• The Journal of the Institute of Internet, Broadcasting and Communication
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• v.21 no.2
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• pp.103-109
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• 2021

About the orthogonal Hadamard matrix announced by Hadamard in France in 1893, Professor Moon Ho Lee newly defined it as Center Weight Hadamard in 1989 and announced it, and discovered the Jacket matrix in 1998. The Jacket matrix is a generalization of the Hadamard matrix. In this paper, we propose a method of obtaining the Symmetric Jacket matrix, analyzing important properties and patterns, and obtaining the Jacket matrix's determinant and Eigenvalue, and proved it using Eigen decomposition. These calculations are useful for signal processing and orthogonal code design. To analyze the matrix system, compare it with DFT, DCT, Hadamard, and Jacket matrix. In the symmetric matrix of Galois Field, the element-wise inverse relationship of the Jacket matrix was mathematically proved and the orthogonal property AB=I relationship was derived.

• The Journal of the Institute of Internet, Broadcasting and Communication
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• v.15 no.5
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• pp.39-49
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• 2015

Some characters and construction theorems of Pseudo Jacket Matrix which is generalized from Jacket Matrix introduced by Jacket Matrices: Construction and Its Application for Fast Cooperative Wireless signal Processing[27] was announced. In this paper, we proposed some examples of Pseudo inverse Jacket matrix, such as $2{\times}4$, $3{\times}6$ non-square matrix for the MIMO channel. Furthermore we derived MIMO singular value decomposition (SVD) pseudo inverse channel and developed application to utilize SVD based on channel estimation of partitioned antenna arrays. This can be also used in MIMO channel and eigen value decomposition (EVD).

• The Journal of Korean Institute of Communications and Information Sciences
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• v.27 no.2A
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• pp.116-123
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• 2002

The jacket matrix is based on the Walsh-Hadamard matrix and an extension of it. While elements of the Walsh-Hadamard matrix are +1, or -1, those of the Jacket matrix are ${\pm}$1 and ${\pm}$$\omega$, which is $\omega$, which is ${\pm}$j and ${\pm}$2$\sub$n/. This matrix has weights in the center part of the matrix and its size is 1/4 of Hadamard matrix, and it has also two parts, sigh and weight. In this paper, instead of the conventional Jacket matrix where the weight is imposed by force, a simple Jacket sequence generation method is proposed. The Jacket sequence is generated by AND and Exclusive-OR operations between the binary indices bits of row and those of column. The weight is imposed on the element by when the product of each Exclusive-OR operations of significant upper two binary index bits of a row and column is 1. Each part of the Jacket matrix can be represented by jacket sequence using row and column binary index bits. Using Distributed Arithmetic (DA), we present a VLSI architecture of the Fast Jacket transform is presented. The Jacket matrix is able to be applied to cryptography, the information theory and complex spreading jacket QPSK modulation for WCDMA.

• Journal of electromagnetic engineering and science
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• v.6 no.4
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• pp.244-252
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• 2006

A block Jacket transform and. its block inverse Jacket transformn have recently been reported in the paper 'Fast block inverse Jacket transform'. But the multiplication of the block Jacket transform and the corresponding block inverse Jacket transform is not equal to the identity transform, which does not conform to the mathematical rule. In this paper, new binary block Jacket transforms and the corresponding binary block inverse Jacket transforms of orders $N=2^k,\;3^k\;and\;5^k$ for integer values k are proposed and the mathematical proofs are also presented. With the aid of the Kronecker product of the lower order Jacket matrix and the identity matrix, the fast algorithms for realizing these transforms are obtained. Due to the simple inverse, fast algorithm and prime based $P^k$ order of proposed binary block inverse Jacket transform, it can be applied in communications such as space time block code design, signal processing, LDPC coding and information theory. Application of circular permutation matrix(CPM) binary low density quasi block Jacket matrix is also introduced in this paper which is useful in coding theory.

