论文标题
算法和界限用于复杂和Quaternionic Lattices，并应用于MIMO传输
Algorithms and Bounds for Complex and Quaternionic Lattices With Application to MIMO Transmission
论文作者
论文摘要
晶格是数学研究中流行的研究领域，但在更实用的领域，例如密码学或多输入/多输出（MIMO）传播。在数学理论中，大多数经常考虑实数的晶格。但是，在通信中，复杂值的处理通常是感兴趣的。此外，通过使用双极化透射以及两个时间插槽或频率的组合，四维（四元化值）方法变得越来越重要。因此，为了解决这一事实，这项工作概括了众所周知的晶格算法和相关概念。为此，对复杂算术的简要审查，包括高斯和爱森斯坦整数的集合，并介绍了四个值的数字，包括Lipschitz和Hurwitz Integers的集合。 On that basis, generalized variants of two important algorithms are derived: first, of the polynomial-time LLL algorithm, resulting in a reduced basis of a lattice by performing a special variant of the Euclidean algorithm defined for matrices, and second, of an algorithm to calculate the successive minima - the norms of the shortest independent vectors of a lattice - and its related晶格点。建立了特定结果质量的广义界限，并评估算法的渐近复杂性。这些发现与传统的实价处理相比，这些发现被广泛比较。结果表明，广义方法在复杂性和/或质量方面的表现优于其实现的对应物。此外，研究了广义算法在MIMO通信中的应用，尤其是在降低晶格和整数的均衡均衡领域。
Lattices are a popular field of study in mathematical research, but also in more practical areas like cryptology or multiple-input/multiple-output (MIMO) transmission. In mathematical theory, most often lattices over real numbers are considered. However, in communications, complex-valued processing is usually of interest. Besides, by the use of dual-polarized transmission as well as by the combination of two time slots or frequencies, four-dimensional (quaternion-valued) approaches become more and more important. Hence, to account for this fact, well-known lattice algorithms and related concepts are generalized in this work. To this end, a brief review of complex arithmetic, including the sets of Gaussian and Eisenstein integers, and an introduction to quaternion-valued numbers, including the sets of Lipschitz and Hurwitz integers, are given. On that basis, generalized variants of two important algorithms are derived: first, of the polynomial-time LLL algorithm, resulting in a reduced basis of a lattice by performing a special variant of the Euclidean algorithm defined for matrices, and second, of an algorithm to calculate the successive minima - the norms of the shortest independent vectors of a lattice - and its related lattice points. Generalized bounds for the quality of the particular results are established and the asymptotic complexities of the algorithms are assessed. These findings are extensively compared to conventional real-valued processing. It is shown that the generalized approaches outperform their real-valued counterparts in complexity and/or quality aspects. Moreover, the application of the generalized algorithms to MIMO communications is studied, particularly in the field of lattice-reduction-aided and integer-forcing equalization.
