.4(b) that achieves the max-flow bound. In this scheme, coding is required at node 3. Finally, we consider the network in Figure 18.5 which has three sink nodes. Figure 18.5(a) shows the capacity of each edge. It is easy to see that maxflow(tl ) = 2 (18.15) 426 18 The Max-Flow Bound 1 s 1 b1 s b2 b2 1 1 2 1 1 b1 2 3 1 1 1 b1 b 1+ b 2 3 b1+ b2 b1+ b2 b2 1 4 1 4 t1 t2 t1 t2 (a) Fig. 18.4. Butterfly network I. (b) for all l. In Figure 18.5(b), we show how to multicast 2 bits b1 and b2 to all the sink nodes. Therefore, the max-flow bound is achieved. Again, it is necessary to code at the nodes in order to multicast the maximum number of bits to all the sink nodes. The network in Figure 18.5 is of special interest in practice because it is a special case of the diversity coding scheme used in commercial disk arrays, which are a kind of fault-tolerant data storage system. For simplicity, assume the disk array has three disks which are represented by nodes 1, 2, and 3 in the network, and the information to be stored are the bits b1 and b2 . The information is encoded into three pieces, namely b1 , b2 , and b1 + b2 , which are stored on the disks represented by nodes 1, 2, and 3, respectively. In the system, there are three decoders, represented by sink nodes t1 , t2 , and t3 , such that each of them has access to a distinct set of two disks. The idea is that when any one disk is out of order, the information can still be recovered from the remaining two disks. For example, if the disk represented by node 1 is out of order, then the information can be recovered by the decoder represented by s 1 1 1 1 1 b1 s b2 b2 b1 b1+b2 1 1 2 1 1 3 1 b1 1 2 b2 3 b1+b2 t1 t2 (a) t3 t1 t2 (b) t3 Fig. 18.5. A diversity coding scheme. 18.3 A Class of Network Codes 427 sink node t3 which has access to the disks represented by node 2 and node 3. When all the three disks are functioning, the information can be recovered by any decoder. 18.3 A Class of Network Codes In this section, we introduce a general class of codes for the point-to-point network defined in Section 18.1. In the next section, the max-flow bound will be proved for this class of network codes. Since the max-flow bound concerns only the values of max-flows from source node s to the sink nodes, we assume without loss of generality that there is no loop in the graph G, i.e., In(i) Out(i) = for all i V , because such edges do not increase the value of a max-flow from node s to a sink node. For the same reason, we assume that there is no input edge at node s, i.e., In(s) = . We consider a block code of length n. Let X denote the information source and assume that x, the outcome of X, is obtained by selecting an index from a set X according to the uniform distribution. The elements in X are called messages. The information sent on an output channel of a node can depend only on the information previously received by that node. This constraint specifies the causality of any coding scheme on the network. An (n, (e : e E), ) network code on the graph G that multicasts information from source node s to sink nodes t1 , t2 , , tL , where n is the block length, is defined by the components listed below; the construction of the code from these components will be described after their definitions are given. 1) A positive integer K. 2) Mappings u : {1, 2, , K} V, v : {1, 2, , K} V, and e : {1, 2, , K} E, ^ such that e(k) Out(u(k)) and e(k) In(v(k)). ^ ^ 3) Index sets Ak = {1, 2, , |Ak |}, 1 k K, such that |Ak | = e , kTe (18.16) (18.17) (18.18) (18.19) where Te = {1 k K : e(k) = e}. ^ (18.20) 428 18 The Max-Flow Bound 4) (Encoding functions). If u(k) = s, then f k : X Ak , where X = {1, 2, , 2n }. If u(k) = s, if Qk = {1 k < k : v(k ) = u(k)} is nonempty, then fk : k Qk (18.21) (18.22) (18.23) (18.24) Ak Ak ; otherwise, let fk be an arbitrary constant taken from Ak . 5) (Decoding functions). Mappings gl : k Wl Ak X (18.25) for l = 1, 2, , L, where Wl = {1 k K : v(k) = tl } such that for all l = 1, 2, , L, gl (x) = x ~ (18.27) (18.26) for all x X , where gl : X X is the function induced inductively by ~ fk , 1 k K and gl , and gl (x) denotes the value of gl as a function of x. ~ The quantity is the rate of the information source X, which is also the rate at which information is multicast from the source node to all the sink nodes. The (n, (e : e E), ) code is constructed from the above components as follows. At the beginning of a coding session, the value of X is available to node s. During the coding session, there are K transactions which take place in chronological order, where each transaction refers to a node sending information to another node. In the kth transaction, node u(k) encodes according to encoding function fk and sends an index in Ak to node v(k). The domain of fk is the set of all possible information that can be received by node u(k) just before the kth transaction, and we distinguish two cases. If u(k) = s, the domain of fk is X . If u(k) = s, Qk gives the time indices of all the previous transactions for which information was sent to node u(k), so the domain of fk is k Qk Ak . The set Te gives the time indices of all the transactions for which information is sent on channel e, so e is the number of possible index tuples that can be sent on channel e during the coding session. Finally, Wl gives the indices of all the transactions for which information is sent to node tl , and gl is the decoding function at node tl which recovers x with zero error. 18.4 Proof of the Max-Flow Bound 429 18.4 Proof of the Max-Flow Bound In this section, we state and prove the max-flow bound for the class of network codes defined in the last section. Definition 18.2. For a graph G with rate constraints R, an information rate 0 is asymptotically achievable if for any > 0, there exists for sufficiently large n an (n, (e : e E), ) network code on G such that n-1 log2 e Re + (18.28) for all e E, where n-1 log2 e is the average bit rate of the code on channel e, and - . (18.29) For brevity, an asymptotically achievable information rate will be referred to as an achievable information rate. Remark It follows from the above definition that if 0 is achievable, then is also achievable for all 0 . Also, if (k) is achievable for all k 1, then it can be shown that = limk (k) , if exists, is also achievable. Therefore, the set of all achievable information rates is closed and fully characterized by the maximum value in the set. Theorem 18.3 (Max-Flow Bound). For a graph G with rate constraints R, if is achievable, then min maxflow(tl ). l (18.30) Proof. It suffices to prove that for a graph G with rate constraints R, if for any > 0 there exists for sufficiently large n an (n, (e : e E), ) code on G such that n-1 log2 e Re + (18.31) for all e E and - , (18.32) then satisfies (18.30). Consider such a code for a fixed and a sufficiently large n, and consider any l = 1, 2, , L and any cut U between node s and node tl . Let ~ wj (x) = (fk (x) : k eIn(j) Te ), (18.33) ~ where x X and fk : X Ak is the function induced inductively by fk , 1 ~ (x) denotes the value of fk as a function of x. The tuple wj (x) is k k, and fk all the information known by node j during the whole coding session when the ~ message is x. Since fk (x) is a function of the information previously received 430 18 The Max-Flow Bound by node u(k), it can be shown by induction (see Problem 3) that wtl (x) is a ~ function of fk (x), k eEU Te , where EU is the set of edges across the cut U as previously defined in (18.7). Since x can be determined at node tl , we have H(X) H(X, wtl (X)) = H(wtl (X)) a) (18.34) (18.35) Te eEU H b) ~ fk (X), k ~ H(fk (X)) (18.36) (18.37) (18.38) eEU kTe c) eEU kTe log2 |Ak | log2 eEU kTe = d) |Ak | (18.39) (18.40) eEU log2 e , where ~ a) follows because wtl (x) is a function of fk (x), k eEU Te ; b) follows from the independence bound for entropy (Theorem 2.39); c) follows from (18.21) and Theorem 2.43; d) follows from (18.19). Thus - n =n eEU -1 -1 (18.41) log2 2 n (18.42) (18.43) (18.44) (18.45) (18.46) (18.47) log2 |X | n-1 log2 e (Re + ) = n-1 H(X) eEU eEU Re + |E| , where (18.45) follows from (18.40). Minimizing the right hand side over all U , we have - min Re + |E| . (18.48) U eEU Problems 431 The first term on the right hand side is the capacity of a min-cut between node s and node tl . By the max-flow min-cut theorem, it is equal to the value of a max-flow from node s to node tl , i.e., maxflow(tl ). Letting 0, we obtain maxflow(tl ). (18.49) Since this upper bound on holds for all l = 1, 2, , L, min maxflow(tl ). l (18.50) The theorem is proved. Remark 1 In proving the max-flow bound, the time evolution and the causality of the network code have been taken into account. Remark 2 Even if we allow an arbitrarily small probability of decoding error in the usual Shannon sense, by modifying our proof by means of a standard application of Fano's inequality, it can be shown that it is still necessary for to satisfy (18.50). The details are omitted here. Chapter Summary Max-Flow Min-Cut Bound: In a point-to-point communication network, if node t receives an information source from node s, then the value of a maximum flow from s to t, or equivalently the capacity of a minimum cut between s to t, is at least equal to the rate of the information source. Problems 1. In a network, for a flow F from a source node s to a sink node t, show that F+ (s) = F- (t) provided that the conservation conditions in (18.4) hold. 2. For the class of codes defined in Section 18.3, show that if the rates (k) are achievable for all k 1, then = limk (k) , if exists, is also achievable (see Definition 18.2). 3. Prove the claim in the proof of Theorem 18.3 that for any cut U between ~ node s and node tl , wtl (x) is a function of fk (x), k eEU Te . Hint: Define ~ wj, (x) = (fk (x) : k eIn(j) Te , k ) and prove by induction on that for all 1 K, (wj, (x) : j U ) is ~ a function of (fk (x) : k eEU Te , k ). 432 18 The Max-Flow Bound 4. Probabilistic network code For a network code defined in Section 18.3, the kth transaction of the coding process is specified by a mapping fk . Suppose instead of a mapping fk , the kth transaction is specified by a transition probability matrix from the domain of fk to the range of fk . Also, instead of a mapping gl , decoding at sink node tl is specified by a transition probability matrix from the domain of gl to the range of gl , 1 l L. Conditioning on the indices received by node u(k) during 1 k < k, the index sent from node u(k) to node v(k) in the kth transaction is independent of all the previously generated random variables. Similarly, conditioning on all the indices received by sink node tl during the whole coding session, the decoding at tl is independent of all the previously generated random variables. We refer to such a code as a probabilistic network code. Since a deterministic network code is a special case of a probabilistic network code, the latter can potentially multicast at a higher rate compared with the former. Prove that this is not possible. 5. Consider a probabilistic network code on the network below. s 1 t Let X = (X1 , X2 ) be uniformly distributed on GF (2)2 , and Z be inde~ pendent of X and uniformly distributed on GF (2). We use Fk to denote the index transmitted in the kth transaction and Wtl to denote ~ (Fk , k eIn(tl ) Te ). The probabilistic network code is specified by the following five transactions: ~ u(1) = s, v(1) = 1, F1 = X1 , ~ u(2) = 1, v(2) = t, F2 = X1 + Z, ~ u(3) = t, v(3) = s, F3 = X1 + Z, ~ u(4) = s, v(4) = 1, F4 = (X1 , X2 + Z), ~ u(5) = 1, v(5) = t, F5 = (X1 , X2 + Z). Note that the fourth transaction is possible because upon knowing X1 and X1 + Z, Z can be determined. a) Determine Wt . b) Verify that X can be recovered from Wt . ~ ~ c) Show that X (F1 , F4 ) Wt does not form a Markov chain. ~1 and F4 are all the random variables sent on edge (s, 1) during ~ Here, F the coding session. Although node t receives all the information through the edge (s, 1), the Markov chain in c) does not hold. (Ahlswede et al. [6].) Problems 433 6. Convolutional network code In the following network, maxflow(s, tl ) = 3 for l = 1, 2, 3. The max-flow bound asserts that 3 bits can be multicast to t0 u2 v2 s t1 v0 u0 t2 v1 u1 all the three sink nodes per unit time. We now describe a network coding scheme which achieve this. Let 3 bits b0 (k), b1 (k), b2 (k) be generated at node s at time k = 1, 2, , where we assume without loss of generality that bl (k) is an element of the finite field GF (2). We adopt the convention that bl (k) = 0 for k 0. At time k 1, information transactions T1 to T11 occur in the following order: T1. s sends bl (k) to vl , l = 0, 1, 2 T2. vl sends bl (k) to ul , tl1 , and tl2 , l = 0, 1, 2 T3. u0 sends b0 (k) + b1 (k - 1) + b2 (k - 1) to u1 T4. u1 sends b0 (k) + b1 (k - 1) + b2 (k - 1) to t2 T5. u1 sends b0 (k) + b1 (k) + b2 (k - 1) to u2 T6. u2 sends b0 (k) + b1 (k) + b2 (k - 1) to t0 T7. u2 sends b0 (k) + b1 (k) + b2 (k) to u0 T8. u0 sends b0 (k) + b1 (k) + b2 (k) to t1 T9. t2 decodes b2 (k - 1) T10. t0 decodes b0 (k) T11. t1 decodes b1 (k) where "" denotes modulo 3 addition and "+" denotes modulo 2 addition. a) Show that the information transactions T1 to T11 can be performed at time k = 1. b) Show that T1 to T11 can be performed at any time k 1 by induction on k. c) Verify that at time k, nodes t0 and t1 can recover b0 (k ), b1 (k ), and b2 (k ) for all k k. 434 18 The Max-Flow Bound d) Verify that at time k, node t2 can recover b0 (k ) and b1 (k ) for all k k, and b2 (k ) for all k k - 1. Note the unit time delay for t2 to recover b2 (k). (Ahlswede et al. [6].) Historical Notes The max-flow bound presented in this chapter was proved by Ahlswede et al. [6], where the point-to-point channels in the network are noiseless. The max-flow bound can be established when the point-to-point channels in the network are discrete memoryless channels. Borade [46] proved the bound with the assumptions that the channels are independent of each other and that the transmissions in the channels are synchronous. Song et al. [342] proved the bound without the latter assumption. These results are network generalizations of the result by Shannon [326] asserting that the capacity of a discrete memoryless channel is not increased by feedback (see Section 7.6), and they imply the asymptotic optimality of separating network coding and channel coding under the corresponding assumptions. 19 Single-Source Linear Network Coding: Acyclic Networks In the last chapter, we have established the max-flow bound as the fundamental bound for multicasting a single information source in a point-to-point communication network. In the next two chapters, we will construct linear network codes that achieve the max-flow bound at various levels of generality. A finite field is a system of symbols on which one can perform operations corresponding to the four operations in arithmetic for real numbers, namely addition, subtraction, multiplication, and division. The set of real numbers together with the associated operations are referred to as the field of real numbers, or simply the real field. Unlike the real field that has an infinite number of elements, a finite field has only a finite number of elements. For finite field theory, we refer the reader to [264]. For our discussions here, since we will not make use of the detailed structural properties of a finite field, the reader may by and large regard the algebra on a finite field and the algebra on the real field as the same. In a linear network code, all the information symbols are regarded as elements of a finite field F called the base field. These include the symbols that comprise the information source as well as the symbols transmitted on the channels. For example, F is taken to be the binary field GF (2) when the information unit is the bit. Furthermore, encoding and decoding are based on linear algebra defined on the based field, so that efficient algorithms for encoding and decoding as well as for code construction can be obtained. In this chapter, we consider acyclic networks, i.e., networks with no directed cycle. We study the network coding problem in which a message consisting of a finite block of symbols is multicast. We make the ideal assumption that the propagation delay in the network, which includes the processing delay at the nodes and the transmission delay over the channels, is zero. In a general setting, a pipeline of messages may be multicast, and the propagation delay may be non-negligible. If the network is acyclic, then the operations in the network can be so synchronized that sequential messages are processed independent of each other. In this way, the network coding problem is inde- 436 19 Single-Source Linear Network Coding: Acyclic Networks pendent of the propagation delay. Therefore, it suffices to study the network coding problem as described. On the other hand, when a network contains directed cycles, the processing and transmission of sequential messages can convolve with together. Then the amount of delay incurred becomes part of the consideration in network coding. This will be discussed in the next chapter. 19.1 Acyclic Networks Denote a directed network by G = (V, E), where V and E are the sets of nodes and channels, respectively. A pair of channels (d, e) E E is called an adjacent pair if there exists a node t V such that d In(t) and e Out(t). A directed path in G is a sequence of channels e1 , e2 , , em (19.1) such that (ei , ei+1 ) is an adjacent pair for all 1 i < m. Let e1 Out(t) and em In(t ). The sequence in (19.1) is called a directed path from e1 to em , or equivalently, a directed path from node t to node t . If t = t , then the directed path is called a directed cycle. A directed network G is cyclic if it contains a directed cycle, otherwise G is acyclic. Acyclic networks are easier to handle because the nodes in the network can be ordered in a way which allows encoding at the nodes to be carried out in a sequential and consistent manner. The following proposition and its proof describe such an order. Proposition 19.1. If G is a finite directed acyclic graph, then it is possible to order the nodes of G in a sequence such that if there is an edge from node i to node j, then node i appears before node j in the sequence. Proof. We partition the set V into subsets V1 , V2 , , such that node i is in Vk if and only if the length of a longest directed path ending at node i is equal to k. We first prove that if node i is in Vk and node j is in Vk such that there exists a directed path from node i to node j, then k < k. Since the length of a longest directed path ending at node i is equal to k and there exists a directed path from node i to node j (with length at least equal to 1), there exists a directed path ending at node j with length equal to k + 1. As node j is in Vk , we have k + 1 k, (19.2) so that k < k. (19.3) Hence, by listing the nodes of G in a sequence such that the nodes in Vk appear before the nodes in Vk if k < k, where the order of the nodes within 19.2 Linear Network Codes 437 each Vk is arbitrary, we obtain an order of the nodes of G with the desired property. Following the direction of the edges, we will refer to an order prescribed by Proposition 19.1 as an upstream-to-downstream order1 . For a given acyclic network, such an order (not unique) is implicitly assumed. The nodes in the network encodes according to this order, referred to as the encoding order. Then whenever a node encodes, all the information needed would have already been received on the input channels of that node. Example 19.2. Consider ordering the nodes in the butterfly network in Figure 17.1 by the sequence s, 2, 1, 3, 4, t2 , t1 . (19.4) It is easy to check that in this sequence, if there is a directed path from node i to node j, then node i appears before node j. 19.2 Linear Network Codes In this section, we formulate a linear network code on an acyclic network G. By allowing parallel channels between a pair of nodes, we assume without loss of generality that all the channels in the network have unit capacity, i.e., one symbol in the base field F can be transmitted on each channel. There exists a unique node s in G, called the source node, where a message consisting of symbols taken from the base field F is generated. To avoid trivially, we assume that every non-source node has at least one input channel. As in Section 18.3, we assume that there is no loop in G, and there is no input channel at node s. To facilitate our discussion, however, we let In(s) be a set of imaginary channels that terminate at node s but have no originating nodes. The reader may think of the symbols forming the message as being received by source node s on these imaginary channels. We emphasize that these imaginary channels are not part of the network, and the number of these channels is context dependent. Figure 19.1(a) illustrates the butterfly network with = 2 imaginary channels appended at source node s. Two directed paths P1 and P2 in G are edge-disjoint if the two paths do not share a common channel. It is not difficult to see from the conservation conditions in (18.4) that for a non-source node t, the maximum number of edge-disjoint paths from node s to node t is equal to maxflow(t). The message generated at source node s, consisting of symbols in the base field F , is represented by a row -vector x F . Based on the value of x, source node s transmits a symbol over each output channel. Encoding at the nodes in the network is carried out according to a certain upstream-todownstream order. At a node in the network, the ensemble of received symbols 1 Also called an ancestral order in graph theory. 438 19 Single-Source Linear Network Coding: Acyclic Networks b1 s t u t b1 w b1 w b1 + b2 b1 + b2 z y x b1 + b2 z b2 b2 b1 s b2 b2 u x y (a) (b) Fig. 19.1. (a) Two imaginary channels are appended to the source node of the butterfly network. (b) A 2-dimensional network code for the butterfly network. is mapped to a symbol in F specific to each output channel, and the symbol is sent on that channel. The following definition of a network code formally describes this mechanism. Since the code so defined is not necessarily linear, the base field F can be regarded in this context as any finite alphabet. Definition 19.3 (Local Description of a Network Code). An -dimensional network code on an acyclic network over a base field F consists of a local encoding mapping ~ ke : F |In(t)| F (19.5) for every channel e in the network, where e Out(t). With the encoding mechanism as described, the local encoding mappings derive recursively the symbols transmitted over all channels e, denoted by ~ fe (x). The above definition of a network code does not explicitly give the ~ values of fe (x), whose mathematical properties are at the focus of the present discussion. Therefore, we also present an equivalent definition below, which describes a network code in terms of both the local encoding mechanisms as ~ well as the recursively derived values fe (x). Definition 19.4 (Global Description of a Network Code). An dimensional network code on an acyclic network over a base field F consists of a local encoding mapping ~ ke : F |In(t)| F and a global encoding mapping (19.6) 19.2 Linear Network Codes 439 ~ fe : F F (19.7) for each channel e in the network, where e Out(t), such that: ~ (19.8) For every node t and every channel e Out(t), fe (x) is uniquely de~ ~ termined by (fd (x) : d In(t)) via the local encoding mapping ke . ~ (19.9) The mappings fe for the imaginary channels e In(s) project F onto the distinct dimensions of F . Example 19.5. Let x = [ b1 b2 ] denote a generic row vector in GF (2)2 . Figure 19.1(b) shows a 2-dimensional binary network code for the butterfly network with the following global encoding mappings: ~ fe (x) = b1 ~ fe (x) = b2 ~ fe (x) = b1 + b2 for e = (o, s), (s, t), (t, w), (t, y) for e = (o, s) , (s, u), (u, w), (u, z) for e = (w, x), (x, y), (x, z), (19.10) (19.11) (19.12) where (o, s) and (o, s) denote the two imaginary channels at node s. The corresponding local encoding mappings are ~ ~ k(s,t) (b1 , b2 ) = b1 , k(s,u) (b1 , b2 ) = b2 , ~ ~ k(t,w) (b1 ) = k(t,y) (b1 ) = b1 , ~ ~ ~ k(u,w) (b2 ) = k(u,z) (b2 ) = b2 , k(w,x) (b1 , b2 ) = b1 + b2 , etc. ~ When a global encoding mapping fe is linear, it corresponds to a column ~ -vector fe such that fe (x) is the product x fe , where the row -vector x is the message generated at node s. Similarly, when a local encoding mapping ~ ke , where e Out(t), is linear, it corresponds to a column |In(t)|-vector ke ~ such that ke (y) = y ke , where y F |In(t)| is the row vector representing the symbols received at node t. In an -dimensional network code on an acyclic network, if all the local encoding mappings are linear, then so are the global encoding mappings since they are functional compositions of the local encoding mappings. The converse is also true: If the global encoding mappings are all linear, then so are the local encoding mappings. We leave the proof as an exercise. In the following, we formulate a linear network code as a network code whose local and global encoding mappings are all linear. Again, both the local and global descriptions are presented even though they are equivalent. The global description of a linear network code will be very useful when we construct such codes in Section 19.4. Definition 19.6 (Local Description of a Linear Network Code). An -dimensional linear network code on an acyclic network over a base field F consists of a scalar kd,e , called the local encoding kernel, for every adjacent pair of channels (d, e) in the network. The |In(t)| |Out(t)| matrix (19.13) (19.14) (19.15) 440 19 Single-Source Linear Network Coding: Acyclic Networks Kt = [kd,e ]dIn(t),eOut(t) is called the local encoding kernel at node t. (19.16) Note that the matrix structure of Kt implicitly assumes an ordering among the channels. Definition 19.7 (Global Description of a Linear Network Code). An -dimensional linear network code on an acyclic network over a base field F consists of a scalar kd,e for every adjacent pair of channels (d, e) in the network as well as a column -vector fe for every channel e such that: (19.17) fe = dIn(t) kd,e fd for e Out(t). (19.18) The vectors fe for the imaginary channels e In(s) form the standard basis of the vector space F . The vector fe is called the global encoding kernel for channel e. We now explain how the global description above specifies the linear network code. Initially, source node s generates a message x as a row -vector. In view of (19.18), the symbols in x are regarded as being received by source node s on the imaginary channels as xfd , d In(s). Starting at source node s, any node t in the network receives the symbols x fd , d In(t), from which it calculates the symbol x fe for sending on each channel e Out(t) via the linear formula x fe = x dIn(t) kd,e fd = dIn(t) kd,e (x fd ), (19.19) where the first equality follows from (19.17). In this way, the symbol x fe is transmitted on any channel e (which may be an imaginary channel) in the network. Given the local encoding kernels for all the channels in an acyclic network, the global encoding kernels can be calculated recursively in any upstream-todownstream order by (19.17), while (19.18) provides the boundary conditions. Remark A partial analogy can be drawn between the global encoding kernels for the channels in a linear network code and the columns of a generator matrix of a linear block code in algebraic coding theory [234][39][378]. The former are indexed by the channels in the network, while the latter are indexed by "time." However, the global encoding kernels in a linear network code are constrained by the network topology via (19.17), while the columns in the generator matrix of a linear block code in general are not subject to any such constraint. The following two examples illustrate the relation between the local encoding kernels and the global encoding kernels of a linear network code. The reader should understand these two examples thoroughly before proceeding to the next section. 19.2 Linear Network Codes f(o,s) = (1 % &0 # ' $ 441 f(o,s)) = (0% &1 # ' $ (1 0% s Kt = !1 1" t Kw = &1 # ' $ (1 % Ks = &0 1# ' $ (0 % &1# ' $ (1% &0 # ' $ Ku= !1 1" u (1% &0 # ' $ (1% &1# '$ (1% &0 # ' $ w (1% &1# '$ (0 % &1# ' $ (0 % &1# ' $ (1% &1# '$ x Kx = !1 1" y z Fig. 19.2. The global and local encoding kernels for the 2-dimensional linear network code in Example 19.8. Example 19.8. The network code in Figure 19.1(b) is in fact linear. Assume the alphabetical order among the channels (o, s), (o, s) , (s, t), , (x, z). Then the local encoding kernels at the nodes are: Ks = 10 1 , Kt = Ku = Kx = 1 1 , Kw = . 01 1 (19.20) The corresponding global encoding kernels are: 1 0 for e = (o, s), (s, t), (t, w), and (t, y) 0 for e = (o, s) , (s, u), (u, w), and (u, z) fe = 1 1 for e = (w, x), (x, y), and (x, z). 1 (19.21) The local/global encoding kernels are summarized in Figure 19.2. In fact, they describe a 2-dimensional linear network code regardless of the choice of the base field. Example 19.9. For a general 2-dimensional linear network code on the network in Figure 19.2, the local encoding kernels at the nodes can be expressed as Ks = ac , Kt = e f , Ku = g h , bd Kw = i , Kx = k l , j (19.22) (19.23) 442 19 Single-Source Linear Network Coding: Acyclic Networks where a, b, c, , l, the entries of the matrices, are indeterminates in the base field F . Starting with f(o,s) = 1 0 and f(o,s) = 0 , 1 (19.24) we can obtain all the global encoding kernels below by applying (19.17) recursively: f(s,t) = a c ae af , f(s,u) = , f(t,w) = , f(t,y) = , b d be bf cg ch aei + cgj , f(u,z) = , f(w,x) = , dg dh bei + dgj aeik + cgjk aeil + cgjl , f(x,z) = . beik + dgjk beil + dgjl (19.25) (19.26) (19.27) f(u,w) = f(x,y) = For example, f(w,x) is obtained from f(t,w) and f(u,w) by f(w,x) = k(t,w),(w,x) f(t,w) + k(u,w),(w,x) f(u,w) =i = ae cg +j be dg aei + cgj . bei + dgj (19.28) (19.29) (19.30) The local/global encoding kernels of the general linear network code are summarized in Figure 19.3. 