Lipschitz Integers for Coded Modulation and Precoding

• Applicants: Professor Dr.-Ing. Robert Fischer; Professor Dr.-Ing. Jürgen Freudenberger
• Subject Area: Electronic Semiconductors, Components and Circuits, Integrated Systems, Sensor Technology, Theoretical Electrical Engineering
• Term: from 2016 to 2020
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 289275110

• Final Report Year: 2020

Final Report Abstract

Today’s digital modulation schemes are mainly based on two-dimensional constellations which are motivated by the equivalent complex-baseband representation of radio-frequency signals. These constellations typically have no algebraic structure that can be exploited for coding or low-complexity detection methods. This research project considered two- and four-dimensional signal constellations which possess an algebraic property and/or can easily be combined with channel coding. This particularly concerns complex-valued finite sets of Gaussian and Eisenstein integers, as well as quaternionvalued sets of Lipschitz and Hurwitz integers. The Eisenstein and Hurwitz constellations are based on the densest two- and four-dimensional lattices. This enables a packing gain and, in some cases, also a shaping gain compared to ordinary QAM constellations. New algebraic constructions for finite Hurwitz constellations were found. These sets have good constellation-figure-of-merit values and can be partitioned into subsets with large distance gains, a useful property for coded-modulation techniques. Four-dimensional signal constellations are of increasing interest in optical communications and wireless communication systems that use antenna polarization or spatial modulation. The algebraic structure of the constellations can be exploited for set-partitioning techniques. These techniques enable low-complexity coded modulation schemes as well as low-complexity detection methods, e.g., by utilizing dependencies between dimensions. In particular, concepts for multilevel coding were developed. For the two-dimensional Eisenstein constellations, the partitioning schemes have a binary or ternary labeling and large distance gains in the subsets. This enables coding with binary or ternary codes. Similarly, a coded-modulation scheme for Hurwitz integers was developed and successfully evaluated in fiber-optical transmission. Moreover, these Hurwitz constellations are useful in multi-user MIMO transmission and spatial modulation. For the MIMO setting, we could show that a four-dimensional adaption of lattice-reduction-aided channel equalization enables both an increased transmission performance and at the same time a decreased computational complexity. In spatial modulation, the proposed Hurwitz constellations can, when combined with sub-optimum detection, even outperform conventional two-dimensional constellations in combination with maximum-likelihood detection.

Publications

• Generalized multistream spatial modulation with signal constellations based on Hurwitz integers and low-complexity detection. IEEE Wireless Communications Letters, 7(3):412–415, June 2018
  J. Freudenberger, D. Rohweder, and S. Shavgulidze
  
• Low-density parity-check codes over finite Gaussian integer fields. In IEEE International Symposium on Information Theory (ISIT), pages 481–485, June 2018
  D. Rohweder, J. Freudenberger, and S. Shavgulidze
  
• Quaternion-valued multi-user MIMO transmission via dual-polarized antennas and QLLL reduction. In 25th International Conference on Telecommunications (ICT), pages 63–69, June 2018
  S. Stern and R. F. H. Fischer
  
• Binary multilevel coding over Eisenstein integers for MIMO broadcast transmission. In 23rd International ITG Workshop on Smart Antennas (WSA), April 2019
  S. Stern, D. Rohweder, J. Freudenberger, and R. F. H. Fischer
  
• Coded modulation using a 512-ary Hurwitz-integer constellation. In 45th European Conference on Optical Communication (ECOC), September 2019
  F. Frey, S. Stern, R. Emmerich, C. Schubert, J. K. Fischer, and R. F. H. Fischer
  
• Low-complexity detection for generalized multistream spatial modulation. In 20th International Workshop on Signal Processing Advances in Wireless Communications (SPAWC), July 2019
  D. Rohweder, S. Stern, R. F. H. Fischer, S. Shavgulidze, and J. Freudenberger
  
• Multilevel coding over Eisenstein integers with ternary codes. In 12th International ITG Conference on Systems, Communications and Coding (SCC), February 2019
  S. Stern, D. Rohweder, J. Freudenberger, and R. F. H. Fischer
  
• Two-stage dimension-wise coded modulation for four-dimensional Hurwitz-integer constellations. In 12th International ITG Conference on Systems, Communications and Coding (SCC), February 2019
  S. Stern, F. Frey, J. K. Fischer, and R. F. H. Fischer
  
• Low-complexity detection for multi-dimensional spatial modulation. In 24rd International ITG Workshop on Smart Antennas (WSA), February 2020
  D. Rohweder, S. Stern, R. F. H. Fischer, S. Shavgulidze, and J. Freudenberger