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Numerische Methoden der Approximationstheorie/Numerical Methods of Approximation Theory pp 207–232Cite as

Birkhäuser

An Application of a Restricted Range Version of the Differential Correction Algorithm to the Design of Digital Systems

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Part of the book series: International Series of Numerical Mathematics ((ISNM,volume 30))

Abstract

The differential correction algorithm of Cheney and Loeb uses linear programming to find good generalized rational approximations on a finite point set. An expositöry discussion of numerical and theoretical results for this algorithm will be given. The application of a restricted range version of the algorithm to the design of digital filters will be considered, with a discussion of numerical results and such topics as continuity of the best approximation operator and degeneracy. A Fortran listing of this weighted, restricted range differential correction program is available upon request.

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References

  1. Barrodale, I., Powell, M. J. D.,and Roberts, F. D. K.: The Differential Correction Algorithm for Rational.£~ Approximation. SIAM J. Numer. Anal., 9 (1972), 493–504.

    MathSciNet  MATH  Google Scholar 

  2. Belford, G. G., and Burkhalter, J. F.: A Differential Correction Algorithm for Exponential Curve Fitting. Technical report UIUC-CAC-73–92, Center for Advanced Computation, Univ. of Illinois, Urbana (1973).

    Google Scholar 

  3. Braess, D.: Die Konstruktion der TschebyscheffApproximierenden bei der Anpassung mit Exponential-summen. J. Approx. Theory, 3 (1970), 261–273.

    Article  MathSciNet  MATH  Google Scholar 

  4. Cheney, E. W.: Introduction to Approximation Theory. McGraw-Hill, New York, 1966.

    MATH  Google Scholar 

  5. Cheney, E. W., and Loeb, H. L.: Two New Algorithms for Rational Approximation. Numer. Math., 3 (1961), 72–75.

    Article  MathSciNet  MATH  Google Scholar 

  6. Cheney, E. W., and Loeb, H. L.: On Rational Chebyshev Approximation. Numer. Math., 4 (1962), 124–127.

    Article  MathSciNet  MATH  Google Scholar 

  7. Cheney, E. W., and Loeb, H. L.: On the Continuity of Rational Approximation Operators. Arch. Rational riech. Anal., 21 (1966), 391–401.

    MathSciNet  MATH  Google Scholar 

  8. Cheney, E. W., and Southard, T. H.: A Survey of Methods for Rational Approximation with Particular Reference to a New Method based on a Formula of Darboux. SIAM Review, 5 (1963), 219–231.

    Article  MathSciNet  Google Scholar 

  9. Deczky, A. G.: Equiripple and Minimax (Chebyshev) Approximations for Recursive Digital Filters. IEEE Trans. Acoust., Speech, and Sig. Proc., ASSP-22 (1974), 98–111.

    Google Scholar 

  10. Dua, S. N., and Loeb, H. L.: Further Remarks on the Differential Correction Algorithm. SIAM J. Numer. Anal., 10 (1973), 123–126.

    Article  MathSciNet  MATH  Google Scholar 

  11. Dudgeon, D. E.: Recursive Filter Design using Differential Correction. IEEE Trans. Acoust., Speech, and Sig. Proc., submitted.

    Google Scholar 

  12. Dunham, C. B.: Rational Chebyshev Approximation on Subsets. J. Approx. Theory, 1 (1968), 484–487.

    Article  MathSciNet  MATH  Google Scholar 

  13. Dunham, C. B.: Varisolvent Approximation on Subsets. Proc. of the International Symposium on Approx. Theory, Austin, Texas (January 1973), edited by G. G. Lorentz, Academic Press, 337–340.

    Google Scholar 

  14. Fraser, W., and Hart, J. F.: On the Computation of Rational Approximations to Continuous Functions. Comm. Assoc. Comput. Mach., 5 (1962), 401–403.

    MATH  Google Scholar 

  15. Gimlin, D. R., Cavin, R. K. III, and Budge, M. C.,Jr.: A Multiple Exchange Algorithm for Best Restricted Approximations. SIAM J. Numer. Anal., 11 (1974), 219–231.

    Article  MathSciNet  MATH  Google Scholar 

  16. Gutknecht, M: Ein Abstiegsverfahren für Gleichmässige Approximation, mit Anwendungen. Thesis, Eidgenössischen Technischen Hochschule Zürich, 1973.

    MATH  Google Scholar 

  17. Helms, H. D.: Digital Filters with Equiripple or Minimax Responses. IEEE Trans. Audio and Electroacoustics, AU-19(1971), 85–93.

    Google Scholar 

  18. Hersey, H. S., Tufts, D. W., and Lewis, J. T.: Interactive Minimax Design of Linear Phase Nonrecursive Digital Filters Subject to Upper and Lower Function Constraints. IEEE Trans. Audio and Electroacoustics, AU-20(1972), 171–173.

    Google Scholar 

  19. Kaufman, E. H.,Jr. and Taylor, G. D.: An Application of Linear Programming to Rational Approximation. Rocky Mtn. J. of Math., 4 (1974), 371–373.