19.3 Desirable Properties of a Linear Network Code We have proved in Section 18.4 that in a communication network represented by a graph G, the rate at which information is transmitted from source node s to any node t cannot exceed maxflow(t), the value of a max-flow from node s to node t. For a collection of non-source nodes T , denote by maxflow(T ) the value of a max-flow from node s to T . Then it is readily seen that the rate at which information is transmitted from source node s to the collection of nodes T cannot exceed maxflow(T ). In the sequel, we adopt the conventional notation for the linear span of a set of vectors. For a node t, let Vt = {fe : e In(t)} and for a collection T of nodes, let VT = tT Vt . (19.32) (19.31) 19.3 Desirable Properties of a Linear Network Code f(o,s) = (1 % &0 # ' $ 443 f(o,s)* = (0% &1 # ' $ (a c% s Kt = !e t Kw = & j # ' $ (ae% &be # ' $ (i% Ks = &b d # ' $ (c % &d # ' $ f " (a % &b # ' $ Ku= !g u h" w ( aei ) cgj % &bei ) dgj # ' $ (af % &bf # ' $ ( cg % &dg # ' $ ( ch % &dh# ' $ ( aeik ) cgjk % &beik ) dgjk # $ ' x Kx = !k l" ( aeil ) cgjl % &beil ) dgjl # ' $ y z Fig. 19.3. Local/global encoding kernels of a general 2-dimensional linear network code. For a collection of channels, let V = {fe : e } , (19.33) with the convention V = {0}, where 0 denotes the zero column -vector. In the next theorem, we first establish a specific form of the max-flow bound which applies to linear network coding. Theorem 19.10 (Max-Flow Bound for Linear Network Coding). For an -dimensional linear network code on an acyclic network, for any collection T of non-source nodes, dim(VT ) min{, maxflow(T )}. (19.34) Proof. Let the acyclic network be G = (V, E). Consider a cut U between source node s and a collection T of non-source nodes, and let EU be the set of edges across the cut U as in (18.7). Then VT is a linear transformation of VEU , where dim(VT ) dim(VEU ) |EU | (19.35) Minimizing over all the cuts between s and T and invoking the max-flow min-cut theorem, we have dim(VT ) maxflow(T ). (19.36) On the other hand, VT is a linear transformation of Vs = F . Therefore, 444 19 Single-Source Linear Network Coding: Acyclic Networks dim(VT ) dim(Vs ) = . Then the proof is completed by combining (19.36) and (19.37). (19.37) For a collection of channels E (i.e., not including the imaginary channels), we denote by maxflow() the value of a max-flow from source node s to . Theorem 19.10 has the following straightforward corollary. Corollary 19.11. For an -dimensional linear network code on an acyclic network, for any collection of channels E, dim(V ) min{, maxflow()}. (19.38) Whether the max-flow bound in Theorem 19.10 or Corollary 19.11 is achievable depends on the network topology, the dimension , and the coding scheme. Three special classes of linear network codes are defined below by the achievement of this bound to three different extents. Definition 19.12. An -dimensional linear network code on an acyclic network qualifies as a linear multicast, a linear broadcast, or a linear dispersion, respectively, if the following hold: (19.39) dim(Vt ) = for every non-source node t with maxflow(t) . (19.40) dim(Vt ) = min{, maxflow(t)} for every non-source node t. (19.41) dim (VT ) = min{, maxflow(T )} for every collection T of non-source nodes. For a set of channels, including possibly the imaginary channels, let F = fe e (19.42) be the || matrix obtained by putting fe , e in juxtaposition. For a node t, the symbols x fe , e In(t) are received on the input channels. Equivalently, the row |In(t)|-vector x FIn(t) (19.43) is received. Obviously, the message x, consisting of information units, can be uniquely determined at the node if and only if the rank of FIn(t) is equal to , i.e., dim(Vt ) = . (19.44) The same applies to a collection T of non-source nodes. For a linear multicast, a node t can decode the message x if and only if maxflow(t) . For a node t with maxflow(t) < , nothing is guaranteed. An application of an -dimensional linear multicast is for multicasting information at rate to all (or some of) those non-source nodes with max-flow at least equal to . 19.3 Desirable Properties of a Linear Network Code 445 For a linear broadcast, like a linear multicast, a node t can decode the message x if and only if maxflow(t) . For a node t with maxflow(t) < , the set of all received vectors, namely {x FIn(t) : x F }, (19.45) form a vector subspace of F with dimension equal to maxflow(t), but there is no guarantee on which such subspace is actually received2 . An application of linear broadcast is for multicasting information on a network at a variable rate (see Problem 14). A random version of linear broadcast (to be discussed in Section 19.4) is also useful for identifying the max-flow of a non-source in an unknown network topology [352]. For a linear dispersion, a collection T of non-source nodes can decode the message x if and only if maxflow(T ) . If maxflow(T ) < , the collection T receives a vector subspace with dimension equal to maxflow(T ). Again, there is no guarantee on which such subspace is actually received. An application of linear dispersion is in a two-tier network system consisting of the backbone network and a number of local area networks (LANs), where each LAN is connected to one or more nodes on the backbone network. An information source with rate , generated at a node s in the backbone network, is to be transmitted to every user on the LANs. With a linear dispersion on the backbone network, every user on a LAN can receive the information source as long as the LAN acquires through the backbone network an aggregated max-flow from node s at least equal to . Moreover, new LANs can be established under the same criterion without modifying the linear dispersion on the backbone network. Note that for all the three classes of linear network codes in Definition 19.12, a sink node is not explicitly identified. Also, it is immediate from the definition that every linear dispersion is a linear broadcast, and every linear broadcast is a linear multicast. The example below shows that a linear broadcast is not necessarily a linear dispersion, a linear multicast is not necessarily a linear broadcast, and a linear network code is not necessarily a linear multicast. Example 19.13. Figure 19.4(a) shows a 2-dimensional linear dispersion on an acyclic network with the global encoding kernels as prescribed. Figure 19.4(b) shows a 2-dimensional linear broadcast on the same network that is not a linear dispersion because maxflow({t, u}) = 2 = , (19.46) while the global encoding kernels of the channels in In(t)In(u) span only a 1dimensional subspace. Figure 19.4(c) shows a 2-dimensional linear multicast that is not a linear broadcast since node u receives no information at all. Finally, the 2-dimensional linear network code in Figure 19.4(d) is not a linear multicast. 2 Here F refers to the row vector space. 446 19 Single-Source Linear Network Coding: Acyclic Networks &1# $ ! %0 " s &0 # $0 ! % " &0 # $ ! %1" &1# $0 ! % " s &0 # $1! % " &1# $0 ! % " t &1# $0 ! % " u &0 # $1! % " t &1# $0 ! % " u &1# $0 ! % " w (a) w (b) &1# $ ! %0 " s &0 # $1! % " &0 # $ ! %0 " &1# $0 ! % " s &1# $0 ! % " t &1# $ ! %0 " u &0 # $ ! %0 " t &0 # $0 ! % " &0 # $1! % " u &0 # $0 ! % " w (c) w (d) Fig. 19.4. (a) A 2-dimensional linear dispersion over an acyclic network. (b) A 2-dimensional linear broadcast that is not a linear dispersion. (c) A 2-dimensional linear multicast that is not a linear broadcast. (d) A 2-dimensional linear network code that is not a linear multicast. Example 19.14. The linear network code in Example 19.8 meets all the criteria (19.39) through (19.41) in Definition 19.12. Thus it is a 2-dimensional linear dispersion, and hence also a linear broadcast and linear multicast, regardless of the choice of the base field. The same applies to the linear network code in Figure 19.4(a). Example 19.15. The general linear network code in Example 19.9 meets the criterion (19.39) for a linear multicast when f(t,w) and f(u,w) are linearly independent; f(t,y) and f(x,y) are linearly independent; f(u,z) and f(x,z) are linearly independent. Equivalently, the criterion says that e, f, g, h, k, l, ad - bc, abei + adgj - baei - bcgj, and daei + dcgj - cbei - cdgj are all nonzero. Example 19.8 has been the special case with a=d=e=f =g=h=i=j=k=l=1 and b = c = 0. (19.48) (19.47) 19.3 Desirable Properties of a Linear Network Code 447 Transformation of a Linear Network Code Consider an -dimensional linear network code C on an acyclic network. Suppose source node s, instead of encoding the message x, encodes x = xA, (19.49) where A is an invertible matrix. Then the symbol sent on a channel e E is given by x fe = (xA) fe = x (A fe ). (19.50) This gives a new linear network code C with respect to the message x with global encoding kernels fe = A fe if e E fe if e In(s). (19.51) Recall the definition of the matrix F in (19.42) for a set of channels . Then (19.17) can be written in matrix form as FOut(t) = FIn(t) Kt (19.52) for all nodes t, where Kt is the local encoding kernel at node t. Similarly, letting F = fe e , (19.53) we obtain from (19.51) that FOut(t) = AFOut(t) for all nodes t, FIn(t) = AFIn(t) for all nodes t = s, and FIn(s) = FIn(s) . For a node t = s, from (19.54), (19.52), and (19.55), FOut(t) = AFOut(t) = A(FIn(t) Kt ) = (AFIn(t) )Kt = FIn(t) Kt . Since fe , e In(s) form the standard basis of F , FIn(s) = FIn(s) = I, (19.61) (19.57) (19.58) (19.59) (19.60) (19.56) (19.55) (19.54) the identity matrix. It then follows from (19.54), (19.52), and (19.61) that 448 19 Single-Source Linear Network Coding: Acyclic Networks FOut(s) = AFOut(s) = A(FIn(s) Ks ) = AKs = FIn(s) (AKs ). (19.62) (19.63) (19.64) (19.65) Comparing (19.60) and (19.65) with (19.52), we see that the local encoding kernels of C are given by Kt = Kt if t = s AKs if t = s. (19.66) The network code C is called the transformation of the network code C by the (invertible) matrix A. In view of Definition 19.12, the requirements of a linear multicast, a linear broadcast, and a linear dispersion are all in terms of the linear independence among the global encoding kernels. We leave it as an exercise for the reader to show that if a network code is a linear multicast, broadcast, or dispersion, then any transformation of it is also a linear multicast, broadcast, or dispersion, respectively. Suppose C is an -dimensional linear multicast and let C be a transformation of C. When the network code C is employed, the message x can be decoded by any node t with maxflow(t) , because from the foregoing C is also a linear multicast. For the purpose of multicasting, there is no difference between C and C , and they can be regarded as equivalent. If C is an -dimensional linear broadcast and C is a transformation of C, then C is also an -dimensional linear broadcast. However, C as a linear broadcast may deliver to a particular node t with maxflow(t) < a certain subset of symbols in the message x, while C may not be able to achieve the same. Then whether C and C can be regarded as equivalent depends on the specific requirements of the application. As an example, the linear network code in Figure 19.1(b) delivers b1 to node t. However, taking a transformation of the network code with matrix A= 10 , 11 (19.67) the resulting network code can no longer deliver b1 to node t, although nodes w, v, and z can continue to decode both b1 and b2 . Implementation of a Linear Network Code In implementation of a linear network code, be it a linear multicast, a linear broadcast, a linear dispersion, or any linear network code, in order that the code can be used as intended, the global encoding kernels fe , e In(t) must be known by each node t if node t is to recover any useful information from the symbols received on the input channels. These global encoding kernels can be made available ahead of time if the code is already decided. Alternatively, they 19.4 Existence and Construction 449 can be delivered through the input channels if multiple usage of the network is allowed. One possible way to deliver the global encoding kernels to node t in a coding session of length n, where n > , is as follows. At time k = 1, 2, , , the source node transmits the dummy message mk , a row -vector with all the components equal to 0 except that the kth component is equal to 1. Note that m1 m2 (19.68) . = I , . . m the identity matrix. At time k = + i, where i = 1, 2, , n - , the source node transmits the message xi . Then throughout the coding session, node t receives m1 m2 I FIn(t) . . x1 x1 FIn(t) . m x2 FIn(t) = (19.69) FIn(t) = x2 FIn(t) x1 . . . . . . x2 xn- FIn(t) xn- . . . xn- on the input channels. In other words, the global encoding kernels of the input channels at node t are received at the beginning of the coding session. This applies to all the sink nodes in the network simultaneously because the dummy messages do not depend on the particular node t. If FIn(t) has full rank, then node t can start to decode x1 upon receiving x1 FIn(t) . Since n - messages are transmitted in a coding session of length n, the utilization of the network is equal to (n - )/n, which tends to 1 as n . That is, the overhead for delivering the global encoding kernels through the network is asymptotically negligible. 19.4 Existence and Construction For a given acyclic network, the following three factors dictate the existence of an -dimensional linear network code with a prescribed set of desirable properties: the value of , the network topology, the choice of the base field F . 450 19 Single-Source Linear Network Coding: Acyclic Networks & y1 # $z ! % 1" s & y4 # $z ! % 4" & y3 # $z ! % 3" u1 u2 & y2 # $z ! % 2" u3 u4 t1 t2 t3 t4 t5 t6 Fig. 19.5. A network with a 2-dimensional ternary linear multicast but without a 2-dimensional binary linear multicast. We begin with an example illustrating the third factor. Example 19.16. On the network in Figure 19.5, a 2-dimensional ternary linear multicast can be constructed by the following local encoding kernels at the nodes: 0111 Ks = and Kui = 1 1 1 (19.70) 1012 for 1 i 4. On the other hand, we can prove the nonexistence of a 2dimensional binary linear multicast on this network as follows. Assuming the contrary that a 2-dimensional binary linear multicast exists, we will derive a contradiction. Let the global encoding kernel f(s,ui ) = [ yi zi ] for 1 i 4. Since maxflow(tk ) = 2 for all 1 k 6, the global encoding kernels for the two input channels to each node tk must be linearly independent. Thus, if node tk is at the downstream of both nodes ui and uj , then the two vectors [ yi zi ] and [ yi zi ] must be linearly independent. As each node tk is at the downstream of a different pair of nodes among u1 , u2 , u3 , and u4 , the four vectors [ yi zi ] , 1 i 4, are pairwise linearly independent, and consequently, must be four distinct vectors in GF (2)2 . Then one of them must be [ 0 0 ] since there are only four vectors in GF (2)2 . This contradicts the pairwise linear independence among the four vectors. In order for the linear network code to qualify as a linear multicast, a linear broadcast, or a linear dispersion, it is required that certain collections of global encoding kernels span the maximum possible dimensions. This is equivalent to certain polynomial functions taking nonzero values, where the indeterminates of these polynomials are the local encoding kernels. To fix ideas, take = 3, consider a node t with two input channels, and put the 19.4 Existence and Construction 451 global encoding kernels of these two channels in juxtaposition to form a 3 2 matrix. Then, this matrix attains the maximum possible rank of 2 if and only if there exists a 2 2 submatrix with nonzero determinant. According to the local description, a linear network code is specified by the local encoding kernels, and the global encoding kernels can be derived recursively in an upstream-to-downstream order. From Example 19.8, it is not hard to see that every component in a global encoding kernel is a polynomial function whose indeterminates are the local encoding kernels. When a nonzero value of a polynomial function is required, it does not merely mean that at least one coefficient in the polynomial is nonzero. Rather, it means a way to choose scalar values for the indeterminates so that the polynomial function is evaluated to a nonzero scalar value. When the base field is small, certain polynomial equations may be unavoidable. For instance, for any prime number p, the polynomial equation z p - z = 0 is satisfied for any z GF (p). The nonexistence of a binary linear multicast in Example 19.16 can also trace its root to a set of polynomial equations that cannot be avoided simultaneously over GF (2). However, when the base field is sufficiently large, every nonzero polynomial function can indeed be evaluated to a nonzero value with a proper choice of the values taken by the set of indeterminates involved. This is formally stated in the following elementary lemma, which will be instrumental in the proof of Theorem 19.20 asserting the existence of a linear multicast on an acyclic network when the base field is sufficiently large. Lemma 19.17. Let g(z1 , z2 , , zn ) be a nonzero polynomial with coefficients in a field F . If |F | is greater than the degree of g in every zj for 1 j n, then there exist a1 , a2 , , an F such that g(a1 , a2 , , an ) = 0. (19.71) Proof. The lemma is proved by induction on n. For n = 0, g is a nonzero constant in F , and the lemma is obviously true. Assume that the lemma is true for n - 1 for some n 1. Express g(z1 , z2 , , zn ) as a polynomial in zn with coefficients in the polynomial ring F [z1 , z2 , , zn-1 ], i.e., g(z1 , z2 , , zn ) = h(z1 , z2 , , zn-1 )zn k + , (19.72) where k is the degree of g in zn and the leading coefficient h(z1 , z2 , , zn-1 ) is a nonzero polynomial in F [z1 , z2 , , zn-1 ]. By the induction hypothesis, there exist a1 , a2 , , an-1 F such that h(a1 , a2 , , an-1 ) = 0. Thus g(a1 , a2 , , an-1 , z) is a nonzero polynomial in z with degree k < |F |. Since this polynomial cannot have more than k roots in F and |F | > k, there exists an F such that g(a1 , a2 , , an-1 , an ) = 0. (19.73) Corollary 19.18. Let g(z1 , z2 , , zn ) be a nonzero polynomial with coefficients in a field F with |F | > m, where m is the highest degree of g in zj for 452 19 Single-Source Linear Network Coding: Acyclic Networks 1 j n. Let a1 , a2 , , an be chosen independently according to the uniform distribution on F . Then Pr{g(a1 , a2 , , an ) = 0} In particular, Pr{g(a1 , a2 , , an ) = 0} 1 as |F | . Proof. The first part of the corollary is proved by induction on n. For n = 0, g is a nonzero constant in F , and the proposition is obviously true. Assume that the proposition is true for n - 1 for some n 1. From (19.72) and the induction hypothesis, we see that Pr{g(z1 , z2 , , zn ) = 0} = Pr{h(z1 , z2 , , zn-1 ) = 0} Pr{g(z1 , z2 , , zn ) = 0| h(z1 , z2 , , zn-1 ) = 0} m Pr{g(z1 , z2 , , zn ) = 0| |F | h(z1 , z2 , , zn-1 ) = 0} 1- m |F | m 1- |F | 1- n-1 n-1 1- m |F | n . (19.74) (19.75) (19.76) (19.77) (19.78) (19.79) (19.80) (19.81) (19.82) = 1- n m |F | . This proves the first part of the corollary. As n is fixed, the lower bound above tends to 1 as |F | . This completes the proof. Example 19.19. Recall the 2-dimensional linear network code in Example 19.9 that is expressed in the 12 indeterminates a, b, c, , l. Place the vectors f(t,w) and f(u,w) in juxtaposition into the 2 2 matrix Lw = ae cg , be dg (19.83) the vectors f(t,y) and f(x,y) into the 2 2 matrix Ly = af aeik + cgjk , bf beik + dgjk (19.84) and the vectors f(u,z) and f(x,z) into the 2 2 matrix LZ = aeil + cgjl ch . beil + dgjl dh (19.85) 19.4 Existence and Construction 453 Clearly, det(Lw ) det(Ly ) det(Lz ) = 0 F [a, b, c, , l]. (19.86) Applying Lemma 19.17 to the polynomial on the left hand side above, we can set scalar values for the 12 indeterminates so that it is evaluated to a nonzero value in F when F is sufficiently large. This implies that the determinants on the left hand side of (19.86) are evaluated to nonzero values in F simultaneously. Thus these scalar values yield a 2-dimensional linear multicast. In fact, det(Lw ) det(Ly ) det(Lz ) = 1 (19.87) when b=c=0 and a = d = e = f = = l = 1. (19.89) Therefore, the 2-dimensional linear network code depicted in Figure 19.2 is a linear multicast, and this fact is regardless of the choice of the base field F . Theorem 19.20. There exists an -dimensional linear multicast on an acyclic network for sufficiently large base field F . Proof. For a directed path P = e1 , e2 , , em , define KP = 1j<m (19.88) kej ,ej+1 . (19.90) Calculating by (19.17) recursively from the upstream channels to the downstream channels, it is not hard to find that fe = dIn(s) P : a path from d to e KP fd (19.91) for every channel e (see Example 19.23 below). Denote by F the polynomial ring over the field F with all the kd,e as indeterminates, where the total number of such indeterminates is equal to t |In(t)||Out(t)|. Thus, every component of every global encoding kernel belongs to F . The subsequent arguments in this proof actually depend on this fact alone but not on the exact form of (19.91). Let t be a non-source node with maxflow(t) . Then there exist edgedisjoint paths from the imaginary channels to distinct channels in In(t). Put the global encoding kernels of these channels in juxtaposition to form an matrix Lt . We will prove that det(Lt ) = 1 for properly set scalar values of the indeterminates. Toward proving this claim, we set (19.92) 454 19 Single-Source Linear Network Coding: Acyclic Networks kd,e = 1 (19.93) for all adjacent pairs of channels (d, e) along any one of the edge-disjoint paths, and set kd,e = 0 (19.94) otherwise. With such local encoding kernels, the symbols sent on the imaginary channels at source node s are routed to node t via the edge-disjoint paths. Thus the columns in Lt are simply the global encoding kernels of the imaginary channels, which form the standard basis of the space F . Then (19.92) follows, and the claim is proved. Consequently, det(Lt ) = 0 F , (19.95) i.e., det(Lt ) is a nonzero polynomial in the indeterminates kd,e . Since this conclusion applies to every non-source node t with maxflow(t) , det(Lt ) = 0 F . t:maxflow(t) (19.96) Applying Lemma 19.17 to the above polynomial when the field F is sufficiently large, we can set scalar values in F for the indeterminates so that det(Lt ) = 0 F, t:maxflow(t) (19.97) which in turns implies that det(Lt ) = 0 F (19.98) for all t such that maxflow(t) . These scalar values then yield a linear network code that meets the requirement (19.39) for a linear multicast. Corollary 19.21. There exists an -dimensional linear broadcast on an acyclic network for sufficiently large base field F . Proof. For every non-source node t in the given acyclic network, install a new node t and input channels to this new node, with min{, maxflow(t)} of them from node t and the remaining - min{, maxflow(t)} from source node s. This constructs a new acyclic network. Now consider an -dimensional linear multicast on the new network whose existence follows from Theorem 19.20. For every node t as described above, dim(Vt ) = because maxflow(t ) = . Moreover, since |In(t )| = , the global encoding kernels fe , e In(t ) are linearly independent. Therefore, dim( {fe : e In(t ) Out(t)} ) = |In(t ) Out(t)| = min{, maxflow(t)}. By (19.17), (19.99) (19.100) 19.4 Existence and Construction 455 {fe : e In(t ) Out(t)} Vt . Therefore, dim(Vt ) dim( {fe : e In(t ) Out(t)} ) = min{, maxflow(t)}. Then by invoking Theorem 19.10, we conclude that dim(Vt ) = min{, maxflow(t)}. (19.101) (19.102) (19.103) (19.104) In other words, an -dimensional linear multicast on the new network incorporates an -dimensional linear broadcast on the original network. Corollary 19.22. There exists an -dimensional linear dispersion on an acyclic network for sufficiently large base field F . Proof. For every nonempty collection T of non-source nodes in the given acyclic network, install a new node uT and maxflow(t) channels from every node t T to this new node. This constructs a new acyclic network with maxflow(uT ) = maxflow(T ) (19.105) for every T . Now consider an -dimensional linear broadcast on the new network whose existence follows from Corollary 19.21. By (19.17), VuT VT . Then dim(VT ) dim(VuT ) = min{, maxflow(uT )} = min{, maxflow(T )}. By invoking Theorem 19.10, we conclude that dim(VT ) = min{, maxflow(T )}. (19.110) (19.107) (19.108) (19.109) (19.106) In other words, an -dimensional linear broadcast on the new network incorporates an -dimensional linear dispersion on the original network. Example 19.23. We now illustrate the formula (19.91) in the proof of Theorem 19.20 with the 2-dimensional linear network code in Example 19.9 which is expressed in the 12 indeterminates a, b, c, , l. The local encoding kernels at the nodes are Ks = ac , Kt = e f , Ku = g h , bd (19.111) 456 19 Single-Source Linear Network Coding: Acyclic Networks Kw = i , Kx = k l . j (19.112) Starting with f(o,s) = [ 1 0 ] and f(o,s) = [ 0 1 ] , we can calculate the global encoding kernels by the formula (19.91). Take f(x,y) as the example. There are two paths from (o, s) to (x, y) and two from (o, s) to (x, y). For these paths, aeik for P = (o, s), (s, t), (t, w), (w, x), (x, y) beik for P = (o, s) , (s, t), (t, w), (w, x), (x, y) KP = (19.113) cgjk for P = (o, s), (s, u), (u, w), (w, x), (x, y) dgjk for P = (o, s) , (s, u), (u, w), (w, x), (x, y). Thus f(x,y) = (aeik)f(o,s) + (beik)f(o,s) + (cgjk)f(o,s) + (dgjk)f(o,s) = aeik + cgjk , beik + dgjk (19.114) (19.115) which is consistent with Example 19.9. The proof of Theorem 19.20 provides an algorithm for constructing a linear multicast that uses Lemma 19.17 as a subroutine to search for scalars a1 , a2 , , an F such that g(a1 , a2 , , an ) = 0 whenever g(z1 , z2 , , zn ) is a nonzero polynomial over a sufficiently large field F . The straightforward implementation of this subroutine is exhaustive search, which is generally computationally inefficient. Nevertheless, the proof of Theorem 19.20 renders a simple method to construct a linear multicast randomly. Corollary 19.24. Consider an -dimensional linear network code on an acyclic network. By choosing the local encoding kernels kd,e for all adjacent pairs of channels (d, e) independently according to the uniform distribution on the base field F , a linear multicast can be constructed with probability tends to 1 as |F | . Proof. This follows directly from Corollary 19.18 and the proof of Theorem 19.20. The technique described in the above theorem for constructing a linear network code is called random network coding. Random network coding has the advantage that the code construction can be done independent of the network topology, making it very useful when the network topology is unknown. A case study for an application of random network coding will be presented in Section 19.7. While random network coding offers a simple construction and more flexibility, a much larger base field is usually required. In some applications, it is necessary to verify that the code randomly constructed indeed possesses the desired properties. Such a task can be computationally non-trivial. 