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block diagonalization and LU-equivalence of hankel matrices

LINEAR ALGEBRA AND ITS APPLICATIONS 2006年第2-3期412卷 247-269页

作者： Ben Atti, N Diaz-Toca, GM Univ Murcia Dpto Matemat Aplicada E-30071 Murcia Spain Univ Franche Comte CNRS UMR 6623 Lab MathUFR Sci & Tech F-25030 Besancon France

This article presents a new algorithm for obtaining a block diagonalization of Hankel matrices by means of truncated polynomial divisions, such that every block is a lower Hankel matrix. In fact, the algorithm generates a block LU-factorization of the matrix. Two applications of this algorithm are also presented. By the one hand, this algorithm yields an algebraic proof of Frobenius' Theorem, which gives the signature of a real regular Hankel matrix by using the signs of its principal leading minors. On the other hand, the close relationship between Hankel matrices and linearly recurrent sequences leads to a comparison with the Berlekamp-Massey algorithm. (c) 2005 Elsevier Inc. All rights reserved.

关键词： Hankel matrix block diagonalization Berlekamp-Massey algorithm

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block diagonalization Precoding and Power Allocation for Multiple-Antenna Systems With Coarsely Quantized Signals

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IEEE TRANSACTIONS ON COMMUNICATIONS 2021年第10期69卷 6793-6807页

作者： Pinto, Silvio F. B. de Lamare, Rodrigo C. Pontifical Catholic Univ Rio de Janeiro Ctr Telecommun Studies CETUC BR-22541041 Rio De Janeiro Brazil Univ York Dept Elect York YO10 5DD N Yorkshire England

In this work, we present block diagonalization and power allocation algorithms for large-scale multiple-antenna systems with coarsely quantized signals. In particular, we develop Coarse Quantization-Aware block diagonalization (CQA-BD) and Coarse Quantization-Aware Regularized block diagonalization (CQA-RBD) precoding algorithms that employ the Bussgang decomposition and can mitigate the effects of low-resolution signals and interference. Moreover, we also devise the Coarse Quantization-Aware Most Advantageous Allocation Strategy (CQA-MAAS) power allocation algorithm to improve the sum rate of precoders that operate with low-resolution signals. An analysis of the sum-rate performance is carried out along with computational complexity and power consumption studies of the proposed and existing techniques. Simulation results illustrate the performance of the proposed CQA-BD and CQA-RBD precoding algorithms, and the proposed CQA-MAAS power allocation strategy against existing approaches.

关键词： Resource management Precoding Quantization (signal) Power demand Nonlinear distortion Signal to noise ratio Receiving antennas Quantization consumption power allocation block diagonalization Bussgang's theorem

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block diagonalization of block Circulant Quaternion Matrices and the Fast Calculation for T-Product of Quaternion Tensors

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JOURNAL OF SCIENTIFIC COMPUTING 2024年第3期100卷 69页

作者： Zheng, Meng-Meng Ni, Guyan North Univ China Sch Math Taiyuan 030051 Shanxi Peoples R China Natl Univ Def Technol Dept Math Changsha 410073 Hunan Peoples R China

With quaternion matrices and quaternion tensors being gradually used in the color image and color video processing, the block diagonalization of block circulant quaternion matrices has become a key issue in the establishment of T-product based methods for quaternion tensors. Out of this consideration, we aim at establishing a fast calculation approach for the block diagonalization of block circulant quaternion matrices with the help of the fast Fourier transform (FFT). We first show that the discrete Fourier matrix Fp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {F_p}$$\end{document} cannot diagonalize pxp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\times p$$\end{document} circulant quaternion matrices, nor can the unitary quaternion matrices Fpj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {F_p}\textbf{j}$$\end{document} and Fp(1+j)/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {F_p}(1+\textbf{j})/\sqrt{2}$$\end{document} with j\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{j}$$\end{document} being an imaginary unit of quaternio

关键词： block circulant quaternion matrix block diagonalization Fast Fourier transform Quaternion tensor Color image compression

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block diagonalization of linear difference equations

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JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS 2002年第2期8卷 177-189页

作者： Siegmund, S Univ Augsburg Dept Math D-86135 Augsburg Germany

A linear autonomous system of difference equations Xk+1 = Ax(k) can be transformed to its Jordan normal form, i.e., the transformed system is in block diagonal form and the blocks correspond to different eigenvalues. We generalize this result to arbitrary nonautonomous linear systems x(k+1) = A(k)x(k) with invertible matrices A(k) epsilon R-NxN, k epsilon {..., -1,0,1....}.

