• Title, Summary, Keyword: LLL algorithm

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Performance Evaluation of Lower Complexity Hybrid-Fix-and-Round-LLL Algorithm for MIMO System

  • Lv, Huazhang
    • 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.

Lattice Reduction Aided Preceding Based on Seysen's Algorithm for Multiuser MIMO Systems (다중 사용자 MIMO 시스템을 위한 Seysen 알고리즘 기반 Lattice Reduction Aided 프리코팅)

  • An, Hong-Sun;Mohaisen, Manar;Chang, Kyung-Hi
    • 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.

ANALYSIS OF THE UPPER BOUND ON THE COMPLEXITY OF LLL ALGORITHM

  • PARK, YUNJU;PARK, JAEHYUN
    • 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.

Systolic Arrays for Lattice-Reduction-Aided MIMO Detection

  • Wang, Ni-Chun;Biglieri, Ezio;Yao, Kung
    • 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.

LLL Algorithm Aided Double Sphere MIMO Detection (LLL 알고리즘 기반 이중 스피어 MIMO 수신기)

  • Jeon, Myeongwoon;Lee, Jungwoo
    • Proceedings of the Korean Society of Broadcast Engineers Conference
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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.

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Lattice Reduction Aided MIMO Detection using Seysen's Algorithm (Seysen 알고리즘을 이용한 Lattice Reduction-aided 다중 안테나 검출기법)

  • An, Hong-Sun;Mohaisen, Manar;Chang, Kyung-Hi
    • 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.

EFFICIENT LATTICE REDUCTION UPDATING AND DOWNDATING METHODS AND ANALYSIS

  • PARK, JAEHYUN;PARK, YUNJU
    • 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.

A Vector-Perturbation Based Lattice-Reduction using look-Up Table (격자 감소 기반 전부호화 기법에서의 효율적인 Look-Up Table 생성 방법)

  • Han, Jae-Won;Park, Dae-Young
    • 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.

Lattice-Reduction-Aided Preceding Using Seysen's Algorithm for Multi-User MIMO Systems (다중 사용자 다중 입출력 시스템에서 Seysen 기법을 이용한 격자 감소 기반 전부호화 기법)

  • Song, Hyung-Joon;Hong, Dae-Sik
    • 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.

Joint Lattice-Reduction-Aided Precoder Design for Multiuser MIMO Relay System

  • Jiang, Hua;Cheng, Hao;Shen, Lizhen;Liu, Guoqing
    • KSII Transactions on Internet and Information Systems (TIIS)
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    • v.10 no.7
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    • pp.3010-3025
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    • 2016
  • Lattice reduction (LR) has been used widely in conventional multiple-input multiple-output (MIMO) systems to enhance the performance. However, LR is hard to be applied to the relay systems which are important but more complicated in the wireless communication theory. This paper introduces a new viewpoint for utilizing LR in multiuser MIMO relay systems. The vector precoding (VP) is designed along with zero force (ZF) criterion and minimum mean square error (MMSE) criterion and enhanced by LR algorithm. This implementable precoder design combines nonlinear processing at the base station (BS) and linear processing at the relay. This precoder is capable of avoiding multiuser interference (MUI) at the mobile stations (MSs) and achieving excellent performance. Moreover, it is shown that the amount of feedback information is much less than that of the singular value decomposition (SVD) design. Simulation results show that the proposed scheme using the complex version of the Lenstra--Lenstra--Lovász (LLL) algorithm significantly improves system performance.