Skip to main content

Roundoff Error Analysis of the Recursive Moving Window Discrete Fourier Transform

Abstract

This paper presents both worst case and average case analysis of roundoff errors occuring in the floating point computation of the recursive moving window discrete Fourier transform (DFT) with precomputed twiddle factors. We show the strong influence of precomputation errors – both within the initial fast Fourier transform (FFT) and the recursion – on the numerical stability. Numerical simulations confirm the theoretical results.

This is a preview of subscription content, access via your institution.

References

  1. J.L. Aravena, Recursive moving window DFT algorithm, IEEE Trans. Comput. 39 (1990) 145–148.

    Google Scholar 

  2. N.J. Higham, Accuracy and Stability of Numerical Algorithms (SIAM, Philadelphia, PA, 1996).

    Google Scholar 

  3. S. Mallat, A Wavelet Tour of Signal Processing (Academic Press, San Diego, CA, 1998).

    Google Scholar 

  4. A.V. Oppenheim and R.W. Schafer, Discrete-Time Signal Processing (Prentice-Hall, Englewood Cliffs, NJ, 1989).

    Google Scholar 

  5. J.A. Rosendo Macías and A. Gómez Expósito, Recursive formulation of short-time discrete trigonometric transforms, IEEE Trans. Circuits Systems 45 (1998) 525–527.

    Google Scholar 

  6. B.G. Sherlock and D.M. Monro, Moving discrete Fourier transform, Proc. IEEE 139 (1992) 279–282.

    Google Scholar 

  7. M. Tasche and H. Zeuner, Roundoff error analysis for fast trigonometric transforms, in: Handbook of Analytic-Computational Methods in Applied Mathematics, ed. G. Anastassiou (CRC Press, Boca Raton, FL, 2000) 357–406.

    Google Scholar 

  8. M. Tasche and H. Zeuner, Worst and average case roundoff error analysis for FFT, BIT 41 (2001) 563–581.

    Google Scholar 

  9. M. Tasche and H. Zeuner, Improved roundoff error analysis for precomputed twiddle factors, J. Comput. Anal. Appl. 4 (2002) 1–18.

    Google Scholar 

  10. C. Van Loan, Computational Frameworks for the Fast Fourier Transform (SIAM, Philadelphia, PA, 1992).

    Google Scholar 

  11. H. Zeuner, Variance of the norm of roundoff error vectors, Preprint, Medical University Luebeck (2000).

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Tasche, M., Zeuner, H. Roundoff Error Analysis of the Recursive Moving Window Discrete Fourier Transform. Advances in Computational Mathematics 18, 65–78 (2003). https://doi.org/10.1023/A:1021254709641

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1021254709641

  • roundoff error analysis
  • worst case study
  • average case study
  • recursive moving window DFT
  • fast Fourier transform
  • discrete windowed Fourier transform