• KSII Transactions on Internet and Information Systems (TIIS)
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• v.12 no.6
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• pp.2554-2580
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• 2018

Lenstra-Lenstra-$Lov{\acute{a}}sz$ (LLL) is an effective receiving algorithm for Multiple-Input-Multiple-Output (MIMO) systems, which is believed can achieve full diversity in MIMO detection of fading channels. However, the LLL algorithm features polynomial complexity and shows poor performance in terms of convergence. The reduction of algorithmic complexity and the acceleration of convergence are key problems in optimizing the LLL algorithm. In this paper, a variant of the LLL algorithm, the Hybrid-Fix-and-Round LLL algorithm, which combines both fix and round measurements in the size reduction procedure, is proposed. By utilizing fix operation, the algorithmic procedure is altered and the size reduction procedure is skipped by the hybrid algorithm with significantly higher probability. As a consequence, the simulation results reveal that the Hybrid-Fix-and-Round-LLL algorithm carries a faster rate of convergence compared to the original LLL algorithm, and its algorithmic complexity is at most one order lower than original LLL algorithm in real field. Comparing to other families of LLL algorithm, Hybrid-Fix-and-Round-LLL algorithm can make a better compromise in performance and algorithmic complexity.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.34 no.9C
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• pp.915-921
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• 2009

Lenstra-Lenstra-Lovasz (LLL) algorithm, which is one of the lattice reduction (LR) techniques, has been extensively used to obtain better bases of the channel matrix. In this paper, we jointly apply Seysen's lattice reduction Algorithm (SA), instead of LLL, with the conventional linear precoding algorithms. Since SA obtains more orthogonal lattice bases compared to those obtained by LLL, lattice reduction aided (LRA) precoding based on SA algorithm outperforms the LRA precoding with LLL. Simulation results demonstrate that a gain of 0.5dB at target BER of $10^{-5}$ is achieved when SA is used instead of LLL or the LR stage.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.20 no.2
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• pp.107-121
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• 2016

We analyze the complexity of the LLL algorithm, invented by Lenstra, Lenstra, and $Lov{\acute{a}}sz$ as a a well-known lattice reduction (LR) algorithm which is previously known as having the complexity of $O(N^4{\log}B)$ multiplications (or, $O(N^5({\log}B)^2)$ bit operations) for a lattice basis matrix $H({\in}{\mathbb{R}}^{M{\times}N})$ where B is the maximum value among the squared norm of columns of H. This implies that the complexity of the lattice reduction algorithm depends only on the matrix size and the lattice basis norm. However, the matrix structures (i.e., the correlation among the columns) of a given lattice matrix, which is usually measured by its condition number or determinant, can affect the computational complexity of the LR algorithm. In this paper, to see how the matrix structures can affect the LLL algorithm's complexity, we derive a more tight upper bound on the complexity of LLL algorithm in terms of the condition number and determinant of a given lattice matrix. We also analyze the complexities of the LLL updating/downdating schemes using the proposed upper bound.

• Journal of the Korean Mathematical Society
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• v.59 no.6
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• pp.1047-1065
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• 2022

For two dimensional lattices, a Gaussian basis achieves all two successive minima. For dimension larger than two, constructing a pairwise Gaussian basis is useful to compute short vectors of the lattice. For three dimensional lattices, Semaev showed that one can convert a pairwise Gaussian basis to a basis achieving all three successive minima by one simple reduction. A pairwise Gaussian basis can be obtained from a given basis by executing Gauss algorithm for each pair of basis vectors repeatedly until it returns a pairwise Gaussian basis. In this article, we prove a necessary and sufficient condition for a pairwise Gaussian basis to achieve the first k successive minima for three dimensional lattices for each k ∈ {1, 2, 3} by modifying Semaev's condition. Our condition directly checks whether a pairwise Gaussian basis contains the first k shortest independent vectors for three dimensional lattices. LLL is the most basic lattice basis reduction algorithm and we study how to use LLL to compute a pairwise Gaussian basis. For δ ≥ 0.9, we prove that LLL(δ) with an additional simple reduction turns any basis for a three dimensional lattice into a pairwise SV-reduced basis. By using this, we convert an LLL reduced basis to a pairwise Gaussian basis in a few simple reductions. Our result suggests that the LLL algorithm is quite effective to compute a basis with all three successive minima for three dimensional lattices.

• Journal of Communications and Networks
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• v.13 no.5
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• pp.481-493
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• 2011

Multiple-input multiple-output (MIMO) technology provides high data rate and enhanced quality of service for wireless communications. Since the benefits from MIMO result in a heavy computational load in detectors, the design of low-complexity suboptimum receivers is currently an active area of research. Lattice-reduction-aided detection (LRAD) has been shown to be an effective low-complexity method with near-maximum-likelihood performance. In this paper, we advocate the use of systolic array architectures for MIMO receivers, and in particular we exhibit one of them based on LRAD. The "Lenstra-Lenstra-Lov$\acute{a}$sz (LLL) lattice reduction algorithm" and the ensuing linear detections or successive spatial-interference cancellations can be located in the same array, which is considerably hardware-efficient. Since the conventional form of the LLL algorithm is not immediately suitable for parallel processing, two modified LLL algorithms are considered here for the systolic array. LLL algorithm with full-size reduction-LLL is one of the versions more suitable for parallel processing. Another variant is the all-swap lattice-reduction (ASLR) algorithm for complex-valued lattices, which processes all lattice basis vectors simultaneously within one iteration. Our novel systolic array can operate both algorithms with different external logic controls. In order to simplify the systolic array design, we replace the Lov$\acute{a}$sz condition in the definition of LLL-reduced lattice with the looser Siegel condition. Simulation results show that for LR-aided linear detections, the bit-error-rate performance is still maintained with this relaxation. Comparisons between the two algorithms in terms of bit-error-rate performance, and average field-programmable gate array processing time in the systolic array are made, which shows that ASLR is a better choice for a systolic architecture, especially for systems with a large number of antennas.