19.4 Existence and Construction 457 The next algorithm constructs a linear multicast deterministically in polynomial time. Unlike the algorithm given in the proof of Theorem 19.20 that assigns values to the local encoding kernels, this algorithm assigns values to the global encoding kernels. Algorithm 19.25 (Jaggi-Sanders Algorithm). This algorithm constructs an -dimensional linear multicast over a finite field F on an acyclic network when |F | > , the number of non-source nodes t in the network with maxflow(t) . Denote these non-source nodes by t1 , t2 , , t . A sequence of channels e1 , e2 , , el is called a path leading to a node tq if e1 In(s), el In(tq ), and (ej , ej+1 ) is an adjacent pair for all 1 j l - 1. For each q, 1 q , there exist edge-disjoint paths Pq,1 , Pq,2 , , Pq, leading to tq . All together there are such paths. The following procedure assigns a global encoding kernel fe for every channel e in the network in an upstream-to-downstream order such that dim(Vtq ) = for 1 q . { // By definition, the global encoding kernels of the // imaginary channels form the standard basis of F . for (q = 1; q ; q + +) for (i = 1; i ; i + +) eq,i = the imaginary channel initiating path Pq,i ; // This initializes eq,i . Subsequently, eq,i will be // dynamically updated by moving down path Pq,i // until it finally becomes a channel in In(tq ). for (every node t, in any upstream-to-downstream order) { for (every channel e Out(t)) { // With respect to this channel e, define a "pair" as a // pair (q, i) of indices such that channel e is on the // path Pq,i . Note that for each q, there exists at most // one pair (q, i). Thus the number of pairs is at least 0 // and at most . Since the nodes t are chosen in // an upstream-to-downstream order, if (q, i) is a pair, // then eq,i In(t) by induction, so that feq,i Vt . For // reasons to be explained in the algorithm verification // below, feq,i {feq,j : j = i} , and therefore // feq,i Vt \ {feq,j : j = i} . Choose a vector w in Vt such that w {feq,j : j = i} for / every pair (q, i); // To see the existence of such a vector w, let // dim(Vt ) = . Then, dim(Vt {feq,j : j = i} ) // - 1 for every pair (q, i) since // feq,i Vt \ {feq,j : j = i} . Thus 458 19 Single-Source Linear Network Coding: Acyclic Networks // |Vt ((q,i): a pair {feq,j : j = i} )| // |F |-1 < |F | = |Vt |. fe = w; // This is equivalent to choosing scalar values for local // encoding kernels kd,e for all d In(t) such that // dIn(t) kd,e fd {feq,j : j = i} for every pair (q, i). / for (every pair (q, i)) eq,i =e; } } } Algorithm Verification. For 1 q and 1 i , the channel eq,i is on the path Pq,i . Initially eq,i is an imaginary channel at source node s. Through dynamic updating, it moves downstream along the path until finally reaching a channel in In(tq ). Fix an index q, where 1 q . Initially, the vectors feq,1 , feq,2 , , feq, are linearly independent because they form the standard basis of F . At the end, they need to span the vector space F . Therefore, in order for the eventually constructed linear network code to qualify as a linear multicast, it suffices to show the preservation of the linear independence among feq,1 , feq,2 , , feq, throughout the algorithm. We need to show the preservation in the generic step inside the "for loop" for each channel e in the algorithm. The algorithm defines a "pair" as a pair (q, i) of indices such that channel e is on path Pq,i . When no (q, i) is a pair for 1 i , the channels eq,1 , eq,2 , , eq, are not changed in the generic step; neither are the vectors feq,1 , feq,2 , , feq, . So we only need to consider the scenario that a pair (q, i) exists for some i. The only change among the channels eq,1 , eq,2 , , eq, is that eq,i becomes e. Meanwhile, the only change among the vectors feq,1 , feq,2 , , feq, is that feq,i becomes a vector w {feq,j : j = i} . / (19.116) This preserves the linear independence among feq,1 , feq,2 , , feq, as desired. Complexity Analysis. There are a total of |E| channels in the network. In the algorithm, the generic step in the "for loop" for each channel e processes at most pairs. Throughout the algorithm, at most |E| such collections of channels are processed. From this, it is not hard to implement the algorithm within a polynomial time in |E| for a fixed . The computational details can be found in [184]. Remark 1 In the Jaggi-Sanders algorithm, all nodes t in the network with maxflow(t) serve as a sink node that receives the message x. The algorithm can easily be modified accordingly if only a subset of such nodes need 19.4 Existence and Construction 459 to serve as a sink node. In that case, the field size requirement is |F | > , where is the total number of sink nodes. Remark 2 It is not difficult to see from the lower bound on the required field size in the Jaggi-Sanders algorithm that if a field much larger than sufficient is used, then a linear multicast can be constructed with high probability by randomly choosing the global encoding kernels. Example 19.26 (Multi-Source Multicast). Consider a network coding problem on an acyclic network G with a set S of source nodes. At node s S, a message xs in the form of a row vector in F s is generated. Let = sS s (19.117) be the total dimension of all the messages, and let x = (xs : s S) (19.118) be referred to as the message. Here, we do not impose the constraint that a node s S has no input channels. Expand the network G into a network G by installing a new node 0, and s channels from node 0 to node s for each s S. Denote the value of a max-flow from node 0 to node t in G by maxflowG (t). Suppose there exists a coding scheme on G such that a node t can decode the message x. Such a coding scheme induces a coding scheme on G for which 1. the message x is generated at node 0; 2. for all s S, the message xs is sent uncoded from node 0 to node s through the s channels from node 0 to node s. Applying the max-flow bound to node t with respect to this coding scheme on G , we obtain maxflowG (t) . (19.119) Thus we have shown that if a node t in G can decode the message x, then (19.119) has to be satisfied. We now show that for a sufficiently large base field F , there exists a coding scheme on G such that a node t satisfying (19.119) can decode the message x. Let be the number of nodes in G that satisfies (19.119). To avoid triviality, assume 1. By Theorem 19.20, there exists an -dimensional linear multicast C on G when the base field is sufficiently large. From the proof of Theorem 19.10, we see that for this linear multicast, the matrix FOut(0) must be invertible, otherwise a node t satisfying (19.119) cannot possibly decode the message x. Transforming C by the matrix [FOut(0) ]-1 , we obtain from (19.54) a linear multicast C with FOut(0) = FOut(0) -1 FOut(0) = I . (19.120) 460 19 Single-Source Linear Network Coding: Acyclic Networks Accordingly, for this linear multicast, the message xs is sent uncoded from node 0 to node s for all s S. Thus a coding scheme on G with the message xs being generated at node s for all s S instead of being received from node 0 is naturally induced, and this coding scheme inherits from the linear multicast C that a node t satisfying (19.119) can decode the message x. Therefore, instead of tackling the multi-source multicast problem on G, we can tackle the single-source multicast problem on G . This has already seen illustrated in Examples 17.1 and 17.2 for the butterfly network. 19.5 Generic Network Codes In the last section, we have seen how to construct a linear multicast by the Jaggi-Sanders algorithm. In light of Corollaries 19.21 and 19.22, the same algorithm can be used for constructing a linear broadcast or a linear dispersion. It is not difficult to see that if the Jaggi-Sanders algorithm is used for constructing a linear broadcast, then the computational complexity of the algorithm remains polynomial in |E|, the total number of channels in the network. However, if the algorithm is used for constructing a linear dispersion, the computational complexity becomes exponential because the number of channels that need to be installed in constructing the new network in Corollary 19.22 grows exponentially with the number of channels in the original network. In this section, we introduce a class of linear network codes called generic network codes. As we will see, if a linear network code is generic, then it is a linear dispersion, and hence also a linear broadcast and a linear multicast. Toward the end of the section, we will present a polynomial-time algorithm that constructs a generic network code. Imagine that in an -dimensional linear network code, the base field F is replaced by the real field . Then arbitrary infinitesimal perturbation of the local encoding kernels would place the global encoding kernels at general positions with respect to one another in the space . General positions of the global encoding kernels maximize the dimensions of various linear spans by avoiding linear dependence in every conceivable way. The concepts of general positions and infinitesimal perturbation do not apply to the vector space F when F is a finite field. However, they can be emulated when F is sufficiently large with the effect of avoiding unnecessary linear dependence. The following definitions of a generic network code captures the notion of placing the global encoding kernels in general positions. In the sequel, for a channel ej E, let ej Out(tj ), and for a collection of channels = {e1 , e2 , , e|| } E, let k = \{ek }. Definition 19.27 (Generic Network Code I). An -dimensional linear network code on an acyclic network is generic if for any nonempty collection of channels = {e1 , e2 , , em } E and any 1 k m, if 19.5 Generic Network Codes 461 a) there is no directed path from tk to tj for j = k, b) Vtk Vk , then fek Vk . Definition 19.28 (Generic Network Code II). An -dimensional linear network code on an acyclic network is generic if for any nonempty collection of channels = {e1 , e2 , , em } E and any 1 k m, if a) there is no directed path from tk to tj for j = k, b) Vtk Vk , c) fe , e k are linearly independent3 , then fek Vk . In Definitions 19.27 and 19.28, if a) does not hold, then fek Vk may not be possible at all as we now explain. Let = {e1 , e2 } and k = 1. Suppose In(t2 ) = {e1 } so that a) is violated. Since node t2 has only e1 as the input channel, fe1 cannot possibly be linear independent of fe2 . The only difference between Definitions 19.27 and 19.28 is the additional requirement c) in the latter. The equivalence between these two definitions of a generic network code can be seen as follows. It is obvious that if a linear network code satisfies Definition 19.27, then it also satisfies Definition 19.28. To prove the converse, suppose a linear network code satisfies Definition 19.28. Consider any collection of channels = {e1 , e2 , , em } E such that there exists 1 k m satisfying a) and b) in Definition 19.27 but not necessarily c) in Definition 19.28. Then we can always find a subset k of k such that fe , e k are linearly independent and V = Vk . Upon letting = {ek } k k and applying Definition 19.28 with in place of , we have fek V = Vk , k (19.121) so the network code also satisfies Definition 19.27. This shows that the two definitions of a generic network code are equivalent. Note that in Definition 19.28, if satisfies all the prescribed conditions, then m because c) and fek Vk together imply that fe , e are linearly independent. Definition 19.28, which has a slightly more complicated form compared with Definition 19.27, will be instrumental in the proof of Theorem 19.32 that establishes various characterizations of a generic network code. Proposition 19.29. For a generic network code, for any collection of m output channels at a node t, where 1 m dim(Vt ), the global encoding kernels are linearly independent. 3 By convention, an empty collection of vectors is linearly independent. 462 19 Single-Source Linear Network Coding: Acyclic Networks Proof. Since the proposition becomes degenerate if dim(Vt ) = 0, we assume dim(Vt ) > 0. In Definition 19.27, let all the nodes tj be node t and let 1 m dim(Vt ). First note that there is no directed path from node t to itself because the network is acyclic. For m = 1, = and V = V = {0}. Since 1 1 dim(Vt ) > 0, we have Vt V . Then fe1 V , which implies fe1 = 0. This 1 1 proves the proposition for m = 1. Assume that the proposition is true for m - 1 some 2 m dim(Vt ). We now prove that the proposition is true for m. By the induction hypothesis, fe1 , fe2 , , fem-1 are linearly independent. Since dim( {fe1 , fe2 , , fem-1 } ) = m - 1 < dim(Vt ), we have Vt {fe1 , fe2 , , fem-1 } . Then by Definition 19.27, fem {fe1 , fe2 , , fem-1 } . (19.124) (19.123) (19.122) Hence, fe1 , fe2 , , fem are linearly independent. The proposition is proved. Corollary 19.30. For a generic network code, if |Out(t)| dim(Vt ) for a node t, then the global encoding kernels of all the output channels of t are linearly independent. A linear dispersion on an acyclic network is not necessarily a generic network code. The following is a counterexample. Example 19.31. The 2-dimensional linear dispersion on the network in Figure 19.6 is not generic because the global encoding kernels of two of the output channels from source node s are equal to [ 1 1 ] , a contradiction to Proposition 19.29. It can be shown, however, that a generic network code on an acyclic network G can be constructed through a linear dispersion on an expanded network G . See Problem 13 for details. Together with Example 19.13, the example above shows that the four classes of linear network codes we have discussed, namely linear multicast, linear broadcast, linear dispersion, and generic network code, achieve the maxflow bound to strictly increasing extents. In the following theorem, we prove two characterizations of a generic network code, each can be regarded as an alternative definition of a generic network code. The reader should understand this theorem before proceeding further but may skip the proof at the first reading. Theorem 19.32. For an -dimensional linear network code on an acyclic network, the following conditions are equivalent: 19.5 Generic Network Codes 463 &1# $0 ! % " &1# $1! %" &1# $0 ! % " x s &0 # $1! % " &1# $1! %" y Fig. 19.6. A 2-dimensional linear dispersion that is not a generic network code. 1) The network code is generic. 2) For any nonempty collection of channels = {e1 , e2 , , em } E, if Vtj V for all 1 j m, then fe , e are linearly independent. j 3) For any nonempty collection of channels E, if || = min{, maxflow()}, then fe , e are linearly independent. Proof. We will prove the theorem by showing that 1) 2) 3) 1). We first show that 1) 2) by using Definition 19.27 as the definition of a generic network code. Assume 1) holds. Consider any m 1 and any collection of channels = {e1 , e2 , , em } E, and assume Vtj V for j all 1 j m. We will show by induction on m that fe , e are linearly independent. The claim is trivially true for m = 1. Assume the claim is true for m - 1 for some 2 m , and we will show that it is true for m. Consider = {e1 , e2 , , em } and assume Vtj V for all 1 j m. We j first prove by contradiction that there exists at least one k such that there is no directed path from tk to tj for all j = k, where 1 j, k m. Assume that for all k, there is at least one directed path from node tk to node tj for some j = k. Starting at any node tk , by traversing such directed paths, we see that there exists a directed cycle in the network because the set {tk : 1 k m} is finite. This leads to a contradiction because the network is acyclic, proving the existence of k as prescribed. Then apply Definition 19.27 to this k to see that fek Vk = {fe : e k } . (19.126) Now for any j = k, since Vtj V and Vk \{ej } = V \{ek } V , we have j j j Vtj Vk \{ej } . (19.127) (19.125) 464 19 Single-Source Linear Network Coding: Acyclic Networks Then apply the induction hypothesis to k to see that fe , e k are linearly independent. It then follows from (19.126) that fe , e are linearly independent. Thus 1) 2). We now show that 2) 3). Assume 2) holds and consider any nonempty collection of channel = {e1 , e2 , , em } E satisfying (19.125). Then m = || = min{, maxflow()}, which implies maxflow() m. (19.129) Therefore, there exist m edge-disjoint paths P1 , P2 , , Pm from source node s to the channels in , where the last channel on path Pj is ej . Denote the length of Pj by lj and let m (19.128) L= j=1 lj (19.130) be the total length of all the paths. We will prove the claim that fe1 , fe2 , , fem are linearly independent by induction on L. For the base case L = m, since m by (19.128), the claim is true by Proposition 19.29 with t = s. Assume that the claim is true for L - 1 for some L m + 1, and we will prove that it is true for L. Let A = {j : lj > 1} and for j A, let j = {e1 , e2 , , ej-1 , ej , ej+1 , , em }, where ej is the channel preceding ej on Pj . Then by the induction hypothesis, fe , e j are linearly independent, which implies that Vtj V . (19.131) j For j A, lj = 1, i.e., tj = s. It follows from (19.128) that m . Then Vtj = Vs V j (19.132) because dim(V ) | | = m - 1 < m . Therefore, (19.131) holds for all j j j, and hence by 2), fe , e are linearly independent. Thus 2) 3). Finally, we show that 3) 1) by using Definition 19.28 as the definition of a generic network code. Assume 3) holds and consider any collection of channels = {e1 , e2 , , em } E, where 1 m , such that a) to c) in Definition 19.28 hold for some 1 k m. Then either tj = s for all 1 j m, or tk = s, because otherwise a) in Definition 19.28 is violated. If tj = s for all 1 j m, then m = || = maxflow(). Since m , we have || = min{, maxflow()}. (19.134) (19.133) Then fe , e are linearly independently by 3), proving that fek Vk . 19.5 Generic Network Codes 465 Otherwise, tk = s. Following b) in Definition 19.28, there exists ek In(tk ) E such that fek and fe , e k are linearly independent. Let k = {e1 , e2 , , ek-1 , ek , ek+1 , , em }. By Corollary 19.11, maxflow(k ) dim(Vk ) = m, (19.135) so e1 , e2 , , ek-1 , ek , ek+1 , em can be traced back to source node s via some edge-disjoint paths P1 , P2 , , Pk-1 , Pk , Pk+1 , , Pm , respectively. Let Pk be obtained by appending ek to Pk . Since there is no directed path from tk to tj and ek = ej for all j = k, P1 , P2 , , Pk-1 , Pk , Pk+1 , , Pm are edge-disjoint. Therefore, maxflow() m. (19.136) On the other hand, maxflow() || = m. Therefore, m = || = maxflow(), (19.138) i.e., (19.133). As before, we can further obtain (19.134). Then by 3), fe , e , are linearly independent, and therefore fek Vk . Thus 3) 1). Hence, the theorem is proved. Corollary 19.33. An -dimensional generic network code on an acyclic network is an -dimensional linear dispersion on the same network. Proof. Consider an -dimensional generic network code on an acyclic network and let T be any collection of non-source nodes. Let m = min{, maxflow(T )}. (19.139) (19.137) Since maxflow(T ) m, there exists m edge-disjoint paths P1 , P2 , , Pm from source node s to T . Let ei be the last channel on path Pi , and let = {e1 , e2 , , em }. Evidently, maxflow() = m. It follows from (19.139) that m . Therefore, || = m = maxflow() = min{, maxflow()}. By Theorem 19.32, fe , e are linearly independent. Then dim(VT ) dim(V ) = m = min{, maxflow(T )}. By Theorem 19.10, we conclude that dim(VT ) = min{, maxflow(T )}. (19.144) (19.143) (19.142) (19.141) (19.140) 466 19 Single-Source Linear Network Coding: Acyclic Networks Hence, we have shown that a generic network code is a linear dispersion. Theorem 19.32 renders the following important interpretation of a generic network code. Consider any linear network code and any collection of channels E. If fe , e are linearly independent, then || = dim(V ). By Corollary 19.11, dim(V ) min{, maxflow()}. Therefore, || min{, maxflow()}. On the other hand, maxflow() ||, which implies min{, maxflow()} ||. Combining (19.147) and (19.149), we see that || = min{, maxflow()} (19.150) (19.149) (19.148) (19.147) (19.146) (19.145) is a necessary condition for fe , e to be linearly independent. For a generic network code, this is also a sufficient condition for fe , e to be linearly independent. Thus for a generic network code, if a set of global encoding kernels can possibly be linearly independent, then it is linear independent. In this sense, a generic network code captures the notion of placing the global encoding kernels in general positions. The condition 2) in Theorem 19.32 is the original definition of a generic network code given in [230]. Unlike 1) and 3), this condition is purely algebraic and does not depend upon the network topology. However, it does not suggest an algorithm for constructing such a code. Motivated by Definition 19.28, we now present an algorithm for constructing a generic network code. The computational complexity of this algorithm is polynomial in |E|, the total number of channels in the network. Algorithm 19.34 (Construction of a Generic Network Code). This algorithm constructs an -dimensional generic network code over a finite field F with |F | > m=1 |E|-1 by prescribing global encoding kernels that constitute m-1 a generic network code. { for (every node t, following an upstream-to-downstream order) { for (every channel e Out(t)) 19.5 Generic Network Codes 467 { Choose a vector w in Vt such that w V , where is any / collection of m - 1 already processed channels, where 1 m , such that fe , e are linearly independent and Vt V ; // To see the existence of such a vector w, denote dim(Vt ) // by . If is any collection of m - 1 channels with Vt V , // then dim(Vt V ) - 1. There are at most m=1 |E|-1 m-1 // such collections . Thus // |Vt ( V )| m=1 |E|-1 |F |-1 < |F | = |Vt |. m-1 fe = w; // This is equivalent to choosing scalar values for the local // encoding kernels kd,e for all d such that dIn(t) kd,e fd // V for every collection of channels as prescribed. / } } } Algorithm Verification. We will verify that the code constructed is indeed generic by way of Condition 3) in Theorem 19.32. Consider any nonempty collection of channels = {e1 , e2 , , em } E satisfying (19.125). Then there exist m edge-disjoint paths P1 , P2 , , Pm from source node s to the channels in , where the last channel on path Pj is ej . Denote the length of Pj by lj and let m L= j=1 lj (19.151) be the total length of all the paths. We will prove the claim that fe , e are linearly independent by induction on L. It is easy to verify that for any set of m channels in Out(s), the global encoding kernels assigned are linearly independent, so the base case L = m is verified. Assume the claim is true for L - 1 for some L m + 1, and we will prove that it is true for L. Let ek be the channel whose global encoding kernel is last assigned among all the channels in . Note that Pk 2 since L m + 1 and the global encoding kernels are assigned by the algorithm in an upstream-to-downstream order. Then let ek be the channel preceding ek on Pk , and let = {e1 , e2 , , ek-1 , ek , ek+1 , , em }. (19.152) By the induction hypothesis, fe , e are linearly independent. Since fek is linearly independent of fe for e \{ek } = k , Vtk Vk . It then follows from the construction that fek Vk because k is one of the collections considered when fek is assigned. Hence, fe , e are linearly independent, verifying that the network code constructed is generic. Complexity Analysis. In the algorithm, the "for loop" for each channel e pro cesses at most m=1 |E|-1 collections of m - 1 channels. The processing m-1 468 19 Single-Source Linear Network Coding: Acyclic Networks includes the detection of those collections as well as the computation of the set Vt \( V ). This can be done, for instance, by Gauss elimination. Throughout the algorithm, the total number of collections of channels processed is at most |E| m=1 |E|-1 , a polynomial in |E| of degree . Thus for a fixed , m-1 it is not hard to implement the algorithm within a polynomial time in |E|. Algorithm 19.34 constitutes a constructive proof for the next theorem. Theorem 19.35. There exists an -dimensional generic network code on an acyclic network for sufficiently large base field F . By noting the lower bound on the required field size in Algorithm 19.34, a generic network code can be constructed with high probability by randomly choosing the global encoding kernels provided that the base field is much larger than sufficient. 19.6 Static Network Codes In our discussion so far, a linear network code has been defined on a network with a fixed topology, where all the channels are assumed to be available at all times. In a real network, however, a channel may fail due to various reasons, for example, hardware failure, cable cut, or natural disasters. With the failure of some subset of channels, the communication capacity of the resulting network is generally reduced. Consider the use of, for instance, an -dimensional multicast on an acyclic network for multicasting a sequence of messages generated at the source node. When no channel failure occurs, a non-source node with the value of a maxflow at least equal to would be able to receive the sequence of messages. In case of channel failures, if the value of a max-flow of that node in the resulting network is at least , the sequence of messages in principle can still be received at that node. However, this would involve the deployment of a network code for the new network topology, which not only is cumbersome but also may cause a significant loss of data during the switchover. In this section, we discuss a class of linear network codes called static network codes that can provide the network with maximum robustness in case of channel failures. To fix ideas, we first introduce some terminology. The status of the network is specified by a mapping : E {0, 1} called a configuration. A channel being in the set -1 (0) = {e E : (e) = 0} (19.153) indicates the failure of that channel, and the subnetwork resulting from the deletion of all the channels in -1 (0) is called the -subnetwork. For the subnetwork, the value of a max-flow from source node s to a non-source node t is denoted by maxflow (t). Likewise, the value of a max-flow from source 19.6 Static Network Codes 469 node s to a collection T of non-source nodes is denoted by maxflow (T ). It is easy to see that the total number of configurations is equal to 2|E| . Definition 19.36. Let be a configuration of the network. For an -dimensional linear network code on the network, the -global encoding kernel of channel e, denoted by fe, , is the column -vector calculated recursively in an upstreamto-downstream order by: (19.154) fe, = (e) dIn(t) kd,e fd, for e Out(t). (19.155) The -global encoding kernels of the imaginary channels are independent of and form the standard basis of the space F . Note that in the above definition, the local encoding kernels kd,e are not changed with the configuration . Given the local encoding kernels, the -global encoding kernels can be calculated recursively by (19.154), while (19.155) serves as the boundary conditions. For a channel e Out(t) with (e) = 0, we see from (19.154) that fe, = 0. (19.156) As before, the message generated at source node s is denoted by a row vector x. When the prevailing configuration is , a node t receives the symbols x fd, , d In(t), from which it calculates the symbol x fe, to be sent on each channel e Out(t) via x fe, = x (e) dIn(t) kd,e fd, kd,e (x fd, ). (19.157) (19.158) = (e) dIn(t) In particular, if (e) = 0, the zero symbol is sent on channel e regardless of the symbols received at node t. In a real network, the zero symbol is not sent on a failed channel. Rather, whenever a symbol is not received on an input channel, the symbol is regarded by the receiving node as being the zero symbol. For a configuration of the network, we let Vt, = {fe, : e In(t)} for a node t, VT, = tT Vt, . for a collection T of nodes, and V, = {fe, : e } , for a collection of channels. (19.161) (19.160) (19.159) 470 19 Single-Source Linear Network Coding: Acyclic Networks Definition 19.37. An -dimensional linear network code on an acyclic network qualifies as a static linear multicast, a static linear broadcast, a static linear dispersion, or a static generic network code, respectively, if the following hold: (19.162) dim(Vt, ) = for every configuration and every non-source node t with maxflow (t) . (19.163) dim(Vt, ) = min{, maxflow (t)} for every configuration and every non-source node t. (19.164) dim(VT, ) = min{, maxflow (T )} for every configuration and every collection T of non-source nodes. (19.165) For any configuration and any nonempty collection of channels E, if || = min{, maxflow ()}, then fe, , e are linearly independent. Here we have adopted Condition 3) in Theorem 19.32 for the purpose of defining a static generic network code. The qualifier "static" in the terms above stresses the fact that, while the configuration varies, the local encoding kernels remain unchanged. The advantage of using a static linear multicast, broadcast, or dispersion is that in case of channel failures, the local operation at every node in the network is affected only at the minimum level. Each receiving node in the network, however, needs to know the configuration before decoding can be done correctly. In implementation, this information can be provided by a separate signaling network. For each class of static network codes in Definition 19.37, the requirement for its non-static version is applied to the -subnetwork for every configuration . Accordingly, a static linear multicast, a static linear broadcast, a static linear dispersion, and a static generic network code are increasingly stronger linear network codes as for the non-static versions. Example 19.38. A 2-dimensional linear network code over GF (5) on the network in Figure 19.7 is prescribed by the local encoding kernels Ks = and 101 011 3 2. 1 (19.166) 1 Kx = 3 1 (19.167) We claim that this is a static generic network code. Denote the three channels in In(x) by c, d, and e and the two channels in Out(x) by g and h. The vectors fg, and fh, for all possible configurations are tabulated in Table 19.1, from which it is straightforward to verify the condition (19.165). The following is an example of a generic network code that does not qualify even as a static linear multicast. 19.6 Static Network Codes &1# $0 ! % " &0 # $1! % " 471 s cd e &1 3# $ ! Kx = $3 2! $1 1 ! % " Ks = $ ! %0 1 1" &1 0 1# g x h y Fig. 19.7. A 2-dimensional GF (5)-valued static generic network code. Example 19.39. On the network in Figure 19.7, a 2-dimensional generic network code over GF (5) is prescribed by the local encoding kernels Ks = and 101 011 1 2. 0 (19.168) 2 Kx = 1 0 For a configuration such that (19.169) (c) = 0 and (d) = (e) = 1, we have the -global encoding kernels (c) (d) (e) 0 0 1 1 1 1 (h) 1 0 1 0 0 3 0 (h) 2 (g) 0 1 1 1 4 1 (h) 3 (g) 1 0 0 1 0 3 (h) 0 (g) 1 0 1 2 1 4 (h) 1 (g) 1 1 0 1 3 3 (h) 2 (g) (19.170) (19.171) 1 1 1 2 4 4 (h) 3 (g) fg, (g) fh, Table 19.1. The vectors fg, and fh, for all possible configurations in Example 19.38. 