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Lasha Ephremidze - One of the best experts on this subject based on the ideXlab platform.
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On the Algorithmization of Janashia-Lagvilava Matrix Spectral Factorization Method
IEEE Transactions on Information Theory, 2018Co-Authors: Lasha Ephremidze, F. Saied, Ilya M. SpitkovskyAbstract:We consider three different ways of algorithmization of the Janashia–Lagvilava Spectral Factorization method. The first algorithm is faster than the second one, however, it is only suitable for matrices of low dimension. The second algorithm, on the other hand, can be applied to matrices of substantially larger dimension. The third algorithm is a superfast implementation of the method, but only works in the polynomial case under the additional restriction that the zeros of the determinant are not too close to the boundary. All three algorithms fully utilize the advantage of the method, which carries out Spectral Factorization of leading principal submatrices step-by-step. The corresponding results of numerical simulations are reported in order to describe the characteristic features of each algorithm and compare them to other existing algorithms.
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On algorithmization of Janashia-Lagvilava matrix Spectral Factorization method
arXiv: Numerical Analysis, 2016Co-Authors: Lasha Ephremidze, F. Saied, I. SpitkovskyAbstract:We consider three different ways of algorithmization of the Janashia-Lagvilava Spectral Factorization method. The first algorithm is faster than the second one, however, it is only suitable for matrices of low dimension. The second algorithm, on the other hand, can be applied to matrices of substantially larger dimension. The third algorithm is a superfast implementation of the method, but only works in the polynomial case under the additional restriction that the zeros of the determinant are not too close to the boundary. All three algorithms fully utilize the advantage of the method which carries out Spectral Factorization of leading principal submatrices step-by-step. The corresponding results of numerical simulations are reported in order to describe the characteristic features of each algorithm and compare them to other existing algorithms.
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Rank-deficient Spectral Factorization and wavelets completion problem
International Journal of Wavelets Multiresolution and Information Processing, 2015Co-Authors: Lasha Ephremidze, I. Spitkovsky, Edem LagvilavaAbstract:A simple constructive proof of polynomial matrix Spectral Factorization theorem is presented in the rank-deficient case. It is then used to provide an elementary solution to the wavelets completion problem.
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an elementary proof of the polynomial matrix Spectral Factorization theorem
Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 2014Co-Authors: Lasha EphremidzeAbstract:A very simple and short proof of the polynomial matrix Spectral Factorization theorem (on the unit circle as well as on the real line) is presented, which relies on elementary complex analysis and linear algebra.
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Matrix Spectral Factorization and wavelets
Journal of Mathematical Sciences, 2013Co-Authors: Gigla Janashia, Edem Lagvilava, Lasha EphremidzeAbstract:In this paper, recently published results on matrix Spectral Factorization is reviewed, and their connection to wavelet matrices is revealed.
Volker Pohl - One of the best experts on this subject based on the ideXlab platform.
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On the Algorithmic Solvability of Spectral Factorization and Applications
IEEE Transactions on Information Theory, 2020Co-Authors: Holger Boche, Volker PohlAbstract:Spectral Factorization is an operation which appears in many different engineering applications. This paper studies whether Spectral Factorization can be algorithmically computed on an abstract machine (a Turing machine). It is shown that there exist computable Spectral densities with very good analytic properties (i.e. smooth with finite energy) such that the corresponding Spectral factor cannot be determined on a Turing machine. Further, it will be proved that it is impossible to decide algorithmically whether or not a given computable density possesses a computable Spectral factor. This negative result has consequences for applications of Spectral Factorization in computer-aided design, because there it is necessary that this problem be decidable. Conversely, this paper will show that if the logarithm of a computable Spectral density belongs to certain Sobolev space of sufficiently smooth functions, then the Spectral factor is always computable. As an application, the paper discusses the possibility of calculating the optimal causal Wiener filter on an abstract machine.
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Boundedness Behavior of the Spectral Factorization for Polynomial Data in the Wiener Algebra
IEEE Transactions on Signal Processing, 2008Co-Authors: Holger Boche, Volker PohlAbstract:Spectral Factorization is of fundamental importance in many areas of signal processing. This paper investigates the boundedness behavior of the Spectral Factorization mapping in the Wiener algebra. Thereby, the focus lies on the Factorization of polynomial Spectral densities with a finite degree N since such spectra are especially important for practical applications. The paper presents a lower and an upper bound on the boundedness behavior which will show that the boundedness constant of the Spectral Factorization mapping gets worse as the degree N of the spectra increases. Therewith, one obtains independently the known result that the Spectral Factorization mapping is unbounded on the Wiener algebra.
