Number Theory Meets Wireless Communications

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Preface Acknowledgements Contents Contributors 1 Number Theory Meets Wireless Communications: An Introduction for Dummies Like Us 1.1 Basic Examples and Fundamentals of Diophantine Approximation 1.1.1 A `baby' Example 1.1.2 Example 1 (Modified `baby' Example) 1.1.3 Badly Approximable Numbers 1.1.4 Probabilistic Aspects 1.1.5 Dirichlet Improvable and Non-improvable Numbers 1.2 A `toddler' Example and Diophantine Approximation in Higher Dimensions 1.2.1 Example 2 1.2.2 Badly Approximable Points 1.2.3 Probabilistic Aspects 1.2.4 The Khintchine-Groshev Theorem and Degrees of Freedom 1.2.5 Dirichlet Improvable and Non-Improvable Points: Achieving Optimal Separation 1.2.6 Singular and Non-Singular Points: The DoF of X-Channel Revisited 1.2.7 Systems of Linear Forms 1.3 A `child' Example and Diophantine Approximation on Manifolds 1.3.1 Example 3 1.3.2 The Khintchine-Groshev Theorem for Manifolds and DoF 1.3.3 Singular and Non-Singular Points on Manifolds References 2 Characterizing the Performance of Wireless Communication Architectures via Basic Diophantine Approximation Bounds 2.1 Introduction 2.1.1 Single-User Gaussian Channels 2.2 Gaussian Multiple-Access Channel Model 2.3 Exploiting Linear Structure 2.4 Universal Bounds via Successive Minima 2.5 Asymptotic Bounds 2.5.1 Small Linear Forms 2.5.2 Independent Channel Gains 2.5.3 Dependent Channel Gains 2.6 Non-Asymptotic Bounds 2.7 Conclusions and Open Problems References 3 On Fast-Decodable Algebraic Space–Time Codes 3.1 Introduction 3.2 Algebraic Tools for Space–Time Coding 3.2.1 Lattices 3.2.2 Algebraic Number Theory 3.2.3 Central Simple Algebras 3.2.3.1 Orders 3.3 Physical Layer Communications 3.3.1 Rayleigh Fading MIMO Channel 3.3.1.1 Performance Parameters of a Wireless Channel 3.3.2 Space–Time Codes 3.3.2.1 Design Criteria for Space–Time Codes 3.3.2.2 Constructions from Cyclic Division Algebras 3.4 Codes with Reduced ML Decoding Complexity 3.4.1 Maximum-Likelihood Decoding 3.4.1.1 Multi-Group Decodable Codes 3.4.1.2 Fast-Group Decodable Codes 3.4.1.3 Block Orthogonal Codes 3.4.2 Inheriting Fast Decodability 3.5 Explicit Constructions 3.5.1 Asymmetric Space–Time Codes 3.5.2 Distributed Space–Time Codes 3.6 Conclusions References 4 Random Algebraic Lattices and Codes for WirelessCommunications 4.1 Introduction 4.1.1 Structure 4.1.2 Summary of Results 4.1.3 Notation 4.2 Classical Methods 4.2.1 Random Lattices 4.2.1.1 Overview 4.2.2 Primitive Points 4.2.3 Linear Improvement 4.3 Random Algebraic Lattices 4.3.1 Eisenstein Integers Z[ω] 4.3.2 Cyclotomic Lattices 4.3.3 Lipschitz and Hurwitz Lattices 4.3.4 A General Construction 4.4 A Glance at Applications to Wireless Communications 4.4.1 Infinite Constellations 4.4.1.1 Classic AWGN Channel 4.4.1.2 Compound Channel Model 4.4.1.3 Block Fading Channel 4.4.2 Power-Constrained General Model 4.4.2.1 Shaping 4.4.3 Lattice Gaussian Distribution 4.5 Achieving Channel Capacity 4.5.1 AWGN Channel 4.5.2 Compound Block Fading Channel 4.5.3 MIMO Fading Channel 4.5.4 Approaching Secrecy Capacity 4.5.4.1 Gaussian Wiretap Channel 4.5.4.2 Fading Wiretap Channel References 5 Algebraic Lattice Codes for Linear Fading Channels 5.1 Introduction 5.2 General Linear Fading Channel 5.3 Lattices and Finite Codes 5.4 Hermite Invariant in the AWGN Channel 5.5 Hermite Invariant in General Linear Fading Models 5.6 Code Design for Diagonal Fading Channels 5.6.1 Codes from Algebraic Number Fields 5.6.2 Codes from Ideals 5.7 Reduced Hermite Invariants as Homogeneous Forms References 6 Multilevel Lattices for Compute-and-Forward and Lattice Network Coding 6.1 Introduction 6.2 Problem Statement 6.3 Background 6.3.1 Algebra 6.3.1.1 Ideal and Principal Ideal Domain 6.3.1.2 Modules Over PID 6.3.2 Lattices and Lattice Codes 6.3.3 Construction A 6.4 Compute-and-Forward and Lattice Network Coding 6.4.1 Compute-and-Forward 6.4.2 Lattice Network Coding 6.5 Multilevel Lattices Evolved from Construction A 6.5.1 Construction D 6.5.2 Construction πA 6.5.3 Multilevel Lattice Network Coding 6.5.4 Elementary Divisor Construction 6.5.5 Nominal Coding Gain and Kissing Number 6.6 Multistage Compute-and-Forward Over Finite Rings 6.6.1 Multistage Compute-and-Forward 6.6.2 Layered Integer Forcing 6.6.3 Multistage Iterative Decoding Algorithm for EDC Lattices 6.6.3.1 Soft Detector for EDC 6.6.4 Simulation Results 6.7 Conclusions Appendix: LIF Quantizer References 7 Nested Linear/Lattice Codes Revisited 7.1 Introduction 7.1.1 System Setup 7.1.2 Structured Codes 7.1.3 Notations 7.2 Preliminaries 7.2.1 Nested Linear Codes 7.2.2 Nested Lattice Codes 7.2.3 Nested Construction A 7.2.4 Results from Number Theory 7.3 Achievable Rate of Nested Linear Codes 7.3.1 Performance Analysis of a Nested Linear Code 7.3.2 Average Performance Analysis of Nested Linear Codes 7.3.2.1 Bounding P(E1) 7.3.2.2 Bounding P(E2(m)) 7.3.2.3 Bounding P(E3(m)) 7.3.2.4 Bounding P(E4(m) ) 7.3.2.5 Putting Everything Together 7.4 Achievable Rate of Nested Lattice Codes 7.4.1 Performance Analysis of a Nested Lattice Code 7.4.2 Average Performance Analysis of a Nested Lattice Codes 7.4.2.1 Bounding P(W e B(re)) 7.4.2.2 Bounding ¶( E3(m) W B(re), Gc = Gc) 7.4.3 Putting Everything Together 7.4.3.1 Spherical Shaping 7.4.3.2 The Selection of Parameters 7.5 Conclusions Appendix 1: Entropy Appendix 2: Typical Sequences References