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- LATIN AMER J SOLIDS STRUCTURES
- INT SCHOOL ADVANCED STUDIES
- Biblioteca Digital da Unicamp
- Elsevier
- Nonlinear Dynamics
- Sociedade Brasileira de Matemática Aplicada e Computacional
- Associação Brasileira de Ciências Mecânicas
- Universidade Nacional da Austrália
- IEEE; USA
- Universidade de Tubinga
- Universidade Rice
- Springer
- Academic Press
- Universidade Cornell
- Institute of Electrical and Electronics Engineers (IEEE Inc)
- Mais Publicadores...

## Power Secant Method applied to natural frequency extraction of Timoshenko beam structures

Fonte: LATIN AMER J SOLIDS STRUCTURES
Publicador: LATIN AMER J SOLIDS STRUCTURES

Tipo: Artigo de Revista Científica

ENG

Relevância na Pesquisa

36.18%

#exact modal analysis#dynamic stiffness matrix#secant method#power deflation#VIBRATIONS#ALGORITHM#Engineering, Civil#Engineering, Mechanical#Mechanics

This work deals with an improved plane frame formulation whose exact dynamic stiffness matrix (DSM) presents, uniquely, null determinant for the natural frequencies. In comparison with the classical DSM, the formulation herein presented has some major advantages: local mode shapes are preserved in the formulation so that, for any positive frequency, the DSM will never be ill-conditioned; in the absence of poles, it is possible to employ the secant method in order to have a more computationally efficient eigenvalue extraction procedure. Applying the procedure to the more general case of Timoshenko beams, we introduce a new technique, named ""power deflation"", that makes the secant method suitable for the transcendental nonlinear eigenvalue problems based on the improved DSM. In order to avoid overflow occurrences that can hinder the secant method iterations, limiting frequencies are formulated, with scaling also applied to the eigenvalue problem. Comparisons with results available in the literature demonstrate the strength of the proposed method. Computational efficiency is compared with solutions obtained both by FEM and by the Wittrick-Williams algorithm.

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## On the generalized eigenvalue method for energies and matrix elements in lattice field theory

Fonte: INT SCHOOL ADVANCED STUDIES
Publicador: INT SCHOOL ADVANCED STUDIES

Tipo: Artigo de Revista Científica

ENG

Relevância na Pesquisa

35.98%

#Lattice QCD#Lattice Quantum Field Theory#Lattice Gauge Field Theories#B-Physics#GAUGE-THEORY#STRING BREAKING#MATTER FIELDS#QCD#COUPLINGS#ADJOINT#Physics, Particles & Fields

We discuss the generalized eigenvalue problem for computing energies and matrix elements in lattice gauge theory, including effective theories such as HQET. It is analyzed how the extracted effective energies and matrix elements converge when the time separations are made large. This suggests a particularly efficient application of the method for which we can prove that corrections vanish asymptotically as exp(-(E(N+1) - E(n))t). The gap E(N+1) - E(n) can be made large by increasing the number N of interpolating fields in the correlation matrix. We also show how excited state matrix elements can be extracted such that contaminations from all other states disappear exponentially in time. As a demonstration we present numerical results for the extraction of ground state and excited B-meson masses and decay constants in static approximation and to order 1/m(b) in HQET.

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## Analise dinamica linear de porticos planos pelo metodo dos elementos finitos; Linear dynamic analysis of plane framework with use of the finite element method

Fonte: Biblioteca Digital da Unicamp
Publicador: Biblioteca Digital da Unicamp

Tipo: Dissertação de Mestrado
Formato: application/pdf

Publicado em 29/06/2006
PT

Relevância na Pesquisa

35.91%

#Metodo dos elementos finitos#Analise modal#Dinamica estrutural#Dynamicas of plane structures#Finite element method#Newmark method#Modal superposition

