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58
Multichannel Blind Deconvolution: Fir Matrix Algebra And Separation Of Multipath Mixtures
, 1996
"... A general tool for multichannel and multipath problems is given in FIR matrix algebra. With Finite Impulse Response (FIR) filters (or polynomials) assuming the role played by complex scalars in traditional matrix algebra, we adapt standard eigenvalue routines, factorizations, decompositions, and mat ..."
Abstract

Cited by 74 (0 self)
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A general tool for multichannel and multipath problems is given in FIR matrix algebra. With Finite Impulse Response (FIR) filters (or polynomials) assuming the role played by complex scalars in traditional matrix algebra, we adapt standard eigenvalue routines, factorizations, decompositions, and matrix algorithms for use in multichannel /multipath problems. Using abstract algebra/group theoretic concepts, information theoretic principles, and the Bussgang property, methods of single channel filtering and source separation of multipath mixtures are merged into a general FIR matrix framework. Techniques developed for equalization may be applied to source separation and vice versa. Potential applications of these results lie in neural networks with feedforward memory connections, wideband array processing, and in problems with a multiinput, multioutput network having channels between each source and sensor, such as source separation. Particular applications of FIR polynomial matrix alg...
Distributed MatrixFree Solution of Large Sparse Linear Systems over Finite Fields
 Algorithmica
, 1996
"... We describe a coarsegrain parallel software system for the homogeneous solution of linear systems. Our solutions are symbolic, i.e., exact rather than numerical approximations. Our implementation can be run on a network cluster of SPARC20 computers and on an SP2 multiprocessor. Detailed timings a ..."
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Cited by 27 (6 self)
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We describe a coarsegrain parallel software system for the homogeneous solution of linear systems. Our solutions are symbolic, i.e., exact rather than numerical approximations. Our implementation can be run on a network cluster of SPARC20 computers and on an SP2 multiprocessor. Detailed timings are presented for experiments with systems that arise in RSA challenge integer factoring efforts. For example, we can solve a 252; 222 \Theta 252; 222 system with about 11.04 million nonzero entries over the Galois field with 2 elements using 4 processors of an SP2 multiprocessor, in about 26.5 hours CPU time. 1 Introduction The problem of solving large, unstructured, sparse linear systems using exact arithmetic arises in symbolic linear algebra and computational number theory. For example the sievebased factoring of large integers can lead to systems containing over 569,000 equations and variables and over 26.5 million nonzero entries, that need to be solved over the Galois field of two...
A minimal polynomial basis solution to residual generation for linear systems
 of IFAC WorldCongress’99
, 1999
"... A fundamental part of a fault diagnosis system is the residual generator. Here anewmethod,theminimal polynomial basis approach, for design of residual generators for linear systems, is presented. The residual generation problem is transformed into a problem of finding polynomial bases for nullspace ..."
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Cited by 19 (11 self)
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A fundamental part of a fault diagnosis system is the residual generator. Here anewmethod,theminimal polynomial basis approach, for design of residual generators for linear systems, is presented. The residual generation problem is transformed into a problem of finding polynomial bases for nullspaces of polynomial matrices. This is a standard problem in established linear systems theory, which means that numerically efficient computational tools are generally available. It is shown that the minimal polynomial basis approach can find all possible residual generators, including those of minimal McMillan degree, and the solution has a minimal parameterization. It is shown that some other well known design methods, do not have these properties.
Model reduction of MIMO systems via tangential interpolation
 SIAM J. Matrix Anal. Appl
, 2004
"... Abstract. In this paper, we address the problem of constructing a reduced order system of minimal McMillan degree that satisfies a set of tangential interpolation conditions with respect to the original system under some mild conditions. The resulting reduced order transfer function appears to be ge ..."
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Cited by 19 (2 self)
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Abstract. In this paper, we address the problem of constructing a reduced order system of minimal McMillan degree that satisfies a set of tangential interpolation conditions with respect to the original system under some mild conditions. The resulting reduced order transfer function appears to be generically unique and we present a simple and efficient technique to construct this interpolating reduced order system. This is a generalization of the multipoint Padé technique which is particularly suited to handle multiinput multioutput systems.
Equivalences of Linear Control Systems
, 2000
"... We show how homological algebra and algebraic analysis allow to give various notions of equivalence for linear control systems which do not depend on their presentations and therefore preserve their structural properties. ..."
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Cited by 15 (9 self)
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We show how homological algebra and algebraic analysis allow to give various notions of equivalence for linear control systems which do not depend on their presentations and therefore preserve their structural properties.
