Algebraic Number Theory and Code Design for Rayleigh Fading Channels

liyuanliu

Fr´ed´erique Oggier
Institut de Math´ematiques Bernoulli
´Ecole Polytechnique F´ed´erale de Lausanne
Lausanne 1015, Switzerland
frederique.oggier@epfl.ch
Emanuele Viterbo
Dipartimento di Elettronica Politecnico di Torino
C.so Duca degli Abruzzi 24
Torino 10129, Italy
[email=viterbo@polito]viterbo@polito[/email].

1 Introduction 1
2 The Communication Problem 5
2.1 The Fading Channel Model 5
2.2 The Transmission System 6
2.3 Signal Space Diversity and Product Distance 8
2.4 Rotated Zn–lattice Constellations 11
3 Some Lattice Theory 15
3.1 First Definitions 15
3.2 Sublattices and Equivalent Lattices 19
3.3 Two Famous Lattices 21
3.4 Lattice Packings and Coverings 23
4 The Sphere Decoder 27
4.1 The Sphere Decoder Algorithm 28
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vi Contents
4.2 The Sphere Decoder with Fading 34
4.3 Conclusions 35
5 First Concepts in Algebraic Number Theory 39
5.1 Algebraic Number Fields 40
5.2 Integral Basis and Canonical Embedding 44
5.3 Algebraic Lattices 48
5.4 Algebraic Lattices over Totally Real Number Fields 53
5.5 Appendix: First Commands in KASH/KANT 54
6 Ideal Lattices 59
6.1 Definition and Minimum Product Distance of an Ideal Lattic59 e
6.2 Zn Ideal Lattices 62
7 RotatedZn–lattices Codes 65
7.1 A Fully Worked Out Example 65
7.2 The Cyclotomic Construction 66
7.3 Mixed Constructions 71
7.4 A Bound on Performance 74
7.5 Some Simulation Results 75
7.6 Appendix: Programming the Lattice Codes 76
8 Other Applications and Conclusions 81
8.1 Dense Lattices for the Gaussian Channel 81
8.2 Complex Lattices for the Rayleigh Fading Channel 82
8.3 Space–Time Block Codes for the Coherent MIMO Chann8e2ls
8.4 Conclusions 84
References 85
TEAM LinG

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