• 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.

• 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.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.35 no.6C
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• pp.523-529
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• 2010

We propose a very low complexity lattice reduction (LR) algorithm for MIMO detection in time varying channels. The proposed scheme reduces the complexity by performing LR in a block-wise manner. The proposed scheme takes advantage of the temporal correlation of the channel matrices in a block and its impact on the unimodular matrices during LR process. From this, the proposed scheme can skip a number of redundant LR processes for consecutive channel matrices and performs a single LR in a block. The simulation results investigated in this letter reveal that the proposed detection scheme requires only 43.4% multiplications and 17.3% divisions of LLL-LR and only 50.2% multiplications and 68.2% divisions of the conventional adaptive LR with almost no performance degradation.

• 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 the Institute of Electronics Engineers of Korea TC
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• v.46 no.5
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• pp.88-94
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• 2009

We propose a scheme which reduce the computational complexity of the lattice reduction (LR) aided detector in MIMO system. The performance of the ML detection algorithm is good but the computational complexity grows exponentially with the number of antenna elements and constellation points. LR aided detector shows the same diversity with the ML scheme with relatively less complexity. But the LR scheme still requires many computations since it involves several iterations of size reduction and column vector exchange. We notice that the LR process depends not on the received signal but only on the channel matrix so we can apply LR process offline and store the results in Look-Up Table (LUT). In this paper we propose an algorithm to generate the LUT which require less memory requirement and we evaluate the performance and complexity of the proposed system. We show that the proposed system requires less computational complexity with similar detection performance compared with the conventional LR aided detector.

• Korean Journal of Materials Research
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• v.16 no.5
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• pp.323-328
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• 2006

In situ electrical conductivity measurements on $V_2O_5WO_3/TiO_2$ catalysts were carried out at between 100 and $300^{\circ}C$ under pure oxygen, NO and $NH_3$ to investigate the reaction mechanism for ammonia SCR (selective catalytic reduction) de NOX. The electrical conductivity of catalysts changed irregularly with supply of NO. It was, however, found that the electrical conductivity change with ammonia supply was regular and the increase of electrical conductivity was mainly caused by reduction of the labile surface oxygen. The electrical conductivity change of catalysts showed close relationship with the conversion rate of NOx. Variation of conversion rate in atmosphere without gaseous oxygen also showed that labile lattice oxygen is indispensable in the initial stage of the de NOx reaction. These results suggest that liable lattice oxygen acts decisive role in the de NOx mechanism. They also support that de NOx reaction occurs through the Eley?Rideal type mechanism. The amount of labile oxygen can be estimated from the measurement of electrical conductivity change for catalysts with ammonia supply. This suggests that measurement of the change can be used as a measure of the de NOx performance.

• Journal of the Institute of Electronics and Information Engineers
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• v.52 no.8
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• pp.40-48
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• 2015

This paper presents a processor that performs QR decomposition (QRD) as well as Lattice Reduction (LR) for multiple-input multiple-output (MIMO) systems. By sharing the operations commonly required in QRD and LR, the hardware complexity of the proposed processor is reduced significantly. In addition, the proposed processor is designed based on a multi-cycle architecture so as to reduce the hardware complexity. The proposed processor is implemented with 139k logic gates in a $0.18-{\mu}m$ CMOS process, and its latency is $5{\mu}s$ for $8{\times}8$ MIMO preprocessing both QRD and LR where the operating frequency is 117MHz.

• Proceedings of the Korea Electromagnetic Engineering Society Conference
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• 2002.11a
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• pp.250-254
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• 2002

In this paper, we design an optimized planar array structure with triangular lattice for side-lobe reduction using a genetic algorithm. A constraint optimization is implemented by optimally removing some outer array elements far from the array center. It is shown that only the proper array shape without optimizing the magnitudes and phases of each array antenna can give low side-lobe level with a slight main beam broadening.

• Journal of Information Processing Systems
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• v.11 no.2
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• pp.184-204
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• 2015

Fuzzy Formal Concept Analysis (FCA) is a mathematical tool for the effective representation of imprecise and vague knowledge. However, with a large number of formal concepts from a fuzzy context, the task of knowledge representation becomes complex. Hence, knowledge reduction is an important issue in FCA with a fuzzy setting. The purpose of this current study is to address this issue by proposing a method that computes the corresponding crisp order for the fuzzy relation in a given fuzzy formal context. The obtained formal context using the proposed method provides a fewer number of concepts when compared to original fuzzy context. The resultant lattice structure is a reduced form of its corresponding fuzzy concept lattice and preserves the specialized and generalized concepts, as well as stability. This study also shows a step-by-step demonstration of the proposed method and its application.

• ETRI Journal
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• v.34 no.4
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• pp.503-510
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• 2012

We analyze the relationship between channel coherence bandwidth and two complexity-reduced lattice reduction aided detection (LRAD) algorithms for multiple-input multiple-output (MIMO) orthogonal frequency division multiplexing (OFDM) systems in correlated fading channels. In both the adaptive LR algorithm and the fixed interval LR algorithm, we exploit the inherent feature of unimodular transformation matrix P that remains the same for the adjacent highly correlated subcarriers. Complexity simulations demonstrate that the adaptive LR algorithm could eliminate up to approximately 90 percent of the multiplications and 95 percent of the divisions of the brute-force LR algorithm with large coherence bandwidth. The results also show that the adaptive algorithm with both optimum and globally suboptimum initial interval settings could significantly reduce the LR complexity, compared with the brute-force LR and fixed interval LR algorithms, while maintaining the system performance.