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
3 - Figures of merit
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
In digital communications, the cubic lattice plays the role of a simple uniform quantizer – for source coding, or an “uncoded” constellation – for channel coding. Better source-channel coding schemes can be associated with more complex lattices. This relation requires a definition of the notion of lattice goodness.
We shall develop two figures of merit of a lattice in the context of digital communication: (i) the normalized second moment – a measure of its goodness as a vector quantizer under a squared-error distortion measure, in Section 3.2, and (ii) the volume to noise ratio – a measure of its goodness as a (coded) constellation for the AWGN channel, in Section 3.3. Before that, in Section 3.1, we introduce two more fundamental quantities associated with lattice goodness for sphere packing and covering. As we may expect, all these quantities are invariant to scaling and rotation of the lattice.
Sphere packing and covering
Which shape minimizes the surface area among all shapes of a given volume in ℝn? Which shape minimizes the diameter or the second moment? The iso-perimetric inequality implies that for all these questions – and many more – the unique solution is a ball. Two of the most fundamental questions about balls (or spheres) are how efficiently they can be packed in the Euclidean space, and how efficiently they can cover it.
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- Lattice Coding for Signals and NetworksA Structured Coding Approach to Quantization, Modulation and Multiuser Information Theory, pp. 39 - 58Publisher: Cambridge University PressPrint publication year: 2014