Lipschitz Integers for Coded Modulation and Precoding

信号星座是数字传输系统的重要组成部分，直接决定了系统的性能。因此，多年来人们一直在构建和分析星座，例如，可以将不同的星座与Forney和魏引入的星座品质因数进行比较。除了直接由QAM信号激励的二维星座外，已经处于早期阶段的高维方法，特别是四维信号集，由于其更高的灵活性而引起了人们的兴趣。目前，四维信号星座在光通信中受到越来越多的关注，最近，一位申请者发现，由Lipschitz整数划分构成的星座比类似的二维QAM星座具有高达10db的优值系数[FS15]。这些显著的收益只在Lipschitz整数的特殊子集上观察到，而不是对于Lipschitz整数本身。然而，到目前为止，只有一些例子存在，有必要对这些星座进行仔细的分析和研究。因此，我们建议在这个项目中分析新颖的四维结构。值得注意的是，最重要的经典二维星座可以解释为Lipschitz整数的特殊子集，这可能导致一种新的星座理论。此外，来自编码调制星座的方法可能有助于构建更好的星座。基于新星座的编码调制可以改善有线、无线和光通信系统。此外，先进的均衡和预编码技术，特别是那些基于格约化概念及其紧密相关的整数强制方法的技术，是基于代数运算的，因此Lipschitz整数及其划分非常适合于新的方法。因此，我们期待着对任何类型的复值信号星座的数字通信系统的未来编码和调制的改进有许多有趣的结果。

Signal constellations are an important ingredient for digital transmission systems, directly determining their performance. Therefore, constellations have been constructed and analyzed for many years, where for instance different constellations can be compared with the constellation figure of merit introduced by Forney and Wei. Besides two-dimensional constellations, immediately motivated by QAM signaling, already at early stages higher-dimensional approaches, in particular four-dimensional signal sets, have been of interest due to their higher flexibility. Nowadays, four-dimensional signal constellations are of increasing interest in optical communications.More recently, one of the applicants found that constellations constructed by partitioning of Lipschitz integers have a figure of merit which is up to 10 dB better than the comparable two-dimensional QAM constellations [FS15]. These remarkable gains are only observed for special subsets of Lipschitz integers and not for Lipschitz integers themselves. However, until now only some examples exist and a careful analysis and study of these constellations is necessary. Therefore, we propose to analyze novel four-dimensional constructions in this project. Noteworthy, the most important classical two-dimensional constellations can be interpreted as special subsets of Lipschitz integers which might lead to a novel theory for constellations. Furthermore, methods from coded modulation constellations might help to construct even better constellations. Coded modulation based on the new constellations can improve wired, wireless, and optical communication systems. In addition, advanced equalization and precoding techniques, in particular those based on the concepts of lattice reduction and its tightly related approach of integer forcing, are based on algebraic operations and thus Lipschitz integers and their partitioning are well suited for novel methods. Thus, we expect many interesting results for the improvement of future coding and modulation for any type of digital communication system with complex-valued signal constellations.