Neste trabalho estuda-se o comportamento de pórticos planos submetidos a ações dinâmicas. Apresenta-se, inicialmente, a Equação de Movimento de Lagrange através das variações das energias cinética, potencial mais o trabalho das forças não conservativas. Em seguida, pelo emprego do Método dos Elementos Finitos são desenvolvidas as matrizes de rigidez, massa e amortecimento para o elemento de pórtico plano. O amortecimento introduzido é o de Rayleigh. Estudam-se dois métodos para a realização da análise dinâmica: o método de Newmark e o Método da Superposição Modal, também sendo realizado um estudo do problema de autovalor e autovetor pelo emprego do Método das Potências e o Método da Deflação de Wielandt. Os autovalores e autovetores fornecerão as freqüências naturais e os modos de vibração da estrutura. Finalmente, são mostrados exemplos numéricos para a análise do comportamento dos pórticos planos; In this work, it is studied the behavior of plane frames submitted to dynamic loads. First of all Lagrange?s Equations of Motion is presented by the kinetic and potential energy variation plus the work of the nonconservative forces. Next, the stiffness, mass and damping matrices for the plane frame element are developed with the use of the Finite Element Method. Damping is introduced from the Rayleigh damping. Both Newmark Method and Modal Superposition Method are studied to carry out the dynamic analysis. It is also carried out a study of the eigenvalue problem by Power Method and Wielandt Deflation Method. Eigenvalues and eigenvectors will provide the natural frequencies and normal modes of the structure...

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## Análise de propagação de incertezas em método de estimação de rigidez estática por dados dinâmicos; Uncertainties propagation analysis through static stiffness estimation method from dynamic data

Fonte: Biblioteca Digital da Unicamp
Publicador: Biblioteca Digital da Unicamp

Tipo: Dissertação de Mestrado
Formato: application/pdf

Publicado em 13/01/2010
PT

Relevância na Pesquisa

35.89%

Este trabalho consiste no estudo de propagação de incertezas aleatórias através de método que permite estimar deflexões de carregamentos estáticos a partir de dados de avaliação dinâmica. Com este propósito, um modelo numérico foi desenvolvido para a realização de simulações decarregamentos estáticos cujas condições de contorno empregadas são similares às usualmente praticadas pela indústria automobilística em avaliações de rigidez de carrocerias. As freqüências naturais e os modos de vibrar também foram calculados pela resolução do problema de autovalor e autovetor das matrizes de massa e rigidez do modelo. Estes últimos dados foram então utilizados pelo método para estimar os mesmos coeficientes de rigidez obtidos da simulação de carregamento estático. Em seguida, incertezas aleatórias devidamente modeladas foram incorporadas aos parâmetros modais. Ferramentas de análise de propagação de incertezas, como Monte Carlo e propagação linear de covariância, foram aplicadas na verificação da incerteza da estimação feita pelo método quando seus parâmetros de entrada são incertos. Por último, ensaios experimentais de carregamento estático e análise modal experimental foram realizados para validação do método frente às incertezas associadas a estas medições. Resultados são apresentados e comentados; This work aims the study of random uncertainties propagation through a method that provides estimation of static loading deflections from dynamic data evaluations. With this purpose...

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## On inverse eigenvalue problems for block Toeplitz matrices with Toeplitz blocks

Fonte: Elsevier
Publicador: Elsevier

Tipo: Artigo de Revista Científica

Publicado em //2010
ENG

Relevância na Pesquisa

36.06%

#Block Toeplitz matrix#Newton method#Inverse eigenvalue problem#Generalized K-centrosymmetric matrix

We propose an algorithm for solving the inverse eigenvalue problem for real symmetric
block Toeplitz matrices with symmetric Toeplitz blocks. It is based upon an algorithm
which has been used before by others to solve the inverse eigenvalue problem for general
real symmetric matrices and also for Toeplitz matrices. First we expose the structure of the
eigenvectors of the so-called generalized centrosymmetric matrices. Then we explore the
properties of the eigenvectors to derive an efficient algorithm that is able to deliver a
matrix with the required structure and spectrum. We have implemented our ideas in a
Matlab code. Numerical results produced with this code are included.; Fundação para a Ciência e a Tecnologia (FCT)

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## The first eigenvalue of the Laplacian and the Conductance of a Compact surface

Fonte: Nonlinear Dynamics
Publicador: Nonlinear Dynamics

Tipo: Artigo de Revista Científica

ENG

Relevância na Pesquisa

36.06%

We present some results with the central theme of is the phenomenon of the first eigenvalue of the Laplacian and
conductance of the dynamical system. Our main tool is a method for studying how the hyperbolic metric on a Riemann surface
behaves under deformation of the surface. With this model, we show variation of the first eigenvalue of the laplacian and the
conductance of the dynamical system, with the Fenchel–Nielsen coordinates, that characterize the surface.

