Journal of Scientific Computing

, Volume 36, Issue 3, pp 333–349 | Cite as

Computing Derivatives of Noisy Signals Using Orthogonal Functions Expansions

  • Adi Ditkowski
  • Abhinav Bhandari
  • Brian W. Sheldon


In many applications noisy signals are measured. These signals have to be filtered and, sometimes, their derivative has to be computed.

In this paper a method for filtering the signals and computing the derivatives is presented. This method is based on expansion onto transformed Legendre polynomials.

Numerical examples demonstrate the efficacy of the method as well as the theoretical estimates.


Noise filtering Spectral methods Orthogonal polynomials 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bhandari, A., Sheldon, B.W., Hearne, S.J.: Competition between tensile and compressive stress creation during constrained thin film island coalescence. J. Appl. Phys. 101, 033528, 157–469 (2007) CrossRefGoogle Scholar
  2. 2.
    Weickert, J.: Anisotropic Diffision in Image Processing. European Consortium for Mathematics in Industry. Teubner, Stuttgart (1998) Google Scholar
  3. 3.
    Gottlieb, D., Orzag, S.A.: Numerical Analysis of Spectral Methods: Theory and Applications. CBMS-NSF, vol. 26. SIAM, Philadelphia (1977) zbMATHGoogle Scholar
  4. 4.
    Hesthaven, J.S., Gottlieb, S., Gottlieb, D.: Spectral Methods for Time-Dependent Problems. Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge (2007) zbMATHGoogle Scholar
  5. 5.
    Gottlieb, D., Shu, C.W.: On the Gibbs phenomenon and its resolution. SIAM Rev. 39, 644–668 (1997) zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Funaro, D.: Polynomial Approximation of Differential Equations. Lecture Notes in Physics, vol. 8. Springer, Berlin (1992) zbMATHGoogle Scholar
  7. 7.
    Szegö, G.: Orthogonal Polynomial. Colloquium Publications, vol. 23. Am. Math. Soc., Providence (1939) Google Scholar
  8. 8.
    Kosloff, D., Tal-Ezer, H.: A modified Chebyshev pseudospectral method with an O(N −1) time step reduction. J. Comput. Phys. 104, 157–469 (1993) CrossRefMathSciNetGoogle Scholar
  9. 9.
    Don, W.S., Solomonoff, A.: Accuracy enhancement for higher derivatives using Chebyshev collocation and a mapping technique. SIAM J. Sci. Comput. 18(4), 1040–1055 (1997) zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Holger, W.: Scattered Data Approximation. Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge (2005) zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Adi Ditkowski
    • 1
  • Abhinav Bhandari
    • 2
  • Brian W. Sheldon
    • 2
  1. 1.School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael
  2. 2.Department of EngineeringBrown UniversityProvidenceUSA

Personalised recommendations