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A new method for the nonlinear approximation of signals. I. The optimal damping factor. (English) Zbl 0619.93079
The following problem is discussed: Given an original signal \(y(t)\), find a substitute signal \(\bar y(t)\) in the form of a real rational function \(\bar Y(s)=\bar M(s)/\bar N(s)\) (the Laplace image of \(\bar y\)), where \(\bar M\) and \(\bar N\) are relatively prime Hurwitz polynomials. The coefficients of \(\bar M\) are denoted by \(b_ i\) and those of \(\bar N\) by \(a_ i\). If the \(b_ i's\) are variated, then the approximation problem is linear while, if the \(a_ i's\) are variated, then the problem is nonlinear. The problem is formulated in the Hilbert space \(L_ 2(0,\infty)\) where the norm of the quadratic error between y and \(\bar y\) must be minimized when the variable parameters are \(a_ i\). The method of finding the solution was previously given by the author [ibid. 19, 491-504 (1983; Zbl 0529.93022)] by modifying the classical Gauss-Newton iteration method. The key of the method consists in introducing at each iteration a parameter (damping factor DF) which improves the convergence. Some theorems concerning the influence of the DF are given. The algorithm will be exposed in a forth-coming paper.
Reviewer: D.Stanomir

93E25 Computational methods in stochastic control (MSC2010)
41A20 Approximation by rational functions
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
Zbl 0529.93022
Full Text: EuDML
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