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## Numerical resolution of cone-constrained eigenvalue problems

Fonte: Sociedade Brasileira de Matemática Aplicada e Computacional
Publicador: Sociedade Brasileira de Matemática Aplicada e Computacional

Tipo: Artigo de Revista Científica
Formato: text/html

Publicado em 01/01/2009
EN

Relevância na Pesquisa

36.08%

#complementarity condition#generalized eigenvalue problem#power iteration method#scaling#projection algorithm

Given a convex cone K and matrices A and B, one wishes to find a scalar λ and a nonzero vector x satisfying the complementarity system K ∋ x ⊥(Ax-λ Bx) ∈ K+. This problem arises in mechanics and in other areas of applied mathematics. Two numerical techniques for solving such kind of cone-constrained eigenvalue problem are discussed, namely, the Power Iteration Method and the Scaling and Projection Algorithm.

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## Power secant method applied to natural frequency extraction of Timoshenko beam structures

Fonte: Associação Brasileira de Ciências Mecânicas
Publicador: Associação Brasileira de Ciências Mecânicas

Tipo: Artigo de Revista Científica
Formato: text/html

Publicado em 01/09/2010
EN

Relevância na Pesquisa

36.18%

This work deals with an improved plane frame formulation whose exact dynamic stiffness matrix (DSM) presents, uniquely, null determinant for the natural frequencies. In comparison with the classical DSM, the formulation herein presented has some major advantages: local mode shapes are preserved in the formulation so that, for any positive frequency, the DSM will never be ill-conditioned; in the absence of poles, it is possible to employ the secant method in order to have a more computationally efficient eigenvalue extraction procedure. Applying the procedure to the more general case of Timoshenko beams, we introduce a new technique, named "power deflation", that makes the secant method suitable for the transcendental nonlinear eigenvalue problems based on the improved DSM. In order to avoid overflow occurrences that can hinder the secant method iterations, limiting frequencies are formulated, with scaling also applied to the eigenvalue problem. Comparisons with results available in the literature demonstrate the strength of the proposed method. Computational efficiency is compared with solutions obtained both by FEM and by the Wittrick-Williams algorithm.

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## Blind channel identification and the eigenvalue problem of structured matrices

Fonte: Universidade Nacional da Austrália
Publicador: Universidade Nacional da Austrália

Tipo: Working/Technical Paper
Formato: 267487 bytes; 356 bytes; application/pdf; application/octet-stream

EN_AU

Relevância na Pesquisa

36.06%

In this paper, we address the problem of restoring a signal from its noisy convolutions with two unknown channels. When the transfer functions of these two channels have no common factors, the blind channel identification problem can be solved by finding the minimum eigenvalue of the Toeplitz-like matrix and its corresponding eigenvector. We present a fast iterative algorithm to solve the numerical solution of the eigenvalue problem for these structured matrices and hence the channel coeficients can be estimated efficiently. Once the channel coefficients are available, they can be used to reconstruct the unknown signal. Preliminary numerical results illustrate the effectiveness of the method.; no

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## Numerical solution of the eigenvalue problem for Hermitian Toeplitz-like matrices

Fonte: Universidade Nacional da Austrália
Publicador: Universidade Nacional da Austrália

Tipo: Working/Technical Paper
Formato: 247168 bytes; 356 bytes; application/pdf; application/octet-stream

EN_AU

Relevância na Pesquisa

36.23%

An iterative method based on displacement structure is proposed for computing eigenvalues and eigenvectors of a class of Hermitian Toeplitz-like matrices which includes matrices of the form T*T where T is arbitrary Toeplitz matrix, Toeplitz-block matrices and block-Toeplitz matrices. The method obtains a specific individual eigenvalue (i.e., the i-th smallest, where i is a specified integer in [1, 2,...,n]) of an n x n matrix at a computational cost of O(n2) operations. An associated eigenvector is obtained as a byproduct. The method is more efficient than general purpose methods such as the QR algorithm for obtaining a small number (compared to n) of eigenvalues. Moreover, since the computation of each eigenvalue is independent of the computation of all other eigenvalues, the method is highly parallelizable. Numerical results illustrate the effectiveness of the method.; no

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## Meshless Eigenvalue analysis for resonant structures based on the radial point interpolation method

Fonte: IEEE; USA
Publicador: IEEE; USA

Tipo: Conference paper

Publicado em //2009
EN

Relevância na Pesquisa

36.02%

#Meshless Methods#Eigenvalues and eigenfunctions#Radial Basis Functions#Radial Point Interpolation Method.

