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Chapter 1 Introduction 1

Contents 1

Before it starts,there is something must be known 1

1.1 What is Information 2

1.2 What's Information Theory? 4

1.2.1 Origin and Development of Information Theory 4

1.2.2 The application and achievement of Information Theory methods 6

1.3 Formation and Development of Information Theory 7

Questions and Exercises 8

Biography of Claude Elwood Shannon 8

Chapter 2 Basic Concepts of Information Theory 11

Contents 11

Preparation knowledge 11

2.1 Self-information and conditional self-information 12

2.1.1 Self-Information 12

2.1.2 Conditional Self-Information 14

2.2 Mutual information and conditional mutual information 14

2.3 Source entropy 16

2.3.1 Introduction of entropy 16

2.3.2 Mathematics description of source entropy 17

2.3.3 Conditional entropy 20

2.3.4 Union entropy(Communal entropy) 20

2.3.5 Basic nature and theorem of source entropy 21

2.4 Average mutual information 26

2.4.1 Definition 26

2.4.2 Physics significance of average mutual information 27

2.4.3 Properties of average mutual information 28

2.5 Continuous source 38

2.5.1 Entropy of the continuous source(also called differential entropy) 39

2.5.2 Mutual information of the continuous random variable 44

Questions and Exercises 44

Additional reading materials 46

Chapter 3 Discrete Source Information 51

Contents 51

3.1 Mathematical model and classification of the source 51

3.2 The discrete source without memory 54

3.3 Multi-marks discrete steady source 60

3.4 Source entropy of discrete steady source and limit entropy 67

3.5 The source redundancy and the information difference 71

3.6 Markov information source 71

Exercise 77

Chapter 4 Channel and Channel Capacity 79

Contents 79

4.1 The model and classification of the channel 79

4.1.1 Channel Models 79

4.1.2 Channel classifications 80

4.2 Channel doubt degree and average mutual information 82

4.2.1 Channel doubt degree 82

4.2.2 Average mutual information 82

4.2.3 Properties of mutual information function 83

4.2.4 Relationship between entropy,channel doubt degree and mutual information 86

4.3 The discrete channel without memory and its channel capacity 88

4.4 Channel capacity 89

4.4.1 Concept of channel capacity 89

4.4.2 Discrete channel without memory and its channel capacity 91

4.4.3 Continuous channel and its channel capacity 99

Chapter 5 Lossless source coding 106

Contents 106

5.1 Lossless coder 106

5.2 Lossless source coding 110

5.2.1 Fixed length coding theorem 110

5.2.2 Unfixed length source coding 113

5.3 Lossless source coding theorems 115

5.3.1 Classification of code and main coding method 115

5.3.2 Kraft theorem 116

5.3.3 Lossless unfixed source coding theorem(Shannon First theorem) 116

5.4 Pragmatic examples of lossless source coding 120

5.4.1 Huffman coding 120

5.4.2 Shannon coding and Fano coding 128

5.5 The Lempel-ziv algorithm 130

5.6 Run-Length Encoding and the PCX format 132

Questions and Exercises 134

Chapter 6 Limited distortion source coding 137

Contents 137

6.1 The start point of limit distortion theory 138

6.2 Distortion measurement 140

6.2.1 Distortion function 140

6.2.2 Average distortion 142

6.3 Information rate distortion function 143

6.4 Property of R(D) 145

6.4.1 Minimum of D and R(D) 145

6.4.2 Dmax and R(Dmax) 151

6.4.3 The under convex function of R(D) 154

6.4.4 The decreasing function of R(D) 154

6.4.5 R(D)is a continuous function of D 155

6.5 Calculation of R(D) 156

6.5.1 Calculation of R(D)of binary symmetric source 156

6.5.2 Calculation of R(D)of Gauss source 158

6.6 Limited distortion source encoding theorem 159

Additional material for this chapter 161

Questions and exercises 168

Chapter 7 Channel Coding Theory 170

Contents 170

7.1 Channel coding theorem for noisy channel 170

7.2 Introduction:the generator and parity-check matrices 174

7.3 Syndrome decoding on q-ary symmetric channels 177

7.4 Hamming geometry and code performance 179

7.5 Hamming codes 180

7.6 Cyclic code 181

7.7 Syndrome decoding on general q-ary channels 191

Questions and exercises 194

Bibliography 197