HKZ and Minkowski Reduction Algorithms for Lattice-Reduction-Aided MIMO Detection (original) (raw)

A Comparison of Two Lattice-Reduction-Based Receivers for MIMO Systems

2008 IEEE Sarnoff Symposium, 2008

In this paper, we compare a new practical lattice reduction method, Seysen's algorithm, with the existing LLL lattice reduction approach. Seysen's algorithm considers all vectors in the lattice simultaneously and performs global search for lattice reduction, while the LLL algorithm concentrates on local optimization to produce a reduced lattice We also study their performance, when combined with linear detectors (Zero Forcing, MMSE and extended MMSE), and successive interference cancellation (SIC) detector. For MIMO digital communications, Seysen-based linear detectors achieve the same diversity order as the optimum ML detector. It outperforms the existing LLL-based linear detectors. Moreover, Seysen requires less computational time than the LLL scheme. However, this gap disappears in SIC scenario: Seysen-based SIC detector functions the same as LLL-based one, due to the efficiency of SIC itself.

Numerical Linear Algebra in Signal Processing Applications

EURASIP Journal on Advances in Signal Processing, 2007

The cross-fertilization between numerical linear algebra and digital signal processing has been very fruitful in the last decades. In particular, signal processing has been making increasingly sophisticated use of linear algebra on both theoretical and algorithmic fronts. The interaction between them has been growing, leading to many new algorithms. In particular, numerical linear algebra tools, such as eigenvalue and singular value decomposition and their higher-extensions, least squares, total least squares, recursive least squares, regularization, orthogonality and projections, are the kernels of powerful and numerically robust algorithms in many signal processing applications.

Integer-forcing MIMO linear receivers based on lattice reduction

IEEE Transactions on Wireless Communications, 2013

A new architecture called integer-forcing (IF) linear receiver has been recently proposed for multiple-input multipleoutput (MIMO) fading channels, wherein an appropriate integer linear combination of the received symbols has to be computed as a part of the decoding process. In this paper, we propose a method based on Hermite-Korkine-Zolotareff (HKZ) and Minkowski lattice basis reduction algorithms to obtain the integer coefficients for the IF receiver. We show that the proposed method provides a lower bound on the ergodic rate, and achieves the full receive diversity. Suitability of complex Lenstra-Lenstra-Lovasz (LLL) lattice reduction algorithm (CLLL) to solve the problem is also investigated. Furthermore, we establish the connection between the proposed IF linear receivers and lattice reduction-aided MIMO detectors (with equivalent complexity), and point out the advantages of the former class of receivers over the latter. For the 2 × 2 and 4 × 4 MIMO channels, we compare the codedblock error rate and bit error rate of the proposed approach with that of other linear receivers. Simulation results show that the proposed approach outperforms the zero-forcing (ZF) receiver, minimum mean square error (MMSE) receiver, and the lattice reduction-aided MIMO detectors. . His research interests are in the broad areas of Signal design for wireless networks, Design, development, and deployment aspects of next-generation wireless communication systems, and Applications of information theory and coding theory to communications.

A Diagonal Lattice Reduction Algorithm for MIMO Detection

IEEE Signal Processing Letters, 2000

Recently, an efficient lattice reduction method, called the effective LLL (ELLL) algorithm, was presented for the detection of multiinput multioutput (MIMO) systems. In this letter, a novel lattice reduction criterion, called diagonal reduction, is proposed. The diagonal reduction is weaker than the ELLL reduction, however, like the ELLL reduction, it has identical performance with the LLL reduction when applied for the sphere decoding and successive interference cancelation (SIC) decoding. It improves the efficiency of the ELLL algorithm by significantly reducing the size-reduction operations. Furthermore, we present a greedy column traverse strategy, which reduces the column swap operations in addition to the size-reduction operations.