Meshless methods are a promising field of numerical methods recently introduced to computational electromagnetics. The potential of conformal and multi-scale modeling and the possibility of dynamic grid refinements are very attractive features that appear more naturally in meshless methods than in classical methods. The Radial Point Interpolation Method (RPIM) uses radial basis functions for the approximation of spatial derivatives. In this publication an eigenvalue solver is introduced for RPIM in electromagnetics. Eigenmodes are calculated on the example of a cylindrical resonant cavity. It is demonstrated that the computed resonance frequencies converge to the analytical values for increasingly fine spatial discretization. The computation of eigenmodes is an important tool to support research on a timedomain implementation of RPIM. It allows a characterization of the method’s accuracy and to investigate stability issues caused by the possible occurrence of non-physical solutions.; Thomas Kaufmann, Christophe Fumeaux, Christian Engstrom and Ruediger Vahldieck

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## A stable cubically convergent GR algorithm and Krylov subspace methods for non-Hermitian matrix eigenvalue problems; A stable cubically convergent GR algorithm and Krylov subspace methods for non-Hermitian matrix eigenvalue problems; Ein stabiles kubisch konvergentes GR-Verfahren und Krylov-Verfahren für nichthermitesche Matrixeigenwertprobleme

Fonte: Universidade de Tubinga
Publicador: Universidade de Tubinga

Tipo: Dissertação

DE_DE

Relevância na Pesquisa

36.06%

#Eigenwert#510#Eigenwert , QR-Algorithmus , GR-Verfahren , Lanczos-Verfahren , Krylov-Verfahren#eigenvalue , QR algorithm , GR algorithm , Lanczos algorithm , Krylov methods

In dieser Dissertation werden Krylov-Verfahren und
Zerlegungsalgorithmen (GR-Algorithmen)
zur Eigenwertberechnung von beliebigen Matrizen untersucht.
Es wird gezeigt, dass das allgemeine restarted Krylov-Verfahren
mathematisch äquivalent zum allgemeinen GR-Algorithmus
ist. Ausgehend von diesem Ergebnis wird ein neues,
numerisch stabiles GR-Verfahren entwickelt.
Es wird bewiesen, dass dieses Verfahren,
angewandt auf eine beliebig gegebene Matrix mit paarweise
verschiedenen Eigenwerten, unter sehr schwachen Voraussetzungen
kubisch konvergiert. Man beachte, dass das QR-Verfahren
unter diesen Voraussetzungen i.a. nur quadratisch konvergiert.; In this thesis Krylov methods and algorithms of decomposition type
(GR algorithms) for the eigenvalue computation of arbitrary matrices are discussed.
It is shown that the general restarted Krylov method is mathematically
equivalent to the general GR algorithm. Using this connection, a new,
numerical stable GR algorithm is developed.
It is proved that this algorithm converges cubically under mild conditions,
when applied to any given matrix with distinct eigenvalues.
Notice, that the QR algorithm converges typically quadratically
under these conditions.

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## Improving the Resolution of Bearing in Passive Sonar Arrays by Eigenvalue Analysis

Fonte: Universidade Rice
Publicador: Universidade Rice

Tipo: Relatório

ENG

Relevância na Pesquisa

36.18%

Tech Report; A method of improving the bearing-resolving capabilities of a passive array is discussed. This method is an adaptive beamforming method, having many similarities to the minimum energy approach. The evaluation of energy in each steered beam is preceded by an eigenvalue-eigenvector analysis of the emperical correlation matrix. Modification of the computations according to the eigenvalue structure result in improved resolution of the bearing of acoustic sources. The increase in resolution is related to the time-bandwidth product of the computation of the correlation matrix. However, this increased resolution is obtained at the expense of array gain.