关键词： block diagonalization linear normal form nonautonomous difference equation

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block diagonalization for Interference Mitigation in Ka-band Backhaul Networks 27

Block Diagonalization for Interference Mitigation in Ka-band...

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27th IEEE Annual International Symposium on Personal, Indoor, and Mobile Radio Communications (PIMRC)

作者： Zetik, Rudolf Ramireddy, Venkatesh Grossmann, Marcus Landmann, Markus Del Galdo, Giovanni Fraunhofer Inst Integrated Circuits IIS D-98693 Ilmenau Germany

ISBN: (纸本)9781509032549

This paper considers the design of precoding and decoding techniques for backhaul networks at Ka band. In this frequency band, large antenna arrays must be employed with appropriate beamforming and precoding techniques to combat the high path loss. Traditional multi-antenna systems use digital baseband beamforming and precoding that is not economical for large antenna arrays due to its cost and power consumption. Recently, hybrid analog-digital solutions were suggested for millimeter wave multiple-input-multiple-output systems. They rely on two steps. The first step consists of an analog only beamforming which aims at the exploitation of the high antenna gain offered by the large-scale antenna array. The second step mitigates the multiuser interference by means of digital precoding. In this paper, we focus on the second step of the hybrid precoding. We propose a solution for interference mitigation in multi-base station scenarios. Our solution is based on a block diagonalization technique and requires full channel state information at each base station of a backhaul network. The performance of the algorithm is compared to a partial block diagonalization which was originally proposed for the single base station downlink scenario.

关键词： backhaul networks multiple base-station scenario interference mitigation hybrid beamforming block diagonalization SVD-based precoding

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block diagonalization with interference control for energy efficient downlink

Block diagonalization with interference control for energy e...

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21st Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC)

作者： Riahi, Anis Kamoun, Mohamed CEA LIST Lab Syst Communicants F-91191 Gif Sur Yvette France

ISBN: (纸本)9781424480166

block diagonalization is a linear precoding technique which eliminates the multi-user interference at the transmitter side by creating parallel orthogonal single user links. In this technique, the transmit diversity order is fixed for each user and only power can be controlled to satisfy SINR requirements. This restricts the degrees of freedom available for resource allocation and limits the energy efficiency of the power allocation schemes performed by the base-station. In this contribution we present a new extension to block diagonalization where spatial degrees of freedom can also be adjusted in addition to power and bandwidth. With this new technique, it is possible to serve multiple users with different rate requirements by tolerating a controllable amount of interference. The amount of interference is controlled through an algebraic parameter that can be tuned in order to allow energy efficient power allocation.

关键词： linear precoding block diagonalization MU-MIMO energy efficient power allocation

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Lattice reduction aided with block diagonalization for multiuser MIMO systems

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EURASIP JOURNAL ON WIRELESS COMMUNICATIONS AND NETWORKING 2015年第1期2015卷 1页

作者： Khan, Md Hashem Ali Chung, Jin-Gyun Lee, Moon Ho Chonbuk Natl Univ Div Elect & Informat Engn Jeonju 54896 South Korea

The block diagonalization (BD) precoding technique is a well-known linear transmit strategy for multiuser multi-input multi-output (MU-MIMO) systems. The MU-MIMO broadcast channel is decomposed into multiple independent parallel single user MIMO (SU-MIMO) channels and achieves the maximum diversity order at high data rates. The lattice reduction-aided decoding (LRAD) features the reduced decoding complexity in MIMO communications. The Lenstra-Lenstra-Lovasz (LLL) algorithm has been extensively used to obtain better bases of the channel matrix while the complex lattice reduction (CLR) is aimed at improving orthogonality of basis vectors and shortening them. The orthogonalization and size reduction work are left for the CLR algorithm so that a modification of the channel matrix is carried out, resulting in better precoding and detection performances. We also derive bounds for lattice decoding. Simulation results show that the bit error rate (BER) performance of our proposed algorithm is better than that of conventional ones and it reduces the complexity compared with the LLL algorithm-based schemes.