• Proceedings of the Korean Society of Broadcast Engineers Conference
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• 2012.07a
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• pp.377-380
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• 2012

격자 감소 (lattice reduction) 알고리즘은 주어진 기저 벡터를 직교에 가까운 기저 벡터로 바꾸어 준다. 그중 대표적인 알고리즘으로 LLL (Lenstra, Lenstra & Lovasz) 알고리즘이 있다. 격자 감소 알고리즘을 이용하여 다중 안테나 입출력 (MIMO) 통신시스템의 선형 수신기(linear detector)의 성능을 향상 시킬 수 있다. 스피어 복호 알고리즘 (sphere decoding algorithm)은 MIMO 통신 시스템에서 사용되는 복호기중 최대 우도 복호기 (Maximum Likelihood Detector)와 비슷한 BER(bit error rate)성능을 가지고 복잡도를 줄일 수 있어서 많이 연구되어 왔다. 이때 스피어의 반지름의 설정이나 트리 검색 구조 방식 등은 복잡도에 큰 영향을 미친다. 본 논문에서는 LLL 알고리즘에 기반하여 스피어의 반지름 설정 및 트리 검색 노드 수를 제한하는 방식으로 스피어 복호 알고리즘의 복잡도를 기존 알고리즘에 비해 크게 낮추면서도 비트 오류률 (BER) 성능 열화를 최소한으로 한 알고리즘을 제안하고 전산 실험을 통해 검증한다.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.34 no.6C
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• pp.642-648
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• 2009

In this paper, we use SA (Seysen's Algorithm) instead of LLL (Lenstra-Lenstra-Lovasz) to perform LRA (Lattice Reduction-Aided) detection. By using SA, the complexity of lattice reduction is reduced and the detection performance is improved Although the performance is improved using SA, there still exists a gap in the performance between SA-LRA and ML detection. To reduce the performance difference, we apply list of candidates scheme to SA-LRA. The list of candidates scheme finds a list of candidates. Then, the candidate with the smallest squared Euclidean distance is considered as the estimate of the transmitted signal. Simulation results show that the SA-LRA detection learn to quasi-ML performance. Moreover, the efficiency of the SA is shown to highly improve the channel matrix conditionality.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.19 no.2
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• pp.171-188
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• 2015

In this paper, the efficient column-wise/row-wise lattice reduction (LR) updating and downdating methods are developed and their complexities are analyzed. The well-known LLL algorithm, developed by Lenstra, Lenstra, and Lov${\acute{a}}$sz, is considered as a LR method. When the column or the row is appended/deleted in the given lattice basis matrix H, the proposed updating and downdating methods modify the preconditioning matrix that is primarily computed for the LR with H and provide the initial parameters to reduce the updated lattice basis matrix efficiently. Since the modified preconditioning matrix keeps the information of the original reduced lattice bases, the redundant computational complexities can be eliminated when reducing the lattice by using the proposed methods. In addition, the rounding error analysis of the proposed methods is studied. The numerical results demonstrate that the proposed methods drastically reduce the computational load without any performance loss in terms of the condition number of the reduced lattice basis matrix.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.36 no.6A
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• pp.551-557
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• 2011

We investigate lattice-reduction-aided precoding techniques using Look-Up table (LUT) for multi-user multiple-input multiple-output(MIMO) systems. Lattice-reduction-aided vector perturbation (VP) gives large sum capacity with low encoding complexity. Nevertheless lattice-reduction process based on the LLL-Algorithm still requires high computational complexity since it involves several iterations of size reduction and column vector exchange. In this paper, we apply the LUT-aided lattice reduction on VP and propose a scheme to generate the LUT efficiently. Simulation results show that a proposed scheme has similar orthogonality defect and Bit-Error-Rate(BER) even with lower memory size.

• Journal of the Institute of Electronics Engineers of Korea TC
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• v.46 no.6
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• pp.86-93
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• 2009

We investigate lattice-reduction-aided precoding techniques for multi-user multiple-input multiple-output (MIMO) channels. When assuming full knowledge of the channel state information only at the transmitter, a vector perturbation (VP) is a promising precoding scheme that approaches sum capacity and has simple receiver. However, its encoding is nondeterministic polynomial time (NP)-hard problem. Vector perturbation using lattice reduction algorithms can remarkably reduce its encoding complexity. In this paper, we propose a vector perturbation scheme using Seysen's lattice reduction (VP-SLR) with simultaneously reducing primal basis and dual one. Simulation results show that the proposed VP-SLR has better bit error rate (BER) and larger capacity than vector perturbation with Lenstra-Lenstra-Lovasz lattice reduction (VP-LLL) in addition to less encoding complexity.