Polar Codes - Emanuele Viterbo

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Résumé :

About the Authors xv

Preface xvii

Acknowledgment xxi

1 Introduction 1

1.1 Reliable Transmission over Noisy Channels 2

1.2 Channel Models 4

1.2.1 Binary Symmetric Channel (BSC) 4

1.2.2 Binary Erasure Channel (BEC) 5

1.2.3 Additive White Gaussian Noise Channel (AWGN) 6

1.2.4 Fading Channel 7

1.3 Fundamentals of Information Theory 7

1.3.1 The Meaning of Information 8

1.3.2 Discrete Entropy 9

1.3.3 Properties of the Discrete Entropy 10

1.3.4 Mutual Information 11

1.3.5 Bhattacharyya Parameter 12

1.4 Channel Capacity and Channel Coding Theorem 13

1.5 Block Error Rate for Finite Length Codes 15

1.6 Shannon Limit on Power Efficiency 17

1.7 Time and Space Complexity 19

1.8 Historical Milestones in Channel Coding 20

1.9 Why Study Polar Codes? 24

1.9.1 How Do Polar Codes Compare with Other Codes? 25

Exercises 26

Bibliographic Notes 27

Bibliography 28

2 Linear Codes 29

2.1 Generator and Parity-Check Matrices 29

2.2 Distance, Weight, and Bounds 32

2.3 Syndrome Decoding 35

2.4 Repetition and Single Parity Check Codes 37

2.5 Reed-Muller Codes 37

2.5.1 Boolean Functions and Boolean Polynomials 38

2.5.1.1 Evaluation of Monomials 38

2.5.1.2 Boolean Products 40

2.5.1.3 Boolean Polynomials 40

2.5.2 Submatrix of Binary Walsh-Hadamard Matrix 41

2.5.3 Plotkin's Construction 42

2.6 Cyclic Codes 45

2.6.1 Arithmetic of Polynomials 45

2.6.2 Generator Polynomial and Encoding 46

2.6.3 Cyclic Redundancy Check (CRC) for Error Detection 50

2.7 Convolutional Codes 52

2.7.1 Encoding of Convolutional Codes 53

2.7.1.1 Encoding with Sequential Logic 53

2.7.1.2 Impulse Response and Convolution 53

2.7.1.3 Generator Matrix 54

2.7.2 Termination, Truncation, and Tail-Biting 55

2.7.3 Tree and Trellis 56

2.8 Concatenated Codes 58

Exercises 61

Bibliographic Notes 63

Bibliography 63

3 Fundamentals of Soft-Decision Decoding 65

3.1 MAP/ML Decoding and Likelihood 66

3.1.1 Log-Likelihood Ratio (LLR) 68

3.1.2 LLR Algebra 69

3.1.3 Maximum-Likelihood (ML) Bound 70

3.1.4 Union Bound 70

3.2 Reliability-Based Universal Decoding 73

3.3 Recursive Decoding 75

3.4 Message-Passing (MP) Decoding 78

3.4.1 Factorization and Factor Graph 78

3.4.2 Message Passing 79

3.4.3 The Effect of Cycles on Message Passing 82

3.4.3.1 Scenario with Cycles 83

3.5 Tree/Trellis-Based Search Decoding 84

3.5.1 Sequential Decoding 84

3.5.1.1 Fano Algorithm 85

3.5.1.2 Stack Decoding 85

3.5.1.3 Path Metric 86

3.5.1.4 Computational Cutoff-Rate 87

3.5.2 List Decoding 88

3.5.3 Trellis-Based Viterbi Decoding 88

3.5.4 Lattice-Based Sphere Decoding 89

Exercises 89

Bibliographic Notes 89

Bibliography 91

4 Polar Codes 95

4.1 Introduction 96

4.2 Basic Channel Transform 97

4.3 Recursive Channel Transformation 99

4.4 Channel Polarization 103

4.5 Code Construction Based on Z (W(i)N) 107

4.6 G N -Coset Codes 110

4.7 Successive Cancellation (SC) Decoding 110

4.8 Performance of Polar Codes Under SC Decoding 112

Exercises 117

Bibliographical Notes 117

Bibliography 118

5 Properties of Polar Codes 121

5.1 Polar Transform 122

5.2 Permutation Matrix 123

5.3 Generator and Parity Check Matrices 125

5.4 Systematic Polar Codes 126

5.5 Reed-Mu...

Biographie:

Mohammad Rowshan, PhD, (Member, IEEE) received the B.Eng. degree (Hons.) in electrical engineering from the University of Nottingham in 2015 (ranked 1), the M.Sc. degree in electrical engineering from The Hong Kong University of Science and Technology in 2016, and the Ph.D. degree in electrical engineering from Monash University in 2021. He is currently an Engineering ECA Fellow with the School of Electrical Engineering and Telecommunications, University of New South Wales, Sydney, Australia, where he serves as a Researcher, a Lecturer, and a Supervisor/Mentor. He also serves as a reviewer of IEEE conferences and journals, and a TPC member of conferences.