关键词： Complex lattice reduction block diagonalization Multiuser MIMO Detection algorithms Proximity factors Low complexity

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Simplified Capacity-Based User Scheduling Algorithm for Multiuser MIMO Systems with block diagonalization

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IEICE TRANSACTIONS ON COMMUNICATIONS 2011年第10期E94B卷 2837-2846页

作者： Chang, Yuyuan Araki, Kiyomichi Tokyo Inst Technol Grad Sch Sci & Engn Tokyo 1528550 Japan

In multiple-input multiple-output (MIMO) systems, the multiuser MIMO (MU-MIMO) systems have the potential to provide. higher channel capacity owing to multiuser and spatial diversity. block diagonalization (BD) is one of the techniques to realize MU-MIMO systems, where multiuser interference can be completely cancelled and therefore several users can be supported simultaneously. When the number of multiantenna users is larger than the number of simultaneously receiving users, it is necessary to select the users that maximize the system capacity. However, computation complexity becomes prohibitive, especially when the number of multiantenna users is large. Thus simplified user scheduling algorithms are necessary for reducing the complexity of computation. This paper proposes a simplified capacity-based user scheduling algorithm, based on analysis of the capacity-based user selection criterion. We find a new criterion that is simplified by using the properties of Gram-Schmidt orthogonalization (GSO). In simulation results, the proposed algorithm provides higher sum rate capacity than the conventional simplified norm-based algorithm;and when signal-to-noise power ratio (SNR) is high, it provides performance similar to that of the conventional simplified capacity-based algorithm, which still requires high complexity. Fairness of the users is also taken into account. With the proportionally fair (PF) criterion, the proposed algorithm provides better performance (sum rate capacity or fairness of the users) than the conventional algorithms. Simulation results also shows that the proposed algorithm has lower complexity of computation than the conventional algorithms.

关键词： MIMO multiuser diversity block diagonalization user scheduling algorithm simplification Gram-schmidt orthogonalization

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A MMSE Vector Precoding with block diagonalization for Multiuser MIMO Downlink

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IEEE TRANSACTIONS ON COMMUNICATIONS 2012年第2期60卷 569-577页

作者： Park, Jungyong Lee, Byungju Shim, Byonghyo Korea Univ Sch Informat & Commun Seoul 136713 South Korea

block diagonalization (BD) algorithm is a generalization of the channel inversion that converts multiuser multi-input multi-output (MIMO) broadcast channel into single-user MIMO channel without inter-user interference. In this paper, we combine the BD technique with a minimum mean square error vector precoding (MMSE-VP) for achieving further gain in performance with minimal computational overhead. Two key ingredients to make our approach effective are the QR decomposition based block diagonalization and joint optimization of transmitter and receiver parameters in the MMSE sense. In fact, by optimizing precoded signal vector and perturbation vector in the transmitter and receiver jointly, we pursue an optimal balance between the residual interference mitigation and the noise enhancement suppression. From the sum rate analysis as well as the bit error rate simulations (both uncoded and coded cases) in realistic multiuser MIMO downlink, we show that the proposed BD-MVP brings substantial performance gain over existing multiuser MIMO algorithms.

关键词： Multiuser MIMO block diagonalization QR decomposition MMSE vector precoding Cholesky factorization

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COUPLED SYLVESTER-TYPE MATRIX EQUATIONS AND block diagonalization

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SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 2015年第2期36卷 580-593页

作者： Dmytryshyn, Andrii Kagstrom, Bo Umea Univ Dept Comp Sci SE-90187 Umea Sweden Umea Univ HPC2N SE-90187 Umea Sweden

We prove Roth-type theorems for systems of matrix equations including an arbitrary mix of Sylvester and star-Sylvester equations, in which the transpose or conjugate transpose of the unknown matrices also appear. In full generality, we derive consistency conditions by proving that such a system has a solution if and only if the associated set of 2 x 2 block matrix representations of the equations are block diagonalizable by (linked) equivalence transformations. Various applications leading to several particular cases have already been investigated in the literature, some recently and some long ago. Solvability of these cases follow immediately from our general consistency theory. We also show how to apply our main result to systems of Stein-type matrix equations.

关键词： matrix equation Sylvester equation Stein equation Roth's theorem consistency block diagonalization

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