• The Journal of the Institute of Internet, Broadcasting and Communication
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• v.15 no.3
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• pp.25-33
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• 2015

As a reversible Jacket is having the compatibility of two sided wearing, the matrix that both the inside and the outside are compatible is called Jacket matrix, and the matrix is having both inside and outside by the processes of element-wise inverse and block-wise inverse. This concept had been completed by one of the authors Moon Ho Lee in 1989, and finally that resultant matrix has been christened as Jacket matrix, in 2000. This is the most generalized extension of the well known Hadamard matrices, which includes both orthogonal and non-orthogonal matrices. This matrix addresses many problems in information and communication theories. we investigate the properties of the Jacket matrix, i.e. determinants, eigenvalues, and kronecker product. These computations are very useful for signal processing and orthogonal codes design. In our proposal, we provide some results to calculate these values by using a very simple mathematical model with less complexity.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.32 no.4C
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• pp.440-446
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• 2007

This paper addresses a new representation of DFT matrix via the Jacket transform based on the element inverse processing. We simply represent the inverse of the DFT matrix following on the factorization way of the Jacket transform, and the results show that the inverse of DFT matrix is only simply related to its sparse matrix and the permutations. The decomposed DFT matrix via Jacket matrix has a strong geometric structure that exhibits a block modulating property. This means that the DFT matrix decomposed via the Jacket matrix can be interpreted as a block modulating process.

• Journal of the Institute of Electronics and Information Engineers
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• v.50 no.7
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• pp.19-26
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• 2013

In this paper we prove that all Jacket matrices are circulant and up to equivalence. This result leads to new constructions of Toeplitz Jacket(TJ) matrices. We present the construction schemes of Toeplitz Jacket matrices and the examples of $4{\times}4$ and $8{\times}8$ Toeplitz Jacket matrices. As a corollary we show that a Toeplitz real Hadamard matrix is either circulant or negacyclic.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.22 no.11
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• pp.2512-2520
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• 1997

The classes of Reverse Jacket matrix [RJ]$_{N}$ and the corresponding Restclass Reverse Jacket matrix ([RRJ]$_{N}$) are defined;the main property of [RJ]$_{N}$ is that the inverse matrices of them can be obtained very easily and have a special structure. [RJ]$_{N}$ is derived from the weighted hadamard Transform corresponding to hadamard matrix [H]$_{N}$ and a basic symmertric matrix D. the classes of [RJ]$_{2}$ can be used as a generalize Quincunx subsampling matrix and serveral polygonal subsampling matrices. In this paper, we will present in particular the systematical block-wise extending-method for {RJ]$_{N}$. We have deduced a new orthorgonal matrix $M_{1}$.mem.[RRJ]$_{N}$ from a nonorthogonal matrix $M_{O}$.mem.[RJ]$_{N}$. These matrices can be used to develop efficient algorithms in IMT2000 signal processing, multidimensional subsampling, spectrum analyzers, and signal screamblers, as well as in speech and image signal processing.gnal processing.g.

• Journal of electromagnetic engineering and science
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• v.7 no.1
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• pp.17-27
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• 2007

Jacket matrices which are defined to be $n{\times}n$ matrices $A=(a_{jk})$ over a field F with the property $AA^+=nI_n$ where $A^+$ is the transpose matrix of elements inverse of A,i.e., $A^+=(a_{kj}^-)$, was introduced by Lee in 1984 and are used for signal processing and coding theory, which generalized the Hadamard matrices and Center Weighted Hadamard matrices. In this paper, some properties and constructions of Jacket matrices are extensively investigated and small orders of Jacket matrices are characterized, also present the full rate and the 1/2 code rate complex orthogonal space time code with full diversity.

• Proceedings of the KIEE Conference
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• 2007.04a
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• pp.252-254
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• 2007

In this paper, we will introduce an encoding algorithm of LDPC Codes in Direct-Sequence UWB systems. We evaluate the performance of the coded systems in an AWGN channel. This new algorithm is based on the Jacket matrics. Mathematically let A = ($a_{kl}$) be a matnx, if $A^{-1}$ = $(a^{-1}_{kl})^r$,then the matrix A is a Jacket matrix. If the Jacket matrices if Low density, the inverse matrices is also Low density which is very important to the introduced encoding algorithm.