Synthesis of a rational filter in the presence of complete alternance

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Abstract

The construction of a rational function that is nonnegative on two intervals of which one is infinite is considered. It is assumed that the maximum deviation of the function from zero on the infinite interval takes the minimum possible value under the condition that the values of the function on the finite interval are within the given bounds. It is assumed that the rational function (fraction) has the complete alternance. In this case, the original problem is reduced to solving a system of nonlinear equations. For solving this system, a two-stage method is proposed. At the first stage, a subsystem is selected and used to find a good approximation for the complete system. At the second stage, the complete system of nonlinear equations is solved. The solution is explained in detail for the case when the order of the fraction is between one and four. Numerical results for a fraction of order ten are presented.

Keywords

rational functions filtering problems complete alternance nonlinear systems of equations 

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References

  1. 1.
    M. Gkhashim and V. N. Malozemov, “Equivalence in the best rational approximation problems,” Vestn. St. Petersburg Univ., Ser. 1, No. 2, 3–8 (1992).MATHGoogle Scholar
  2. 2.
    E. I. Zolotarev, Complete Set of Works (Akad. Nauk SSSR, Leningrad, 1932), Vol. 2, pp. 1–59.Google Scholar
  3. 3.
    N. I. Akhiezer, Elements of the Theory of Elliptic Functions (American Mathematical Society, Providence, R.I., 1990).MATHGoogle Scholar
  4. 4.
    M. Gkhashim and V. N. Malozemov, “An invariant of a filtering problem,” Vestn. St. Petersburg Univ., Ser. 1, No. 3, 89–91 (1992).MATHGoogle Scholar
  5. 5.
    N. I. Akhiezer, “On a Zolotarev’s problem,” Izv. Akad. Nauk SSSR, No. 10, 919–931 (1929).Google Scholar

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© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.St. Petersburg State UniversitySt. PetersburgRussia

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