Algorithms and bounds for complex and quaternionic lattices with application to MIMO transmission
File(s)
Accepted version
Author(s)
Stern, Sebastian
Ling, Cong
Fischer, Robert FH
Type
Journal Article
Abstract
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.
Date Issued
2022-07-01
Date Acceptance
2022-01-19
Citation
IEEE Transactions on Information Theory, 2022, 68 (7), pp.4491-4517
ISSN
0018-9448
Publisher
Institute of Electrical and Electronics Engineers
Start Page
4491
End Page
4517
Journal / Book Title
IEEE Transactions on Information Theory
Volume
68
Issue
7
Copyright Statement
© 2022 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.
Sponsor
Engineering & Physical Science Research Council (EPSRC)
Identifier
http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000812529200026&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=1ba7043ffcc86c417c072aa74d649202
Grant Number
EP/S021043/1
Subjects
Science & Technology
Technology
Computer Science, Information Systems
Engineering, Electrical & Electronic
Computer Science
Engineering
Lattices
MIMO communication
Computational complexity
Time-frequency analysis
Quaternions
Quantization (signal)
Numerical simulation
lattice reduction
LLL algorithm
successive minima
Gaussian integers
Eisenstein integers
quaternions
Lipschitz integers
Hurwitz integers
MIMO
lattice-reduction-aided equalization
integer-forcing equalization
REDUCTION
DIVERSITY
CODES
ANTENNA
Publication Status
Published
Date Publish Online
2022-02-11