Numerical stability of the levinson algorithm, generalized for Toeplitz-like systems, is studied. Arguments based on the analytic results of an error analysis for floating point arithmetic produce an upper bound on th...
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Numerical stability of the levinson algorithm, generalized for Toeplitz-like systems, is studied. Arguments based on the analytic results of an error analysis for floating point arithmetic produce an upper bound on the norm of the residual vector, which grows exponentially with respect to the size of the problem. The base of such an exponential function can be small for diagonally dominant Toeplitz-like matrices. Numerical experiments show that, for these matrices, Gaussian elimination by row and the levinson algorithm have residuals of the same order of magnitude. As expected, the empirical results point out that the theoretical bound is too pessimistic.
Periodic signals arc encountered in many applications. Such signals can be modelled by a weighted sum of sinusoidal components whose frequencies are integer multiples of a fundamental frequency. Given a data set, the ...
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ISBN:
(纸本)9780992862633
Periodic signals arc encountered in many applications. Such signals can be modelled by a weighted sum of sinusoidal components whose frequencies are integer multiples of a fundamental frequency. Given a data set, the fundamental frequency can be estimated in many ways including a maximum likelihood (ML) approach. Unfortunately, the ML estimator has a very high computational complexity, and the more inaccurate, but faster correlation-based estimators are therefore often used instead. In this paper, we propose a fast algorithm for the evaluation of the ML cost function for complex-valued data over all frequencies on a Fourier grid and up to a maximum model order. The proposed algorithm significantly reduces the computational complexity to a level not far from the complexity of the popular harmonic summation method which is an approximate ML estimator.
This paper presents a new version for the classical levinson algorithm for solution of a symmetric (Hermitian) Toeplitz set of equations. The new version has the property that for a Toeplitz matrix with (Gaussian) int...
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ISBN:
(纸本)9781424414833
This paper presents a new version for the classical levinson algorithm for solution of a symmetric (Hermitian) Toeplitz set of equations. The new version has the property that for a Toeplitz matrix with (Gaussian) integer entries the algorithm is carried out entirely over integers. The new algorithm has a low binary complexity with a near-linear integer growth rate. The integer preserving property provides an immediate means to control the numerical accuracy of the solution and its associated triangular factorization. It is also more attractive for symbolic computation.
Starting from the Durbin algorithm in polynomial space with an inner product defined by the signal autocorrelation matrix, an isometric transformation is defined that maps this vector space into another one where the ...
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Starting from the Durbin algorithm in polynomial space with an inner product defined by the signal autocorrelation matrix, an isometric transformation is defined that maps this vector space into another one where the levinson algorithm is performed. Alternatively, for iterative algorithms such as discrete all-pole (DAP), an efficient implementation of a Gohberg-Semencul (GS) relation is developed for the inversion of the autocorrelation matrix which considers its centrosymmetry. In the solution of the autocorrelation equations, the levinson algorithm is found to be less complex operationally than the procedures based on GS inversion for up to a minimum of five iterations at various linear prediction (LP) orders.
The paper gives a self-contained survey of fast algorithms for solving linear systems of equations with Toeplitz or Hankel coefficient matrices. It is written in the style of a *** of levinson-type and Schur-type are ...
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The paper gives a self-contained survey of fast algorithms for solving linear systems of equations with Toeplitz or Hankel coefficient matrices. It is written in the style of a *** of levinson-type and Schur-type are discussed. Their connections with triangular factorizations, Pade recursions and Lanczos methods are demonstrated. In the case in which the matrices possess additional symmetry properties, split algorithms are designed and their relations to butterfly factorizations are developed. (C) 2010 Elsevier Inc. All rights reserved.
A causal digital autoregressive filter is completely defined by a polynomial P. In the general case, and particularly for stable filters, it is well-known that P can be expressed recursively in terms of the so-called ...
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A causal digital autoregressive filter is completely defined by a polynomial P. In the general case, and particularly for stable filters, it is well-known that P can be expressed recursively in terms of the so-called reflection coefficients k(j) appearing in the lattice representation of the associated filter. This letter proposes an explicit expression for polynomial P in terms of the coefficients k(j).
Split levinson and Schur algorithms for the inversion of symmetric Toeplitz matrices are presented that work, in contrast to previous algorithms, without additional conditions like strong non-singularity. Copyright (c...
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Split levinson and Schur algorithms for the inversion of symmetric Toeplitz matrices are presented that work, in contrast to previous algorithms, without additional conditions like strong non-singularity. Copyright (c) 2004 John Wiley & Sons, Ltd.
Split levinson-type and Schur-type algorithms for the solutions of linear systems with a non-singular skewsymmetric Toeplitz matrix are designed. In contrast to previous ones, the algorithms work for any nonsingular s...
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Split levinson-type and Schur-type algorithms for the solutions of linear systems with a non-singular skewsymmetric Toeplitz matrix are designed. In contrast to previous ones, the algorithms work for any nonsingular skewsymmetric Toeplitz matrix. Moreover, generalizations of ZW- and WZ-factorizations of skewsymmetric Toeplitz: matrices related to the new split algorithms are presented. (C) 2004 Elsevier B.V. All rights reserved.
Split algorithms for Toeplitz matrices exploit besides the Toeplitz structure additional symmetry properties to reduce the number of operations. In this paper split levinson and Schur algorithms for hermitian Toeplitz...
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Split algorithms for Toeplitz matrices exploit besides the Toeplitz structure additional symmetry properties to reduce the number of operations. In this paper split levinson and Schur algorithms for hermitian Toeplitz matrices are presented that work, in contrast to previous algorithms, without additional conditions like strong nonsingularity. The main contribution is the generalization of the split levinson-type algorithms of B. Krishna/H. Krishna and H. Krishna/S. Morgera to general nonsingular hermitian Toeplitz matrices. Furthermore, a Schur-type counterpart of this algorithm is presented that is also new in the strongly nonsingular case. Some auxiliary considerations concerning the kernel structure of hermitian Toeplitz matrices might be of independent interest. (C) 2004 Elsevier Inc. All rights reserved.
In this article we deal with the Arov-Grossman functional model to describe all the solutions of the Covariance Extension Problem for q-variate stationary stochastic processes and we find the density that maximizes th...
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In this article we deal with the Arov-Grossman functional model to describe all the solutions of the Covariance Extension Problem for q-variate stationary stochastic processes and we find the density that maximizes the Burg Multivariate Entropy. This description is based on a one-to-one correspondence between the set of all solutions of the Covariance Extension Problem and the set of all contractive analytic functions H from the open unit disk with values on the space of q x q matrices. With this correspondence, the density that maximizes the Burg Multivariate Entropy corresponds to the function H equivalent to 0. Also, from the information that the Arov-Grossman functional model provides we obtain a version of the levinson algorithm. The partial autocorrelation coefficient matrices are computed directly from levinson's recursions.
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