Minimal statespace realization in linear system theory: An overview
 JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
, 2000
"... ..."
Proper Solutions Of Polynomial Equations
, 1999
"... The linear equation AX + BY = C is studied, where A; B; are given polynomial matrices such that A 1 B is a strictly proper rational matrix, and C ensures existence of a solution X 0 ; Y 0 that gives rise to a proper rational matrix Y 0 X 1 0 . Then all polynomial matrix solution pairs X; Y such ..."
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Cited by 9 (0 self)
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The linear equation AX + BY = C is studied, where A; B; are given polynomial matrices such that A 1 B is a strictly proper rational matrix, and C ensures existence of a solution X 0 ; Y 0 that gives rise to a proper rational matrix Y 0 X 1 0 . Then all polynomial matrix solution pairs X; Y such that Y X 1 is a proper rational matrix are parametrized. The study is motivated by the pole placement techniques in the design of linear control systems. Keywords: Linear control systems; pole assignment; polynomial methods, rational matrices; transfer functions.
Computation of JInnerOuter Factorizations of Rational Matrices
, 1998
"... this paper we address the problem to compute (if possible) a factorization of the TFM G as a product of a square and proper Jlossless factor G i with a stable, minimumphase factor G o , that is G = G i G o , where J is an indefinite signature matrix. In certain conditions, the algorithm to compute ..."
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Cited by 8 (4 self)
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this paper we address the problem to compute (if possible) a factorization of the TFM G as a product of a square and proper Jlossless factor G i with a stable, minimumphase factor G o , that is G = G i G o , where J is an indefinite signature matrix. In certain conditions, the algorithm to compute this factorization can be also employed to determine a more general factorization of G, where the factor G i is Jinner (not constrained to be Jlossless). Notice that for a surjective G, these factorizations are the usual Jlosslesouter and Jinnerouter factorizations of G, respectively. Important applications of these factorizations and of the closely related Jspectral factorization encompass the H# control theory 6, 9, 15, 18 and also model reduction
Minimal Degree Coprime Factorization of Rational Matrices
, 1999
"... Given a rational matrix G with complex coe#cients and a domain # in the closed complex plane, both arbitrary, we develop a complete theory of coprime factorizations of G over #, with denominators of McMillan degree as small as possible. The main tool is a general pole displacement theorem which give ..."
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Cited by 8 (6 self)
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Given a rational matrix G with complex coe#cients and a domain # in the closed complex plane, both arbitrary, we develop a complete theory of coprime factorizations of G over #, with denominators of McMillan degree as small as possible. The main tool is a general pole displacement theorem which gives conditions for an invertible rational matrix to dislocate by multiplication a part of the poles of G. We apply this result to obtain the parametrized class of all coprime factorizations over # with denominators of minimal McMillan degree n b the number of poles of G outside #. Specific choices of the parameters and of # allow us to determine coprime factorizations, as for instance, with polynomial, proper, or stable factors. Further, we consider the case in which the denominator has a certain symmetry, namely it is J allpass with respect either to the imaginary axis or to the unit circle. We give necessary and su#cient solvability conditions for the problem of coprime factorization with J allpass denominator of McMillan degree n b and, when a solution exists, we give a construction of the class of coprime factors. When no such solution exists, we discuss the existence of, and give solutions to, coprime factorizations with J allpass denominators of minimal McMillan degree (>n b ). All the developments are carried out in terms of descriptor realizations associated with rational matrices, leading to explicit and computationally e#cient formulas.
Computation of General InnerOuter and Spectral Factorizations
 IEEE TRANS. AUTO. CONTR
, 2000
"... In this paper we solve two problems in linear systems theory: the computation of the innerouter and spectral factorizations of a continuoustime system considered in the most general setting. We show that these factorization problems rely essentially on solving for the stabilizing solution a stan ..."
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Cited by 8 (3 self)
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In this paper we solve two problems in linear systems theory: the computation of the innerouter and spectral factorizations of a continuoustime system considered in the most general setting. We show that these factorization problems rely essentially on solving for the stabilizing solution a standard algebraic Riccati equation of order usually much smaller than the McMillan degree of the transfer function matrix of the system. The proposed procedures are completely general being applicable for a polynomial /proper/improper system whose transfer function matrix could be rank deficient and could have poles/zeros on the imaginary axis or at infinity. As an application we discuss the extension to rational matrices of the complete orthogonal decomposition of a constant matrix. Numerical refinements are discussed in detail. To illustrate the proposed approach several numerical examples are also given.