472 19 Single-Source Linear Network Coding: Acyclic Networks fg, = and fh, = 0 1 0 , 2 (19.172) (19.173) and therefore dim(Vy, ) = 1. On the other hand, maxflow (y) = 2. Hence, this generic network code is not a static linear multicast. Recall that in Algorithm 19.34 for constructing a generic network code, the key step chooses for a channel e Out(t) a vector in Vt to be the global encoding kernel fe such that fe V , / (19.174) where is any collection of m - 1 channels as prescribed with 1 m . This is equivalent to choosing scalar values for the local encoding kernels kd,e for all d In(t) such that kd,e fd V . / dIn(t) (19.175) Algorithm 19.34 is adapted below for the construction of a static generic network code. Algorithm 19.40 (Construction of a Static Generic Network Code). This algorithm constructs an -dimensional static generic network code over a finite field F on an acyclic network with |F | > 2|E| m=1 |E|-1 . m-1 { for (every node t, following an upstream-to-downstream order) { for (every channel e Out(t)) { Choose scalar values for kd,e for all d In(t) such that for any configuration , dIn(t) kd,e fd V, , where is any / collection of m - 1 already processed channels such that fe, , e are linearly independent and Vt, V, ; // To see the existence of such values kd,e , denote // dim(Vt, ) by . For any collection of channels // with Vt, V, , dim(Vt, V, ) < . Consider // the linear mapping [kd,e ]dIn(t) dIn(t) kd,e fd, // from F |In(t)| to F . The nullity of this linear // mapping is |In(t)| - , so the pre-image of // the space (Vt, V, ) has dimension less than // |In(t)|. Thus the pre-image of , (Vt, V, ) // contains at most 2|E| m=1 |E|-1 |F ||In(t)|-1 m-1 19.7 Random Network Coding: A Case Study 473 // elements, which are fewer than |F ||In(t)| if // |F | > 2|E| m=1 |E|-1 . m-1 for (every configuration ) fe, = (e) dIn(t) kd,e fd, ; } } } Algorithm Verification. The explanation for the code constructed by Algorithm 19.40 being a static generic network code is exactly the same as that given for Algorithm 19.34. The details are omitted. Algorithm 19.40 constitutes a constructive proof for the next theorem. By noting the lower bound on the required field size in the algorithm, we see that a generic network code can be constructed with high probability by randomly choosing the local encoding kernels provided that the base field is much larger than sufficient. Theorem 19.41. There exist an -dimensional static linear multicast, a static linear broadcast, a static linear dispersion, and a static generic network code on an acyclic network for sufficiently large base field F . The requirements (19.162) through (19.165) in Definition 19.37 refer to all the 2|E| possible configurations. Conceivably, a practical application may only need to deal with a certain collection {1 , 2 , , } of configurations, where 2|E| . Thus we may define, for instance, an {1 , 2 , , }-static linear multicast and an {1 , 2 , , }-static linear broadcast by replacing the conditions (19.162) and (19.163), respectively by (19.176) dim(Vt, ) = for every configuration {1 , 2 , , } and every non-source node t with maxflow (t) . (19.177) dim(Vt, ) = min{, maxflow (t)} for every configuration {1 , 2 , , } and every non-source node t. Algorithm 19.34 has been converted into Algorithm 19.40 by modifying the key step in the former. In a similar fashion, Algorithm 19.25 can be adapted for the construction of an {1 , 2 , , }-static linear multicast or broadcast. This will lower the threshold for the sufficient size of the base field as well as the computational complexity. The details are left as an exercise. 19.7 Random Network Coding: A Case Study We have seen in Corollary 19.24 that if the local encoding kernels of a linear network code are randomly chosen, a linear multicast can be obtained with high probability provided that the base field is sufficiently large. Since the 474 19 Single-Source Linear Network Coding: Acyclic Networks code construction is independent of the network topology, the network code so constructed can be used when the network topology is unknown. In this section, we study an application of random network coding in peer-to-peer (P2P) networks. The system we will analyze is based on a prototype for large scale content distribution on such networks proposed in [133]. 19.7.1 How the System Works A file originally residing on a single server is to be distributed to a large number of users through a network. The server divides the file into k data blocks, B1 , B2 , , Bk , and uploads coded versions of these blocks to different users according to some protocol. These users again help distributing the file by uploading blocks to other users in the network. By means of such repeated operations, a logical network called an overlay network is formed by the users as the process evolves. On this logical network, henceforth referred to as the network, information can be dispersed very rapidly, and the file is eventually delivered to every user in the network. Note that the topology of the network is not known ahead of time. In the system, new users can join the network as a node at any time as long as the distribution process is active. Upon arrival, a new user will contact a designated node called the tracker that provides a subset of the other users already in the system forming the set of neighboring nodes of the new user. Subsequent information flow in the network is possible only between neighboring nodes. For the purpose of coding, the data blocks B1 , B2 , , Bk are represented as symbols in a large finite field F referred to as the base field4 . At the beginning of the distribution process, a Client A contacts the server and receives a number of encoded blocks. For example, the server uploads two encoded blocks E1 and E2 to Client A, where for i = 1, 2, Ei = ci B1 + ci B2 + + ci Bk , 1 2 k (19.178) with ci , 1 j k being chosen randomly from the base field F . Note that j each E1 and E2 is some random linear combination of B1 , B2 , , Bk . In general, whenever a node needs to upload an encoded block to a neighboring node, the block is formed by taking a random linear combination of all the blocks possessed by that node. Continuing with the above example, when Client A needs to upload an encoded block E3 to a neighboring Client B, we have E3 = c3 E1 + c3 E2 , (19.179) 1 2 where c3 and c3 are randomly chosen from F . Substituting (19.178) into 1 2 (19.179), we obtain 4 In the system proposed in [133], the size of the base field is of the order 216 . 19.7 Random Network Coding: A Case Study k 475 E3 = j=1 (c3 c1 + c3 c2 )Bj . 1 j 2 j (19.180) Thus E3 and in general every encoded block subsequently uploaded by a node in the network is some random linear combination of the data blocks B1 , B2 , , Bk . The exact strategy for downloading encoded blocks from the neighboring nodes so as to avoid receiving redundant information depends on the implementation. The main idea is that downloading from a neighboring node is necessary only if the neighboring node has at least one block not in the linear span of all the blocks possessed by that particular node. Upon receiving enough linearly independent encoded blocks, a node is able to decode the whole file. Compared with store-and-forward, the application of network coding as described in the above system can reduce the file download time because an encoded block uploaded by a node contains information about every block possessed by that node. Moreover, in case some nodes leave the system before the end of the distribution process, it is more likely that the remaining nodes have the necessary information to recover the whole file if network coding is used. In the following, we will give a quantitative analysis to substantiate these claimed advantages of network coding. 19.7.2 Model and Analysis Let V be the set of all the nodes in the system. In implementation, blocks of data are transmitted between neighboring nodes in an asynchronous manner, and possibly at different speeds. To simplify the analysis, we assume that every transmission from one node to a neighboring node is completed in an integral number of time units. Then we can unfold the network of nodes in discrete time into a graph G = (V , E ) with the node set V = {it : i V and t 0}, (19.181) where node it V corresponds to node i V at time t. The edge set E specified below is determined by the strategy adopted for the server as well as for all the other nodes in V to request uploading of data blocks from the neighboring nodes. Specifically, there are two types of edges in E : 1. There is an edge with capacity m from node it to node jt , where t < t , if m blocks are transmitted from node i to node j, starting at time t and ending at time t . 2. For each i V and t 0, there is an edge with infinite capacity from node it to node it+1 . An edge of the second type models the assumption that the blocks, once possessed by a node, are retained in that node indefinitely over time. Without 476 19 Single-Source Linear Network Coding: Acyclic Networks t= 0 Server S Client A Client B Client C 4 2 t= 1 1 2 1 t= 2 t= 3 1 2 1 ... Fig. 19.8. A illustration of the graph G . loss of generality, we may assume that all the blocks possessed by nodes il , l t are transmitted uncoded on the edge from node it to node it+1 . An illustration of the graph G up to t = 3 with V consisting of the server S and three clients A, B, and C is given in Figure 19.8, where the edges with infinite capacities are lightened for clarity. Note that the graph G is acyclic because each edge is pointed in the positive time direction and hence a cycle cannot be formed. Denote the server S by node s V and regard node s0 in G as the source node generating the whole file consisting of k data blocks and multicasting it to all the other nodes in G via random linear network coding, with the coefficients in the random linear combinations forming the encoded blocks being the local encoding kernels of the network code. Note that random network coding is applied on G , not the logical network formed by the user nodes. Also note that in order to simplify our description of the system, we have omitted the necessity of delivering the global encoding kernels to the nodes for the purpose of decoding. We refer the reader to the discussion toward the end of Section 19.3 for this implementation detail. We are now ready to determine the time it takes for a particular node i V to receive the whole file. Denote the value of a max-flow from node s0 to a node v G other than s0 by maxflow(v). When the base field is sufficiently large, by Corollary 19.24, with probability close to 1, the network code generated randomly during the process is a linear multicast, so that those nodes it with maxflow(it ) k (19.182) can receive the whole file. In other words, with high probability, the time it takes a node i V to receive the whole file is equal to t , the minimum t that satisfies (19.182). Obviously, this is a lower bound on the time it takes a node i V to receive the whole file, and it is achievable with high probability by the system under investigation. In the rare event that node i cannot decode at time t , it can eventually decode upon downloading some additional encoded blocks from the neighboring nodes. 19.7 Random Network Coding: A Case Study s t u 477 Fig. 19.9. A simple packet network. When some nodes leave the system before the end of the distribution process, an important question is whether the remaining nodes have the necessary information to recover the whole file. To be specific, assume that a subset of users U c V leave the system after time t, and we want to know whether the users in U = V \U c have sufficient information to recover the whole file. If they do, by further exchanging information among themselves, every user in U can eventually receive the whole file (provided that no more nodes leave the system). Toward this end, again consider the graph G . Let Ut = {ut : u U } (19.183) and denote the value of a max-flow from node s0 to the set of nodes Ut by maxflow(Ut ). If maxflow(Ut ) k, (19.184) then the users in U with high probability would have the necessary information to recover the whole file. This is almost the best possible performance one can expect from such a system, because if maxflow(Ut ) < k, (19.185) it is simply impossible for the users in U to recover the whole file even if they are allowed to exchange information among themselves. Thus we see that random network coding provides the system with both maximum bandwidth efficiency and maximum robustness. However, additional computational resource is required compared with store-and-forward. These are engineering tradeoffs in the design of such systems. We conclude this section by an example further demonstrating the advantage of random network coding when it is applied to packet networks with packet loss. Example 19.42. The random network coding scheme discussed in this section can be applied to packet networks with packet loss. Consider the network depicted in Figure 19.9 consisting of three nodes, s, t, and u. Data packets are sent from node s to node u via node t. Let the packet loss rates of channels (s, t) and (t, u) be , i.e., a fraction of packets are lost during their transmission through the channel. Then the fraction of packets sent by node s that are eventually received at node u is (1 - )2 . To fix idea, assume the packet size is sufficiently large and one packet is sent on each channel per unit time. To remedy the problem of packet loss, a fountain code [51] can be employed at node s. This would allow data packets 478 19 Single-Source Linear Network Coding: Acyclic Networks to be sent from node s to node u reliably at an effective rate equal to (1 - )2 . In such a scheme, node t simply forwards to node u the packets it receives from node s. On the other hand, by using the random network coding scheme we have discussed, data packets can be sent from node s to node u reliably at an effective rate equal to 1 - , which is strictly higher than (1 - )2 whenever > 0. This can be proved by means of the analysis presented in this section. The details are left as an exercise. While a fountain code can remedy the problem of packet loss between the source node and the sink node, it cannot prevent the packet loss rate from accumulating when packets are routed through the network. On the other hand, the use of random network coding allows information to be transmitted from the source node to the sink node at the maximum possible rate, namely the min-cut between the source node and the sink node after the packet loss in the channels has been taken into account. Chapter Summary Linear Network Code: kd,e is the local encoding kernel of the adjacent pair of channels (d, e). fe is the global encoding kernel of channel e. fe = dIn(t) kd,e fd for e Out(t). fe , e In(s) form the standard basis of F . Channel e transmits the symbol x fe , where x F is the message generated at source node s. Linear Multicast, Broadcast, and Dispersion: An -dimensional linear network code is a linear multicast if dim(Vt ) = for every node t = s with maxflow(t) . linear broadcast if dim(Vt ) = min{, maxflow(t)} for every node t = s. linear dispersion if dim (VT ) = min{, maxflow(T )} for every collection T of non-source nodes. Generic Network Code: An -dimensional linear network code is generic if for any nonempty collection of channels = {e1 , e2 , , em } E, where ej Out(tj ), and any 1 k m, if a) there is no directed path from tk to tj for j = k, b) Vtk Vk , where k = \{ek }, then fek Vk . A generic network code is a linear dispersion and hence a linear broadcast and a linear multicast. Characterizations of Generic Network Code: Each of the following is a necessary and sufficient condition for an -dimensional linear network code to be generic: Problems 479 1) For any nonempty collection of channels = {e1 , e2 , , em } E, where ej Out(tj ), if Vtj V for all 1 j m, then fe , e are j linearly independent. 2) For any nonempty collection of channels E, if || = min{, maxflow()}, then fe , e are linearly independent. Static Network Code: For a given linear network code and a configuration of the network, the -global encoding kernel fe, of channel e is calculated recursively by: fe, = (e) dIn(t) kd,e fd, for e Out(t). fe, , e In(s) are independent of and form the standard basis of F . An -dimensional linear network code is a static linear multicast if dim(Vt, ) = for every and every node t = s with maxflow (t) . linear broadcast if dim(Vt, ) = min{, maxflow (t)} for every and every node t = s. linear dispersion if dim(VT, ) = min{, maxflow (T )} for every and every collection T of non-source nodes. generic network code if for every and any nonempty collection of channels E, if || = min{, maxflow ()}, then fe, , e are linearly independent. Lemma: Let g(z1 , z2 , , zn ) be a nonzero polynomial with coefficients in a field F . If |F | is greater than the degree of g in every zj for 1 j n, then there exist a1 , a2 , , an F such that g(a1 , a2 , , an ) = 0. Existence and Construction: All the linear network codes defined in this chapter exist and can be constructed either deterministically or randomly (with high probability) when the base field is sufficiently large. Problems In the following, let G = (V, E) be the underlying directed acyclic network on which the linear network code is defined, and let s be the unique source node in the network. 1. Show that in a network with the capacities of all the edges equal to 1, the number of edge-disjoint paths from source node s to a non-source node t is equal to maxflow(t). 2. For the network code in Definitions 19.4 and 19.6, show that if the global encoding mappings are linear, then so are the local encoding mappings. (Yeung et al. [408].) 3. Network transfer matrix Consider an -dimensional linear network code. a) Prove (19.91). 480 19 Single-Source Linear Network Coding: Acyclic Networks b) Fix an upstream-to-downstream order for the channels in the network and let K be the |E| |E| matrix with the (d, e)th element equal to kd,e if (d, e) is an adjacent pair of channels and equal to 0 otherwise. Let A be the |E| matrix obtaining by appending |E| - |Out(s)| columns of zeroes to Ks , and Be be the |E|-column vector with all the components equal to 0 except that the eth component is equal to 1. Show that fe = A(I - K)-1 Be for all e E. The matrix M = (I -K)-1 is called the network transfer matrix. (Koetter and Mdard [202].) e Apply Lemma 19.17 to obtain a lower bound on the field size for the existence of a 2-dimensional linear multicast on the butterfly network. Show that |E| m=1 |E|-1 is a polynomial in |E| of degree . This is m-1 the lower bound on the required field size in Algorithm 19.34. Verify that the network code in Example 19.38 is a generic network code. Simplified characterization of a generic network code Consider an dimensional generic network code on a network for which |Out(s)| . a) Show that Condition 3) in Theorem 19.32 can be modified to restricting the cardinality of to . Hint: If || < , expand by including a certain subset of the channels in Out(s). b) Simplify Algorithm 19.34 and tighten the lower bound on the required field size accordingly. (Tan et al. [348].) For the network below, prove the non-existence of a two-dimensional binary generic network code. 4. 5. 6. 7. 8. &1# $0 ! % " &0 # $1! % " s x y 9. Show that for 2, a linear multicast can be constructed by the JaggiSanders algorithm provided that |F | . Hint: Two vector subspaces intersect at the origin. 10. Modify the Jaggi-Sanders algorithm for the construction of a static linear multicast. Problems 481 11. Obtain a lower bound on the required field size and determine the computational complexity when Algorithm 19.40 is adapted for the construction of an {1 , 2 , , }-static generic network code. 12. Show that a transformation of a static generic network code is also a static generic network code. 13. A generic network code as a linear dispersion Expand the network G into a network G = (V , E ) as follows. For an edge e E, let the edge be from node ve to node we . Install a new node te and replace edge e by two new edges e and e , where e is from node ve to node te and e is from node te to node we . Show that a linear dispersion on G is equivalent to a generic network code on G. Hint: Use Theorem 19.32. (Kwok and Yeung [217], Tan et al. [348].) 14. Multi-rate linear broadcast Consider a network on which an -dimensional linear network code over a base field F is defined. For all e E, let fe = [ I b ] fe , where I is the ( - 1) ( - 1) identity matrix and b is an ( - 1)-column vector. a) Show that fe , e E constitute the global encoding kernels of an (-1)dimensional linear network code on the same network. b) Show that the ( - 1)-dimensional linear network code in a) and the original -dimensional linear network code have the same local encoding kernels for all the non-source nodes. It was shown in Fong and Yeung [115] that an ( - 1)-dimensional linear broadcast can be constructed from any -dimensional linear broadcast by choosing a suitable vector b, provided |F | |V |. This implies that multirate linear multicast/broadcast can be supported on a network without changing the local encoding kernels of the non-source nodes. 15. Let a message x F be generated at source node s in a network for which maxflow(t) for all non-source nodes t. Show that x can be multicast to all the non-source nodes by store-and-forward. In other words, for this special case, network coding has no advantage over store-and-forward if complete information on the network topology is known ahead of time. This result is implied by a theorem on directed spanning tree packing by Edmonds [101] (see also Wu et al. [386]). 16. Let L be the length of the message x generated at source node s, where L is divisible by maxflow(t) for all non-source nodes t. Allowing multiple usage of the network, devise a linear network coding scheme such that each nonsource node t can receive x in L/maxflow(t) units of time. Such a scheme enables each non-source node in the network to receive the message within the shortest possible time. 17. Consider distributing a message of 5 data blocks on a P2P network with 4 nodes, Server S and Clients A, B, and C, by the system discussed in Section 19.7. Assume each data block is sufficiently large. The following transmissions take place during the process. 482 19 Single-Source Linear Network Coding: Acyclic Networks From To Start Time End Time # Blocks S A 0 1 2 0 1 3 S B S C 0 1 2 1 2 1 B A C B 1 3 2 2 3 1 S B B C 2 3 2 a) Which client is the first to receive the whole message? b) If Client B leaves the system after t = 3, do Clients A and C have sufficient information to reconstruct the whole message? c) Suppose the hard disk of Client B crashes at t = 1.5 and loses 2 blocks of data. Repeat b) by making the assumption that the transmissions by Client B starting at t 1 are not affected by the disk failure. 18. Prove the claim in Example 19.42 that by using random network coding, data packets can be sent from node s to node u at an effective rate equal to 1 - . Historical Notes The achievability of the max-flow bound by linear network codes was proved by Li et al. [230] using a vector space approach and then by Koetter and Mdard [202] using a matrix approach. These two approaches correspond ree spectively to the notions of global encoding kernel and local encoding kernel discussed here. Neither the construction in [230] for a generic network code nor the construction in [202] for a linear multicast is a polynomial-time algorithm. Jaggi and Sanders et al. [184] obtained a polynomial-time algorithm for constructing a linear multicast by modifying the construction of a generic network code in [230]. A polynomial-time algorithm for constructing a generic network code was subsequently obtained in Yeung et al. [408]. In [202], static network code was introduced and its existence was proved. An explicit construction of such codes was given in [408]. The optimality of random network coding was proved in Ahlswede et al. [6]. Ho et al. [162] proved the optimality of random linear network coding and proposed the use of such codes on networks with unknown topology. A tight upper bound on the probability of decoding error for random linear network coding has recently been obtained by Balli et al. [23]. Implementation issues of network coding were discussed in Chou et al. [70]. The application of random network coding in peer-to-peer networks discussed in Section 19.7 is due to Gkantsidis and Rodriguez [133]. Cai and Yeung have generalized the theory of single-source network coding on acyclic networks to network error correction [53][405][54] and secure network coding [52]. Network error correction subsumes classical algebraic Historical Notes 483 coding, while secure network coding subsumes secret sharing in cryptography. The presentation in this chapter is largely based on the tutorial paper by Yeung et al. [408]. The various characterizations of a generic network code is due to Tan et al. [348]. The analysis of a large scale content distribution system with network coding is due to Yeung [403]. 20 Single-Source Linear Network Coding: Cyclic Networks A directed network is cyclic if it contains at least one directed cycle. In Chapter 19, we have discussed network coding over an acyclic network, for which there exists an upstream-to-downstream order on the nodes. Following such an order, whenever a node encodes, all the information needed would have already been received on the input channels of that node. For a cyclic network, such an order of the nodes does not exists. This makes network coding over a cyclic network substantially different from network coding over an acyclic network. 20.1 Delay-Free Cyclic Networks When we discussed network coding over an acyclic network in Chapter 19, we assumed that there is no propagation delay in the network. Based on this assumption, a linear network code can be specified by either the local description in Definition 19.6 or the global description in Definition 19.7. The local and global descriptions of a linear network code are equivalent over an acyclic network because given the local encoding kernels, the global encoding kernels can be calculated recursively in any upstream-to-downstream order. In other words, the equation (19.17) has a unique solution for the global encoding kernels in terms of the local encoding kernels, while (19.18) serves as the boundary conditions. If these descriptions are applied to a cyclic network, it is not clear whether for any given set of local encoding kernels, there exists a unique solution for the global encoding kernels. In the following, we give one example with a unique solution, one with no solution, and one with multiple solutions. Example 20.1. Consider the cyclic network in Figure 20.1. Let (s, t) precede (v, t) in the ordering among the channels. Similarly, let (s, t ) precede (v, t ). Given the local encoding kernels 486 20 Single-Source Linear Network Coding: Cyclic Networks (1% &0 # ' $ (0 % &1# ' $ s (1% &0 # ' $ (1% Kt = &0# ' $ Ks = &0 1# $ ' (1% '$ (0 % &1# ' $ (1% &1# '$ (0 % &1# ' $ (1 0% Ku = & # 1 u (1% &0 # ' $ (1% &1# '$ (1% &1# '$ t t' Kt' = &0# ' $ (1% v Kv = !1 1" Fig. 20.1. A 2-dimensional linear broadcast on a cyclic network. Ks = 10 1 1 , Kt = Kt = , Ku = , Kv = 1 1 , 01 0 1 (20.1) the equation (19.17) yields the following unique solution for the global encoding kernels: f(s,t) = f(t,u) = 1 0 , f(s,t ) = f(t ,u) = 0 1 1 . 1 (20.2) (20.3) f(u,v) = f(v,t) = f(v,t ) = These global encoding kernels are shown in Figure 20.1, and they in fact define a 2-dimensional linear broadcast regardless of the choice of the base field. Since k(v,t),(t,u) = 0 (20.4) and k(v,t ),(t ,u) = 0, (20.5) information looping in the directed cycles (t, u), (u, v), (v, t) and (t , u), (u, v), (v, t ) is prevented. Example 20.2. An arbitrarily prescribed set of local encoding kernels on a cyclic network is unlikely to be compatible with any global encoding kernels. In Figure 20.2(a), a local encoding kernel is prescribed at each node in a cyclic network. Had a global encoding kernel fe existed for each channel e, the requirement (19.17) would imply the equations 20.1 Delay-Free Cyclic Networks &1 # $0 ! % " &0 # $1 ! % " 487 a &1 0 # $0 1 ! % " b s s &1 # $0 ! % " Ks = &0 # $1 ! % " a p = a+r b f(x,y) = $ Kx = &1# $1! %" x &1 # ! %0 " + f(w,x) y Ky = &1# $1! %" x r=q y q = b+p f(w,x) = f(y,w) w Kw = '1( f(y,w) = &0 # $1! + f(x,y) % " w (a) (b) Fig. 20.2. An example of a cyclic network and local encoding kernels that do not render a solution for the global encoding kernels. f(x,y) = f(y,w) = 1 + f(w,x) 0 0 + f(x,y) 1 (20.6) (20.7) (20.8) f(w,x) = f(y,w) , which sum up to 1 0 = , 0 1 (20.9) a contradiction. The nonexistence of compatible global encoding kernels can also be interpreted in terms of the message transmission. Let the message x = [ a b ] be a generic vector in F 2 , where F denotes the base field. The symbol transmitted on channel e, given by x fe , are shown in Figure 20.2(b). In particular, the symbols transmitted on channels (x, y), (y, w), and (w, x), namely p, q, and r, are related through p = a+r q = b+p r = q. These equalities imply that a + b = 0, (20.13) a contradiction to the independence between the two components a and b of a generic message. (20.10) (20.11) (20.12) 488 20 Single-Source Linear Network Coding: Cyclic Networks Example 20.3. Let F be an extension field of GF (2)1 . Consider the same prescription of the local encoding kernels at the nodes as in Example 20.2 except that 11 KS = . (20.14) 00 The following three sets of global encoding kernels meet the requirement (19.17) in the definition of a linear network code: f(s,x) = f(s,y) = f(s,x) = f(s,y) = f(s,x) = f(s,y) = 1 0 1 , f(x,y) = , f(y,w) = f(w,x) = ; 0 0 0 1 1 0 , f(x,y) = , f(y,w) = f(w,x) = ; 0 0 0 1 0 1 , f(x,y) = , f(y,w) = f(w,x) = . 0 1 1 (20.15) (20.16) (20.17) 20.2 Convolutional Network Codes In a real network, the propagation delay, which includes the processing delay at the nodes and the transmission delay over the channels, cannot be zero. For a cyclic network, this renders the implementation non-physical because the transmission on an output channel of a node can only depend on the information received on the input channels of that node. Besides, technical difficulties as described in the last section arise even with the ideal assumption that there is no propagation delay. In this section, we introduce the unit-delay network as a model for network coding on a cyclic network G = (V, E), where V and E are the sets of nodes and channels of the network, respectively. In this model, a symbol is transmitted on every channel in the network at every discrete time index, with the transmission delay equal to exactly one time unit. Intuitively, this assumption on the transmission delay over a channel ensures no information looping in the network even in the presence of a directed cycle. The results to be developed in this chapter, although discussed in the context of cyclic networks, apply equally well to acyclic networks. As a time-multiplexed network in the combined space-time domain, a unitdelay network can be unfolded with respect to the time dimension into an indefinitely long network called a trellis network . Corresponding to a physical node t is a sequence of nodes t0 , t1 , t2 , in the trellis network, with the subscripts being the time indices. A channel ej in the trellis network represents the transmission on the physical channel e between times j and j + 1. When the physical channel e is from node t to node u, the channel ej in the trellis 1 In an extension field of GF (2), the arithmetic on the symbols 0 and 1 are modulo 2 arithmetic. 