Four-Dimensional Hurwitz Signal Constellations With Convenient Bit Mapping and Set Partitioning

Abstract:
The construction of four-dimensional (quaternionic) Voronoi-based signal constellations is considered for the application in dual-polarized transmission. These constellations are drawn from the set of Hurwitz integers, an isomorphic representation of the four-dimensional checkerboard lattice, and enable both a packing and a shaping gain over conventional quadratureamplitude modulation (QAM) per polarization. Based on a basisexpansion approach in which pairs of bits are jointly mapped to quaternion numbers, an isomorphism is established in which the addition of constellation points corresponds to the bit-wise addition (XOR) with carry in the corresponding bit labeling. Moreover, this basis expansion defines a set partitioning in which the minimum squared distance grows in every second bit level. The proposed signal constellations are predestined for the use in physical-layer network coding and in lattice-reduction-aided multiple-input/multiple-output (MIMO) communication where linear combinations of signal points have to be decoded.