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## Computing matrix symmetrizers. Part 2: new methods using eigendata and linear means; a comparison

Fonte: Elsevier
Publicador: Elsevier

Tipo: info:eu-repo/semantics/acceptedVersion; info:eu-repo/semantics/article

Publicado em 10/07/2015
ENG

Relevância na Pesquisa

35.86%

#Symmetric matrix factorization#symmetrizer#symmetrizer computation#eigenvalue method#linear equation#principal subspace computation#matrix optimization#numerical algorithm#MATLAB code#Matemáticas

Over any field F every square matrix A can be factored into the product of two symmetric matrices as A = S1 . S2 with S_i = S_i^T ∈ F^(n,n) and either factor can be chosen nonsingular, as was discovered by Frobenius in 1910. Frobenius’ symmetric matrix factorization has been lying almost dormant for a century. The first successful method for computing matrix symmetrizers, i.e., symmetric matrices S such that SA is symmetric, was inspired by an iterative linear systems algorithm of Huang and Nong (2010) in 2013 [29, 30]. The resulting iterative algorithm has solved this computational problem over R and C, but at high computational cost. This paper develops and tests another linear equations solver, as well as eigen- and principal vector or Schur Normal Form based algorithms for solving the matrix symmetrizer problem numerically. Four new eigendata based algorithms use, respectively, SVD based principal vector chain constructions, Gram-Schmidt orthogonalization techniques, the Arnoldi method, or the Schur Normal Form of A in their formulations. They are helped by Datta’s 1973 method that symmetrizes unreduced Hessenberg matrices directly. The eigendata based methods work well and quickly for generic matrices A and create well conditioned matrix symmetrizers through eigenvector dyad accumulation. But all of the eigen based methods have differing deficiencies with matrices A that have ill-conditioned or complicated eigen structures with nontrivial Jordan normal forms. Our symmetrizer studies for matrices with ill-conditioned eigensystems lead to two open problems of matrix optimization.; This research was partially supported by the Ministerio de Economía y Competitividad of Spain through the research grant MTM2012-32542.

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## The spectral method and numerical continuation algorithm for the von Kármán problem with postbuckling behaviour of solutions

Fonte: Springer
Publicador: Springer

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

36.02%

#Keywords: Bifurcation phenomenon#Eigenvalue problems#Iterative scheme#Nonlinear partial differential equations#Numerical continuation algorithm#Spectral method#Von Kármán plates

In this paper a spectral method and a numerical continuation algorithm for solving eigenvalue problems for the rectangular von Kármán plate with different boundary conditions (simply supported, partially or totally clamped) and physical parameters are i

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## An efficient method for computing eigenvalues of a real normal matrix

Fonte: Academic Press
Publicador: Academic Press

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

35.9%

#Keywords: Eigenvalue decomposition#Jacobi algorithm and QR algorithm#Normal matrix#Parallel computing

Jacobi-based algorithms have attracted attention as they have a high degree of potential parallelism and may be more accurate than QR-based algorithms. In this paper we discuss how to design efficient Jacobi-like algorithms for eigenvalue decomposition of

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## On the generalized eigenvalue method for energies and matrix elements in lattice field theory

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

35.98%

We discuss the generalized eigenvalue problem for computing energies and
matrix elements in lattice gauge theory, including effective theories such as
HQET. It is analyzed how the extracted effective energies and matrix elements
converge when the time separations are made large. This suggests a particularly
efficient application of the method for which we can prove that corrections
vanish asymptotically as $\exp(-(E_{N+1}-E_n) t)$. The gap $E_{N+1}-E_n$ can be
made large by increasing the number $N$ of interpolating fields in the
correlation matrix. We also show how excited state matrix elements can be
extracted such that contaminations from all other states disappear
exponentially in time. As a demonstration we present numerical results for the
extraction of ground state and excited B-meson masses and decay constants in
static approximation and to order $1/m_b$ in HQET.; Comment: (1+28) pages, 9 figures; minor corrections to table 1 and figures,
main results unaffected

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## Eigenvalue method to compute the largest relaxation time of disordered systems

