英文版《信息论基础》第二版概览

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"信息论基础 (英文版）是第二版，由Thomas M. Cover和Joy A. Thomas合著，由John Wiley & Sons, Inc.出版，共计774页。本书涵盖了信息论的基本概念和理论，是该领域的经典教材。" 在信息论中，主要研究的是信息的度量、编码、传输和处理等问题。以下是一些核心知识点： 1. **信息熵（Entropy）**：信息熵是衡量一个随机变量不确定性或信息量的度量，通常用比特（bits）表示。熵越大，不确定性越高，包含的信息也越多。公式为 H(X) = -∑ P(x) log2(P(x))，其中X是随机变量，P(x)是x发生的概率。 2. **互信息（Mutual Information）**：互信息用来衡量两个随机变量之间的关联程度，表示从一个变量中获取的信息对另一个变量的不确定性减少的程度。公式为 I(X; Y) = H(X) - H(X|Y)，它非负且等于零当且仅当X和Y独立。 3. **信源编码（Source Coding）**：信源编码的目标是用尽可能少的符号来表示信息，无损编码保证了原始信息能完全恢复，而有损编码则允许一定程度的信息损失以换取更高的压缩效率。 4. **信道容量（Channel Capacity）**：由香农定理定义，信道容量是给定信道条件下，能够无错误传输的最大信息速率。对于一个离散无记忆信道，信道容量可以用C = max_{p(x)} [I(X; Y)] 表示，其中X是输入，Y是输出，I(X; Y)是互信息。 5. **噪声与信噪比（SNR）**：信道中的噪声会影响信息传输的准确性。信噪比是信号功率与噪声功率之比，通常用来衡量信道质量。高信噪比意味着更好的传输效果。 6. **编码理论（Coding Theory）**：包括纠错码和检错码，如汉明码、循环冗余校验（CRC）、卷积码和涡轮码等，用于检测和纠正传输过程中出现的错误。 7. **最大似然解码（Maximum Likelihood Decoding）**：是一种常见的解码策略，旨在找到最可能生成接收到的信号的发送序列。 8. **信息博弈论（Game Theory in Information）**：信息博弈论研究在有信息不对称的情况下，决策者如何进行策略选择，如在通信网络中的资源分配问题。 9. **信息理论与数据压缩**：如LZ77、LZW算法和JPEG、PNG、MP3等数据压缩标准都是基于信息论原理设计的。 10. **信息理论与通信**：信息论在无线通信、数字通信、卫星通信等领域有着广泛的应用，如CDMA、OFDM等通信系统的设计。以上只是信息论基础的冰山一角，实际内容还包括信道容量的计算、信道编码定理、率失真理论等复杂主题。通过深入学习这本书，读者可以全面理解信息论的基础理论，并为在通信、数据科学、密码学等相关领域的工作打下坚实的基础。

PREFACE TO THE

SECOND EDITION

In the years since the publication of the ﬁrst edition, there were many

aspects of the book that we wished to improve, to rearrange, or to expand,

but the constraints of reprinting would not allow us to make those changes

between printings. In the new edition, we now get a chance to make some

of these changes, to add problems, and to discuss some topics that we had

omitted from the ﬁrst edition.

The key changes include a reorganization of the chapters to make

the book easier to teach, and the addition of more than two hundred

new problems. We have added material on universal portfolios, universal

source coding, Gaussian feedback capacity, network information theory,

and developed the duality of data compression and channel capacity. A

new chapter has been added and many proofs have been simpliﬁed. We

have also updated the references and historical notes.

The material in this book can be taught in a two-quarter sequence. The

ﬁrst quarter might cover Chapters 1 to 9, which includes the asymptotic

equipartition property, data compression, and channel capacity, culminat-

ing in the capacity of the Gaussian channel. The second quarter could

cover the remaining chapters, including rate distortion, the method of

types, Kolmogorov complexity, network information theory, universal

source coding, and portfolio theory. If only one semester is available, we

would add rate distortion and a single lecture each on Kolmogorov com-

plexity and network information theory to the ﬁrst semester. A web site,

http://www.elementsoﬁnformationtheory.com, provides links to additional

material and solutions to selected problems.