Near-optimum vector perturbation precoding using a candidate list

Wireless Communications and …, 2009

An improvement to lattice-reduction-aided (LRA) vector perturbation precoding for multi-user MIMO downlink is introduced. Closest lattice point approximation by means of lattice reduction techniques can significantly lower the complexity of the closest point search compared to using a sphere encoder, but the performance of the system is also impaired. In this paper, we propose a new technique improving the suboptimal LRA closest-point approximation in a subsequent stage. This stage consists of a low-complexity candidate list generation of also likely approximations, and an evaluation step of this list. We present simulation results showing that our improvement to the LRA closest-point approximation can achieve near-optimum performance.

A Fast Least-Squares Solution-Seeker Algorithm for Vector-Perturbation

2008

Finding the least-squares solution to a system of linear equations where the unknown vector is comprised of integers, but the matrix coefficient and given vector are comprised of real or complex numbers is a problem equivalent to finding the closest lattice-point to a given point and is well known that the search is hard. However, in communications applications the given vector is not arbitrary but rather is an unknown lattice-point that has been perturbed by an additive offset vector whose statistical properties are known, making it relatively easier to decode. In this paper we will discuss the vector- perturbation technique proposed for solving this problem and analyse a possible solution for overcome the complexity issues.

Low-dimensional lattice basis reduction revisited

ACM Transactions on Algorithms, 2009

Lattice reduction is a geometric generalization of the problem of computing greatest common divisors. Most of the interesting algorithmic problems related to lattice reduction are NP-hard as the lattice dimension increases. This article deals with the low-dimensional case. We study a greedy lattice basis reduction algorithm for the Euclidean norm, which is arguably the most natural lattice basis reduction algorithm, because it is a straightforward generalization of an old two-dimensional algorithm of Lagrange, usually known as Gauss' algorithm, and which is very similar to Euclid's gcd algorithm. Our results are two-fold. From a mathematical point of view, we show that up to dimension four, the output of the greedy algorithm is optimal: the output basis reaches all the successive minima of the lattice. However, as soon as the lattice dimension is strictly higher than four, the output basis may be arbitrarily bad as it may not even reach the first minimum. More importantly, from a computational point of view, we show that up to dimension four, the bit-complexity of the greedy algorithm is quadratic without fast integer arithmetic, just like Euclid's gcd algorithm. This was already proved by Semaev up to dimension three using rather technical means, but it was previously unknown whether or not the algorithm was still polynomial in dimension four. We propose two different analyzes: a global approach based on the geometry of the current basis when the length decrease stalls, and a local approach showing directly that a significant length decrease must occur every O(1) consecutive steps. Our analyzes simplify Semaev's analysis in dimensions two and three, and unify the cases of dimensions two to four. Although the global approach is much simpler, we also present the local approach because it gives further information on the behavior of the algorithm. 2 · P. Q. Nguyen and D. Stehlé 2008; and in practice for high-dimensional lattices are based on a repeated use of low-dimensional HKZ-reduction.

Notes on Numerical Linear Algebra

This paper describes many of the standard numerical methods used in Linear Algebra. Topics include Gaussian Elimination, LU and QR Factorizations, The Singular Value Decomposition, Eigenvalues and Eigenvectors via the QR Method with Shifts or the Divide-and-Conquer Method, and the Conjugate Gradient and Lanczos Iterative Methods.

An Experimental Comparison of Some LLL-Type Lattice Basis Reduction Algorithms

International Journal of Applied and Computational Mathematics, 2015

In this paper we experimentally compare the performance of the L 2 lattice basis reduction algorithm, whose importance recently became evident, with our own Gram-based lattice basis reduction algorithm, which is a variant of the Schnorr-Euchner algorithm. We conclude with observations about the algorithms under investigation for lattice basis dimensions up to the theoretical limit. We also reexamine the notion of "buffered transformations" and its impact on performance of lattice basis reduction algorithms. We experimentally compare four different algorithms directly in the Sage Mathematics Software: our own algorithm, the L 2 algorithm and "buffered" versions of them resulting in a total of four algorithms.