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

46.18%

We consider the dynamics of finite-size disordered systems as defined by a
master equation satisfying detailed balance. The master equation can be mapped
onto a Schr\"odinger equation in configuration space, where the quantum
Hamiltonian $H$ has the generic form of an Anderson localization tight-binding
model. The largest relaxation time $t_{eq}$ governing the convergence towards
Boltzmann equilibrium is determined by the lowest non-vanishing eigenvalue
$E_1=1/t_{eq}$ of $H$ (the lowest eigenvalue being $E_0=0$). So the relaxation
time $t_{eq}$ can be computed {\it without simulating the dynamics} by any
eigenvalue method able to compute the first excited energy $E_1$. Here we use
the 'conjugate gradient' method to determine $E_1$ in each disordered sample
and present numerical results on the statistics of the relaxation time $t_{eq}$
over the disordered samples of a given size for two models : (i) for the random
walk in a self-affine potential of Hurst exponent $H$ on a two-dimensional
square of size $L \times L$, we find the activated scaling $\ln t_{eq}(L) \sim
L^{\psi}$ with $\psi=H$ as expected; (ii) for the dynamics of the
Sherrington-Kirkpatrick spin-glass model of $N$ spins, we find the growth $\ln
t_{eq}(N) \sim N^{\psi}$ with $\psi=1/3$ in agreement with most previous
Monte-Carlo measures. In addition...

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## Complex-band-structure eigenvalue method adapted to Floquet systems: topological superconducting wires as a case study

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

46.05%

For systems that can be modeled as a single-particle lattice extended along a
privileged direction as, e.g., quantum wires, the so-called eigenvalue method
provides full information about the propagating and evanescent modes as a
function of energy. This complex-band structure method can be applied either to
lattices consisting of an infinite succession of interconnected layers
described by the same local Hamiltonian or to superlattices: Systems in which
the spatial periodicity involves more than one layer. Here, for time-dependent
systems subject to a periodic driving, we present an adapted version of the
superlattice scheme capable of obtaining the Floquet states and the Floquet
quasienergy spectrum. Within this scheme the time periodicity is treated as
existing along spatial dimension added to the original system. The solutions at
a single energy for the enlarged artificial system provide the solutions of the
original Floquet problem. The method is suited for arbitrary periodic
excitations including strong and anharmonic drivings. We illustrate the
capabilities of the methods for both time-independent and time-dependent
systems by discussing: (a) topological superconductors in multimode quantum
wires with spin-orbit interaction and (b) microwave driven quantum dot in
contact with a topological superconductor.; Comment: 14 pages...

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## A Quick simulation method for fading communications channels using a novel eigenvalue importance sampling technique

Fonte: Institute of Electrical and Electronics Engineers (IEEE Inc)
Publicador: Institute of Electrical and Electronics Engineers (IEEE Inc)

Tipo: Conference paper

Relevância na Pesquisa

36.25%

#Keywords: Channel capacity#Computer simulation#Eigenvalues and eigenfunctions#Matrix algebra#Probability density function#Rayleigh fading#Receiving antennas#Sampling#Vectors#White noise#Eigenvalue importance sampling

In this paper, we introduce a quick simulation method for fading communications channels using a novel eigenvalue importance sampling technique. Our approach is motivated by the fact that many performance analyses involve metrics which are functions of the eigenvalues of the channel correlation matrix. More specifically in Rayleigh fading we often require the eigenvalues of the Wishart matrix HH† where H is the matrix of channel gains. Hence we propose direct simulation of the Wishart eigenvalues rather than simulation of the full channel matrix. If H is nR × nT then this idea in itself reduces simulation time since m = min(nR,nT) eigenvalues are required rather than the 2 × nR × nT real Gaussians. However, direct generation of the eigenvalues is complicated. Therefore we introduce a novel eigenvalue importance sampling technique which generates the eigenvalues from a simple biased density which "mimics" the real density. We call our approach Eigenvalue Importance Sampling (EVIS). Secondly, we try to reduce rare event simulation time by using biased eigenvalue densities to encourage the rare event of interest. We denote this approach Rare event Eigenvalue Importance Sampling (REVIS). Both methods are demonstrated via the example of simulating capacity outages and values for a MIMO system. Results show that considerable savings are offered by this novel approach even with simple implementations and small scale systems (m ≤ 4).

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