• Honam Mathematical Journal
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• v.45 no.3
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• pp.418-432
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• 2023

In this study, we consider the Hankel and Gram matrices which are defined by the elements of special number sequences. Firstly, the eigenvalues, determinant, and norms of the Hankel matrix defined by special number sequences are obtained. Afterwards, using the relationship between the Gram and Hankel matrices, the eigenvalues, determinants, and norms of the Gram matrices defined by number sequences are given.

• Journal of Communications and Networks
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• v.17 no.6
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• pp.634-646
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• 2015

Assuming perfect channel state information (CSI) at the transmitter and receiver, the optimization problem of maximizing the minimum Euclidean distance between two received signals by a linear precoder is considered for multiple-input multiple-output (MIMO) systems with arbitrary dimensions and arbitraryary quadrature amplitude modulation (QAM) input. A general precoding framework is first presented based on the Gram matrix, which is shown for 2-dimensional (2-D) and 3-dimensional (3-D) MIMO systems when employing the ellipse expanding method (EEM). An extended precoder for high-dimensional MIMO system is proposed following the precoding framework, where the Gram matrix for high-dimensional precoding matrix can be generated through those chosen from 2-D and 3-D results in association with a permutation matrix. A complexity-reduced maximum likelihood detector is also obtained according to the special structure of the proposed precoder. The analytical and numerical results indicate that the proposed precoder outperforms the other precoding schemes in terms of both minimum distance and bit error rate (BER).

• Kyungpook Mathematical Journal
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• v.9 no.1_2
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• pp.67-69
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• 1969

• The Journal of the Acoustical Society of Korea
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• v.38 no.5
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• pp.522-533
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• 2019

The compressive sensing model for range-Doppler estimation can be expressed as an under-determined linear system y = Ax. To find the solution of the linear system with the compressive sensing method, matrix A should be sufficiently incoherent and x to be sparse. In this paper, we propose a transmission waveform design method that maintains the bandwidth required by the sonar system while lowering the mutual coherence of the matrix A so that the matrix A is incoherent. The proposed method combines two methods of optimizing the sensing matrix with the alternating projection and suppressing unwanted frequency bands using the DFT (Discrete Fourier Transform) matrix. We compare range-Doppler estimation performance of existing waveform LFM(Linear Frequency Modulated) and designed waveform using the matched filter and the compressive sensing method. Simulation shows that the designed transmission waveform has better detection performance than the existing waveform LFM.

• 제어로봇시스템학회:학술대회논문집
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• 2004.08a
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• pp.1694-1700
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• 2004

In this paper, we propose the AWLS/MFT (Adaptive Weighed Least Squares/ Modulation Function Technique) devised by A. E. Pearson et al. for the transfer function estimation of a modal system and investigate the performance of several algorithms, the Gram matrix method, a Luenberger Observer (LO), Least Squares (LS), and Recursive Least Squares (RLS), for the estimation of initial conditions. With the benefit of the Modulation Function Technique (MFT), we can separate the estimation problem into two phases: the transfer function parameters are estimated in the first phase, and the initial conditions are estimated in the second phase. The LO method produces excellent IC estimates in the noise free case, but the other three methods show better performance in the noisy case. Finally, we compared our result with the Prony based method. In the noisy case, the AWLS and one of the three methods - Gram matrix, LS, and RLS- show better performance in the output Signal to Error Ratio (SER) aspect than the Prony based method under the same simulation conditions.

• Communications for Statistical Applications and Methods
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• v.2 no.1
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• pp.48-54
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• 1995

A direct method for fitting analysis-of-variance models to unbalanced data is presented. This method exploits sparsity and rank deficiency of the matrix and is based on Gram-Schmidt orthogonalization of a set of sparse columns of the model matrix. The computational algorithm of the sum of squares for testing estmable hyphotheses is given.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.24 no.9B
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• pp.1785-1794
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• 1999

The affine projection algorithm has known t require less computational complexity than RLS but have much faster convergence than NLMS for speech-like input signals. But the affine projection algorithm is still much more computationally demanding than the LMS algorithm because it requires the matrix inversion. In this paper, we show that the affine projection algorithm can be realized with the Gram-Schmidt orthogonalizaion of input vectors. Using the derived relation, we propose an approximate but much more efficient implementation of the affine projection algorithm. Simulation results show that the proposed algorithm has the convergence speed that is comparable to the affine projection algorithm with only a slight extra calculation complexity beyond that of NLMS.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.4 no.1
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• pp.109-119
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• 2000

It was shown that if Q is a fully indecomposable $n{\times}n$ orthogonal matrix then Q has at least 4n-4 nonzero entries in 1993. In this paper, we show that for each integer p with $4n-4{\leq}p{\leq}n^2$, there exist a fully indecomposable $n{\times}n$ orthogonal matrix with exactly p nonzero entries. Furthermore, we obtain a method of construction of a fully indecomposable $n{\times}n$ orthogonal matrix which has exactly 4n-4 nonzero entries. This is a part of the study in sparse matrices.

• Proceedings of the Korean Society of Broadcast Engineers Conference
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• 2015.11a
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• pp.4-7
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• 2015

In this paper, we propose a matrix completion algorithm for Internet of Things (IoT) localization. The proposed algorithm recovers the Gram matrix of sensors by performing optimization over the Riemannian manifold of fixed-rank positive semidefinite matrices. We compute and show the closed forms of all the differentially geometric components required for applying nonlinear conjugate gradients combined with Armijo line search method. The numerical experiments show that the performance of the proposed algorithm in solving IoT localization is outstanding compared with the state-of-the-art matrix completion algorithms both in noise and noiseless scenarios.

• Proceedings of the Korean Information Science Society Conference
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• 2005.07b
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• pp.715-717
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• 2005

Support vector machine(SVM)은 최근 각광받는 기계학습 방법 중 하나로서, kernel function 이라는 사상(mapping)을 이용하여 입력 공간의 벡터를 classification이 용이한 특징 (feature) 공간의 벡터로 변환하는 것을 근간으로 한다. SVM은 이러한 특징 공간에서 두 클래스를 구분 짓는 hyperplane을 일련의 최적화 방법론을 사용하여 찾아내며, 주어진 문제가 convex problem 인 경우 항상 global optimal solution 을 보장하는 등의 장점을 지닌다. 한편 bioinformatics 연구에서 주로 사용되는 데이터는 측정 오류 등 일련의 오류를 포함하고 있으며, 이러한 오류는 기계학습 방법론이 어떤 decision boundary를 찾아내는가에 영향을 끼치게 된다. 특히 SVM의 경우 이러한 오류는 특징 공간 벡터간의 관계를 나타내는 Gram matrix를 변화로 나타나게 된다. 본 연구에서는 입력 공간에 오류가 발생할 때 그것이 SVM 의 decision boundary를 어떻게 변화시키는가를 대표적인 두 가지 kernel function, 즉 linear kernel과 Gaussian kernel에 대해 분석하였다. Wisconsin대학의 유방암(breast cancer) 데이터에 대해 실험한 결과, 데이터의 오류에 따른 SVM 의 classification 성능 변화 양상을 관찰하여 커널의 종류에 따라 SVM이 어떠한 특성을 보이는가를 밝혀낼 수 있었다. 또 흥미롭게도 어떤 조건 하에서는 오류가 크더라도 오히려 SVM 의 성능이 향상되는 것을 발견했는데, 이것은 바꾸어 생각하면 Gram matrix 의 일부를 변경하여 SVM 의 성능 향상을 꾀할 수 있음을 나타낸다.