Constellations Designed for the Rayleigh Fading Channel

Fading channels: information-theoretic and communications aspects

“…1) Signal-Space Coding: Design of multidimensional constellations aimed at optimality on the Rayleigh fading channel has been recently developed into an active research area (see [362]- [364], [370], [389], [385], [429], [432], [398], and [431]). …”

“…Two approaches were proposed to construct high modulation-diversity constellations (see [362], [363], [389], [385], [429], [432], [364], and [370]). The first was based on the design of high-diversity lattice constellations by applying the canonical embedding to the ring of integers of an algebraic number field.…”

“…Rearranging (4.3.9) we obtain (4.3.10) where is the number of points at -product distance from and with different components, By analogy with the lattice theta series, is called the product kissing number. This shows that the error probability is determined asymptotically by the diversity order , the minimum product distance , and the kissing number In particular, good signal sets have high and , and small High-diversity integral lattices from algebraic number fields: The algebraic approach [363], [389], [386], [385], [429], [364], [365], [388], [370], [366], [372], [390], [378], [413], [415] allows one to build a generator matrix exhibiting a guaranteed diversity.…”

Algebraic Number Theory and Code Design for Rayleigh Fading Channels

“…Finally, these signal constellations may be used either in a concatenated scheme with an outer code or in a coded modulation scheme using set partitioning [34,29,31,30,16,14,13].…”

Constructions of algebraic lattices

“…Algebraic lattices defined over algebraic number fields have been studied in several papers and from different points of view [1][2][3][4][5][6][7]. Giraud and Belfiore [8] proposed a technique for constructing signal sets suitable for the Rayleigh fading channel. The basic idea was to use lattice rotations to increase diversity, that is, the number of different values in the components of any two distinct points of the constellation.…”