NON-BINARY ERROR CONTROL CODING FOR WIRELESS COMMUNICATION

Rolando Antonio Carrasco

Newcastle University, UK

Martin Johnston

Newcastle University, UK

John Wiley & Sons, Ltd

Contents

Biographies xiii

Preface xv

Acknowledgements xvii

1 Information, Channel Capacity and Channel Modelling 1 1.1 Introduction l

1.1.1 Information Theory [ 1 ] 2 1.1.2 Definition of Information [1,2] 2

1.2 Measure of Information [1-3] 2 1.2.1 Average Information 5 1.2.2 The Entropy of a Binary Source 6 1.2.3 Mutual Information 9

1.3 Channel Capacity 11 1.3.1 Binary Symmetric Input Channel 12 1.3.2 Binary Erasure Channel (ВЕС) 13 / .3.3 The 4-ary Symmetric Channel [1,5] 15 1.3.4 Binary Input Capacity of a Continuous Channel (Channel Capacity) 17 1.3.5 Channel Capacity in Fading Environments 20

1.4 Channel Modelling 22 1.4.1 Classification of Models 22

1.5 Definition of a Communications Channel and its Parameters 24 1.5.1 Doppler Shift 26 1.5.2 Scattering 28 1.5.3 Angle of Arrival and Spectra 30 / .5.4 Multipath Channel and Tapped Delay Line Model 31 1.5.5 Delay Spread 33 1.5.6 The Fixed Broadband Wireless Access Channel 34 1.5.7 UMTS Mobile Wireless Channel Model 35 1.5.8 Simulating the Fixed Broadband Wireless Access

Channel Model 37 1.5.9 Simulating the UMTS Mobile Radio Channel 38

1.6 (Multiple-Input-Multiple-Output) (MIMO) Channel 40

viii CONTENTS

1.7 Magnetic Storage Channel Modelling 43 1.7.1 Longitudinal Recording 44 1.7.2 (Partial Response Maximum Likelihood) (PRML) Detection 45

1.8 Conclusions 48 References 48

2 Basic Principles of Non-Binary Codes 51 2.1 Introduction to Algebraic Concepts 51

2.1.1 Groups 51 2.1.2 Rings 52 2.1.3 Ring of Polynomials 53 2.1.4 Fields 54 2.1.5 Primitive Polynomials 56 2.7.6 Minimal Polynomials and Cyclotomic Cosets 58 2.1.7 Subfields 59

2.2 Algebraic Geometry Principles 60 2.2.1 Projective and Affine Space 60 2.2.2 Projective and Affine Curves 60 2.2.3 Finding Points on an Affine Curve 61 2.2.4 Rational Functions on Curves 64 2.2.5 Riemann-Roch Theorem 67 2.2.6 The Zero Order of a Monomial 68 2.2.7 AG Code Parameters 70

2.3 Conclusions 71 References 71

3 Non-Binary Block Codes 73 3.1 Introduction 73 3.2 Fundamentals of Block Codes 74

3.2.1 Generator and Parity Check Matrices 11 3.2.2 Decoding Block Codes 79

3.3 Cyclic Codes 80 3.3.1 Polynomials 80 3.3.2 Systematic Cyclic Codes 82

3.4 Bose-Chaudhuri-Hocquenghem (BCH) Codes 84 3.5 Reed-Solomon Codes 86 3.6 Decoding Reed-Solomon Codes 88

3.6.1 Euclid's Algorithm 89 3.6.2 Вerlekamp-Massey s Algorithm 93 3.6.3 Determining the Error Magnitudes Using Forney's Formula 95

3.7 Coded Modulation 96 3.7.1 Block Coded Modulation (BCM) Codes 96 3.7.2 Multi-Level Block Coding 97

CONTENTS ix

3.7.3 Set Partitioning 99 3.7.4 Construction of a Block Code Trellis 100 3.7.5 Non-Binary BCM Codes 103

3.8 Conclusions 106 References 106

4 Algebraic-Geometric Codes 109 4.1 Introduction 109 4.2 Construction of Algebraic-Geometric Codes 110

4.2.1 Simplified Construction of AG Codes 113 4.2.2 Comparison of AG Codes with Reed-Solomon Codes 115 4.2.3 Systematic Algebraic-Geometric Codes 115