    MathSciNet  MATH  Google Scholar 

  20. Kaufman, E. H.,Jr. and Taylor, G. D.: Uniform Rational Approximation of Functions of Several Variables. Numerical Methods in Engineering, 9 (1975), 297–323.

    Article  MathSciNet  MATH  Google Scholar 

  21. Kaufman, E. H., Jr. and Taylor, G. D.: Quadratic Convergence of the Differential Correction Algorithm. In preparation.

    Google Scholar 

  22. Lee, C. M. and Roberts, F. D. K.: A Comparison of Algorithms for Rational L Approximation. Math. Comp., 27 (1973), 111–121.

    MathSciNet  MATH  Google Scholar 

  23. Lewis, J. T.: Restricted Range Approximation and Its Application to Digital Filter Design. Math. Comp., to appear.

    Google Scholar 

  24. Loeb, H. L.: Approximation by Generalized Rationals. J. SIAM Numer. Anal., 3 (1966), 34–55.

    Article  MathSciNet  MATH  Google Scholar 

  25. Loeb, H. L. and Moursund, D. G.: Continuity of the Best Approximation Operator for Restricted Range Approximations. J. Approx. Theory, 1 (1968), 391–400.

    Article  MathSciNet  MATH  Google Scholar 

  26. Loeb, H. L., Moursund, D. G. and Taylor, G. D.: Uniform Rational Weighted Approximations Having Restricted Ranges. J. Approx. Theory, 1 (1968), 401–411.

    Article  MathSciNet  MATH  Google Scholar 

  27. Maehly, H. J. and Witzgall, C.: Tschebyscheff-Approximationen in Kleinen Intervallen II. Numer. Math., 2 (1960), 293–307.

    Article  MathSciNet  Google Scholar 

  28. Rabiner, L. R., Graham, N. Y., and Helms, H. D.: Linear Programming Design of IIR Digital Filters with arbitrary Magnitude Functions. IEEE Trans. Acoust., Speech, and Sig. Proc., ASSP-22 (1974), 117–123.

    Google Scholar 

  29. Rice, J. R.: The Approximation of Functions, II. Addison-Wesley, Reading, 1969.

    Google Scholar 

  30. Taylor, G. D.: Approximation by Functions Having Restricted Ranges III. J. Math. Anal. App1.,. 27 (1969), 241–248.

    Article  MATH  Google Scholar 

  31. Taylor, G. D. and Winter, M. J.: Calculation of Best Restricted Approximations. SIAM J. Numer. Anal., 7 (1970), 248–255.

    Article  MathSciNet  MATH  Google Scholar 

  32. Taylor, K. A.: Contributions to the Theory of Restricted Polynomial and Rational Approximation. Thesis, Michigan State University, 1970.

    Google Scholar 

  33. Thajchayapong, P. and Rayner, P. J. W.: Recursive Digital Filter Design by Linear Programming. IEEE Trans. Audio and Electro-acoustics. AU-21(1973), 107–112.

    Google Scholar 

  34. Werner, H.: Tschebyscheff-Approximation in Bereich der Rationalen Funktionen bei Vorliegen einer Guten Ausgangsnaherung. Arch. Rat. Mech. Anal., 10 (1962), 205–219.

    Article  MATH  Google Scholar 

  35. Werner, H.: On the Rational Tschebyscheff Operator. Math. Zeitschr, 86 (1964), 317–326.

    Article  MATH  Google Scholar 

  36. Werner, H.: Diskretisierung bei Tschebyscheff-Approximation mit Verallgemeinerten Rationalen Funktionen. ISNM, 9 (1968), 381–390.

    Google Scholar 

  37. Werner, H. Theoretische und Praktische Ergebnisse auf dem Gebeit der Tschebyscheff-Approximationen mit Rationalen Funktionen. Vortrag gehalten in Grenoble (20 Marz 1968 ).

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Central Michigan University, Mount Pleasant, MI, 48859, USA

    E. H. Kaufman Jr. & G. D. Taylor

  2. Department of Mathematics, Colorado State University, Fort Collins, CO, 80523, USA

    E. H. Kaufman Jr. & G. D. Taylor

Authors
  1. E. H. Kaufman Jr.
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  2. G. D. Taylor
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Editor information

Editors and Affiliations

  1. Hamburg, Germany

    L. Collatz

  2. Münster, Germany

    H. Werner

  3. Erlangen, Germany

    G. Meinardus

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© 1976 Springer Basel AG

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Kaufman, E.H., Taylor, G.D. (1976). An Application of a Restricted Range Version of the Differential Correction Algorithm to the Design of Digital Systems. In: Collatz, L., Werner, H., Meinardus, G. (eds) Numerische Methoden der Approximationstheorie/Numerical Methods of Approximation Theory. International Series of Numerical Mathematics, vol 30. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7692-6_11

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  • DOI: https://doi.org/10.1007/978-3-0348-7692-6_11

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-7643-0824-7

  • Online ISBN: 978-3-0348-7692-6

  • eBook Packages: Springer Book Archive

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