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Continuity versus boundedness of the Spectral Factorization mapping
Studia Mathematica, 2008Co-Authors: Holger Boche, Volker PohlAbstract:This paper characterizes the Banach algebras of continuous functions on which the Spectral Factorization mapping S is continuous or bounded. It is shown that S is continuous if and only if the Riesz projection is bounded on the algebra, and that S is bounded only if the algebra is isomorphic to the algebra of continuous functions. Conse- quently, S can never be both continuous and bounded, on any algebra under consideration. 1. Introduction. The operation by which a functionf on the unit circle is written as f( ) = f+( )f+( ) for all 2 T :=fz2 C :jzj = 1g, with an outer functionf+, is known as Spectral Factorization. This operation arises in many dierent
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On the Boundedness of the Spectral Factorization Mapping on Decomposing Banach Algebras
SIAM Journal on Control and Optimization, 2008Co-Authors: Holger Boche, Volker PohlAbstract:In [SIAM J. Control Optim., 40 (2001), pp. 88-106], Jacob and Partington studied the continuity and boundedness of the Spectral Factorization mapping on decomposing Banach algebras. Many function spaces considered in systems theory are decomposing Banach algebras. The most well known example is the Wiener algebra, the space of all absolutely convergent Fourier series. Jacob and Partington showed in the above paper that the Spectral Factorization is locally Lipschitz continuous on all decomposing algebras, but unbounded on the most important examples of decomposing algebras. Our paper gives an extension of this result and shows that the Spectral Factorization mapping is unbounded on every decomposing algebra.
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EUSIPCO - On the boundedness behavior of the Spectral Factorization in the Wiener algebra for FIR data
2007Co-Authors: Holger Boche, Volker PohlAbstract:It is known that the Spectral Factorization mapping is unbounded in the Wiener algebra, in general. However in applications, the given data are often polynomials. For such finite dimensional spectra, the Spectral Factorization mapping is bounded, of course, but the boundedness constant depends on the degree of the given polynomial spectra. This paper presents lower and upper bounds for this boundedness constant depending on the degree N of the given data.
Jovan Stefanovski - One of the best experts on this subject based on the ideXlab platform.
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canonical form of para hermitian pencils generalized Spectral Factorization and optimal control over frequency region
International Journal of Robust and Nonlinear Control, 2013Co-Authors: Jovan StefanovskiAbstract:SUMMARY We generalize the J-Spectral Factorization of para-Hermitian proper rational matrix in the case when it has no constant inertia on the imaginary axis. This result and the presented numerical algorithm are based on a canonical form of para-Hermitian matrix pencils. We apply the new Spectral Factorization to the optimal control of the proper plant by dynamic measurement feedback over a frequency region. Copyright © 2012 John Wiley & Sons, Ltd.
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Transformation of J-Spectral Factorization of improper matrices to proper matrices
Systems & Control Letters, 2010Co-Authors: Jovan StefanovskiAbstract:We reduce the general problem of J-Spectral Factorization with improper matrices, given by a descriptor system realization, to the problem of J-Spectral Factorization with proper matrices.
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Spectral Factorization of non-symmetric polynomial matrices
Linear Algebra and its Applications, 2006Co-Authors: Jovan StefanovskiAbstract:The topic of the paper is Spectral Factorization of rectangular and possibly non-full-rank polynomial matrices. To each polynomial matrix we associate a matrix pencil by direct assignment of the coefficients. The associated matrix pencil has its finite generalized eigenvalues equal to the zeros of the polynomial matrix. The matrix dimensions of the pencil we obtain by solving an integer linear programming (ILP) minimization problem. Then by extracting a deflating subspace of the pencil we come to the required Spectral Factorization. We apply the algorithm to most general-case of inner–outer Factorization, regardless continuous or discrete time case, and to finding the greatest common divisor of polynomial matrices.
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brief polynomial j Spectral Factorization in minimal state space
Automatica, 2003Co-Authors: Jovan StefanovskiAbstract:By a specific choice of matrices in optimal LQ return difference equality, we develop a simple state-space algorithm for polynomial J-Spectral Factorization. For this purpose we solve a minimal-order algebraic Riccati equation, the order obtained by solving an Integer Linear Programming (ILP) problem. An example that cannot be solved by the existing algorithms illustrates our algorithm.
Martin Vetterli - One of the best experts on this subject based on the ideXlab platform.
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sparse Spectral Factorization unicity and reconstruction algorithms
International Conference on Acoustics Speech and Signal Processing, 2011Co-Authors: Martin VetterliAbstract:Spectral Factorization is a classical tool in signal processing and communications. It also plays a critical role in X-ray crystallography, in the context of phase retrieval. In this work, we study the problem of sparse Spectral Factorization, aiming to recover a one-dimensional sparse signal from its autocorrelation. We present a sufficient condition for the recovery to be unique, and propose an iterative algorithm that can obtain the original signal (up to a sign change, time-shift and time-reversal). Numerical simulations verify the effectiveness of the proposed algorithm.
