A Tutorial on GrÖbner Bases With Applications in Signals and Systems

Lin, Zhiping; Xu, Li; Bose, N.K.

doi:10.1109/tcsi.2007.914007

Cited by 50 publications

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References 114 publications

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“…of the denominators of the rational entries of K 0 ð Þ ) and inequalities that can be brought to polynomial equations. The resulting polynomial set of equations can be solved (parametrically) by using the Gröbner basis method (see e.g., [49][50][51][52]). By applying the Gröbner basis method, one would get an indication to the existence of solutions and in case that solutions do exist, it would tell what are the free parameters and how other parameters depend on the free parameters.…”

Section: Parametrizations Of All the Static Output Feedbacksmentioning

confidence: 99%

Control Systems in Engineering and Optimization Techniques

2022

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signals-frequency and amplitude of the stator voltage generated by various algorithms, the frequency and amplitude of the rotor current, the natural slip, and the rotational speed of the motor rotor. This made it possible to assess the effectiveness of interpreting asynchronous electric drives and methods of their regulation as objectively as possible. Over the past 25-30 years, numerous articles on this topic have not provided such results.The book can be used as a reference book for several interdisciplinary research areas. We express our sincere thanks to all the contributors, managing editors, and typesetters.

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Section: Parametrizations Of All the Static Output Feedbacksmentioning

confidence: 99%

Control Systems in Engineering and Optimization Techniques

2022

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“…For multivariate polynomials, there is a GCD (since the ring is a unique factorization domain) but the GCD is not necessarily a linear combination of the polynomials. The theory of Gröbner bases has been introduced to compute with multivariate polynomials [6], [7] and the theory is widely used in multidimensional signal processing [8], [9], [10], [11], [12], [13]. Methods using Gröbner bases techniques for testing the invertibility of and for computing a particular inverse of an N × 1 multivariate polynomial matrix H(z) were proposed in [14], [15].…”

Section: Introductionmentioning

confidence: 99%

Generic Invertibility of Multidimensional FIR Filter Banks and MIMO Systems

Law

Fossum

Minh

2009

IEEE Trans. Signal Process.

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Abstract-We study the invertibility of M -variate Laurent polynomial N × P matrices. Such matrices represent multidimensional systems in various settings such as filter banks, multiple-input multiple-output systems, and multirate systems. Given an N × P Laurent polynomial matrix H (z1, ..., zM ) of degree at most k, we want to find a P × N Laurent polynomial left inverse matrix G(z) of H (z) such that G(z)H (z) = I. We provide computable conditions to test the invertibility and propose algorithms to find a particular inverse.The main result of this paper is to prove that H (z) is generically invertible when N − P ≥ M ; whereas when N − P < M , then H (z) is generically noninvertible. As a result, we propose an algorithm to find a particular inverse of a Laurent polynomial matrix that is faster than current algorithms known to us.

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