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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

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