Emanuele Viterbo, PhD, is Professor in the Department of Electrical and Computer Systems Engineering, Monash University, Melbourne, Australia. He is a Fellow of the IEEE and an ISI Highly Cited Researcher (2009, Thompson Reuters). He has published extensively on a range of subjects related to channel coding and wireless communication....

Sommaire:

About the Authors xv

Preface xvii

Acknowledgment xxi

1 Introduction 1

1.1 Reliable Transmission over Noisy Channels 2

1.2 Channel Models 4

1.2.1 Binary Symmetric Channel (BSC) 4

1.2.2 Binary Erasure Channel (BEC) 5

1.2.3 Additive White Gaussian Noise Channel (AWGN) 6

1.2.4 Fading Channel 7

1.3 Fundamentals of Information Theory 7

1.3.1 The Meaning of Information 8

1.3.2 Discrete Entropy 9

1.3.3 Properties of the Discrete Entropy 10

1.3.4 Mutual Information 11

1.3.5 Bhattacharyya Parameter 12

1.4 Channel Capacity and Channel Coding Theorem 13

1.5 Block Error Rate for Finite Length Codes 15

1.6 Shannon Limit on Power Efficiency 17

1.7 Time and Space Complexity 19

1.8 Historical Milestones in Channel Coding 20

1.9 Why Study Polar Codes? 24

1.9.1 How Do Polar Codes Compare with Other Codes? 25

Exercises 26

Bibliographic Notes 27

Bibliography 28

2 Linear Codes 29

2.1 Generator and Parity-Check Matrices 29

2.2 Distance, Weight, and Bounds 32

2.3 Syndrome Decoding 35

2.4 Repetition and Single Parity Check Codes 37

2.5 Reed-Muller Codes 37

2.5.1 Boolean Functions and Boolean Polynomials 38

2.5.1.1 Evaluation of Monomials 38

2.5.1.2 Boolean Products 40

2.5.1.3 Boolean Polynomials 40

2.5.2 Submatrix of Binary Walsh-Hadamard Matrix 41

2.5.3 Plotkin's Construction 42

2.6 Cyclic Codes 45

2.6.1 Arithmetic of Polynomials 45

2.6.2 Generator Polynomial and Encoding 46

2.6.3 Cyclic Redundancy Check (CRC) for Error Detection 50

2.7 Convolutional Codes 52

2.7.1 Encoding of Convolutional Codes 53

2.7.1.1 Encoding with Sequential Logic 53

2.7.1.2 Impulse Response and Convolution 53

2.7.1.3 Generator Matrix 54

2.7.2 Termination, Truncation, and Tail-Biting 55

2.7.3 Tree and Trellis 56

2.8 Concatenated Codes 58

Exercises 61

Bibliographic Notes 63

Bibliography 63

3 Fundamentals of Soft-Decision Decoding 65

3.1 MAP/ML Decoding and Likelihood 66

3.1.1 Log-Likelihood Ratio (LLR) 68

3.1.2 LLR Algebra 69

3.1.3 Maximum-Likelihood (ML) Bound 70

3.1.4 Union Bound 70

3.2 Reliability-Based Universal Decoding 73

3.3 Recursive Decoding 75

3.4 Message-Passing (MP) Decoding 78

3.4.1 Factorization and Factor Graph 78

3.4.2 Message Passing 79

3.4.3 The Effect of Cycles on Message Passing 82

3.4.3.1 Scenario with Cycles 83

3.5 Tree/Trellis-Based Search Decoding 84

3.5.1 Sequential Decoding 84

3.5.1.1 Fano Algorithm 85

3.5.1.2 Stack Decoding 85

3.5.1.3 Path Metric 86

3.5.1.4 Computational Cutoff-Rate 87

3.5.2 List Decoding 88

3.5.3 Trellis-Based Viterbi Decoding 88

3.5.4 Lattice-Based Sphere Decoding 89

Exercises 89

Bibliographic Notes 89

Bibliography 91

4 Polar Codes 95

4.1 Introduction 96

4.2 Basic Channel Transform 97

4.3 Recursive Channel Transformation 99

4.4 Channel Polarization 103

4.5 Code Construction Based on Z (W(i)N) 107

4.6 G N -Coset Codes 110

4.7 Successive Cancellation (SC) Decoding 110

4.8 Performance of Polar Codes Under SC Decoding 112

Exercises 117

Bibliographical Notes 117

Bibliography 118

5 Properties of Polar Codes 121

5.1 Polar Transform 122

5.2 Permutation Matrix 123

5.3 Generator and Parity Check Matrices 125

5.4 Systematic Polar Codes 126

5.5 Reed-Mu...