Four-Dimensional Hurwitz Signal Constellations, Set Partitioning, Detection, and Multilevel Coding
- Resource Type
- Authors
- Daniel Rohweder; Robert F. H. Fischer; Sebastian Stern; Sergo Shavgulidze; Juergen Freudenberger
- Source
- IEEE Transactions on Communications. 69:5079-5090
- Subject
- Discrete mathematics
Modulo operation
Hamming bound
Constellation diagram
symbols.namesake
Additive white Gaussian noise
Signal-to-noise ratio
Eisenstein integer
symbols
Electrical and Electronic Engineering
Quaternion
Finite set
Computer Science::Information Theory
Mathematics
- Language
- ISSN
- 1558-0857
0090-6778
The Hurwitz lattice provides the densest four-dimensional packing. This fact has motivated research on four-dimensional Hurwitz signal constellations for optical and wireless communications. This work presents a new algebraic construction of finite sets of Hurwitz integers that is inherently accompanied by a respective modulo operation. These signal constellations are investigated for transmission over the additive white Gaussian noise (AWGN) channel. It is shown that these signal constellations have a better constellation figure of merit and hence a better asymptotic performance over an AWGN channel when compared with conventional signal constellations with algebraic structure, e.g., two-dimensional Gaussian-integer constellations or four-dimensional Lipschitz-integer constellations. We introduce two concepts for set partitioning of the Hurwitz integers. The first method is useful to reduce the computational complexity of the symbol detection. This suboptimum detection approach achieves near-maximum-likelihood performance. In the second case, the partitioning exploits the algebraic structure of the Hurwitz signal constellations. We partition the Hurwitz integers into additive subgroups in a manner that the minimum Euclidean distance of each subgroup is larger than in the original set. This enables multilevel code constructions for the new signal constellations.