20.2 Convolutional Network Codes j=0 j=1 j=2 j=3 j=4 j=5 489 j=6 x0 0 x1 a0 x2 a1 x3 a1 a3 b3 s3 b1 b2 a2 x4 a2+b0 a4 b4 s4 b3 a3 x5 a0+ a3+b1 a5 b5 a4 x6 0 a0 b0 s0 0 0 a1 b1 s1 b0 a0 a2 b2 s2 s5 b4 s6 y0 0 y1 0 y2 0 y3 b0 y4 a0+b1 y5 a1+b2 y6 0 0 b0 a0+b1 a1+b2 a2+b0+b3 w0 w1 w2 w3 w4 w5 w6 Fig. 20.3. The trellis network depicting a convolutional network code defined on the physical network in Figure 20.2. network is from node tj to node uj+1 . Note that the trellis network is acyclic regardless of the topology of the physical network, because all the channels are pointing in the forward time direction so that a directed cycle cannot be formed. Example 20.4. Regard the network in Figure 20.2 as a unit-delay network. For each channel e in the network, the scalar values in the base field F transmitted on the channels ej , j 0 in the corresponding trellis network are determined by the local encoding kernels. This is illustrated in Figure 20.3. For instance, the channels (x, y)j , j 0 carry the scalar values 0, 0, a0 , a1 , a2 + b0 , a0 + a3 + b1 , a1 + a4 + b2 , , (20.18) respectively. This constitutes an example of a convolutional network code to be formally defined in Definition 20.6. Let cj be the scalar value in F transmitted on a particular channel in the network at time j. A succinct mathematical expression for the sequence of scalars c0 , c1 , c2 , is the z-transform cj z j = c0 + c1 z + c2 z 2 + , j=0 (20.19) where the power j of the dummy variable z represents discrete time. The pipelining of scalars transmitted over a time-multiplexed channel can thus be 490 20 Single-Source Linear Network Coding: Cyclic Networks regarded as the transmission of a power series over the channel. For example, the transmission of a scalar value on the channel (x, y)j for each j 0 in the trellis network in Figure 20.3 translates into the transmission of the power series a0 z 2 + a1 z 3 + (a2 + b0 )z 4 + (a0 + a3 + b1 )z 5 + (a1 + a4 + b2 )z 6 + (20.20) over the channel (x, y) in the network in Figure 20.2. The z-transform in (20.19) is a power series in the dummy variable z, which would be regarded as either a real number or a complex number in the context of signal analysis. However, in the context of convolutional coding, such a power series should not be regarded as anything more than a representation of the sequence of scalars c0 , c1 , c2 , . Specifically, the dummy variable z is not associated with any value, and there is no notion of convergence. Such power series are called formal power series. Given a field F , consider rational functions of a dummy variable z of the form p(z) , (20.21) 1 + zq(z) where p(z) and q(z) are polynomials. The following properties of such a function are relevant to our subsequent discussion: 1. The denominator has a constant term, so the function can be expanded into a power series by long division (see Example 20.5). 2. If p(z) is not the zero polynomial, the inverse function, namely 1 + zq(z) , p(z) exists. Note that the rational function in (20.22) does not represent a power series if p(z) contains the factor z, or equivalently, does not contain a constant term. The ring of power series over F is conventionally denoted by F [[z]]. Rational functions of the form (20.21) will be called rational power series which constitute a ring denoted by F z [408]. It follows directly from the definitions that F z is a subring of F [[z]]. We refer the reader to [124] for a comprehensive treatment of abstract algebra. In the following, we illustrate the concepts of rational power series through a few simple examples. Example 20.5. If z is a complex number, then we can write 1 = 1 + z + z2 + z3 + 1-z (20.23) (20.22) 20.2 Convolutional Network Codes 491 provided that |z| < 1, where we have interpreted the coefficients in the power series on the right hand side as real (or complex) numbers. If |z| > 1, the above expression is not meaningful because the power series diverges. However, if we do not associate z with a value but regard the coefficients in the power series as elements in a commutative ring, we can always write (1 - z)(1 + z + z 2 + z 3 + ) = (1 + z + z 2 + z 3 + ) - (z + z 2 + z 3 + ) = 1. (20.24) (20.25) In this sense, we say that 1 - z is the reciprocal of the power series 1 + z + z 2 + z 3 + and write 1 = 1 + z + z2 + z3 + . 1-z (20.26) 1 We also say that 1 + z + z 2 + z 3 + is the power series expansion of 1-z . In fact, the power series on the right hand side can be readily obtained by dividing 1 by 1 - z using long division. Alternatively, we can seek the inverse of 1 - z by considering the identity (1 - z)(a0 + a1 z + a2 z 2 + ) = 1. By equating the powers of z on both sides, we have a0 = 1 -a0 + a1 = 0 -a1 + a2 = 0 . . . Then by forward substitution, we immediately obtain 1 = a0 = a1 = a2 = , (20.27) (20.28) (20.29) (20.30) (20.31) (20.32) which gives exactly the power series obtained by long division. The reader can easily verify that long division indeed mimics the process of forward substitution. For polynomials p(z) and q(z) where q(z) is not the zero polynomial, we can always expand the rational function p(z) into a series. However, such a q(z) series is not always a power series. For example, 1 1 1 = z - z2 z 1-z 1 = (1 + z + z 2 + ) z = z -1 + 1 + z + z 2 + . (20.33) (20.34) (20.35) 492 20 Single-Source Linear Network Coding: Cyclic Networks The above is not a power series because of the term involving a negative power of z. In fact, the identity (z - z 2 )(a0 + a1 z + a2 z 2 + ) = 1 (20.36) has no solution for a0 , a1 , a2 , since there is no constant term on the left 1 hand side. Therefore, z-z2 indeed does not have a power series expansion. From the above example, we see that p(z) represents a rational power q(z) series if and only if q(z) has a nonzero constant term, or equivalently, does not contain the factor z. Definition 20.6 (Convolutional Network Code). An -dimensional convolutional network code on a unit-delay network over a base field F consists of an element kd,e (z) F z for every adjacent pair of channels (d, e) in the network as well as a column -vector fe (z) over F z for every channel e such that: (20.37) fe (z) = z dIn(t) kd,e (z)fd (z) for e Out(t). (20.38) The vectors fe (z) for the imaginary channels e In(s) consist of scalar components that form the standard basis of the vector space F . The vector fe (z) is called the global encoding kernel for channel e, and kd,e (z) is called the local encoding kernel for the adjacent pair of channels (d, e). The |In(t)| |Out(t)| matrix Kt (z) = [kd,e (z)]dIn(t),eOut(t) is called the local encoding kernel at node t. The constraint (20.37) is the time-multiplexed version of (19.17), with the factor z in the equation indicating a unit-time delay that represents the transmission delay over a channel. In the language of electronic circuit theory, for an adjacent pair of channels (d, e), the "gain" from channel d to channel e is given by zkd,e (z). A convolutional network code over a unit-delay network can be viewed as a discrete-time linear time-invariant (LTI) system defined by the local encoding kernels, where the local encoding kernel kd,e (z) specifies the impulse response of an LTI filter from channel d to channel e. The requirement that kd,e (z) is a power series corresponds to the causality of the filter. The additional requirement that kd,e (z) is rational ensures that the filter is implementable by a finite circuitry of shift registers. Intuitively, once the local encoding kernels are given, the global encoding kernels are uniquely determined. This is explained as follows. Write (20.39) fe (z) = j=0 fe,j z j = fe,0 + fe,1 z + fe,2 z 2 + (20.40) 20.2 Convolutional Network Codes 493 and kd,e (z) = j=0 kd,e,j z j = kd,e,0 + kd,e,1 z + kd,e,2 z 2 + , (20.41) where fe,j is a column -vector in F and kd,e,j is a scalar in F . Then the equation in (20.37) can be written in time domain as the convolutional equation j-1 fe,j = dIn(t) u=0 kd,e,u fd,j-1-u (20.42) for j 0, with the boundary conditions provided by (20.38): The vectors fe,0 , e In(t) form the standard basis of the vector space F . The vectors fe,j , e In(t) are the zero vector for all j 1. For j = 0, the summation in (20.42) is empty, so that fe,0 vanishes. For j 0, the right hand side of (20.42) involves the vectors fd,i only for 0 i j - 1. Thus the vectors fe,j , j 1 can be calculated recursively via (20.42) with the boundary condition fd,0 = 0 for all d E. (20.43) Together with fe,0 = 0, the global encoding kernel fe (z) is determined (cf. (20.40)). In other words, in a convolutional network code over a unit-delay network, the global encoding kernels are determined once the local encoding kernels are given. From (20.40), we see that the components of fe (z) are power series in z, so fe (z) is a column -vector over F [[z]]. In Theorem 20.9, we will further establish that the components of the global encoding kernels are in fact rational functions in z, proving that fe (z) is indeed a column -vector over f z as required in Definition 20.6 for a convolutional network code. Example 20.7. In Figure 20.2, denote the two imaginary channels by (o, s) and (o, s) . A convolutional network code is specified by the prescription of a local encoding kernel at every node as shown in the figure: Ks (z) = 10 1 , Kx (z) = Ky (z) = , Kw (z) = 1 , 01 1 (20.44) and a global encoding kernel for every channel: f(o,s) (z) = 1 0 , f(o,s) (z) = 0 1 10 01 10 01 1 z = 0 0 0 0 = 1 z (20.45) f(s,x) (z) = z f(s,y) (z) = z (20.46) (20.47) 494 20 Single-Source Linear Network Coding: Cyclic Networks &1 # $0 ! % " &0 # $1 ! % " s &z# $0 ! % " Ks = &1 0 # $0 1 ! % " & z 2 /(1 ' z 3 )# $ 4 3 ! % z /(1 ' z )" &0 # $z! % " Kx = &1# $1! %" x & z 4 /(1 ' z 3 )# $ 3 3 ! % z /(1 ' z ) " y Ky = &1# $1! %" & z 3 /(1 ' z 3 ) # $ 2 3 ! % z /(1 ' z )" w Kw = (1) Fig. 20.4. The local and global encoding kernels of the convolutional network code in Example 20.7. f(x,y) (z) = f(y,w) (z) = f(w,x) (z) = z 2 /(1 - z 3 ) z 4 /(1 - z 3 ) z 3 /(1 - z 3 ) z 2 /(1 - z 3 ) z 4 /(1 - z 3 ) , z 3 /(1 - z 3 ) (20.48) (20.49) (20.50) where the last three global encoding kernels have been solved from the following equations: f(x,y) (z) = z f(s,x) (z) f(w,x) (z) f(y,w) (z) = z f(s,y) (z) f(x,y) (z) 1 1 = z2 + z f(w,x) (z) 1 0 1 0 = z2 + z f(x,y) (z) 1 1 (20.51) (20.52) (20.53) f(w,x) (z) = z(f(y,w) (z)) 1 = z f(y,w) (z). These local and global encoding kernels of a 2-dimensional convolutional network code are summarized in Figure 20.4. Represent the message generated at source node s at time j, where j 0, by a row -vector xj F . Equivalently, source node s generates the message 20.2 Convolutional Network Codes 495 pipeline represented by the z-transform x(z) = j=0 xj z j , (20.54) which is a row -vector over F [[z]], the ring of power series over F . Here, x(z) is not necessarily rational. Through a convolutional network code, each channel e carries the power series x(z) fe (z). Write x(z) fe (z) = j=0 me,j z j , (20.55) where me,j = j xu fe,j-u . u=0 (20.56) For e Out(t), from the equation in (20.37), we obtain x(z) fe (z) = x(z) z dIn(t) kd,e (z) fd (z) kd,e (z) [ x(z) fd (z) ] , (20.57) (20.58) =z dIn(t) or equivalently in time domain, j-1 me,j = dIn(t) u=0 kd,e,u md,j-1-u . (20.59) The reader should compare (20.59) with (20.42). Note that the scalar values me,j , j 1 can be calculated recursively via (20.59) with the boundary condition md,0 = 0 for all d E. (20.60) Thus a node t calculates the scalar value me,j for transmitting on each output channel e at time j from the cumulative information it has received on all the input channels up to time j - 1. The convolutional equation (20.59) can be implemented by a finite circuit of shift-registers in a causal manner because the local encoding kernels belong to F z , the ring of rational power series over F (cf. Definition 20.6). Example 20.8. Consider the convolutional network code in Example 20.7. Let source node s pipelines the message 496 20 Single-Source Linear Network Coding: Cyclic Networks x(z) = bj z j . (20.61) aj z j j=0 j=0 Then the five channels (s, x), (s, y), (x, y), (y, w), and (w, x) carry the following power series, respectively: x(z) f(s,x) (z) = j=0 aj z j+1 bj z j+1 j=0 (20.62) x(z) f(s,y) (z) = x(z) f(x,y) (z) = (20.63) bj z j+4 /(1 - z 3 ) (20.64) aj z j+2 + j=0 j=0 = bj z j+4 aj z j+2 + j=0 j=0 z 3j j=0 (20.65) = a0 z 2 + a1 z 3 + (a2 + b0 )z 4 +(a0 + a3 + b1 )z 5 + (20.66) (20.67) x(z) f(y,w) (z) = j=0 aj z j+3 + j=0 bj z j+2 /(1 - z 3 ) bj z j+3 /(1 - z 3 ). j=0 x(z) f(w,x) (z) = aj z j+4 + j=0 (20.68) At each time j 0, the source generates a message xj = [ aj bj ]. Thus channel (s, x) carries the scalar 0 at time 0 and the scalar aj-1 at time j 1. Similarly, channel (s, y) carries the scalar 0 at time 0 and the scalar bj-1 at time j 1. For every channel e, write x(z) fe (z) = j=0 me,j z j (20.69) as in (20.55). The actual encoding process at node x is as follows. At time j, node x has received the sequence md,0 , md,1 , , md,j-1 for d = (s, x) and (w, x). Accordingly, at time j 1, channel (x, y) transmits the scalar value j-1 m(x,y),j = u=0 k(s,x),(x,y),u m(s,x),j-1-u j-1 + u=0 k(w,x),(x,y),u m(w,x),j-1-u (20.70) (20.71) = m(s,x),j-1 + m(w,x),j-1 . 20.2 Convolutional Network Codes 497 Similarly, channels (y, w) and (w, x) transmit the scalar values m(y,w),j = m(s,y),j-1 + m(x,y),j-1 and m(w,x),j = m(y,w),j-1 , (20.73) respectively. The values m(x,y),j , m(y,w),j , and m(w,x),j for j 1 can be calculated recursively by the above formulas with the boundary condition me,0 = 0 for all e E, (20.74) (20.72) and they are shown in the trellis network in Figure 20.3 for small values of j. For instance, the channel (x, y) carries the scalar values m(x,y),0 = 0, m(x,y),1 = 0, m(x,y),2 = a0 , m(x,y),3 = a1 , m(x,y),4 = a2 + b0 , m(x,y),5 = a0 + a3 + b1 , . The z-transform of this sequence is (20.75) bj z j+4 /(1 - z 3 ), (20.76) x(z) f(x,y) (z) = j=0 aj z j+2 + j=0 as calculated in (20.66). In the discussion following Definition 20.6, we have shown that once the local encoding kernels of a convolutional network code over a unit-delay network are given, the global encoding kernels are determined. The proof of the next theorem further provides a simple closed-form expression for the global encoding kernels fe (z), from which it follows that the entries in fe (z) indeed belong to F z as required in Definition 20.6. Theorem 20.9. Let F be the base field and kd,e (z) F z be given for every adjacent pair of channels (d, e) on a unit-delay network. Then there exists a unique -dimensional convolutional network code over F with kd,e (z) as the local encoding kernel for every (d, e). Proof. Let the unit-delay network be represented by a directed graph G = (V, E). Let [kd,e (z)] be the |E||E| matrix in which both the rows and columns are indexed by E, with the (d, e)th entry equal to the given kd,e (z) if (d, e) is an adjacent pair of channels, and equal to zero otherwise. Denote the global encoding kernel of channel e by fe (z) if exists. Let [fe (z)] be the |E| matrix obtained by putting the global encoding kernels fe (z), e E in juxtaposition. Let Hs (z) be the |E| matrix obtained by appending |E|-|Out(s)| columns of zeroes to the local encoding kernel Ks (z). The requirements (20.37) and (20.38) in Definition 20.6 can be written as [fe (z)] = z[fe (z)] [kd,e (z)] + zIHs (z), (20.77) 498 20 Single-Source Linear Network Coding: Cyclic Networks where I in the above denotes the identity matrix representing the global encoding kernels fe (z), e In(s) in juxtaposition. Rearranging the terms in (20.77), we obtain [fe (z)](I - z[kd,e (z)]) = zHs (z). (20.78) In the matrix z[kd,e (z)], the diagonal elements are equal to zero because (e, e) does not form an adjacent pair of channels for all e E, while the non-zero off-diagonal elements all contain the factor z. Therefore, det(I - z[kd,e (z)]) has the form 1 + zq(z), (20.79) where q(z) F z , so that it is invertible inside F z because [det(I - z[kd,e (z)])]-1 = is a rational power series. It follows that (I - z[kd,e (z)])-1 (20.81) 1 1 + zq(z) (20.80) exists and is a matrix over F z . Then the unique solution for [fe (z)] in (20.78) is given by [fe (z)] = zHs (z)(I - z[kd,e (z)])-1 . (20.82) With the two matrices [kd,e (z)] and Hs (z) representing the given local encoding kernels and the matrix [fe (z)] representing the global encoding kernels, (20.82) is a closed-form expression for the global encoding kernels in terms of the local encoding kernels. In particular, [fe (z)] is a matrix over F z because all the matrices on the right hand side of (20.82) are over F z . Thus we conclude that all the components of the global encoding kernels are in F z . Hence, the given local encoding kernels kd,e (z) for all adjacent pairs (d, e) together with the associated global encoding kernels fe (z), e In(s) E constitute a unique convolutional network code over the unit-delay network G. In view of Definition 19.7 for the global description of a linear network code over an acyclic network, Definition 20.6 can be regarded as the global description of a convolutional network code over a unit-delay network, while Theorem 20.9 renders a local description by specifying the local encoding kernels only. 20.3 Decoding of Convolutional Network Codes For a node t, let Ft (z) = [fe (z)]eIn(t) (20.83) be the |In(t)| matrix obtained by putting the global encoding kernels fe (z), e In(t) in juxtaposition. In the following, we define a convolutional 20.3 Decoding of Convolutional Network Codes 499 multicast, the counterpart of a linear multicast defined in Chapter 19, for a unit-delay cyclic network. The existence of a convolutional multicast will also be established. Definition 20.10 (Convolutional Multicast). An -dimensional convolutional network code on a unit-delay network qualifies as an -dimensional convolutional multicast if for every non-source node t with maxflow(t) , there exists an |In(t)| matrix Dt (z) over F z and a positive integer such that Ft (z) Dt (z) = z I, (20.84) where > 0 depends on node t and I is the identity matrix. The matrix Dt (z) and the integer are called the decoding kernel and the decoding delay at node t, respectively. Source node s generates the message pipeline x(z) = j=0 xj z j , (20.85) where xj is a row -vector in F and x(z) is a row -vector over F [[z]]. Through the convolutional network code, a channel e carries the power series x(z) fe (z). The power series x(z) fe (z) received by a node t from the input channels e In(t) form the row |In(t)|-vector x(z) Ft (z) over F [[z]]. If the convolutional network code is a convolutional multicast, node t can use the decoding kernel Dt (z) to calculate (x(z) Ft (z))Dt (z) = x(z)(Ft (z)Dt (z)) = x(z)(z I) = z x(z). (20.86) (20.87) (20.88) The row -vector z x(z) of power series represents the message pipeline generated by source node s delayed by time units. Note that > 0 because the message pipeline x(z) is delayed by one time unit at node s. Example 20.11. Consider the network in Figure 20.4. Again let source node s pipelines the message x(z) = j=0 aj z j j=0 bj z j . (20.89) For node x, we have Fx (z) = Let z z 4 /(1 - z 3 ) . 0 z 3 /(1 - z 3 ) (20.90) 500 20 Single-Source Linear Network Coding: Cyclic Networks Dx (z) = Then z 2 -z 3 . 0 1 - z3 (20.91) Fx (z)Dt (z) = z 3 I2 (20.92) (I2 is the 2 2 identity matrix). From channels (s, x) and (w, x), node x receives the row vector j+4 j+3 aj z + bj z (20.93) x(z)Fx (z) = aj z j+1 1 - z3 j=0 j=0 and decodes the message pipeline as aj z j+4 + bj z j+3 z 2 -z 3 z 3 x(z) = aj z j+1 . 0 1 - z3 1 - z3 j=0 j=0 (20.94) Decoding at node y is similar. Thus the 2-dimensional convolutional network code is a convolutional multicast. Toward proving the existence of a convolutional multicast, we first observe that Lemma 19.17 can be strengthened as follows with essentially no change in the proof. Lemma 20.12. Let g(y1 , y2 , , ym ) be a nonzero polynomial with coefficients ~ ~ ~ ~ in a field F . For any subset E of F , if |E| is greater than the degree of g in ~ every yj , then there exist a1 , a2 , , am E such that g(a1 , a2 , , am ) = 0. (20.95) In the above lemma, the values a1 , a2 , , am can be found by exhaustive ~ ~ ~ ~ search in E provided that E is finite. If E is infinite, simply replace E by a ~ sufficiently large finite subset of E. Theorem 20.13. There exists an -dimensional convolutional multicast over any base field F . Furthermore, the local encoding kernels of the convolutional multicast can be chosen in any sufficiently large subset of F z . Proof. Recall the equation (20.82) in the proof of Theorem 20.9: [fe (z)] = zHs (z)(I - z[kd,e (z)])-1 . (20.96) In this equation, the |E| matrix [fe (z)] on the left hand side represents the global encoding kernels, while the |E| matrix Hs (z) and the |E| |E| matrix [kd,e (z)] on the right hand side represent the local encoding kernels. Analogous to the proof of Theorem 19.20, denote by (F z ) the polynomial ring over F z with all the kd,e (z) as indeterminates. 20.3 Decoding of Convolutional Network Codes 501 Let t be a non-source node with maxflow(t) . Then there exist edgedisjoint paths from the imaginary channels to distinct channels in In(t). Put the global encoding kernels of these channels in juxtaposition to form the matrix Lt (z) over (F z ). We will show that det(Lt (z)) = 0 (F z ). Toward proving (20.97), it suffices to show that det(Lt (z)) = 0 F z (20.98) (20.97) when the determinant is evaluated at some particular values for the indeterminates kd,e (z). Analogous to the proof of Theorem 19.20, we set kd,e (z) = 1 (20.99) for all adjacent pairs of channels (d, e) along any one of the edge-disjoint paths, and set kd,e (z) = 0 (20.100) otherwise. Then with a suitable indexing of the columns, the matrix Lt (z) becomes diagonal with all the diagonal entries being powers of z. Hence, det(Lt (z)) is equal to some positive power of z, proving (20.98) for this particular choice of the indeterminates kd,e (x) and hence proving (20.97). As the conclusion (20.97) applies to every non-source node t with maxflow(t) , it follows that det(Lt (z)) = 0 (F z ). (20.101) t:maxflow(t) Let F (z) be the conventional notation for the field of rational functions in z over the given base field F . The ring F z of rational power series is a subset of F (z). Then any subset of F z is also a subset of F (z). Note that the ring F z is infinite. Then for any sufficiently large subset of F z , we can ~ ~ apply Lemma 20.12 to the polynomial in (20.101) with F = F (z) and E = to see that we can choose a value ad,e (z) F z for each of the indeterminates kd,e (z) so that det(Lt (z)) = 0 F z (20.102) t:maxflow(t) when evaluated at kd,e (z) = ad,e (z) for all (d, e), which in turn implies that det(Lt (z)) = 0 F z (20.103) for all nodes t such that maxflow(t) . Henceforth, the local encoding kernel kd,e (z) will be fixed at the appropriately chosen value ad,e (z) for all (d, e) as prescribed above. Without loss of generality, we assume that Lt (z) consists of the first columns of Ft (z). From (20.103), we can write 502 20 Single-Source Linear Network Coding: Cyclic Networks det(Lt (z)) = z 1 + zq(z) , p(z) (20.104) where p(z) and q(z) are polynomials over F and p(z) is not the zero polynomial. Note that the right hand side of (20.104) is the general form for a nonzero rational function in z. In this particular context, since the columns of Lt (z) are global encoding kernels as prescribed by (20.82), each containing the factor z in the numerator, we see that > 0. Denote by Jt (z) the adjoint matrix2 of Lt (z). Take the matrix p(z) Jt (z) 1 + zq(z) (20.105) and append to it |In(t)| - rows of zeroes to form an |In(t)| matrix Dt (z). Then Ft (z)Dt (z) = Lt (z) 0 = p(z) 1+zq(z) Jt (z) (20.106) (20.107) (20.108) (20.109) 0 p(z) Lt (z)Jt (z) 1 + zq(z) p(z) det(Lt (z))I = 1 + zq(z) = z I, where the last equality follows from (20.104). Hence, the matrix Dt (z) qualifies as a decoding kernel at node t in Definition 20.10. This proves the existence of the convolutional multicast as required. The proof of Theorem 20.13 constitutes an algorithm for constructing ~ a convolutional multicast. By noting the lower bound on the size of E in Lemma 20.12, a convolutional multicast can be constructed with high probability by randomly choosing the local encoding kernels in the subset of F z provided that is much larger than sufficient. Example 20.14. When the base field F is sufficiently large, Theorem 20.13 can be applied with = F so that the local encoding kernels of the convolutional multicast can be chosen to be scalars. This special case is the convolutional counterpart of Theorem 19.20 for the existence of a linear multicast over an acyclic network. In this case, the local encoding kernels can be found by exhaustive search over F . More generally, by virtue of Lemma 20.12, the same exhaustive search applies to any large enough subset of F z . For example, F can be GF (2) and can be the set of all binary polynomials up to a sufficiently large degree. 2 For a matrix B whose entries are elements in a ring, denote by Adj(B) the adjoint matrix of B. Then Adj(B)B = BAdj(B) = det(A)I. Chapter Summary 503 Chapter Summary Algebraic Structures: F [[z]] denotes the ring of power series in z over F . F z denotes the ring of rational power series in z over F . F (z) denotes the field of rational functions in z over F . Convolutional Network Code: kd,e (z) F z is the local encoding kernel of the adjacent pair of channels (d, e). fe (z) (F z ) is the global encoding kernel of channel e. fe (z) = z dIn(t) kd,e (z)fd (z) for e Out(t). fe (z), e In(s) form the standard basis of F . Channel e carries the power series x(z) fe (z), where x(z) (F [[z]]) is the message pipeline generated at source node s. Uniqueness of Convolutional Network Code: For given kd,e (z) for every adjacent pair of channels (d, e) on a unit-delay network, there exists a unique convolutional network code with kd,e (z) as the local encoding kernel for every (d, e). The global encoding kernels fe (z) can be expressed in terms of the local encoding kernels kd,e (z) as [fe (z)] = zHs (z)(I - z[kd,e (z)])-1 , where Hs (z) is determined by the local encoding kernel at source node s. Convolutional Multicast: A convolutional network code is a convolutional multicast if for every node t = s with maxflow(t) , there exists an |In(t)| matrix Dt (z) over F z and a positive integer such that Ft (z) Dt (z) = z I, where Ft (z) = [fe (z)]eIn(t) and > 0 depends on node t. The matrix Dt (z) and the integer are called the decoding kernel and the decoding delay at node t, respectively. Existence and Construction: A convolutional multicast on a unit-delay network with the local encoding kernels chosen in a subset of F z exists and can be constructed randomly (with high probability) when is sufficiently large. For example, can be the base field F provided that F is sufficiently large, or can be the set of all binary polynomials up to a sufficiently large degree. 504 20 Single-Source Linear Network Coding: Cyclic Networks Problems 1. Show that the right hand side of (20.104) is the general form for a nonzero rational function in z. 2. A formal Laurent series over a field F has the form a-m z -m + a-(m-1) z -(m-1) + + a-1 z -1 + a0 + a1 z + a2 z 2 + , where m is a nonnegative integer. Show that for any formal Laurent series f (z) over F , there exists a unique formal Laurent series g(z) over F such that f (z)g(z) = 1. 3. Verify the following series expansion: 1 = -z -1 - z -2 - z -3 - . 1-z Can you obtain this series by long division? 4. Construct a finite circuit of shift-registers that implements a discrete-time LTI system with transfer function a0 + a1 z + + an z n , b0 + b1 z + + bn z n where ai and bi are elements in a finite field and b0 = 0. 5. Consider the convolutional network code in Figure 20.4. a) Is it a convolutional multicast? b) If your answer in a) is positive, give the decoding kernel at node y with minimum decoding delay. 11 c) Change Ks to and determine the corresponding global encoding 01 kernels. d) Instead of a convolutional multicast, can you construct a linear multicast on the network? Historical Notes The asymptotic achievability of the max-flow bound for unit-delay cyclic networks was proved by Ahlswede et al. [6], where an example of a convolutional network code achieving this bound was given. Li et al. [230] conjectured the existence of convolutional multicasts on such networks. This conjecture was subsequently proved by Koetter and Mdard [202]. Construction and decode ing of convolutional multicast have been studied by Erez and Feder [105][106], Fragouli and Soljanin [122], and Barbero and Ytrehus [24]. The unifying treatment of convolutional codes here is based on Li and Yeung [229] (see also Yeung et al. [408]). Li and Ho [228] recently obtained a general abstract formulation of convolutional network codes based on ring theory. 