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Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Acknowledgements
- List of notation
- 1 Introduction
- 2 Lattices
- 3 Figures of merit
- 4 Dithering and estimation
- 5 Entropy-coded quantization
- 6 Infinite constellation for modulation
- 7 Asymptotic goodness
- 8 Nested lattices
- 9 Lattice shaping
- 10 Side-information problems
- 11 Modulo-lattice modulation
- 12 Gaussian networks
- 13 Error exponents
- Appendix
- References
- Index
1 - Introduction
Published online by Cambridge University Press: 05 August 2014
- Frontmatter
- Dedication
- Contents
- Preface
- Acknowledgements
- List of notation
- 1 Introduction
- 2 Lattices
- 3 Figures of merit
- 4 Dithering and estimation
- 5 Entropy-coded quantization
- 6 Infinite constellation for modulation
- 7 Asymptotic goodness
- 8 Nested lattices
- 9 Lattice shaping
- 10 Side-information problems
- 11 Modulo-lattice modulation
- 12 Gaussian networks
- 13 Error exponents
- Appendix
- References
- Index
Summary
Roughly speaking, a lattice is a periodic arrangement of points in the n-dimensional Euclidean space. It reflects the “geometry of numbers” – in the words of the late nineteenth century mathematician Hermann Minkowski. Except for the onedimensional case (where all lattices are equivalent up to scaling), there are infinitely many shapes of lattices in each dimension. Some of them are better than others.
Good lattices form effective structures for various geometric and coding problems. Crystallographers look for symmetries in three-dimensional lattices, and relate them to the physical properties of common crystals. A mathematician's classical problem is to pack high-dimensional spheres – or cover space with such spheres – where their centers form a lattice. The communication engineer and the information theorist are interested in using lattices for quantization and modulation, i.e., as a means for lossy compression (source coding) and noise immunity (channel coding). Although these problems seem different, they are in fact closely related.
The effectiveness of good lattices – as well as the complexity of describing or using them for coding – increases with the spatial dimension. Such lattices tend to be “perfect” in all aspects as the dimension goes to infinity. But what does “goodness” mean in dimensions 2, 3, 4, …?
- Type
- Chapter
- Information
- Lattice Coding for Signals and NetworksA Structured Coding Approach to Quantization, Modulation and Multiuser Information Theory, pp. 1 - 10Publisher: Cambridge University PressPrint publication year: 2014