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ICASSP - Sparse Spectral Factorization: Unicity and reconstruction algorithms
2011 IEEE International Conference on Acoustics Speech and Signal Processing (ICASSP), 2011Co-Authors: Martin VetterliAbstract:Spectral Factorization is a classical tool in signal processing and communications. It also plays a critical role in X-ray crystallography, in the context of phase retrieval. In this work, we study the problem of sparse Spectral Factorization, aiming to recover a one-dimensional sparse signal from its autocorrelation. We present a sufficient condition for the recovery to be unique, and propose an iterative algorithm that can obtain the original signal (up to a sign change, time-shift and time-reversal). Numerical simulations verify the effectiveness of the proposed algorithm.
Holger Boche - One of the best experts on this subject based on the ideXlab platform.
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On the Algorithmic Solvability of Spectral Factorization and Applications
IEEE Transactions on Information Theory, 2020Co-Authors: Holger Boche, Volker PohlAbstract:Spectral Factorization is an operation which appears in many different engineering applications. This paper studies whether Spectral Factorization can be algorithmically computed on an abstract machine (a Turing machine). It is shown that there exist computable Spectral densities with very good analytic properties (i.e. smooth with finite energy) such that the corresponding Spectral factor cannot be determined on a Turing machine. Further, it will be proved that it is impossible to decide algorithmically whether or not a given computable density possesses a computable Spectral factor. This negative result has consequences for applications of Spectral Factorization in computer-aided design, because there it is necessary that this problem be decidable. Conversely, this paper will show that if the logarithm of a computable Spectral density belongs to certain Sobolev space of sufficiently smooth functions, then the Spectral factor is always computable. As an application, the paper discusses the possibility of calculating the optimal causal Wiener filter on an abstract machine.
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Boundedness Behavior of the Spectral Factorization for Polynomial Data in the Wiener Algebra
IEEE Transactions on Signal Processing, 2008Co-Authors: Holger Boche, Volker PohlAbstract:Spectral Factorization is of fundamental importance in many areas of signal processing. This paper investigates the boundedness behavior of the Spectral Factorization mapping in the Wiener algebra. Thereby, the focus lies on the Factorization of polynomial Spectral densities with a finite degree N since such spectra are especially important for practical applications. The paper presents a lower and an upper bound on the boundedness behavior which will show that the boundedness constant of the Spectral Factorization mapping gets worse as the degree N of the spectra increases. Therewith, one obtains independently the known result that the Spectral Factorization mapping is unbounded on the Wiener algebra.
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Continuity versus boundedness of the Spectral Factorization mapping
Studia Mathematica, 2008Co-Authors: Holger Boche, Volker PohlAbstract:This paper characterizes the Banach algebras of continuous functions on which the Spectral Factorization mapping S is continuous or bounded. It is shown that S is continuous if and only if the Riesz projection is bounded on the algebra, and that S is bounded only if the algebra is isomorphic to the algebra of continuous functions. Conse- quently, S can never be both continuous and bounded, on any algebra under consideration. 1. Introduction. The operation by which a functionf on the unit circle is written as f( ) = f+( )f+( ) for all 2 T :=fz2 C :jzj = 1g, with an outer functionf+, is known as Spectral Factorization. This operation arises in many dierent
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On the Boundedness of the Spectral Factorization Mapping on Decomposing Banach Algebras
SIAM Journal on Control and Optimization, 2008Co-Authors: Holger Boche, Volker PohlAbstract:In [SIAM J. Control Optim., 40 (2001), pp. 88-106], Jacob and Partington studied the continuity and boundedness of the Spectral Factorization mapping on decomposing Banach algebras. Many function spaces considered in systems theory are decomposing Banach algebras. The most well known example is the Wiener algebra, the space of all absolutely convergent Fourier series. Jacob and Partington showed in the above paper that the Spectral Factorization is locally Lipschitz continuous on all decomposing algebras, but unbounded on the most important examples of decomposing algebras. Our paper gives an extension of this result and shows that the Spectral Factorization mapping is unbounded on every decomposing algebra.
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EUSIPCO - On the boundedness behavior of the Spectral Factorization in the Wiener algebra for FIR data
2007Co-Authors: Holger Boche, Volker PohlAbstract:It is known that the Spectral Factorization mapping is unbounded in the Wiener algebra, in general. However in applications, the given data are often polynomials. For such finite dimensional spectra, the Spectral Factorization mapping is bounded, of course, but the boundedness constant depends on the degree of the given polynomial spectra. This paper presents lower and upper bounds for this boundedness constant depending on the degree N of the given data.