21 Multi-Source Network Coding In Chapters 19 and 20, we have discussed single-source network coding in which an information source is multicast in a point-to-point communication network. The maximum rate at which information can be multicast has a simple characterization in terms of the maximum flows in the graph representing the network. In this chapter, we consider the more general multi-source network coding problem in which more than one mutually independent information sources are generated at possibly different nodes, and each of the information sources is multicast to a specific set of nodes. The achievable information rate region of a multi-source network coding problem, which will be formally defined in Section 21.4, refers to the set of all possible rates at which multiple information sources can be multicast simultaneously on a network. In a single-source network coding problem, we are interested in characterizing the maximum rate at which information can be multicast from the source node to all the sink nodes. In a multi-source network coding problem, we are interested in characterizing the achievable information rate region. As discussed in Section 17.3, source separation is not necessarily optimal for multi-source network coding. It is therefore not a simple extension of singlesource network coding. Unlike the single-source network coding problem which has an explicit solution, the multi-source network coding problem has not been completely solved. In this chapter, by making use of the tools we have developed for information inequalities in Chapter 13 to Chapter 15, we will develop an implicit characterization of the achievable information rate region for multi-source network coding on acyclic networks. 21.1 The Max-Flow Bounds The max-flow bound, which fully characterizes the maximum rate of an information source that can be multicast in a network, plays a central role in single-source network coding. We now revisit this bound in the context of 506 21 Multi-Source Network Coding X1X2 1 [X1] b1 1 b2 2 4 [X1X2] (a) 3 [X2] 2 b1 3 4 (b) b2 Fig. 21.1. A network which achieves the max-flow bound. multi-source network coding. In the following discussion, the unit of information is the bit. Consider the graph in Figure 21.1(a). The capacity of each edge is equal to 1. Two independent information sources X1 and X2 with rates 1 and 2 , respectively are generated at node 1. Suppose we want to multicast X1 to nodes 2 and 4 and multicast X2 to nodes 3 and 4. In the figure, an information source in square brackets is one which is to be received at that node. It is easy to see that the values of a max-flow from node 1 to node 2, from node 1 to node 3, and from node 1 to node 4 are respectively 1, 1, and 2. At node 2 and node 3, information is received at rates 1 and 2 , respectively. At node 4, information is received at rate 1 + 2 because X1 and X2 are independent. Applying the max-flow bound at nodes 2, 3, and 4, we have 1 1 2 1 and 1 + 2 2, (21.3) respectively. We refer to (21.1) to (21.3) as the max-flow bounds. Figure 21.2 is an illustration of all (1 , 2 ) which satisfy these bounds, where 1 and 2 are obviously nonnegative. We now show that the rate pair (1, 1) is achievable. Let b1 be a bit generated by X1 and b2 be a bit generated by X2 . In the scheme in Figure 21.1(b), b1 is received at node 2, b2 is received at node 3, and both b1 and b2 are received at node 4. Thus the multicast requirements are satisfied, and the information rate pair (1, 1) is achievable. This implies that all (1 , 2 ) which satisfy the max-flow bounds are achievable because they are all inferior to (1, 1) (see Figure. 21.2). In this sense, we say that the max-flow bounds are achievable. (21.1) (21.2) 21.1 The Max-Flow Bounds 2 507 (0,2) (1,1) (2,0) 1 Fig. 21.2. The max-flow bounds for the network in Figure 21.1. Suppose we now want to multicast X1 to nodes 2, 3, and 4 and multicast X2 to node 4 as illustrated in Figure 21.3. Applying the max-flow bound at either node 2 or node 3 gives 1 1, (21.4) and applying the max-flow bound at node 4 gives 1 + 2 2. (21.5) Figure 21.4 is an illustration of all (1 , 2 ) which satisfy these bounds. We now show that the information rate pair (1, 1) is not achievable. Suppose we need to send a bit b1 generated by X1 to nodes 2, 3, and 4 and send a bit b2 generated by X2 to node 4. Since b1 has to be recovered at node 2, the bit sent to node 2 must be an invertible transformation of b1 . This implies that the bit sent to node 2 cannot not depend on b2 . Similarly, the bit sent to node 3 also cannot depend on b2 . Therefore, it is impossible for node 4 to recover b2 because both the bits received at nodes 2 and 3 do not depend on X1X2 1 [ X1 ] 2 4 [ X1 X 2 ] Fig. 21.3. A network which does not achieve the max-flow bounds. 3 [X1] 508 21 Multi-Source Network Coding 2 (0,2) (1,1) (2,0) 1 Fig. 21.4. The max-flow bounds for the network in Figure 21.3. b2 . Thus the information rate pair (1, 1) is not achievable, which implies that the max-flow bounds (21.4) and (21.5) are not achievable. From this example, we see that the max-flow bounds do not always fully characterize the achievable information rate region. We leave it as an exercise for the reader to show that for this example, source separation is in fact optimal. 21.2 Examples of Application Multi-source network coding is a very rich model which encompasses many communication situations arising from fault-tolerant network communication, disk array, satellite communication, etc. In this section, we discuss some applications of the model. 21.2.1 Multilevel Diversity Coding Let X1 , X2 , , XK be K information sources in decreasing order of importance. These information sources are encoded into pieces of information. There are a number of users, each of them having access to a certain subset of the information pieces. Each user belongs to a level between 1 and K, where a Level k user can decode X1 , X2 , , Xk . This model, called multilevel diversity coding, finds applications in fault-tolerant network communication, disk array, and distributed data retrieval. Figure 21.5 shows a graph which represents a 3-level diversity coding system. The graph consists of three layers of nodes. The top layer consists of a node at which information sources X1 , X2 , and X3 are generated. These information sources are encoded into three pieces, each of which is stored in a distinct node in the middle layer. A dummy node is associated with such a 21.2 Examples of Application X1X2X3 509 Information Sources Storage nodes Users [X1] [X1] [X1] [X1X2] [X1X2] [X1X2] [X1X2X3] Level 1 Level 2 Level 3 Fig. 21.5. A 3-level diversity coding system. node to model the effect that the same information is retrieved every time the node is accessed (see the discussion in Section 17.2). The nodes in the bottom layer represent the users, each of them belonging to one of the three levels. Each of the three Level 1 users has access to a distinct node in the second layer (through the associated dummy node) and decodes X1 . Similarly, each of the three Level 2 users has access to a distinct set of two nodes in the second layer and decodes X1 and X2 . There is only one Level 3 user, who has access to all the three nodes in the second layer and decodes X1 , X2 , and X3 . The model represented by the graph in Figure 21.5 is called symmetrical 3-level diversity coding because the model is unchanged by permuting the nodes in the middle layer. By degenerating information sources X1 and X3 , the model is reduced to the diversity coding model discussed in Section 18.2. In the following, we describe two applications of symmetrical multilevel diversity coding: Fault-Tolerant Network Communication In a computer network, a data packet can be lost due to buffer overflow, false routing, breakdown of communication links, etc. Suppose the packet carries K messages, X1 , X2 , , XK , in decreasing order of importance. For improved reliability, the packet is encoded into K sub-packets, each of which is sent over a different channel. If any k sub-packets are received, then the messages X1 , X2 , , Xk can be recovered. Disk Array Consider a disk array which consists of K disks. The data to be stored in the disk array are segmented into K pieces, X1 , X2 , , XK , in decreasing order of importance. Then X1 , X2 , , XK are encoded into K pieces, each of which is stored on a separate disk. When any k out of the K disks are functioning, the data X1 , X2 , , Xk can be recovered. 510 21 Multi-Source Network Coding Transmitter Receiver Fig. 21.6. A satellite communication network. 21.2.2 Satellite Communication Network In a satellite communication network, a user is at any time covered by one or more satellites. A user can be a transmitter, a receiver, or both. Through the satellite network, each information source generated at a transmitter is multicast to a certain set of receivers. A transmitter can transmit to all the satellites within the line of sight, while a receiver can receive from all the satellites within the line of sight. Neighboring satellites may also communicate with each other. Figure 21.6 is an illustration of a satellite communication network. The satellite communication network in Figure 21.6 can be represented by the graph in Figure 21.7 which consists of three layers of nodes. The top layer represents the transmitters, the middle layer consists of nodes representing the satellites as well as the associated dummy nodes modeling the broadcast nature of the satellites, and the bottom layer represents the receivers. If a satellite is within the line-of-sight of a transmitter, then the corresponding pair of nodes are connected by a directed edge. Likewise, if a receiver is within the line-of-sight of a satellite, then the corresponding pair nodes are connected by a directed edge. An edge between two nodes in the middle layer represent the communication links between two neighboring satellites. Each information source is multicast to a specified set of receiving nodes as shown. 21.3 A Network Code for Acyclic Networks Let G = (V, E) denote an acyclic point-to-point communication network, where V and E are the set of nodes and the set of channels, respectively. We assume that each channel e E is error-free with rate constraint Re . As in 21.3 A Network Code for Acyclic Networks X1X2 X3 X4X5X6 X7 X8 511 Transmitters Satellites Receivers [X1X8] [X2X7] [X4] [X5X6] [X3X5] [X3X7X8] Fig. 21.7. A graph representing a satellite communication network. our previous discussions, we let In(t) and Out(t) be the set of input channels and the set of output channels of node t, respectively. Let S V be the set of source nodes and T V be the set of sink nodes. Without loss of generality, we assume G has the structure that a source node has no input channel and a sink node has no output channel. Accordingly, S and T are disjoint subsets of V . An information source represented by a random variable Xs is generated at a source node s S, where Xs takes values in Xs = {1, 2, , 2ns } (21.6) according to the uniform distribution, where s is the rate of the information source. The information sources Xs , s S are assumed to be mutually independent. To simplify the notation, we will denote (Xs : s A) by XA , sA Xs by XA , etc. At a sink node t T , the set of information sources X(t) , where (t) S, is received. We assume that each information source is received at at least one sink node, i.e., for every s S, s (t) for some t T . In the case when (t) = S for all t T , the problem is reduced to the single-source network coding problem. Definition 21.1. An (n, (e : e E), (s : s S)) block code of length n on a given communication network is defined by 1) for all source node s S and all channels e Out(s), a local encoding function ke : Xs {0, 1, , e }; (21.7) 2) for all node i V \ (S T ) and all channels e Out(i), a local encoding function 512 21 Multi-Source Network Coding ke : {0, 1, , d } {0, 1, , e }; dIn(i) (21.8) 3) for all sink node t T , a decoding function gt : {0, 1, , d } X(t) . dIn(t) (21.9) The nodes in V are assumed to be ordered in an upstream-to-downstream manner as prescribed in Proposition 19.1. This defines a coding order among the nodes such that whenever a node encodes, all the information needed would have already been received on the input channels of that node. For all sink node t T , define t = Pr gt (XS ) = X(t) , ~ (21.10) where gt (XS ) denotes the value of gt as a function of XS . t is the probability ~ that the set of information sources X(t) is decoded incorrectly at sink node t. Throughout this chapter, all the logarithms are in the base 2. Definition 21.2. An information rate tuple = (s : s S), where 0 (componentwise) is asymptotically achievable if for any > 0, there exists for sufficient large n an (n, (e : e E), (s : s S)) code such that n-1 log e Re + , e E s s - , s S t , t T. (21.11) (21.12) (21.13) For brevity, an asymptotically achievable information rate tuple will be referred to as an achievable information rate tuple. 21.4 The Achievable Information Rate Region In this section, we define the achievable information rate region and give a characterization of this region. Definition 21.3. The achievable information rate region, denoted by R, is the set of all achievable information rate tuples . Remark It follows from the definition of the achievability of an information rate vector that if is achievable, then is achievable for all 0 . Also, if (k) , k 1 are achievable, then it can be proved by techniques similar to those in the proof of Theorem 8.12 that = lim (k) k (21.14) 21.4 The Achievable Information Rate Region 513 is also achievable, i.e., R is closed. The details are omitted here. Consider the set of all information rate tuples such that there exist auxiliary random variables {Ys , s S} and {Ue , e E} which satisfy the following conditions: H(Ys ) s , s S H(YS ) = sS (21.15) (21.16) (21.17) (21.18) (21.19) (21.20) H(Ys ) H(UOut(s) |Ys ) = 0, s S H(UOut(i) |UIn(i) ) = 0, i V \ (S T ) H(Ue ) Re , e E H(Y(t) |UIn(t) ) = 0, t T, where YS denotes (Ys : s S), UOut(s) denotes (Ue : e Out(s)), etc. Here, Ys is an auxiliary random variable associated with the information source Xs , and Ue is an auxiliary random variable associated with the codeword sent on channel e. The interpretations of (21.15) to (21.20) are as follows. The inequality in (21.15) says that the entropy of Ys is greater than or equal to s , the rate of the information source Xs . The equality in (21.16) says that Ys , s S are mutually independent, which corresponds to the assumption that the information sources Xs , s S are mutually independent. The equality in (21.17) says that UOut(s) is a function of Ys for s S, and the equality in (21.18) says that UOut(i) is a function of UIn(i) for i V \ (S T ). These correspond to the requirement that the codewords sent out by a source node s are functions of the information source Xs , and that the codewords sent out by a non-source node i are functions of the codewords received by node i. The inequality in (21.19) says that the entropy of Ue is less than or equal to Re , the rate constraint for channel e. The equality in (21.20) says that Y(t) is a function of UIn(t) for t T , which corresponds to the requirement that the information sources to be received at a sink node t can be decoded from the codewords received at node t. For a given multi-source network coding problem, let N = {Ys : s S; Ue : e E} (21.21) be a collection of discrete random variables whose joint distribution is unspecified, and let QN = 2N \ {} (21.22) with cardinality 2|N | - 1. Let HN be the |QN |-dimensional Euclidean space with the coordinates labeled by hA , A QN . A vector h = (hA : A QN ) (21.23) in HN is said to be finitely entropic if there exists a joint distribution for all X N , where |X | < for all X N and 514 21 Multi-Source Network Coding hA = H(X : X A) (21.24) for all A QN . Note that h HN is entropic if it is finitely entropic, but not vice versa. We then define the region N = {h HN : h is finitely entropic}. (21.25) To simplify notation, for any nonempty A, A QN , define hA|A = hAA - hA , (21.26) where we have used juxtaposition to denote the union of two sets. In using the above notation, we do not distinguish elements and singletons of N , i.e., for a random variable Z N , hZ is the same as h{Z} . We now define the following regions in HN : L1 = h HN : hYS = sS hYs (21.27) (21.28) (21.29) (21.30) (21.31) L2 = h HN : hUOut(s) |Ys = 0, s S L3 = h HN : hUOut(i) |UIn(i) = 0, i V \ (S T ) L4 = { h HN : hUe Re , e E } L5 = h HN : hY(t) |UIn(t) = 0, t T . Evidently, (21.27) to (21.31) are the regions in HN corresponding to (21.16) to (21.20), respectively. We further denote i Li by L for {1, 2, 3, 4, 5}. We now introduce a few notations. For a vector h HN , let hYS = (hYs : s S). For a subset B of HN , let 1. projYS (B) = {hYS : h B} be the projection of the set B on the coordinates hYs , s S; 2. (B) = {h HN : 0 h h for some h B}; 3. con(B) be the convex hull of B; 4. B be the closure of B. Note that a vector h 0 is in (B) if and only if it is inferior to some vector h in B. The following theorem gives a characterization of the achievable information rate region R in terms of the region N . Definition 21.4. Define the region R = projYS con(N L123 ) L4 L5 . (21.32) Theorem 21.5. R = R . This theorem, which characterizes the achievable information rate region R, will be proved in Sections 21.6 and 21.7. In the next section, we first discuss how more explicit inner and outer bounds on R can be obtained. 21.5 Explicit Inner and Outer Bounds 515 21.5 Explicit Inner and Outer Bounds Theorem 21.5 gives a characterization of the achievable information rate re gion R in terms of the region N . However, so far there exists no complete characterization of N . Therefore, the region R cannot be evaluated explicitly. In the definition of R in (21.32), if N is replaced by an inner bound (outer bound) on N , then an inner bound (outer bound) on R is obtained. The results in [261] and [259] which are beyond the scope of our discussion here, provide explicit constructions of inner bounds on N . We now discuss how an explicit outer bound on R can be obtained. To facilitate our discussion, we further define iA;A = hA - hA|A and iA;A |A = hA|A - hA|A A (21.34) for A, A , A QN . Let N be the set of h HN such that h satisfies all the basic inequalities involving some or all of the random variables in N , i.e., for all A, A , A QN , hA 0 hA|A 0 iA;A 0 iA;A |A 0. (21.33) (21.35) (21.36) (21.37) (21.38) We know from Section 14.2 that N N . Then upon replacing N by N in (21.32), we obtain an outer bound on Rout . This outer bound, called the LP bound (LP for linear programming), is given by RLP = projYS con(N L123 ) L4 L5 = projYS (N L12345 ) , (21.39) (21.40) where the last equality follows because both N and L123 are closed and convex. Since RLP involves only a finite number of linear constraints, RLP can be evaluated explicitly. Using the technique in [411], it can be proved that RLP is tight for most special cases of multi-source network coding on an acyclic network for which the achievable information region is known. In addition to single-source network coding, these include the models described in [160] [400] [308] [410] [411] [341]. Since RLP encompasses all Shannon-type information inequalities and the converse proofs of the achievable information rate region for all these special cases do not involve non-Shannon-type inequalities, the tightness of RLP for all these cases is expected. 516 21 Multi-Source Network Coding However, there exist multi-source network coding problems that requires non-Shannon-type inequalities for the characterization of the achievable information rate region [99] [63]. As new non-Shannon-type inequalities are discovered from time to time, improved outer bounds on R can be obtained by incorporating these inequalities. 21.6 The Converse In this section, we establish the converse part of Theorem 21.5, namely R projYS con(N L123 ) L4 L5 =R. (21.41) Let k be a sequence such that 0 < k < 1 for all k and k monotonically decreases to 0 as k . Consider an achievable information rate tuple R. Then for all k, for all sufficiently large n, there exists an (k) (k) n, (e : e E), (s : s S) (21.42) code satisfying (k) n-1 log e Re + (k) s (k) t (k) k, k, eE sS (21.43) (21.44) (21.45) s - k, t T, where t denotes the decoding error probability at sink node t (cf. (21.10)). We now fix k to be any positive integer and temporarily suppress all the superscripts involving k. For all e E, let Ue be the codeword sent on channel e and denote the alphabet of Ue by Ue . The following lemma, whose proof will be deferred to the end of the section, is a consequence of Fano's inequality. Lemma 21.6. For all n and k, for all t T , H(X(t) |UIn(t) ) nt (n, where 1. t (n, 2. t (n, 3. t (n, k) k) k ), (21.46) is bounded; 0 as n, k ; k ) is monotonically decreasing in both n and k. Since the information source Xs , s S are mutually independent, H(XS ) = sS H(Xs ). (21.47) 21.6 The Converse 517 For any s S, since UOut(s) is a function of Xs , H(UOut(s) |Xs ) = 0. Similarly, for all i V \ (S T ), H(UOut(i) |UIn(i) ) = 0. For all e E, H(Ue ) log |Ue | = log(e + 1) n(Re + 2 k ) (21.50) (21.51) (21.52) (21.49) (21.48) where (21.52) follows from (21.43) assuming that n is sufficiently large. For all t T , from Lemma 21.6, we have H(X(t) |UIn(t) ) nt (n, For all s S, from (21.44), H(Xs ) = log |Xs | = log 2ns ns n(s - k ). k ). (21.53) (21.54) By letting Ys = Xs for all s S, we then obtain from (21.47) to (21.49) and (21.52) to (21.54) that H(YS ) = sS H(Ys ) (21.55) (21.56) (21.57) (21.58) (21.59) (21.60) H(UOut(s) |Ys ) = 0, s S H(UOut(i) |UIn(i) ) = 0, i V \ (S T ) H(Ue ) n(Re + 2 k ), e E H(Y(t) |UIn(t) ) nt (n, H(Ys ) n(s - k ), t T k ), s S. Now define the following two regions in HN : Ln 4, Ln 5, k k = {h HN : hUe n(Re + 2 k ), e E} = {h HN : hY(t) |UIn(t) nt (n, k ), t (21.61) (21.62) T }. Note that all the auxiliary random variables Ys , s S and Ue , e E have finite alphabets, because |Ys | = |Xs | = 2ns < and log |Ue | n(Re + 2 k ) < (21.64) (21.63) 518 21 Multi-Source Network Coding (cf. (21.50) through (21.52)). Then we see from (21.55) to (21.60) that there exists h(k) N (21.65) such that h(k) L123 Ln 4, and hYs n(s - h(k) N L123 Ln 4, (k) k) k Ln 5, k (21.66) (21.67) for all s S. From (21.65) and (21.66), we obtain k Ln k . 5, (21.68) Upon dividing by n, (21.67) becomes n-1 hYs s - (k) k. (21.69) Since N L123 contains the origin in HN , we see that n-1 h(k) con(N L123 ) L4, k L5, k , (21.70) where L4, L5, k k = {h HN : hUe Re + 2 k , e E} = {h HN : hY(t) |UIn(t) t (n, k ), t T }. (21.71) (21.72) Note that the region L5, k depends on n though it is not indicated explicitly. For all n and k, define the set B (n,k) = {h con(N L123 ) L4, k L5, k : (21.73) hYs s - k for all s S}. Lemma 21.7. For all n and k, the set B (n,k) is compact1 . Again, the proof of this lemma is deferred to the end of the section. Now from Lemma 21.6, t (n, k ) is monotonically decreasing in both n and k, so for all n and k, B (n+1,k) B (n,k) (21.74) and B (n,k+1) B (n,k) . (21.75) For any fixed k and all sufficiently large n, we see from (21.69) and (21.70) that B (n,k) is nonempty. Since B (n,k) is compact by Lemma 21.7, n 1 lim B (n,k) = n=1 B (n,k) (21.76) A subset of the Euclidean space is compact if and only if it is closed and bounded. 21.6 The Converse 519 is both compact and nonempty. By the same argument, we conclude that k n lim lim B (n,k) = k=1 n=1 B (n,k) (21.77) is also nonempty. Now the set k n lim lim B (n,k) (21.78) is equal to h con(N L123 ) L4 L5 : hYs s for all s S . (21.79) Hence, there exists h satisfying h con(N L123 ) L4 L5 (21.80) (21.81) and hYs s , s S. Let r = projYS (h ). Then we have r projYS con(N L123 ) L4 L5 (21.82) (21.83) and r componentwise. By (21.82) and (21.83), we finally conclude that projYS con(N L123 ) L4 L5 . (21.84) This completes the proof of the converse part of Theorem 21.5. Proof of Lemma 21.6. For any t T , by Fano's inequality, we have H(X(t) |UIn(t) ) 1 + t log |X(t) | = 1 + t H(X(t) ) 1+ k H(X(t) ), (21.85) (21.86) (21.87) where (21.86) follows because Xs is distributed uniformly on Xs and Xs , s S are mutually independent, and (21.87) follows from (21.45). Then H(X(t) ) = I(X(t) ; UIn(t) ) + H(X(t) |UIn(t) ) a) (21.88) (21.89) (21.90) (21.91) I(X(t) ; UIn(t) ) + 1 + H(UIn(t) ) + 1 + b) k H(X(t) ) k H(X(t) ) eIn(t) log e + 1 + n(Re + k) k H(X(t) ) c) eIn(t) +1+ k H(X(t) ), (21.92) 520 21 Multi-Source Network Coding where a) follows from (21.87); b) follows from Theorem 2.43; c) follows from (21.43). Rearranging the terms in (21.92), we obtain n H(X(t) ) (Re + 1- k eIn(t) 1 . k) + n (21.93) Substituting (21.93) into (21.87), we have 1 k H(X(t) |UIn(t) ) < n + n 1- k = nt (n, where 1 k t (n, k ) = + n 1- k k ), eIn(t) 1 (Re + k ) + n (21.94) (21.95) eIn(t) 1 (Re + k ) + . n k (21.96) monotonically Invoking the assumption that 0 < k < 1 for all k and decreases to 0 as k , it is evident that 1. t (n, 2. t (n, 3. t (n, k) k) is bounded for all n and k; 0 as n, k ; k ) is monotonically nonincreasing in both n and k. The lemma is proved. Proof of Lemma 21.7. We need to show that the set B (n,k) is both closed and bounded. The closedness of B (n,k) is immediate from its definition. To establish the boundedness of B (n,k) , we need to show that for any h B (n,k) , all the components of h are bounded. Consider any h B (n,k) . Since B (n,k) L4, k , we see from (21.71) that hUe are bounded for all e E. Since B (n,k) L5, k , we see from (21.72) that for every t T , hY(t) hY(t) UIn(t) = hY(t) |UIn(t) + hUIn(t) t (n, t (n, k) k) (21.97) (21.98) (21.99) (21.100) (21.101) hUe eIn(t) + hUIn(t) + (21.102) 21.7 Achievability 521 where (21.100) and the boundedness of t (n, k ) follow from Lemma 21.6. This shows that hY(t) is bounded for all t T . In our model, for every s S, there exists at least one t T such that s (t). Then the boundedness of hY(t) for all t T implies the boundedness of hYs for all s S. Finally, the boundedness of all the other components of h is established by invoking the independence bound for entropy. The lemma is proved. 21.7 Achievability In this section, we establish the direct part of Theorem 21.5, namely R = projYS con(N L123 ) L4 L5 R. (21.103) Before we proceed, we first prove an alternative form of R that will be used in constructing the random code. For a subset B of HN , let D(B) = {h : h B and 0 1}. Define the two subsets A1 = con(N L123 ) (21.104) (21.105) (21.106) and A2 = D(N L123 ) of HN . Lemma 21.8. A1 = A2 . Proof. Since the origin of HN is in N , it is also in N L123 because L123 is a linear subspace of HN . Upon observing that for 0 1, h = (1 - )0 + h is a convex combination of 0 and h, we obtain D(N L123 ) con(N L123 ). (21.107) (21.108) It follows that A2 A1 . To prove that A1 A2 , it suffices to show that A2 is convex because 1. (N L123 ) A2 , where A2 is closed; 2. A1 is the smallest closed convex set containing N L123 . Toward this end, consider any h1 , h2 A2 and any 0 1. We will show that h = h1 + (1 - )h2 A2 . (21.109) 522 21 Multi-Source Network Coding Here, we can assume without loss of generality that h1 , h2 = 0, because otherwise (21.109) holds by the definition of A2 . Since h1 , h2 A2 , there exist hk , hk D(N L123 ) such that hk h1 and hk h2 . Again, we can 1 2 1 2 assume without loss of generality that hk , hk = 0 for all k because h1 , h2 = 0. 1 2 Since hk , hk D(N L123 ), we can write 1 2 k^ hk = 1 hk 1 1 (21.110) and k^ hk = 2 hk , 2 2 (21.111) k k k k ^ ^ where hk , hk N L123 and 0 < 1 , 2 1. Note that 1 and 2 are 1 2 k k k k strictly positive because h1 , h2 = 0. Now let n1 and n2 be integer sequences such that nk , nk and 1 2 k nk 2 1 , k k 1- n2 1 (21.112) and let ^ ^ ^ hk = nk hk + nk hk . 1 1 2 2 It can be seen from Lemma 15.3 and Corollary 15.4 that ^ hk N . (21.113) (21.114) ^ ^ ^ Furthermore, since hk , hk L123 and L123 is a linear subspace, hk L123 . 1 2 Therefore, ^ hk N L123 . (21.115) Let hk = k k 1 2 ^ hk . k + nk 1 2 k nk 2 1 (21.116) k k Since 1 , 2 1 and nk , nk , for sufficiently large k, 1 2 k k 1 2 1, k k nk 2 + nk 1 1 2 (21.117) and therefore hk D(N L123 ) D(N L123 ) = A2 . (21.118) Substituting (21.113), (21.110), and (21.111) into (21.116), we obtain hk = k nk 2 nk k 1 hk + k k 2 1 k k hk . 1 k k nk 2 + nk 1 n1 2 + n2 1 2 1 2 (21.119) It can readily be seen from (21.112) that 21.7 Achievability k nk 2 1 k + nk 1 2 523 k nk 2 1 (21.120) and k nk 1 2 1 - . k k nk 2 + nk 1 1 2 (21.121) Since hk h1 and hk h2 , we see from (21.119) and (21.109) that hk h. 1 2 Finally, since hk A2 and A2 is closed, we conclude that h A2 . Therefore, A2 is convex, and hence A1 A2 . The lemma is proved. By virtue of this lemma, we can write R = projYS D(N L123 ) L4 L5 , (21.122) and we will establish R R by proving that projYS D(N L123 ) L4 L5 R. (21.123) By the remark following Definition 21.3, we only need to show the achievability of the region projYS D(N L123 ) L4 L5 . (21.124) Consider any in this region. Then there exists h D(N L123 ) L4 L5 (21.125) (21.126) (21.127) (21.128) (21.129) (21.130) such that = projYS (h). Since h D(N L123 ), there exist a sequence h(k) D(N L123 ) such that h = lim h(k) . k Let (k) = projYS (h(k) ). It then follows from (21.129) that k lim (k) = . (21.131) By (21.128), ^ h(k) = (k) h(k) (21.132) 524 21 Multi-Source Network Coding ^ where h(k) N L123 and 0 (k) 1. (21.133) ^ Note that h(k) is an entropy function because it is in N , but h(k) and h ^ are not necessarily entropy functions. Since h(k) N L123 , there exists a collection of random variables with finite alphabets (k) N (k) = (Ys(k) : s S), (Ue : e E) (21.134) such that (k) (k) H Ys(k) = s , s S (21.135) (21.136) (21.137) (21.138) H YS (k) (k) = sS H Ys(k) H UOut(s) Ys(k) = 0, s S H UOut(i) UIn(i) = 0, i V \ (S T ), (k) (k) where (21.135) is implied by (21.130). Furthermore, since h L4 L5 , it follows from (21.129) and (21.132) that (k) (k) H Ue Re + (k) , e E (21.139) (21.140) (k) H Y(t) UIn(t) (k) , t T, (k) (k) where (k) , (k) 0 as k . In the rest of the section, we will prove the achievability of (k) for all sufficiently large k. Then the closedness of R implies the achievability of by the remark following Definition 21.3. 