 | Zamir R. Lattice coding for signals and networks : a structured coding approach to quantization, modulation and multi-user information theory / with contributions by B.Nazer, Y.Kochman. - Cambridge: Cambridge university press, 2014. - xx, 437 p.: ill., tab. - Bibliogr.: p.408-424. - Ind.: p.425-437. - ISBN 978-0-521-76698-2
Шифр: (И/З.81-Z21) 02
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Preface ...................................................... xiii
Acknowledgements ............................................... xv
List of notation ............................................ xviii
1 Introduction ................................................. 1
1.1 Source and channel coding ............................... 4
1.2 The information theoretic view .......................... 6
1.3 Structured codes ........................................ 7
1.4 Preview ................................................. 8
2 Lattices .................................................... 11
2.1 Representation ......................................... 11
2.2 Partition .............................................. 17
2.3 Equivalent cells and coset leaders ..................... 21
2.4 Transformation and tiling .............................. 25
2.5 Algebraic constructions ................................ 29
Summary ..................................................... 34
Problems .................................................... 35
Interesting facts about lattices ............................ 37
3 Figures of merit ............................................ 39
3.1 Sphere packing and covering ............................ 39
3.2 Quantization: normalized second moment ................. 46
3.3 Modulation: volume to noise ratio ...................... 49
Summary ..................................................... 56
Problems .................................................... 56
Historical notes ............................................ 57
4 Dithering and estimation .................................... 59
4.1 Crypto lemma ........................................... 61
4.2 Generalized dither ..................................... 66
4.3 White dither spectrum .................................. 71
4.4 Wiener estimation ...................................... 74
4.5 Filtered dithered quantization ......................... 78
Summary ..................................................... 80
Problems .................................................... 80
Historical notes ............................................ 82
5 Entropy-coded quantization .................................. 84
5.1 The Shannon entropy .................................... 84
5.2 Quantizer entropy ...................................... 85
5.3 Joint and sequential entropy coding* ................... 89
5.4 Entropy-distortion trade-off ........................... 92
5.5 Redundancy over Shannon ................................ 94
5.6 Optimum test-channel simulation ........................ 98
5.7 Comparison with Lloyd's conditions .................... 101
5.8 Is random dither really necessary? .................... 102
3.1 Universal quantization* ............................... 103
Summary .................................................... 106
Problems ................................................... 106
Historical notes ........................................... 108
6 Infinite constellation for modulation ...................... 110
6.1 Rate per unit volume .................................. 110
6.2 ML decoding and error probability ..................... 112
6.3 Gap to capacity ....................................... 114
6.4 Non-AWGN and mismatch ................................. 117
6.5 Non-equiprobable signaling ............................ 119
6.6 Maximum a posteriori decoding* ........................ 128
Summary .................................................... 131
Problems ................................................... 132
Historical notes ........................................... 133
7 Asymptotic goodness ........................................ 134
7.1 Sphere bound s ........................................ 137
7.2 Sphere-Gaussian equivalence ........................... 142
7.3 Good covering and quantization ........................ 147
7.4 Does packing imply modulation? ........................ 150
7.5 The Minkowski-Hlawka theorem .......................... 152
7.6 Good packing .......................................... 154
7.7 Good modulation ....................................... 156
7.8 Non-AWGN .............................................. 162
7.9 Simultaneous goodness ................................. 164
Summary .................................................... 173
Problems ................................................... 174
Historical notes ........................................... 176
8 Nested lattices ............................................ 178
8.1 Definition and properties ............................. 179
8.2 Cosets and Voronoi codebooks .......................... 181
8.3 Nested linear, lattice and trellis codes .............. 185
8.4 Dithered codebook ..................................... 189
8.5 Good nested lattices .................................. 192
Summary .................................................... 194
Problems ................................................... 195
Historical notes ........................................... 196
9 Lattice shaping ............................................ 197
9.1 Voronoi modulation .................................... 199
9.2 Syndrome dilution scheme .............................. 204
9.3 The high SNR case ..................................... 206
9.4 Shannon meets Wiener (at medium SNR) .................. 212
9.5 The mod Λ channel ..................................... 219
9.6 Achieving Cawgn for all SNR ........................... 226
9.7 Geometric interpretation .............................. 233
9.8 Noise-matched decoding ................................ 234
9.9 Is the dither really necessary? ....................... 237
9.10 Voronoi quantization ................................. 240
Summary .................................................... 242
Problems ................................................... 243
Historical notes ........................................... 245
10 Side-information problems .................................. 247
10.1 Syndrome coding ....................................... 250
10.2 Gaussian multi-terminal problems ...................... 259
10.3 Rate distortion with side information ................. 262
10.4 Lattice Wyner-Ziv coding .............................. 267
10.5 Channels with side information ........................ 279
10.6 Lattice dirty-paper coding ............................ 283
Summary .................................................... 289
Problems ................................................... 290
Historical notes ........................................... 292
11 Modulo-lattice modulation .................................. 295
11.1 Separation versus JSCC ................................ 296
11.2 Figures of merit for JSCC ............................. 298
11.3 Joint Wyner-Ziv/dirty-paper coding .................... 299
11.4 Bandwidth conversion .................................. 305
Summary .................................................... 309
Problems ................................................... 310
Historical notes ........................................... 311
12 Gaussian networks .......................................... 313
12.1 The two-help-one problem .............................. 314
12.2 Dirty multiple-access channel ......................... 326
12.3 Lattice network coding ................................ 335
12.4 Interference alignment ................................ 355
12.5 Summary and outlook ................................... 362
Summary .................................................... 364
Problems ................................................... 366
Historical notes ........................................... 368
13 Error exponents ............................................ 372
13.1 Sphere packing exponent ............................... 373
13.2 Measures of lattice to noise density .................. 376
13.3 Threshold-decoding exponent ........................... 377
13.4 Nearest-neighbor decoding exponent .................... 380
13.5 Distance spectrum and pairwise errors ................. 383
13.6 Minimum-distance exponent ............................. 385
13.7 The expurgated MHS ensemble ........................... 386
13.8 Error exponents of Voronoi codes ...................... 388
Summary .................................................... 396
Problems ................................................... 396
Historical notes ........................................... 398
Appendix ...................................................... 400
A.l Entropy and mutual information ........................ 400
A.2 Success-threshold exponent ............................ 402
A.3 Coset density and entropy ............................. 403
A.4 Convolution of log-concave functions .................. 404
A.5 Mixture versus Gaussian noise ......................... 405
A.6 Lattice-distributed noise ............................. 406
References .................................................... 408
Index ......................................................... 425
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