Universal lattice decoding: principle and recent advances

Abstract

SummaryThe idea of formulating the detection of a lattice-type modulation, such as M-PAM and M-QAM, transmitted over a linear channel as the so-called universal lattice decoding problem dates back to at least the early 1990s.The applications of such lattice decoders have proliferated in the last few years because of the growing importance of some linear channel models, such as multipleantenna fading channels and multi-user CDMA channels. The principle of universal lattice decoding can trace its roots back to the theory and algorithms developed for solving the shortest/closest lattice vector problem for integer programming and cryptoanalysis applications. In this semi-tutorial paper, such a principle as well as some related recent advances will be reviewed and extended. It will be shown that the lattice basis reduction algorithm of Lenstra, Lenstra and Lovász can significantly improve the performance of suboptimal lattice decoders, such as the zero-forcing and VBLAST detectors. In addition, new implementation of the optimal lattice decoder that is particularly efficient at moderate to high signal-to-noise ratios will also be presented.

A Digital Signal Processing Architecture for Soft-Output MIMO Lattice Reduction Aided Detection

“…H and µ are computed using a modified GSO procedure [11] 2. Size reduction aims to make basis vectors shorter and more orthogonal by asserting the condition that |ℜ(µ k,j )| ≤ 0.5 and |ℑ(µ k,j )| ≤ 0.5 for all j < k 3.…”

Solving Box-Constrained Integer Least Squares Problems

“…In the original BGBF algorithm, the bounds for all integer parameters are identical, i.e., l 1 = · · · = l n and u 1 = · · · = u n . But we can easily extend the algorithm to deal with the more general constraint B given in (2). The following is the description of the BGBF algorithm (with slight modifications).…”

“…In channel coding, the ILS problem is referred to as decoding. The ILS problem may arise from many applications, such as communications, cryptograph, lattice design, Monte Carlo second-moment estimation, radar imaging, and global positioning systems etc, see, e.g., [1], [2], [3] and references therein. It is well know that the ILS problem (1) is NP-hard [4].…”

Algorithms for the Shortest and Closest Lattice Vector Problems

“…We refer the interested reader to the surveys [17] and [48]. They are also very frequent in communications theory [60,1,34,61]. The cryptographic role of SVP and CVP is twofold.…”

“…Also connected to CVP is the Bounded Distance Decoding problem (BDD): in BDD, the target vector is guaranteed to be within a distance r ≤ λ 1 (L)/2 of the lattice L. Several algorithms [45,50] have been specifically designed for such CVP instances. BDD is of particular significance for MIMO communications [61] and cryptography [71]. We also refer to [51] for reductions of several lattice problems to and from BDD.…”