21.7.1 Random Code Construction Fix k and > 0, and let be a small positive quantity to be specified later. We first construct a random (k) (k) (n, (e : e E), (s : s S)) (21.141) (21.142) (21.143) code with for all e E and (k) s - (k) (k) (k) e 2n( (k) (k) (k) H(Ue )+e ) 2 (k) (k) s s - , 3 where e > 0 and e 0 as 0, by the steps below. For the sake of simplicity, we temporarily suppress all the superscripts involving k. In light of the heavy notation, we list in Table 21.1 the symbols involved in the description of the random code construction. 21.7 Achievability 525 1. Let n = n . ^ (21.144) Here n is the block length of the random code we will construct, while n is ^ the length of a sequence of the typical sets that we will use for constructing the random code. For each source s S, let s = 2ns (21.145) n ^ and construct a codebook Cs by generating s codewords in Ys rann ^ domly and independently according to p (ys ). Denote these sequences by Ys (1), Ys (2), , Ys (s ), and let Ys (0) be an arbitrary constant sequence n ^ in Ys . 2. Reveal the codebook Cs , s S to all the nodes in the network. 3. At a source node s S, the information source Xs is generated according to the uniform distribution on Xs = {1, 2, , s }. (21.146) n ^ 4. Let T[Ue ] denote the set of strongly typical sequences2 with respect to the distribution p(ue ). Let n ^ e = |T[Ue ] |. (21.147) By the strong AEP and (21.144), n n ^ Xs Xs Ys Cs Ys (0) s n ^ T[Ue ] Ue (0) ke ue ^ Ce gt the block length of the random code the length of a sequence in the typical sets used in the code construction the information source generated at source node s the alphabet of information source Xs , equal to {1, 2, , s } the auxiliary random variable associated with Xs the codebook for information source Xs consisting of the codewords n ^ Ys (1), Ys (2), , Ys (s ) Ys n ^ an arbitrary constant sequence in Ys the common size of the alphabet Xs and the codebook Cs equal to {Ue (1), Ue (2), , Ue (e )} n ^ an arbitrary constant sequence in Ue the local encoding function for channel e the function defined by Ue = ue (Ys ) for e Out(s) and Ue = ue (UIn(i) ) ^ ^ for e Out(i) the index sent on channel e, i.e., the value taken by the local encoding function ke the decoding function at sink node t Table 21.1. The list of symbols involved in the random code construction. 2 Strong typicality applies because all the random variables in N (k) have finite alphabets. 526 21 Multi-Source Network Coding ^ e 2n(H(Ue )+e /(2)) 2n(H(Ue )+e /2) , (21.148) where e 0 as 0. For all channels e E, choose an e satisfying 2n(H(Ue )+e /2) e 2n(H(Ue )+e ) . (21.149) n ^ Denote the sequences in T[Ue ] by Ue (1), Ue (2), , Ue (e ), and let Ue (0) n ^ be an arbitrary constant sequence in Ue . a) Let the outcome of Xs be xs for a source node s. For a channel e Out(s), define the local encoding function ke : Xs {0, 1, , e } (21.150) as follows. By (21.137), for each channel e Out(s), there exists a function ue such that ^ Ue = ue (Ys ), ^ (21.151) i.e., Pr{Ue = ue (y)|Ys = y} = 1 ^ (21.152) for all y Ys . By the preservation property of strong typicality (Theorem 6.8), if n ^ Ys (xs ) T[Ys ] , (21.153) then n ^ ue (Ys (xs )) T[Ue ] , ^ (21.154) where in ue (Ys (xs )), the function ue is applied to Ys (xs ) componen^ ^ n ^ twise. If so, let ke (xs ) be the index of ue (Ys (xs )) in T[Ue ] , i.e., ^ Ue (ke (xs )) = ue (Ys (xs )). ^ (21.155) Otherwise, let ke (xs ) be 0. Note that ke is well-defined because e e (21.156) by (21.148) and (21.149). b) For a channel e Out(i), where i V \(S T ), define the local encoding function ke : {0, 1, , d } {0, 1, , e } dIn(i) (21.157) as follows. By (21.138), there exists a function ue such that ^ Ue = ue (UIn(i) ). ^ (21.158) Let Ce be the index sent on channel e, i.e., the value taken by the local encoding function ke . With a slight abuse of notation, we write 21.7 Achievability 527 UE (CE ) = (Ud (Cd ) : d E ) for E E, and YS (xS ) = (Ys (xs ) : s S ) (21.159) (21.160) for S S. By the preservation property of strong typicality, if n ^ UIn(i) (CIn(i) ) T[UIn(i) ] , (21.161) then n ^ ue (UIn(i) (CIn(i) )) T[Ue ] . ^ (21.162) n ^ If so, let ke (CIn(i) ) be the index of ue (UIn(i) (CIn(i) )) in T[Ue ] , i.e., ^ Ue (ke (CIn(i) )) = ue (UIn(i) (CIn(i) )). ^ (21.163) Otherwise, let ke (CIn(i) ) be 0. Again, ke is well-defined because (21.156) holds. 5. For a sink node t T , define the decoding function gt : {0, 1, , d } X(t) dIn(t) (21.164) as follows. If the received index Cd on channel d is nonzero for all d In(t) and there exists a unique tuple x(t) X(t) such that n ^ (Y(t) (x(t) ), UIn(i) (CIn(t) )) T[UIn(t) Y(t) ] , (21.165) (21.166) then let gt (CIn(t) ) be x(t) . Otherwise, declare a decoding error. 21.7.2 Performance Analysis Let us reinstate all the superscripts involving k that were suppressed when we described the construction of the random code. Our task is to show that for any sufficiently large k and any > 0, the random code we have constructed satisfies the following when n is sufficiently large: (k) n-1 log e Re + , e E (k) (k) s s - , s S (k) t (21.167) (21.168) (21.169) , t T. For e E, consider 528 21 Multi-Source Network Coding (k) (k) (k) n-1 log e (k) H(Ue ) + e (21.170) (21.171) Re + (k) + (k) e , where the first inequality follows from the upper bound in (21.149) and the second inequality follows from (21.139). Since (k) 0 as k , we can let k be sufficiently large so that (k) < . (21.172) With k fixed, since e have (k) 0 as 0, by letting be sufficiently small, we (k) (k) + e , (21.173) and (21.167) follows from (21.171). For s S, from the lower bound in (21.143), we have (k) (k) s s - , (21.174) proving (21.168). The proof of (21.169), which is considerably more involved, will be organized into a few lemmas. For the sake of presentation, the proofs of these lemmas will be deferred to the end of the section. For i S and i V \(S T ), the function ue , where e Out(i), has been ^ defined in (21.151) and (21.158), respectively. Since the network is acyclic, we see by induction that all the auxiliary random variables Ue , e E are functions of the auxiliary random variables YS . Thus there exists a function ue such that ~ Ue = ue (YS ). ~ (21.175) Equating the above with (21.151) and (21.158), we obtain ue (Ys ) = ue (YS ) ^ ~ and ue (UIn(i) ) = ue (YS ), ^ ~ (21.177) respectively. These relations will be useful subsequently. In the rest of the section, we will analyze the probabilities of decoding (k) error for the random code we have constructed for a fixed k, namely t , for t T . With a slight abuse of notation, we write uE () = (~d () : d E ) ~ u (21.178) (21.176) for E E. Again, we temporarily suppress all the superscripts invoking k. Lemma 21.9. Let XS = xS YS (xS ) = yS n ^ T[YS ] , (21.179) (21.180) 21.7 Achievability 529 and for e E, let Ce take the value ce , which by the code construction is a function of xS and yS . Then UIn(t) (cIn(t) ) = uIn(t) (yS ). ~ and n ^ (yS , UIn(t) (cIn(t) )) T[YS UIn(t) ] (21.181) (21.182) for all t T . Let Errt = {gt (CIn(t) ) = X(t) } = {~t (XS ) = X(t) } g be the event of a decoding error at sink node t, i.e., Pr{Errt } = t (21.184) (21.183) (cf. (21.10)). In the following, we will obtain an upper bound on Pr{Errt }. Consider Pr{Errt } = x(t) X(t) Pr{Errt |X(t) = x(t) } Pr{X(t) = x(t) }, (21.185) and for S S, let 1S = (1, 1, , 1) . |S | (21.186) Since Pr{Errt |X(t) = x(t) } are identical for all x(t) by symmetry in the code construction, from (21.185), we have Pr{Errt } = Pr{Errt |X(t) = 1(t) } x(t) X(t) Pr{X(t) = x(t) } (21.187) (21.188) = P r{Errt |X(t) = 1(t) }. In other words, we can assume without loss of generality that X(t) = 1(t) . To facilitate our discussion, define the event n ^ ES = {YS (1S ) T[YS ] .} (21.189) Following (21.188), we have Pr{Errt } = P r{Errt |X(t) = 1(t) , ES } Pr{ES |X(t) = 1(t) } c c +P r{Errt |X(t) = 1(t) , ES } Pr{ES |X(t) = 1(t) } (21.190) (21.191) (21.192) (21.193) = P r{Errt |X(t) = 1(t) , ES } Pr{ES } c c +P r{Errt |X(t) = 1(t) , ES } Pr{ES } c P r{Errt |X(t) = 1(t) , ES } 1 + 1 Pr{ES } P r{Errt |X(t) = 1(t) , ES } + , 530 21 Multi-Source Network Coding where the last inequality follows from the strong AEP and 0 as 0. Upon defining the event ES = {X(t) = 1(t) } ES , we have Pr{Errt } P r{Errt |ES } + . (21.195) We now further analyze the conditional probability in (21.195). For x(t) X(t) , define the event n ^ Et (x(t) ) = (Y(t) (x(t) ), UIn(t) (CIn(t) )) T[Y(t) UIn(t) ] . (21.194) (21.196) Since X(t) = 1(t) , decoding at sink node t is correct if the received indices CIn(t) is decoded to 1(t) . This is the case if and only if Et (1(t) ) occurs but Et (x(t) ) does not occur for all x(t) = 1(t) . It follows that c Errt = Et (1(t) ) x(t) =1(t) Et (x(t) )c , (21.197) (21.198) or Errt = Et (1(t) )c x(t) =1(t) Et (x(t) ) , which implies Pr{Errt |ES } = Pr Et (1(t) )c x(t) =1(t) Et (x(t) ) ES . By the union bound, we have Pr{Errt |ES } Pr{Et (1(t) )c |ES } + x(t) =1(t) (21.199) Pr{Et (x(t) )|ES } (21.200) (21.201) = x(t) =1(t) Pr{Et (x(t) )|ES }, where the last step follows because Pr{Et (1(t) )c |ES } in (21.200) vanishes by Lemma 21.9. The next two lemmas will be instrumental in obtaining an upper bound on Pr{Et (x(t) )|ES } in (21.201). For any proper subset of (t), let = {x(t) = 1(t) : xs = 1 if and only if s }. (21.202) Note that { } is a partition of the set X(t) \{1(t) }. For x(t) , x(t) and 1(t) are identical for exactly the components indexed by . Lemma 21.10. For x(t) , where is a proper subset of (t), Pr{Et (x(t) )|ES } 2-n(H(Y(t)\ )-H(Y(t) |UIn(t) )-t ) , where t 0 as n and 0. (21.203) 21.7 Achievability 531 Lemma 21.11. For all sufficiently large n, | | 2n(H(Y(t)\ )- /4) . (21.204) We now reinstate all the superscript involving k that have been suppressed. By Lemma 21.10, Lemma 21.11, and (21.201), Pr{Errt |ES } x(t) =1(t) Pr{Et (x(t) )|ES } Pr{Et (x(t) )|ES } x(t) -n(k) H(Y(t)\ -H Y(t) |UIn(t) -t (k) (k) (k) (k) (21.205) (21.206) 2|E| 2 2 n (k) H Y(t)\ - /4 -n /4- (k) (21.207) - (k) = 2|E| 2 2 |E| H (k) (k) Y(t) |UIn(t) (k) t (21.208) (21.209) (21.210) 2 -n( /4- /4- (k) - t ) 2|E| 2-n( (k) -t ) where (21.209) follows from (21.140) and (21.210) follows from (21.133). Then from (21.184), (21.195), and (21.210), we have t (k) 2|E| 2-n( /4- (k) -t ) + (21.211) We now choose k, n, and to make the upper bound above smaller than any prescribed > 0. Since (k) 0 as k , we can let k to be sufficiently large so that (k) < /4. (21.212) Then with k fixed, since t 0 as n and 0, and 0 as 0, by letting n be sufficiently large n and be sufficiently small, we have 1. (k) + t < /4, so that 2|E| 2-n( (k) 2. t . /4- (k) -t ) 0 as n ; This completes the proof of (21.169). Hence, we have proved the achievability of (k) for all sufficiently large k. Then the closedness of R implies the achievability of = limk (k) , where R . The achievability of R is established. Proof of Lemma 21.9. We first prove that given 532 21 Multi-Source Network Coding XS = xS and n ^ YS (xS ) = yS T[YS ] , (21.213) (21.214) the following hold for all non-source nodes i (i.e., i V \S): n ^ i) UIn(i) (cIn(i) ) T[UIn(i) ] ; ii) ke (cIn(i) ) = 0, e Out(i); iii) Ue (ke (cIn(i) )) = ue (yS ), e Out(i). ~ Note that for i T , Out(i) = in ii) and iii). By the consistency of strong typicality (Theorem 6.7), n ^ ys T[Ys ] (21.215) for all s S. Then according to the construction of the code, for all e Out(s), ke (xs ) = 0 and Ue (ke (xs )) = ue (ys ). ^ (21.217) We now prove i) to iii) by induction on the non-source nodes according to any given coding order. Let i1 be the first non-source node to encode. Since In(i1 ) S, for all d In(i1 ), d Out(s) for some s S. Then Ud (cd ) = Ud (kd (xs )) = ud (ys ) ^ = ud (yS ), ~ (21.219) (21.220) (21.221) (21.218) (21.216) where (21.220) and (21.221) follows from (21.217) and (21.176), respectively. Thus UIn(i1 ) (cIn(i1 ) ) = uIn(i1 ) (yS ). ~ (21.222) Since UIn(i1 ) is a function of YS , in light of (21.180), n ^ UIn(i) (cIn(i) ) T[UIn(i) ] (21.223) by the preservation property of strong typicality, proving i). According to the code construction, this also implies ii). Moreover, Ue (ke (cIn(i1 ) )) = ue (UIn(i1 ) (cIn(i1 ) )) ^ = ue (yS ), ~ (21.224) (21.225) where the last equality is obtained by replacing in (21.177) the random variable Ud by the sequence Ud (cd ) and the random variable Ys by the sequence YS (xS ) = yS , proving iii). 21.7 Achievability 533 We now consider any non-source node i in the network. Assume that i) to iii) are true for all the nodes upstream to node i. For d In(i), if d Out(s) for some s S, we have already proved in (21.221) that Ud (cd ) = ud (yS ). ~ Otherwise, d Out(i ), where node i is upstream to node i. Then Ud (cd ) = Ud (kd (cIn(i ) )) = ud (UIn(i ) (cIn(i ) )) ^ = ud (yS ), ~ (21.227) (21.228) (21.229) (21.226) In the above, (21.228) follows from ii) for node i by the induction hypothesis and the code construction, and (21.229) follows from (21.177). Therefore, (21.226) is valid for all d In(i). Hence, UIn(i) (cIn(i) ) = uIn(i) (yS ). ~ (21.230) which is exactly the same as (21.222) except that i1 is replaced by i. Then by means of the same argument, we conclude that i), ii), and iii) hold for node i. As (21.230) holds for any non-source node i, it holds for any sink node t. This proves (21.181) for all t T . Furthermore, since UIn(t) is a function of YS , (YS , UIn(t) ) is also a function of YS . Then in view of (21.181), (21.182) follows from the preservation property of strong typicality. This completes the proof of the lemma. Proof of Lemma 21.10. Consider Pr{Et (x(t) )|ES } = n ^ yS T[Y S ] Pr{Et (x(t) )|YS (1S ) = yS , ES } Pr{YS (1S ) = yS |ES }. (21.231) To analyze Pr{Et (x(t) )|YS (1S ) = yS , ES } in the above summation, let us n ^ condition on the event {YS (1S ) = yS , ES }, where yS T[YS ] . It then follows from (21.181) in Lemma 21.9 that UIn(t) (cIn(t) ) = uIn(t) (yS ). ~ Therefore, the event Et (x(t) ) is equivalent to n ^ (y , Y(t)\ (x(t)\ ), uIn(t) (yS )) T[Y Y(t)\ UIn(t) ] ~ (21.232) (21.233) (cf. (21.196)), or n ^ Y(t)\ (x(t)\ ) T[Y(t)\ |Y UIn(t) ] (y , uIn(t) (yS )). ~ (21.234) 534 21 Multi-Source Network Coding Thus Pr{Et (x(t) )|YS (1S ) = yS , ES } = ^ y(t)\ T n [ Y(t)\ |Y UIn(t) ] (y ,~In(t) (yS )) u Pr{Y(t)\ (x(t)\ ) = y(t)\ |YS (1S ) = yS , ES }. (21.235) Since xs = 1 for s (t)\ , Y(t)\ (x(t)\ ) is independent of the random sequences YS (1S ) and the event ES by construction. Therefore, Pr{Y(t)\ (x(t)\ ) = y(t)\ |YS (1S ) = yS , ES } = Pr{Y(t)\ (x(t)\ ) = y(t)\ }. By the consistency of strong typicality, if n ^ ~ y(t)\ T[Y(t)\ |Y UIn(t) ] (y , uIn(t) (yS )), (21.236) (21.237) then n ^ y(t)\ T[Y(t)\ ] . (21.238) Since Y(t)\ (x(t)\ ) are generated i.i.d. according to the distribution of Y(t)\ , by the strong AEP, n Pr{Y(t)\ (x(t)\ ) = y(t)\ } 2-^ (H(Y(t)\ )-) , (21.239) where 0 as 0. Combining (21.236) and (21.239), we have n Pr{Y(t)\ (x(t)\ ) = y(t)\ |YS (1S ) = yS , ES } 2-^ (H(Y(t)\ )-) . (21.240) By the strong conditional AEP, n ^ ^ |T[Y(t)\ |Y UIn(t) ] (y , uIn(t) (yS ))| 2n(H(Y(t)\ |Y UIn(t) )+) , ~ (21.241) where 0 as n and 0. It then follows from (21.235), (21.240), ^ and (21.241) that Pr{Et (x(t) )|YS (1S ) = yS , ES } ^ n 2n(H(Y(t)\ |Y UIn(t) )+) 2-^ (H(Y(t)\ )-) (21.242) (21.243) (21.244) (21.245) =2 -^ (H(Y(t)\ )-H(Y(t)\ |Y UIn(t) )--) n -^ (H(Y(t)\ )-H(Y(t) |UIn(t) )--) n 2 2-n(H(Y(t)\ )-H(Y(t) |UIn(t) )-t ) where (21.244) is justified by 21.7 Achievability 535 H(Y(t)\ |Y UIn(t) ) H(Y(t)\ |Y UIn(t) ) + H(Y |UIn(t) ) = H(Y(t),\ , Y |UIn(t) ) = H(Y(t) |UIn(t) ), (21.245) follows from (21.144), and t 0 as n and 0. In (21.231), Pr{YS (1S ) = yS |ES } = Pr{YS (1S ) = yS |X(t) = 1(t) , ES } = Pr{YS (1S ) = yS |ES } = Pr{YS (1S ) = yS |YS (1S ) n ^ T[YS ] }. (21.246) (21.247) (21.248) (21.249) (21.250) (21.251) Hence, it follows from (21.231) and (21.245) that Pr{Et (x(t) )|ES } 2-n(H(Y(t)\ )-H(Y(t) |UIn(t) )-t ) n ^ Pr{YS (1S ) = yS |YS (1S ) T[YS ] } n ^ yS T[Y S ] (21.252) (21.253) (21.254) =2 =2 -n(H(Y(t)\ )-H(Y(t) |UIn(t) )-t ) -n(H(Y(t)\ )-H(Y(t) |UIn(t) )-t ) 1 . The lemma is proved. Proof of Lemma 21.11. Let n be sufficiently large. Consider | | = s(t)\ |Xs | s s(t)\ (21.255) (21.256) (21.257) /3) = = s(t)\ c) b) a) 2ns 2n(s - s(t)\ s(t)\ (21.258) (21.259) /4) 2n(s - /4) = s(t)\ d) 2n(H(Ys )- n n s(t)\ (21.260) (21.261) (21.262) =2 =2 (H(Ys )- /4) H(Ys )-(|(t)|-| |) /4 s(t)\ 536 21 Multi-Source Network Coding = 2n[H(Y(t)\ )-(|(t)|-| |) 2n(H(Y(t)\ )- where e) /4) /4] (21.263) (21.264) , a) follows from (21.146); b) follows from (21.145); c) follows from (21.143); d) follows from (21.135); e) follows because is a proper subset of (t). The lemma is proved. Chapter Summary A Multi-source Network Coding Problem: A point-to-point communication network is represented by an acyclic graph consisting of a set of nodes V and a set of channels E. The set of source nodes is denoted by S. At a source node s, an information source Xs is generated. The rate constraint on a channel e is Re . The set of sink nodes is denoted by T . At a sink node t, the information sources Xs , s (t) are received, where (t) S depends on t. Achievable Information Rate Region R: An information rate tuple = (s : s S) is achievable if for a sufficiently large block length n, there exists a network code such that 1. at a source node s, the rate of the information source Xs is at least s - ; 2. the rate of the network code on a channel e is at most Re + ; 3. at a sink node t, the information sources Xs , s (t) can be decoded with negligible probability of error. The achievable information rate region R is the set of all achievable information rate tuples . Characterization of R: Let N = {Ys : s S; Ue : e E} and HN be the entropy space for the random variables in N . Then R = projYS con(N L123 ) L4 L5 , where N = {h HN : h is finitely entropic} and Problems 537 L1 = h HN : hYS = sS hYs L2 = h HN : hUOut(s) |Ys = 0, s S L3 = h HN : hUOut(i) |UIn(i) = 0, i V \ (S T ) L4 = { h HN : hUe Re , e E } L5 = h HN : hY(t) |UIn(t) = 0, t T . An Explicit Outer Bound on R (LP Bound): RLP = projYS con(N L123 ) L4 L5 . Problems 1. Show that source separation is optimal for the networking problem depicted in Figure 21.3. 2. By letting S = {s} and (t) = {s} for all t T , the multi-source network coding problem described in Section 21.3 becomes a single-source network coding problem. Write = s . a) Write out the achievable information rate region R. b) Show that if s R, then s maxflow(t) for all t T . 3. Consider the following network. X1 X2 2 1 2 2 1 2 1 1 2 1 1 [X1X2] [ X1 X2 ] [X1X2] [X1] [X1] a) Let i be the rate of information source Xi . Determine and illustrate the max-flow bounds. b) Are the max-flow bounds achievable? c) Is source separation always optimal? 538 21 Multi-Source Network Coding X 1 X2 [X1X2] [X1] [X1X2] [X1] 4. Repeat Problem 3 for the above network in which the capacities of all the edges are equal to 1. 5. Consider a disk array with 3 disks. Let X1 , X2 , and X3 be 3 mutually independent pieces of information to be retrieved from the disk array, and let S1 , S2 , and S3 be the data to be stored separately in the 3 disks. It is required that X1 can be retrieved from Si , i = 1, 2, 3, X2 can be retrieved from (Si , Sj ), 1 i < j 3, and X3 can be retrieved from (S1 , S2 , S3 ). a) Prove that for i = 1, 2, 3, H(Si ) = H(X1 ) + H(Si |X1 ). b) Prove that for 1 i < j 3, H(Si |X1 ) + H(Sj |X1 ) H(X2 ) + H(Si , Sj |X1 , X2 ). c) Prove that H(S1 , S2 , S3 |X1 , X2 ) = H(X3 ). d) Prove that for i = 1, 2, 3, H(Si ) H(X1 ). e) Prove that H(Si ) + H(Sj ) 2H(X1 ) + H(X2 ) + H(Si , Sj |X1 , X2 ). f) Prove that 2Si + Si1 + Si2 4H(X1 ) + 2H(X2 ) + H(X3 ), where i = 1, 2, 3 and ij = for 1 i, j 3. i+j if i + j 3 i + j - 3 if i + j > 3 Historical Notes 539 g) Prove that 3 H(S1 ) + H(S2 ) + H(S3 ) 3H(X1 ) + H(X2 ) + H(X3 ). 2 Parts d) to g) give constraints on H(S1 ), H(S2 ), and H(S3 ) in terms of H(X1 ), H(X2 ), and H(X3 ). It was shown in Roche et al. [308] that these constraints are the tightest possible. 6. Generalize the setup in Problem 5 to K disks and show that K H(Si ) K i=1 H(X ) . =1 K Hint: Use the inequalities in Problem 18 in Chapter 2 to prove that for s = 0, 1, , K - 1, K H(Si ) nK i=1 H(X ) + =1 s K K s+1 T :|T |=s+1 H(ST |X1 , X2 , , Xs ) s+1 by induction on s, where T is a subset of {1, 2, , K}. 7. Write out the achievable information rate region R for the network in Problem 3. 8. Show that if there exists an (n, (ij : (i, j) E), (s : s S)) code which satisfies (21.11) and (21.13), then there always exists an (n, (ij : (i, j) E), (s : s S)) code which satisfies (21.11) and (21.13), where s s for all s S. Hint: use a random coding argument. Historical Notes Multilevel diversity coding was studied by Yeung [400], where it was shown that source separation is not always optimal. Roche et al. [308] showed that source separation is optimal for symmetrical three-level diversity coding. This result was extended to any level by Yeung and Zhang [410] with a painstaking proof. Hau [160] studied all the one hundred configurations of a threeencoder diversity coding systems and found that source separation is optimal for eighty-six configurations. 540 21 Multi-Source Network Coding Yeung and Zhang [411] introduced the distributed source coding model discussed in Section 21.2.2 which subsumes multilevel diversity coding. The region of all entropy functions previously introduced by Yeung [401] for studying information inequalities enabled them to obtain inner and outer bounds on the achievable information rate region for a variety of networks. Distributed source coding is equivalent to multi-source network coding on a special class of acyclic networks. The inner and outer bounds on the achievable information rate region in [411] were generalized to arbitrary acyclic networks by Song et al. [341]. The gap between these bounds was finally closed by Yan et al. [391]. 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Index a posterior distribution, 214 Abel, N.H., 390 Abelian group, 390, 402, 406, 407 Abrahams, J., 100, 541 Abramson, N., 80, 541 absolutely integrable, 241 abstract algebra, 387, 490 Abu-Mostafa, Y.S., 541 acyclic network, X, 435479, 485, 488, 502, 540, see also directed graph Aczl, J., 541 e additive colored Gaussian noise, 287 additive noise, worst, IX, 270, 289, 297 additive white Gaussian noise (AWGN), 280, 281 adjacent pair of channels, 436, 439, 440, 454, 457, 480, 492, 498 adjoint matrix, 502 Ahlswede, R., 98, 180, 419, 432, 434, 482, 504, 541 Ahmed, E., 552 algebraic coding, 440, 483 Algoet, P., 541 algorithm, 435, 456, 457, 466, 466, 472 exponential-time, 460 polynomial-time, X, 457, 458, 460, 466, 468, 482 almost everywhere (a.e.), 244 almost perfect reconstruction, 104, 106 alternating optimization algorithm, 212214, 217 convergence, 222 Amari, S., 49, 542 Anantharam, V., XIII, 542 ancestral order, 437, see upstream-todownstream order Anderson, J.B., 542 applied mathematics, 3 applied probability, 3 Argawal, A., 542 Arimoto, S., 166, 181, 228, 542, see also Blahut-Arimoto algorithms arithmetic, 435 arithmetic mean, 151 ascendant, 94 Ash, R.B., 542 asymptotic equipartiion property (AEP) for continuous random variables, VIII asymptotic equipartition property (AEP) for continuous random variables, 245, 245247 typical sequence, 245, 246 typical set, 245 strong, see also strong asymptotic equipartition property weak, see also weak asymptotic equipartition property asymptotically reliable communication, 140 atom of a field, 52 weight of, 311 audio compact disc, 1 audio signal, 184 audio source, 382 562 Index binary symmetric channel (BSC), 137, 146, 176, 177, 180 binomial formula, 306, 355, 356 bit, 3, 13, 424 Blackwell, D., 542 Blahut, R.E., XIII, 181, 210, 228, 543, 557 Blahut-Arimoto algorithms, VIII, 149, 200, 211228 channel capacity, 214218, 227 convergence, 225226 rate-distortion function, 219222, 227 Blakley, G.R., 543 block code, 104 linear, 440 block length, 104, 139, 151, 183, 211, 287, 427 Blundo, C., 80, 543 Bollobs, B., 543 a Bondy, J.A., 314, 543 Borade, S., 434, 543 Bose, R.C., 543, see also BCH code bottleneck, 423 boundary condition, 440, 485, 493, 495 brain, 382 branching probabilities, 94 Breiman, L., 112, 542, 543, see also Shannon-McMillan-Breiman theorem broadcast, 415 constraint, 416 broadcast constraint, 415 Bruck, J., 551 BSC, see binary symmetric channel Burrows, M., 543 butterfly network, X, 412415, 419, 425, 460, 480 Byers, J., 543 cable cut, 468 Cai, N., XIII, 117, 419, 420, 432, 434, 482, 483, 504, 540, 541, 543, 551, 556, 559 Caire, G., 543 Calderbank, R., 543 Capocelli, R.M., 79, 543 Cartesian product, 212 cascade of channels, 177 Cauchy distribution, 255 autocorrelation function, 280, 285 auxiliary random variable, 370, 371, 383, 536 average distortion, 184, 185, 187, 201, 205, 211 expected, 201 average input constraint, 268, 269, 296 average probability of error, 150, 180, 261 Ayanoglu, E., 420, 542 backbone network, 445 Balkenhol, B., 98, 541 Balli, H., 482, 542 bandlimited signal, 283, 284, 287 orthonormal basis, 283, 284 bandpass filter, 281, 288 bandwidth, 281, 412, 477 downlink, 417 Barbero, A.I., 504, 542 Barron, A.R., 542 base field, 437442, 446, 449, 451, 453456, 459, 460, 468, 473, 474, 476, 481, 486, 487, 489, 492, 497, 500502 basic inequalities, 27, 2628, 59, 131, 313, 324, 339361, 372, 373, 381383, 385, 387, 402 Bassalygo, L.A., 542 Bayesian network, 11, 152, 351, 352 intersection, 11 BCH (Bose-Chaudhuri-Hocquenghem) code, 166 Beethoven's violin concerto, 1 Bell Telephone Laboratories, 2 Berger, T., XIII, 78, 134, 209, 210, 228, 542, 547, 559 Berlekamp, E.R., 542, 556 Berrou, C., 166, 542 Berstel, J., 542 Bertsekas, D., 542 beta distribution, 255 Bhargava, V.K., XIII, 558 biased coin, 81 binary arbitrarily varying channel, 180 binary covering radius, 208 binary entropy function, 13, 33, 34, 197 binary erasure channel, 148, 171 Index causality, 177, 427, 431, 492, 495 CDF, see cumulative distribution function Cesro mean, 39 a chain rule for conditional entropy, 21 conditional mutual information, 22 differential entropy, 243 entropy, 21, 29 mutual information, 22, 30 Chan, A.H., XIII Chan, T.H., XIII, 12, 131, 384, 385, 408, 540, 543, 544 Chan, V.W.S., XIII channel capacity computation of, VIII channel characteristics, 137 channel code, 137, 151, 175 probability of error, 138 rate, 140, 151 with feedback, 167, 178 without feedback, 149 channel coding, 411, 415 channel coding theorem, VIII, 134 for continuous memoryless channel, 261, 260261 achievability, 265269 converse, 262265 random code, 265 for discrete memoryless channel, 3, 50, 151, 149151, 181 achievability, 149, 157163 converse, 149, 171 random code, 159 strong converse, 157, 181 channel failure, X, 468 channel with memory, 172, 176, 178, 179 Charikar, M., 542 Chatin, G.J., 544 Chekuri, C., 544 Chernoff bound, 117 Chernoff, H., 544 Cheung, K.W., XIII Chiang, M., 544 child, 94 Chinese University of Hong Kong, The, XIII Chiu, D.M., 544 563 Chou, P.A., 420, 482, 544, 549, 551 chronological order, 428 Chuang, I.L., 386, 553 Chung, K.L., 112, 544 cipher text, 71, 78 classical entropy, 385 closure, of a group, 388 CMC, see continuous memoryless channel code alphabet, 82 code tree, 86, 8697, 100 pruning of, 95 codebook, 149, 187, 202, 260, 267 codeword, 104, 149, 187, 202, 260 coding session, 428, 449 transaction, 428, 433 coding theory, 166 coefficient of correlation, 254 Cohen, A.S., 544 coloring filter, 297 column space, 334 combinatorics, 127 commodity, 411, 425 commodity flow, 419 communication, 420 communication channel, 281, 412 capacity, 412 error-free, 415 receiver, 282 communication complexity, 420 communication engineer, 3 communication engineering, 137, 234, 270 communication system, 1, 3, 239, 257, 411, 416 design of, 415 discrete-time, 142 practical, 137, 166 Shannon's model, 2 communication theory, 3 commutative, 390, 391, 491 commutative group, see Abelian group compact set, 146, 225 complex conjugate, 280 complex number, 490 composite function, 391 compound source, 209 computation, 420 computational 564 Index convolutional multicast, 499, 500, 502, 504 convolutional network code, X, 433, 492, 488502, 504 decoding, 498502 decoding delay, 499, 504 decoding kernel, 499, 502, 504 multicast, 499 correlation matrix, 230, 255, 289291, 294 coset left, 391, 392394 right, 391 Costa, M.H.M., 544 Costello, Jr., D.J., 543, 552, 557 countable alphabet, 33, 35, 41, 48, 110, 112, 132135, 230 covariance, 230 covariance matrix, 230, 232, 233, 251, 254, 255, 277 Cover, T.M., XIII, 228, 297, 541, 544, 545 cross-correlation function, 280, 281, 284 cross-spectral density, 281 crossover probability, 137, 177, 180, 198 cryptography, 483 Csiszr, I., XIII, 49, 80, 135, 228, 338, a 541, 545, 548, 557 cumulative distribution function (CDF), 229 conditional, 230 joint, 230, 274 marginal, 230 cyclic network, X, 485502, 504, see also directed graph delay-free, 485488 D-adic distribution, 88, 93 D-ary source code, 82 D-it, 13, 84, 97 Dmax , 186, 191, 193, 197 dmax , 201 Dana, A.F., 545 Dantzig, G.B., 545 Darczy, Z., 541 o data block, 474 aysnchronous transmission, 475 encoded, 474, 475 data communication, 167, 411 complexity, 458, 460, 466, 467, 473, 481 procedure, 343, 346 resource, 477 computer communication, 166 computer science, 13, 420 computer storage systems, 166 concavity, 46, 49, 67, 70, 222225, 259, 292 conditional branching distribution, 94 conditional distribution, 240 conditional entropy, 7, 14 conditional independence, IX, 8, 321, 336337, 351 elemental forms, 358 structure of, 12, 385 conditional mutual independence, 300309 conditional mutual information, 7, 16, 19, 242 conditional probability of error, 150, 261 configuration, 468, 470 constant sequence, 187 constant term, 490, 492 continuous extension, 19 continuous memoryless channel (CMC), IX, 258, 258269, 296, 297 achievable rate, 261 average input constraint, 259 capacity, 259, 261 continuous partial derivatives, 212, 217, 224, 225 continuous-valued channels, IX, 257294 convergence, 490 in L1 , 19 in L2 , 20, 45 in divergence, 26, 48 in probability, 101, 102, 108, 266 in variational