4.3 Decoding Algebraic-Geometric Codes 119 4.3.1 Modification of the Sets F and G 123

4.4 Majority Voting 125 4.5 Calculating the Error Magnitudes 126 4.6 Complete Hard-Decision Decoding Algorithm for

Hermitian Codes 128 4.7 Simulation Results 140 4.8 Conclusions 144

References 144

5 List Decoding 147 5.1 Introduction 147 5.2 List Decoding of Reed-Solomon Codes Using the

Guruswami-Sudan Algorithm 148 5.2.1 Weighted Degrees and Lexicographic Orders 148 5.2.2 Interpolation Theorem 150 5.2.3 Iterative Polynomial Construction 152 5.2.4 Complexity Reduced Modification 154 5.2.5 Factorization 159 5.2.6 Recursive Coefficient Search 160

5.3 Soft-Decision List Decoding of Reed-Solomon Codes Using the Kötter-Vardy Algorithm 164 5.3.1 Mapping Reliability Values into Multiplicity Values 164 5.3.2 Solution Analysis for Soft-Decision

List Decoding 168 5.3.3 Simulation Results 171

5.4 List Decoding of Algebraic-Geometric Codes 171 5.5 Determining the Corresponding Coefficients 178 5.6 Complexity Reduction Interpolation 181 5.7 General Factorization 187

X CONTENTS

5.8 Soft-Decision List Decoding of Hermitian Codes 192 5.8.1 System Solution 193 5.8.2 Simulation Results 199

5.9 Conclusions 199 References 200

6 Non-Binary Low-Density Parity Check Codes 201 6.1 Introduction 201 6.2 Construction of Binary LDPC Codes - Random and Structured

Methods 201 6.2.1 Tanner Graph Representation 202 6.2.2 Structured Construction Methods 203 6.2.3 LDPC Codes from Class 1 Böse B1BD 204 6.2.4 Constructing the Generator Matrix of a Binary LDPC Code 206

6.3 Decoding of Binary LDPC Codes Using the Belief Propagation Algorithm 208 6.3.1 Initialization of the Belief Propagation Algorithm 208 6.3.2 The Horizontal Step 210 6.3.3 The Vertical Step 211 6.3.4 Reducing the Decoding Complexity Using Fast Fourier

Transforms 216 6.4 Construction of Non-Binary LDPC Codes Defined Over

Finite Fields 218 6.4.1 Construction of Non-Binary LDPC Codes from

Reed-Solomon Codes 219 6.5 Decoding Non-Binary LDPC Codes with the Sum-Product

Algorithm 221 6.5.7 Received Symbol Likelihoods 222 6.5.2 Permutation of Likelihoods 223 6.5.3 Factor Graph of a Non-Binary LDPC Code 224 6.5.4 The Fast Fourier Transform for the Decoding of Non-Binary

LDPC Codes 224 6.6 Conclusions 234

References 235

7 Non-Binary Convolutional Codes 237 7.1 Introduction 237

7.1.1 The Viterbi Algorithm 243 7.1.2 Trellis Coded Modulation 243 7.1.3 TCM Based on Rings of Integers 246 7.1.4 Ring-TCM Codes for M-PSK 247 7.1.5 Ring-TCM Codes Using Quadrature Amplitude Modulation 250 7.1.6 Searching for Good Ring-TCM Codes 251

CONTENTS x\

7.1.7 Genetic AIgorithm 253 7.1.8 Performance of Ring-TCM Codes on Urban Fading Channels 261

7.2 Space-Time Coding Modulation 262 7.2.1 Introduction 262 7.2.2 Space-Time Codes Model 264 7.2.3 Performance ofST-RTCM Codes Using QPSKfor MIMO

Urban Environments 266 7.2.4 Ideal CS! and Slow Fading 267 7.2.5 Log-Likelihood Function 268 7.2.6 Maximum-Likelihood Sequence Detection 269 7.2.7 ST Codes Pairwise Error Probability 269 7.2.8 ST Code Design Criteria for Slow Fading Channels 273 7.2.9 ST Code Design Criteria for Fast Fading Channels 274 7.2.10 Space-Time Ring-TCM Codes for QAM in Fading Channels 21A

7.3 Conclusions 278 References 278

8 Non-Binary Turbo Codes 281 8.1 Introduction 281 8.2 The Turbo Encoder 281

8.2.1 Notation 282 8.3 The Turbo Decoder 284

8.3.1 Log-Likelihood Ratios (LLRs) 285 8.3.2 Maximum A Posteriori (MAP) Decoding 287 8.3.3 Max-Log MAP 290 8.3.4 Log MAP Algorithm 292 8.3.5 Iterative Log MAP Decoding 292

8.4 Non-Binary Turbo Codes 294 8.4.1 Multi-Dimensional Log-Likelihood Ratios 296 8.4.2 Non-Binary Iterative Turbo Decoding 297

8.5 Conclusions 298 References 299

Index 301