In the years since the ﬁrst edition of the book, information theory

celebrated its 50th birthday (the 50th anniversary of Shannon’s original

paper that started the ﬁeld), and ideas from information theory have been

applied to many problems of science and technology, including bioin-

formatics, web search, wireless communication, video compression, and

PREFACE TO THE

FIRST EDITION

This is intended to be a simple and accessible book on information theory.

As Einstein said, “Everything should be made as simple as possible, but no

simpler.” Although we have not veriﬁed the quote (ﬁrst found in a fortune

cookie), this point of view drives our development throughout the book.

There are a few key ideas and techniques that, when mastered, make the

subject appear simple and provide great intuition on new questions.

This book has arisen from over ten years of lectures in a two-quarter

sequence of a senior and ﬁrst-year graduate-level course in information

theory, and is intended as an introduction to information theory for stu-

dents of communication theory, computer science, and statistics.

There are two points to be made about the simplicities inherent in infor-

mation theory. First, certain quantities like entropy and mutual information

arise as the answers to fundamental questions. For example, entropy is

the minimum descriptive complexity of a random variable, and mutual

information is the communication rate in the presence of noise. Also,

as we shall point out, mutual information corresponds to the increase in

the doubling rate of wealth given side information. Second, the answers

to information theoretic questions have a natural algebraic structure. For

example, there is a chain rule for entropies, and entropy and mutual infor-

mation are related. Thus the answers to problems in data compression

and communication admit extensive interpretation. We all know the feel-

ing that follows when one investigates a problem, goes through a large

amount of algebra, and ﬁnally investigates the answer to ﬁnd that the

entire problem is illuminated not by the analysis but by the inspection of

the answer. Perhaps the outstanding examples of this in physics are New-

ton’s laws and Schr

odinger’s wave equation. Who could have foreseen the

awesome philosophical interpretations of Schr

odinger’s wave equation?

In the text we often investigate properties of the answer before we look

at the question. For example, in Chapter 2, we deﬁne entropy, relative

entropy, and mutual information and study the relationships and a few

xvii

xviii PREFACE TO THE FIRST EDITION

interpretations of them, showing how the answers ﬁt together in various

ways. Along the way we speculate on the meaning of the second law of

thermodynamics. Does entropy always increase? The answer is yes and

no. This is the sort of result that should please experts in the area but

might be overlooked as standard by the novice.

In fact, that brings up a point that often occurs in teaching. It is fun

to ﬁnd new proofs or slightly new results that no one else knows. When

one presents these ideas along with the established material in class, the

response is “sure, sure, sure.” But the excitement of teaching the material

is greatly enhanced. Thus we have derived great pleasure from investigat-

ing a number of new ideas in this textbook.

Examples of some of the new material in this text include the chapter

on the relationship of information theory to gambling, the work on the uni-

versality of the second law of thermodynamics in the context of Markov

chains, the joint typicality proofs of the channel capacity theorem, the

competitive optimality of Huffman codes, and the proof of Burg’s theorem

on maximum entropy spectral density estimation. Also, the chapter on

Kolmogorov complexity has no counterpart in other information theory

texts. We have also taken delight in relating Fisher information, mutual

information, the central limit theorem, and the Brunn–Minkowski and

entropy power inequalities. To our surprise, many of the classical results

on determinant inequalities are most easily proved using information the-

oretic inequalities.

Even though the ﬁeld of information theory has grown considerably

since Shannon’s original paper, we have strived to emphasize its coher-

ence. While it is clear that Shannon was motivated by problems in commu-

nication theory when he developed information theory, we treat informa-

tion theory as a ﬁeld of its own with applications to communication theory

and statistics. We were drawn to the ﬁeld of information theory from

backgrounds in communication theory, probability theory, and statistics,

because of the apparent impossibility of capturing the intangible concept

of information.

Since most of the results in the book are given as theorems and proofs,

we expect the elegance of the results to speak for themselves. In many

cases we actually describe the properties of the solutions before the prob-

lems. Again, the properties are interesting in themselves and provide a

natural rhythm for the proofs that follow.

One innovation in the presentation is our use of long chains of inequal-

ities with no intervening text followed immediately by the explanations.

By the time the reader comes to many of these proofs, we expect that he

or she will be able to follow most of these steps without any explanation

and will be able to pick out the needed explanations. These chains of

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