distance, 19, 20, 26, 45, 48 convex closure, 398, 401 convex cone, 366 convexity, 24, 46, 47, 69, 188, 191, 193, 201, 212, 216, 221, 326, 337, 368, 515 convolution, 282 convolutional code, 166 convolutional coding, 490 Index data packet, 411, 477, 482 data processing theorem, 31, 73, 156, 262, 349, 382 Davisson, L.D., 545 Dawid, A.P., 545 De Santis, A., 79, 80, 543 De Simone, R., 80, 543 decoder, 104, 159, 202 decoding function, 110, 149, 167, 187, 260, 428 decorrelation, 233, 278 deep space communication, 166 delay processing, 435, 488 propagation, 435, 485, 488 transmission, 435, 488, 492 Dembo, A., 420, 555 Dempster, A.P., 545 denominator, 490 dependency graph, 152153, 168, 179, 180 directed edge, 152 dotted edge, 152 parent node, 152 solid edge, 152 descendant, 87, 95 destination, 2 determinant, 231, 239, 451 deterministic distribution, 48 diagonal element, 232234, 278, 279, 289, 296, 498 diagonal matrix, 232, 234, 239, 255, 279, 296, 501 diagonalization, 232, 233, 234, 239, 278, 296 differential entropy, VIII, 235, 229255, 271, 385 conditional, 240, 257 joint, 238 scaling, 237, 238 translation, 236, 238, 271 Diggavi, S.N., 297, 545 digital, 2 digital communication system, XI, 3 directed cycle, 435, 436, 485, 486, 488, 489 directed graph, 412, 421 acyclic, 435, 436 cut, 422 565 capacity of, 423 cyclic, 436, 485 directed cycle, 435, 436 directed path, 436 edge, 412, 421 max-flow, 422 min-cut, 423 node, 412, 421 non-source node, 421 rate constraint, 422, 423 sink node, 411, 422, 424 source node, 411, 421, 424 directed network, 436, 485, see also directed graph directed path, 436, 453, 463 longest, 436 directional derivative, 224 discrete alphabet, 140, 142 discrete channel, 140145 noise variable, 141, 142 discrete memoryless channel (DMC), 140149, 167, 176, 177, 179, 211, 257, 287, 434 achievable rate, 151 capacity, 3, 145, 137181, 211 computation of, 149, 181, 214218, 222, 227 feedback, 166172, 176, 178, 434 generic noise variable, 144, 145, 258 symmetric, 177 discrete-time continuous channel, 257, 258 noise variable, 257 discrete-time linear system, XI discrete-time stochastic system, 142 disk array, 426, 508, 538 distortion measure, 183210 average, 184 context dependent, 185 Hamming, 185, 196, 199 normalization, 185, 195 single-letter, 184 square-error, 185 distortion-rate function, 191 distributed source coding, 420, 540 divergence, 7, 20, 23, 2326, 46, 48, 49, 247, 247248, 255, 378 convexity of, 47 566 Index English, entropy rate of, 109 ensemble average, 108 entropic, 326, 365, 395 entropies, linear combination of, 12, 28, 80, 326, 341 entropy, 3, 7, 12, 19, 49, 271, 393, 417 concavity of, 67 relation with groups, 387408 entropy bound, VIII, 84, 8285, 88, 90, 93, 96, 97 for prefix code, 93 entropy function, 326, 361, 366, 368, 372, 382, 385, 393 continuity of, 49, 401 discontinuity of, 48 group characterization, 396, 393397 entropy rate, VIII, 7, 38, 3841, 183 of English, 109 entropy space, 326, 329, 341, 361, 393 Ephremides, A., XIII equivalence relation, 312 erasure probability, 148 Erez, E., 504, 546 ergodic, 108 ergodic stationary source, 108, 112, 172 entropy rate, 109 Estrin, D., 549 Euclidean distance, 20, 216, 369 Euclidean space, 326, 362 expectation-maximization (EM) algorithm, 228 expected distortion, 186, 219 minimum, 187, 197 exponential distribution, 254 extreme direction, 365, 374, 380 facsimile, 1 fair bits, 106, 235, 411 almost fair bits, 106, 107 fair coin, 81 Fan, B., 544 Fano's inequality, 34, 3236, 50, 107, 156, 171, 174, 181, 431, 516, 519 simplified version, 35 tightness of, 48 Fano, R.M., 50, 181, 546 fault-tolerant data storage system, 381, 426 divergence inequality, 24, 26, 47, 215, 247, 378 diversity coding, 426, 509 DMC, see discrete memoryless channel Dobrushin, R.L., 243, 542, 545 double infimum, 213, 220 double supremum, 212, 213, 216, 217 Dougherty, R., 385, 540, 545 duality theorem, 345 dual, 345 primal, 345 Dueck, G., 546 dummy message, 449 dummy node, 416, 417 dyadic distribution, 88 dyadic expansion, 235 ear drum, 382 east-west direction, 213 eavesdropper, 71 edge-disjoint paths, 437, 453, 457, 464, 465, 467, 479, 501 Edmonds, J., 481, 546 efficient source coding, 106107 Effros, M., 482, 540, 545, 548, 549, 553 Egner, S., 482, 549 eigenvalue, 232, 233, 254 eigenvector, 232, 254 electronic circuit, 492 elemental inequalities, 341, 339341, 345, 361, 362 -inequalities, 353357 -inequalities, 353357 minimality of, 353357 Elias, P., 546 EM algorithm, 228 emotion, 1 empirical differential entropy, 246 empirical distribution, 111, 133 joint, 133, 205 empirical entropy, 102, 104, 111, 113 encoder, 104, 159, 202 encoding function, 110, 149, 167, 187, 260, 428 Encyclopedia Britannica, 3, 543 energy constraint, 271 energy signal, 280 engineering, 3, 175 engineering tradeoff, 477 Index fault-tolerant network communication, 508 FCMI, see full conditional mutual independencies Feder, M., 504, 546 feedback, 140, 176, 179181 Feinstein, A., 181, 546 Feldman, J., 546 Feller, W., 546 ferromagnetic material, 314, 321 field, 504, see also finite field real, 435, 460 field size, 456, 457, 459, 466, 468, 472, 473, 480, 481 field, in measure theory, 52 file, 474 finite alphabet, 18, 33, 35, 45, 49, 107, 110, 112, 122, 132, 133, 145, 159, 172, 184, 203, 211, 401, 412, 421 finite duration, 286 finite field, 435, 460, 504 algebra, 435 extension field, 488 finite group, 389, 387408 finite resolution, 1 finite-dimensional maximization, 211 Fitingof, B.M., 546 fix-free code, 98 flow on a directed graph, 422 conservation conditions, 422 value of, 422 Fong, S.L., XIII, 481, 546 Ford, Jr., L.K., 546 Forney, Jr., G.D., 546 forward substitution, 491 fountain code, 477 Fourier transform, 280, 282, 284, 286 inverse, 280, 285 Fragouli, C., 420, 504, 544, 546 Fraleigh, J.B., 547 Freiling, C., 385, 540, 545 frequency, 280 frequency band, 281, 287 frequency component, 282 frequency of error, 185 frequency response, 288 frequency spectrum, 281 Frey, B.J., 550 Fu, F., 74, 547 567 Fubini' theorem, 241 Fujishige, S., 360, 547 Fulkerson, D.K., 546 full conditional independence, IX full conditional mutual independencies, 300, 309313, 351 axiomatization, 321 image of, 310, 312 set-theoretic characterization, 321 functional dependence, 336 fundamental inequality, VIII, 23, 85, 244, 248 fundamental limits, 3 Gcs, P., 541, 544 a Gallager, R.G., XIII, 166, 177, 181, 287, 297, 542, 547, 555, 556 Galois field, see finite field gamma distribution, 255 n , 341361 n , 326, 342, 361, 387 n , 365 group characterization of, 398401 Gargano, L., 79, 543 Gauss elimination, 468 Gaussian channel, 270 bandlimited colored Gaussian channel, IX, 287289, 297 capacity, 289, 297 bandlimited white Gaussian channel, IX, 279287, 297 capacity, 282, 297 bandpass white Gaussian channel, 287, 297 capacity, 287 correlated Gaussian channels, IX, 277279, 294, 297 capacity, 279, 289, 296 noise variable, 277, 278 memoryless Gaussian channel, IX, 270, 269272, 282, 286, 288 capacity, 270272, 274, 296 parallel Gaussian channels, IX, 272277, 279, 297 capacity, 277, 279, 296 noise variable, 272 Gaussian distribution, 231, 236, 237, 286 568 Index axioms of, 388 closure, 390, 391, 393 identity, 388, 390394 inverse, 388391 order of, 387, 389, 391 group inequalities, 387401, 405, 408 group theory, IX, 135, 365 relation with information theory, 387408 group-characterizable entropy function, 396, 393397 Guiasu, S., 547 Gutman, M., 546 Hadamard's inequality, 255 Hadamard, J., 547 Hagenauer, J., XIII Hajek, B., XIII, 547 half-space, 329 Hammer, D., 386, 547 Hamming ball, 208 Hamming code, 166 Hamming distance, 49, 180 Hamming distortion measure, 185 Hamming, R.V., 547 Han, T.S., XIII, 47, 50, 80, 338, 369, 547, 548, 557, 558 Hanly, S.V., 557 hard disk, 482 hardware failure, 468 Hardy, G.H., 49, 548 Harremos, P., 548 e Harvey, N., 548 Hassibi, B., 545, 548 Hau, K.P., 420, 539, 548, 555 Heegard, C., 548 heuristic argument, IX, 282, 287 hiker, 213 Ho, S.-W., 50, 111, 134, 135, 548, 549 Ho, S.T., XIII, 480, 481, 483, 504, 551, 557 Ho, S.W., XIII Ho, T., 420, 482, 540, 548, 552, 553 Hocquenghem, A., 549, see also BCH code home entertainment systems, 166 Horibe, Y., 549 Hu, G.-D., 80, 549 Huang, J., 544 multivariate, 231, 233, 239, 250, 254, 255, 278, 285, 291, 385 zero-mean, 280, 285 Gaussian noise, 270, 297 independent, 297 process, 280, 285, 287 zero-mean, IX, 274, 277, 280, 287, 289294, 297 Ge, Y., 321, 547 generator matrix, 440 generic continuous channel, 258, 259, 270 generic discrete channel, 143, 158, 211, 214 generic message, 487 generic network code, 460, 460468, 471, 480482 alternative definition, 462 construction, 466 simplified characterization, 480 static, 470, 473, 481 construction, 472 transformation of, 481 geometric distribution, 38 Gersho, A., 547 Gitlin, R.D., 420, 542 Gkantsidis, C., 482, 547 Glavieux, A., 542 global encoding kernel, 440, 445, 448, 450, 451, 453, 454, 457, 459, 461, 462, 466, 467, 471, 472, 481, 482, 485, 486, 488, 492, 493, 497, 500, 502, 504 general positions, 460, 466 global Markov property, 314, 320 Goldman, S., 547 Goldsmith, A., 547, 550 Goli, J.Dj., 369, 547 c Golomb, S.W., 547 Govindan, R., 549 Gowaikar, R., 545 gradient, 224 Grant, A., 540, 544 graph theory, 314, 412, 415, 421, 422, 437 graphical models, VIII, 321 Gray, R.M., 544, 547 group, 388, 387408, 540 associativity, 388, 389391, 393 Index Huffman code, 88, 8893 expected length, 90, 92 optimality of, 90 Huffman procedure, 88, 8893 dummy symbols, 89 Huffman, D.A., 100, 549 Hui, J.Y., 549 human factor, 1 Humboldt Foundation, Alexander von, XIII hypergraph, 321 hyperplane, 328, 332, 336, 365, 380 Hyvarinen, L.P., 549 I, C.-L., 420, 542 I-Measure, VIII, 58, 5180, 154, 299321, 361, 368 empty atom, 56 Markov chain, 6067, 74, 317319 Markov structures, 299321 negativity of, 5960 nonempty atom, 56 uniqueness, 58, 63 universal set, 53, 56 i.i.d. source, 104, 107, 111, 113, 184, 209, 211 bivariate, 122, 124 Ibinson, B., 549 identity matrix, 447, 481, 498, 499 Ihara, S., 255, 297, 549 image, 184 imaginary channel, 437, 444, 453, 457, 458, 492, 493, 501 imperfect secrecy theorem, 71 implication problem, IX, 336337, 351353, 385 involves only FCMI's, 312, 336 impulse response, 281, 297, 492 inclusion-exclusion formula, 55 a variation of, 74 incomplete data, 228 incompressible, 106 independence bound for differential entropy, 245, 264 entropy, 29, 106 independence of random variables, 712 mutual, 8, 29, 30, 39, 45, 62, 78, 203, 278, 331, 350 pairwise, 8, 45, 59, 363 569 indeterminate, 442, 450455, 500, 501 inferior, 506, 514 infinite group, 389 infinitesimal perturbation, 460 Information Age, 3 information diagram, VIII, 61, 5180, 347, 352, 372 Markov chain, 6367, 74, 153, 317319 information expressions, 323 canonical form, 326329 alternative, 338 uniqueness, 327, 338 nonlinear, 338 symmetrical, 338 information identities, VIII, 28, 80, 323 constrained, 332, 344345 unconstrained, 329 information inequalities, VIII, X, 28, 67, 323, 387, 401405, 540 constrained, 330332, 344345 equivalence of, 333335, 338 framework for, IX, 323338 machine-proving, ITIP, 325, 347350 non-Shannon-type, IX, 28, 361386 Shannon-type, IX, 339360 symmetrical, 359 unconstrained, 329, 343344, 365, 388, 401 information looping, 486, 488 information rate-distortion function, 192, 202 continuity of, 202 properties of, 193 information source, 2, 38, 82, 183, 265, 411, 417, 428, 435 informational divergence, see divergence Ingleton inequality, 385, 407 Ingleton, A.W., 549 input distribution, 145, 158, 211, 214, 220, 265 strictly positive, 217, 227 input power, 274 input power allocation, 275, 277, 279, 288, 296 input power constraint, 270, 272, 277, 279, 282, 286, 287, 296 Intanagonwiwat, C., 549 intermediate value theorem, 33 570 Index Koga, H., 548 Kolmogorov complexity, 386, 408 Kolmogorov, A.N., 255, 550 Krner, J., 80, 135, 338, 541, 545, 546 o Kraft inequality, VIII, 82, 84, 85, 87, 88, 92, 98, 99 Kraft, L.G., 550 Kramer, G., 550 Kschischang, F.R., 550 Kuhn, H.W., 276, 551, see also Karush-Kuhn-Tucker (KKT) condition Kullback, S., 49, 551 Kullback-Leibler distance, see divergence Kung, S.-Y., 481, 558 Kurose. J.F., 551 Kushilevitz, E., 551 Kwok, P.-W., 481, 551 L1 -convergence, 19 L2 -convergence, 20, 45 Lagrange multipliers, 217, 275 Lagrange's theorem, 393 Laird, N.M., 545 Landau, H.J., 551 Langberg, M., 549, 551 Langdon, G.G., 551 Lapidoth, A., 544, 551 Laplace distribution, 254 large scale content distribution, 474, 483 lattice theory, 80 Lau, L.C., 551 Laurent series, formal, 504 Lauritzen, S.L., 551 laws of information theory, IX, 324, 385 Le Boudec, J.-Y., 546 leaf, 86, 8697 Lebesgue measure, 230, 247, 293, 338, 365 Lebesgue-Stieltjes integration, 229 Lee, T.T., 321, 559 left-continuous, 259, 260, 267 Lehman, E., 540, 555 Leibler, R.A., 49, 551 Lempel, A., 560 Leong, B., 548 letter, 38 internal node, 86, 8697 conditional entropy of, 94 inverse function, 490 invertible matrix, 279, 447, 498 Ising model, 314, 321 iterated integral, 241, 293 iterative algorithm, 181, 210, 211, 220, 228 ITIP, IX, 347350, 360, 369, 383, 385 efficient implementation, 353 Jacquet, P., 549 Jaggi, S., 482, 549 Jaggi-Sanders algorithm, 457, 458460, 480 Jain, K., 481, 482, 544, 549, 558 Jaynes, E.T., 50, 549, 550 Jelinek, F., 550 Jensen's inequality, 201, 292 Jensen, J.L.W.V., 550 Jerohin, V.D., 208, 550 Jewell, W.S., 556 Jindal, N., 550 Johnsen, O., 98, 550 Johnson, R.W., 556 joint entropy, 14, 325 joint source-channel coding, 174, 175 Jones, G.A., 550 Jones, J.M., 550 Kakihara, Y., 550 Karger, D.R., 482, 540, 548, 552, 553 Karush, J., 99, 276, 550 Karush-Kuhn-Tucker (KKT) condition, 276 Katabi, D., 549 Katona, G.O.H., 548, 557 Katti, S., 549 Kawabata, T., 80, 321, 550 key, of a cryptosystem, 71, 78 Khachatrian, L., 98, 541 Khinchin, A.I., 550 Kieffer, J.C., 550, 559 Kindermann, R., 550 King, R., 544 Kleinberg, R., 548 Kobayashi, K., 548 Koetter, R., XIII, 480, 482, 504, 548, 550, 552 Index Leung, S.K., 544 Li, B., 551 Li, J., 551 Li, M., 551 Li, S.-Y.R., XIII, 419, 420, 432, 434, 482, 483, 504, 541, 551, 559 Li, Z., 551 Liang, X.-B., 552 Lieb, E.H., 385, 552 Lin, S., 552 Linden, N., 385, 549, 552 Linder, T., 100, 552 line of sight, 510 linear algebra, XI linear broadcast, 444, 445, 448, 450, 454, 460, 462, 486 multi-rate, 445, 481 static, 470, 473 linear code, 166 linear constraints, 330, 333 linear dispersion, 444, 445, 448, 450, 455, 460, 462, 465, 481 static, 470, 473 linear mapping nullity, 472 pre-image, 472 linear multicast, 444, 445, 448, 450, 453, 459, 460, 462, 473, 476, 480, 482, 499, 502, 504 construction, see Jaggi-Sanders algorithm random, 456 static, 470, 473, 480 transformation of, 459 linear network code, X, 437479, 482, 488 base field, 435 dimension, 439 global description, 440, 485 implementation of, 448449, 476 overhead, 449 linear broadcast, 444 linear dispersion, 444 linear multicast, 444 local description, 439, 451, 485 transformation of, 447448 linear network coding, 421 matrix approach, 482 vector space approach, 482 571 linear programming, IX, 339, 341345, 347, 360 linear span, 442 linear subspace, 331, 374, 380 linear time-invariant (LTI) system, 492, 504 causality, 492 linear transformation, 443 invertible, 328 linear transformation of random variables, 231234, 254, 278 Littlewood, J.E., 49, 548 Lnnika, R., 385, 552 e c local area network (LAN), 445 local encoding kernel, 439, 440, 450, 451, 454, 455, 471, 473, 481, 482, 485, 486, 488, 489, 492, 493, 495, 497, 500, 502 local Markov property, 320 local redundancy, 96 local redundancy theorem, 9697 Loeliger, H.-A., 550 log-optimal portfolio, 228 log-sum inequality, 25, 47, 226 logical network, 474 Lok, T.M., XIII long division, 490, 491, 504 Longo, G., 542 lossless data compression, 3, 100 Lovsz, L., 552 a low-density parity-check (LDPC) code, 166 Luby, M., 543 Lun, D.S., 548, 552 MacKay, D.J.C., 166, 552 majority vote, 139 majorization, 49 Makarychev, K., 385, 552 Makarychev, Y., 385, 552 Malkin, T., 546 Malvestuto, F.M., 321, 552 Mann, H.B., 542 Mansuripur, M., 552 mapping approach, 210 marginal distribution, 266, 295, 370, 371 Marko, H., 180, 552 572 Index membership table, 394 Menger, K., 553 Merhav, N., 546 message, 425 message pipeline, 435, 495, 499 message set, 137, 149, 150, 260, 261 method of types, 117, 135 microelectronics, 166 min-cut, 423, 431, 478 minimization, 343345 minimum distance decoding, 180 Mittelholzer, T., 543, 557 Mitter, S., 557 Mitzenmacher. M., 543, 553 mixing random variable, 68, 264 modulo 2 addition, 389390, 396, 397, 412, 433 modulo 2 arithmetic, 488 modulo 3 addition, 433 Mohan, S., 542 most likely sequence, 104 Moulin, P., 553 Moy, S.C., 553 , see I-Measure multi-dimensional direction, 213 multi-source multicast, 459 multi-source network coding, X, XI, 408, 418, 505540 achievable information rate region, 505 LP bound, 515 information rate region, 540 insufficiency of linear coding, 540 network code for acyclic network, 510 source separation, 537, 539 multicast, X, 411, 412, 421, 435, 505540 multigraph, 421 multilevel diversity coding, 508509, 539 symmetrical, 509, 539 multiple descriptions, 74 multiple unicasts, 414 multiterminal source coding, 135, 209 Munich University of Technology, XIII Murty, U.S.R., 314, 543 mutual information, 7, 15, 219, 241, 242, 255, 265 Markov chain, IX, 7, 9, 30, 63, 66, 67, 70, 72, 74, 79, 143, 153, 171, 177, 179, 258, 262, 299, 300, 314, 317319, 335, 347, 349, 370, 371, 382, 385 information diagram, 6367, 74, 153, 317319 Markov graph, 314 Markov random field, IX, 67, 300, 314316, 321 hypergraph characterization of, 321 Markov star, 321 Markov structures, IX, 299321 Markov subchain, 10 Marshall, A.W., 552 Marton, K., 552 Massey, J.L., XIII, 71, 180, 552, 553 Mathai, A.M., 553 matroid, 385, 540 Mat, F., 369, 385, 553 us Maurer, U.M., 543, 553, 557 max-flow, 422, 431, 442, 444, 445, 459, 468, 476, 477 collection of edges, 423 collection of non-source nodes, 423 max-flow bound, X, 429, 421431, 435, 459, 462, 482, 504, 505 for linear network coding, 443 max-flow bounds, 505508, 537 max-flow min-cut theorem, 423, 431, 443 maximal probability of error, 150, 158, 173, 180, 261 maximization, 276 maximum differential entropy, VIII, 248251 maximum entropy, VIII, IX, 3638, 255 maximum likelihood decoding, 180 Mazo, J., 420, 542 McEliece, R.J., 553 McGill, W.J., 80, 553 McLaughlin, S.W., 558 McMillan, B., 99, 112, 553, see also Shannon-McMillan-Breiman theorem mean ergodic, 108 mean-square error, 185 meaningful information, 1 measure theory, 52, 108, 229, 255 Index between more than two random variables, 60 concavity of, 70, 259, 265, 296 convexity of, 69, 194 mutual typicality, 265266 typical sequence, 266 typical set, 265 mutually independent information sources, 505536 Mdard, M., 480, 482, 504, 540, e 548550, 552, 553 Nair, C., 548 Narayan, P., XIII, 117, 545, 551 nat, 13, 236 natural disasters, 468 neighborhood, 246 neighboring node, 415 nerve impulse, 382 network code deployment, 468 global description, 438 global encoding mapping, 438, 479 local description, 438 local encoding mapping, 438, 479 network coding, XI, 505540 advantage of, 412415, 419 source separation, 417418, 420 Network Coding Homepage, 420 network communication, 411, 412, 415, 417 network error correction, 482 network topology, 449, 468, 481 unknown, 445, 456, 474, 482 network transfer matrix, 479 Ng, W.-Y., XIII Nielsen, M.A., 386, 553 Nisan, N., 551 Nobel, A., 420, 555 noise energy, 270 noise power, 270, 274, 277, 281, 286 noise process, 281, 285, 287, 288 noise source, 2 noise variable, 257, 272, 277, 278 noise vector, 278, 289, 294, 296, 297 noisy channel, 3, 137, 164 noisy environment, 1 non-decreasing, 259 non-increasing, 191 573 non-Shannon-type inequalities, IX, 28, 324, 325, 347, 361386, 540 constrained, 374380 unconstrained, 369374, 388, 404, 540 nonlinear optimization, 211 nonnegative linear combination, 345, 346 nonnegative orthant, 326, 328, 337, 341, 362, 369 normal distribution, see Gaussian distribution north-south direction, 213 null space, 333 numerator, 502 numerical computation, 149, 199, 211228 Nyquist, H., 553 O'Sullivan, J.A., 553 off-diagonal element, 498 Olkin, I., 552 Omura, J.K., 554, 558 Ooi, J.M., 554 optimal coding scheme, 3 Ordentlich, E., 117, 554, 558 order of a node, 87 ordinate, 219 Orlitsky, A., XIII, 554 Ornstein, D.S., 554 orthogonal complement, 333, 334 orthogonal matrix, 232, 234 orthogonal transformation, 233, 234 orthonormal basis, 283, 284, 287 orthonormal set, 284 orthonormal system, 232 output channel, 412 overlay network, 474 Oxley, J.G., 554 Ozarow, L.H., 554 P2P, see peer-to-peer network packet loss, 477 rate, 477 packet network, 477 Palanki, R., 545 Papadimitriou, C.H., 554 Papoulis, A., 80, 554 parallel channels, 177, 359 parity check, 167 574 Index power spectral density, 281, 281, 285, 287 prefix code, VIII, 82, 86, 8697 entropy bound, VIII existence of, 87 expected length, 95 random coding, 99 redundancy, VIII, 9397 prefix-free code, see prefix code Preston, C., 321, 554 prime number, 451 probabilistic coding, 110, 176 probability density function (pdf), 229, 240 conditional, 230, 239241 joint, 230, 290, 291, 294 probability distribution, IX rational, 398 strictly positive, 7, 11, 214, 359 factorization of, 12 with zero masses, 7, 214, 320 probability of error, 34, 104, 138, 157, 185, 192 probability theory, XI, 336, 381 product measure, 243 product source, 208, 359 projection, 514 prolate spheroidal wave functions, 287 pyramid, 341, 342, 345 quantization, 242 quantized samples, 184 quantum information theory, 386 quantum mechanics, 385 quasi-uniform structure, 395 asymptotic, 130, 395 Rabin, M.O., 420, 554 Radon-Nikodym derivative, 243 random code, 159, 202, 268, 539 random coding error exponent, 181 random linear combination, 474, 476 random network coding, 456, 473478, 482 robustness, 477 random noise, 137 random variable, real, 36, 229297 continuous, VIII, 229 discrete, 229 partition, 312 pdf, see probability density function Pearl, J., 554 peer-to-peer (P2P) network, 474, 482 client, 474 neighboring node, 474476 server, 474 tracker, 474 Peile, R.E., 547 perceived distortion, 184 Perez, A., 554 perfect secrecy theorem, Shannon's, 71 permutation, 49, 390 Perrin, D., 542 physical network, 489 physical system, 3 Pierce, J.R., 554 Pinkston, J.T., 209, 554 Pinsker's inequality, 26, 47, 49, 50, 117 Pinsker, M.S., 50, 255, 297, 542, 554 Pippenger, N., 385, 554 plain text, 71, 78 point-to-point channel, 137 noiseless, 411, 421, 434 capacity, 421 point-to-point communication network, 412, 421, 424, 435, 505 point-to-point communication system, 2, 417 Pollak, H.O., 551, 556 Polya, G., 49, 548 polymatroid, 356, 360, 385 polynomial, 338, 450, 451, 480, 490, 491, 500502 equation, 451 degree, 451 nonzero, 451, 454 root, 451 polynomial ring, 451, 453, 500 positive definite matrix, 231, 254, 290 positive semidefinite matrix, 231, 233, 255 postal system, 411 power series, 490 expansion, 491 formal, 490 rational, 490, 492, 498, 501 ring of, 490, 495 ring of, 490, 495 Index mixed, 229 second moment, 234, 286 rank function, 408 rank of a matrix, 333 full, 334, 335, 449 Rasala Lehman, A., 540, 554, 555 rate constraint, 412, 421, 422, 424, 429 rate-distortion code, 187, 183211 rate-distortion function, 191, 187192, 202, 209, 211 binary source, 196 forward channel description, 208 reverse channel description, 198 computation of, VIII, 200, 210, 219222, 227 normalization, 209 product source, 208, 359 properties of, 191 Shannon lower bound, 208 rate-distortion pair, 188, 200 rate-distortion region, 188, 191, 219 rate-distortion theorem, VIII, 134, 193, 192200, 209 achievability, 202207 converse, 200202 random code, 202 relation with source coding theorem, 199 rate-distortion theory, VIII, 183210 Rathie, P.N., 553 rational function, 490, 502 field of, 501 rational number, 189, 398 Ratnakar, N., 552 Ray-Chaudhuri, D.K., 543, see also BCH code Rayleigh's energy theorem, 283 reaching probability, 94, 95 real number, 435 receiver, 2 reciprocal, 491 rectangular lattice, 314, 321 reduced code tree, 91 reduced probability set, 91 redundancy of prefix code, VIII, 9397 of uniquely decodable code, 85 Reed, I.S., 555 Reed-Solomon code, 166 575 relative entropy, see divergence relative frequency, 113, 122 relay node, 416, 417 Rnyi, A., 555 e repetition code, 139 replication of information, 425 reproduction alphabet, 184, 196, 203 reproduction sequence, 183185, 190, 203 reservoir, 277, 289 resultant flow, 422 Reza, F.M., 80, 555 right-continuous, 229 Riis, S., 540, 555 ring, 451, 453, 490, 502 commutative, 491 ring theory, 504 Rissanen, J., 555 Roche, J.R., 420, 539, 555 Rockafellar, R.T., 555 Rodriguez, P.R., 482, 547 Roman, S., 555 Romashchenko, A., 385, 386, 408, 547, 552, 555 Rose, K., 210, 555 Ross, K.W., 551 routing, 411, 411, 412, 425 row space, 333, 334 Rubin, D.B., 545 Rudin, W., 241, 555 Ruskai, M.B., 385, 552 Ruskey, F., 555 Russian, 80 Rustin, R., 546 sampling theorem, 282, 284, 286 bandpass, 287 sampling time, 285 Sanders, P., 482, 549, see also Jaggi-Sanders algorithm Santhanam, N.P., 554 Sason, I., 555 satellite communication, X, 412, 415417, 419 satellite communication network, 508, 510 Savari, S.A., 550, 555 Scholtz, R.A., 547 Schur-concave function, 49 576 Index machine-proving, ITIP, 339, 347350, 361 unconstrained, 343344 Shen, A., 386, 408, 547, 555, 556 Shen, S.-Y., 556 Shenvi, S., XIII Shields, P.C., 556 shift-register, 492, 495, 504 Shore, J.E., 556 Shtarkov, Y.M., 558 Shunsuke, I., 556 sibling, 90 side-information, 209, 417 signal, 137 signal analysis, 280, 282, 490 signal-to-noise ratio, 272 signaling network, 470 signed measure, 52, 58, 59 Simmons, G.J., 552 Simonnard, M., 556 simplex method, 344, 345 optimality test, 344, 345 sinc function, 283, 285 single-input single-output system, 137, 140, 142 single-letter characterization, 211 single-letter distortion measure, 211 single-source network code a class of, 427 causality, 427, 431 single-source network coding, XI, 420, 421, 505 achievable information rate, 429 acyclic network, 435478 cyclic network, 485502 one sink node, 424 three sink nodes, 425 two sink nodes, 425 sink node, 411, 422, 424, 425, 459 Slepian, D., 209, 556 Slepian-Wolf coding, 209 Sloane, N.J.A., 543, 556 Snell, J., 550 Soljanin, E., 420, 504, 544, 546 Solomon, G., 555, see also ReedSolomon code Song, L., 434, 540, 556 sound wave, 382 source code, 82, 175, 183 science, 3 science of information, the, 3 secret key cryptosystem, 71, 78 secret sharing, 79, 80, 349, 483 access structure, 79 information-theoretic bounds, 79, 350 participants, 79 secure network coding, 483 security level of cryptosystem, 71 self-information, 16 semi-graphoid, 352, 360 axioms of, 351 separation of network and channel coding, 434 source and channel coding, 140, 172175, 209, 411 Seroussi, G., 117, 558 Servedio, R.A., 546 set function, 326 additive, 52, 74, 309 set identity, 55, 80, 305 set operations, 51, 52 set theory, VIII, 51 Shadbakht, S., 548 Shamai, S., XIII, 543, 555, 558 Shamir, A., 555 Shannon code, 93 Shannon's information measures, VIII, 7, 1218, 28, 51 continuity of, 1820 discontinuity of, 20 elemental forms, 340, 358 irreducible, 339, 357 linear combination of, 323 reducible, 339, 357 set-theoretic structure of, see I-Measure Shannon's papers, collection of, 3 Shannon, C.E., VII, 2, 49, 99, 104, 112, 181, 208, 209, 255, 297, 360, 385, 546, 555, 556 Shannon-McMillan-Breiman theorem, VIII, 41, 108, 107109, 173 Shannon-type identities constrained, 344345 Shannon-type inequalities, IX, 324, 325, 339360, 369, 540 constrained, 344345 Index source coding theorem, VIII, 3, 104, 104105, 112, 183, 191, 199 coding rate, 104 converse, 105 direct part, 104 general block code, 110 source node, 411, 421, 424 super, 415 source random variable, 208 source sequence, 183185, 190, 202 space-time domain, 488 spanning tree packing, 481 Spitzer, F., 321, 556 Sprintson, A., 551 standard basis, 440, 454, 458, 492 standard deviation, 250 static network code, X, 469, 468473, 482 configuration, 468 generic, 470, see also generic network code, static linear broadcast, 470, see also linear broadcast, static linear dispersion, 470, see also linear dispersion, static linear multicast, 470, see also linear multicast, static robustness, 468 stationary source, VIII, 39, 108 entropy rate, 7, 3841 Steiglitz, K., 554 Stein, C., 546 Steinberg, Y., 557, 558 still picture, 2 Stinson, D.R., 556 Stirling's approximation, 125 stock market, 228 store-and-forward, X, 411, 419, 475, 477, 481 strong asymptotic equipartition property (AEP), 101, 114, 113121, 132, 204 conditional, 125, 204, 398, 534 strong law of large numbers, 108 strong typicality, VIII, 113135, 203, 245 alternative definition, 133 consistency, 122, 158, 204 joint, 122130 577 joint AEP, 124 joint typicality array, 129, 395 jointly typical sequence, 122 jointly typical set, 122 typical sequence, 113 typical set, 113 vs. weak typicality, 121 Studen, M., 80, 352, 360, 385, 552, y 553, 557 sub-channel, 287 subcode, 190 subgroups, IX, 387408 intersection of, 387, 393 membership table, 394 subnetwork, 468 subring, 490 substitution of symbols, 55 suffix code, 98 summit of a mountain, 213 support, 7, 13, 23, 45, 111, 113, 184, 229, 291, 297, 364, 396 supremum, 259, 274 switching theory, 411 symmetric group, 390, 400 symmetric matrix, 231, 233, 254, 290 synchronous transmission, 434 Szpankowski, W., 549, 557 Tan, M., XIII, 480, 481, 483, 557 Taneja, I.J., 557 tangent, 219 Tardos, G., 548, 557 Tarokh, V., 100, 552 Tatikonda, S., 557 Telatar, I.E., 557 telephone conversation, 166 telephone line, 1, 167 television broadcast channel, 2 ternary channel, 178 thermodynamics, 49 Thitimajshima, P., 542 Thomas, J.A., 545 Thomasian, A.J., 542 time average, 108 time domain, 282, 493, 495 time-sharing, 189 Tjalkens, T.J., 558 Toledo, A.L., 557 Tolhuizen, L., 482, 549 578 Index Vaccaro, U., 79, 80, 543 van der Lubbe, J.C.A., 557 van der Meulen, E.C., 557 van Dijk, M., 79, 557 Varaiya, P.P., 547 variable-length channel code, 171 variance, 230, 233, 234, 249, 285 variational dist