Generalized polynomial operators for nonlinear systems analysis

Halme, Aarne; Orava, J.

doi:10.1109/tac.1972.1099931

Cited by 28 publications

(18 citation statements)

References 10 publications

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“…However, under certain conditions, the fixed point solution may be obtained by invoking the classical contraction mapping theorem (CMT). This is closely related to the approaches followed in [8] and [17]. Before going into the problem formulation, we briefly discuss the similarities and differences between the work in this paper and that of the previous papers.…”

Section: Volterra Filter Equalizationmentioning

confidence: 83%

“…An elegant theory for local polynomial (Volterra) system inversion was pioneered in [8] and [17] and the references therein. The mathematical underpinning of this theory is based on contraction mappings and was formulated in a general Banach space setting.…”

Section: Equalization Problem Statementmentioning

confidence: 99%

“…Halme et al [8] consider the special case of the local inverse for polynomial operators on general Banach spaces, emphasizing a special construction of the local inverse. This special construction is quite different from the successive approximation construction used for the equalizers studied in this paper.…”

Section: Volterra Filter Equalizationmentioning

confidence: 99%

“…Using the method of successive approximation to obtain and then applying to leads, naturally, to a Volterra filter realization for the postfilter equalizer . Note that both the prefilter equalization (4) and the postfilter inversion (7) are represented by the following fixed point equation: (8) In the prefilter equalization, , and . A fixed point of satisfies or equivalently (9) For postfilter equalization, and .…”

Section: A Fixed Point Formulation Of Equalization Problemmentioning

confidence: 99%

“…Note that both the prefilter equalization (4) and the postfilter inversion (7) are represented by the following fixed point equation: (8) In the prefilter equalization, , and . A fixed point of satisfies or equivalently (9) For postfilter equalization, and . The fixed point of then satisfies or equivalently (10) Conditions are established in the next section under which the fixed point is unique, and therefore, .…”

Section: A Fixed Point Formulation Of Equalization Problemmentioning

confidence: 99%

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Volterra filter equalization: a fixed point approach

Nowak

Veen

1997

IEEE Trans. Signal Process.

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Abstract-One important application of Volterra filters is the equalization of nonlinear systems. Under certain conditions, this problem can be posed as a fixed point problem involving a contraction mapping. In this paper, we generalize the previously studied local inverse problem to a very broad class of equalization problems. We also demonstrate that subspace information regarding the response behavior of the Volterra filters can be incorporated to improve the theoretical analysis of equalization algorithms. To this end, a new "windowed" signal norm is introduced. Using this norm, we show that the class of allowable inputs is increased and the upper bounds on the convergence rate are improved when subspace information is exploited.

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Section: Volterra Filter Equalizationmentioning

confidence: 83%

Section: Equalization Problem Statementmentioning

confidence: 99%

Section: Volterra Filter Equalizationmentioning

confidence: 99%

Section: A Fixed Point Formulation Of Equalization Problemmentioning

confidence: 99%

Section: A Fixed Point Formulation Of Equalization Problemmentioning

confidence: 99%

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Volterra filter equalization: a fixed point approach

Nowak

Veen

1997

IEEE Trans. Signal Process.

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Criteria for the Global Existence of Functional Expansions for Input/Output Maps*

Sandberg

1985

At&T Technical Journal

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Much has been learned in recent years about the existence, determination, and properties of power-series-like expansions for expressing a nonlinear system's outputs in terms of its inputs. In particular, the existence and local convergence of expansions, and of certain "associated expansions," for impor tant large classes of systems are now well established. While the focus of attention has been on questions such that the size of the inputs for which convergence is guaranteed is not the main issue, some related material has appeared that bears on the problem of determining the extent of the region of convergence. The result most closely related to this paper is a recent theorem that gives necessary and sufficient conditions under which/ -1 has a generalized power-series expansion when / is an invertible locally-Lipshitz map between certain general subsets of two complex Banach spaces. In applications involv ing nonlinear models, ordinarily only real spaces of inputs and outputs are of direct interest. A "complexification" involving a certain solvability condition in complex spaces has to be able to be carried out to use the theorem referred to above. This paper reports on pertinent general results concerning invertible maps between subsets of real Banach spaces, with their complex extensions, and with generalized power-series expansions in both real and complex spaces. It focuses on questions concerning expansions for inverses of maps defined in real spaces. The results show that for a very large class of systems that have input/output maps, the ability to complexify is not just a useful sufficient * This paper was presented at the Conference on Information Sciences and Systems, Johns Hopkins University, March 27-29, 1985. f AT&T Bell Laboratories.Copyright © 1985 AT&T. Photo reproduction for noncommercial use is permitted with out payment of royalty provided that each reproduction is done without alteration and that the Journal reference and copyright notice are included on the first page. The title and abstract, but no other portions, of this paper may be copied or distributed royalty free by computer-based and other information-service systems without further permis sion. Permission to reproduce or republish any other portion of this paper must be obtained from the Editor. 1639condition for expandability, but is in fact the key condition for an input/ output map to be representable by a generalized power-series expansion.

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