Skip to main content
SpringerLink
Log in

Roundoff error analysis of fast DCT algorithms in fixed point arithmetic

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

Discrete cosine transforms (DCT) are essential tools in numerical analysis and digital signal processing. Processors in digital signal processing often use fixed point arithmetic. In this paper, we consider the numerical stability of fast DCT algorithms in fixed point arithmetic. The fast DCT algorithms are based on known factorizations of the corresponding cosine matrices into products of sparse, orthogonal matrices of simple structure. These algorithms are completely recursive, are easy to implement and use only permutations, scaling, butterfly operations, and plane rotations/rotation-reflections. In comparison with other fast DCT algorithms, these algorithms have low arithmetic costs. Using von Neumann–Goldstine’s model of fixed point arithmetic, we present a detailed roundoff error analysis for fast DCT algorithms in fixed point arithmetic. Numerical tests demonstrate the performance of our results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Acharya, T., Tsai, P.-S.: JPEG2000 Standard for Image Compression. Wiley, New Jersey (2005)

    Google Scholar 

  2. Ansari, R., Memon, N.: The JPEG lossy image compression standard. Handbook of Image and Video Processing, pp. 513–526. Academic, New York (2000)

    Google Scholar 

  3. Higham, N.J.: Accuracy and Stability of Numerical Algorithms, 2nd edn. SIAM, Philadelphia (2002)

    MATH  Google Scholar 

  4. Knight, W.R., Kaiser, R.: A simple fixed-point error bound for the fast Fourier transform. IEEE Trans. Acoust. Speech Signal Process. 27, 615–620 (1979)

    Article  Google Scholar 

  5. Meyer-Bäse, U.: Schnelle Digitale Signalverarbeitung, pp. 95–96. Springer, Berlin (2000)

    MATH  Google Scholar 

  6. von Neumann, J., Goldstine, H.H.: Numerical inverting of matrices of high order. Bull. Amer. Math. Soc. 53, 1021–1099 (1947)

    Article  MATH  MathSciNet  Google Scholar 

  7. Plonka, G., Tasche, M.: Fast and numerically stable algorithms for discrete cosine transforms. Linear Algebra Appl. 394, 309–345 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  8. Primbs, M.: Worst-case error analysis of lifting-based fast DCT-algorithms. IEEE Trans. Signal Process. 53, 3211–3218 (2005)

    Article  MathSciNet  Google Scholar 

  9. Rao, K.R., Yip, P.: Discrete Cosine Transform: Algorithms, Advantages, Applications. Academic, Boston (1990)

    MATH  Google Scholar 

  10. Schreiber, U.: Fast and numerically stable trigonometric transforms (in German). Thesis, Univ. of Rostock (1999)

  11. Tasche, M., Zeuner, H.: Roundoff error analysis for fast trigonometric transforms. In: Anastassiou, G (ed.) Handbook of Analytic-Computational Methods in Applied Mathematics. Chapman & Hall (2000)

  12. Vaidyanathan, P.P.: Multirate Systems and Filterbanks. Prentice Hall, Englewood Cliffs, NJ (1993)

    Google Scholar 

  13. Van Loan, C.F.: Computational Framework for the Fast Fourier Transform. SIAM, Philadelphia (1992)

    Google Scholar 

  14. Wang, Z.: Fast algorithms for the discrete W transform and the discrete Fourier transform. IEEE Trans. Acoust. Speech Signal Process. 32, 803–816 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  15. Wilkinson, J.H.: Rounding Errors in Algebraic Processes. Prentice Hall, London (1963)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Mathematics, University of Rostock, 18051, Rostock, Germany

    Katja Ihsberner

Authors
  1. Katja Ihsberner
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Katja Ihsberner.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ihsberner, K. Roundoff error analysis of fast DCT algorithms in fixed point arithmetic. Numer Algor 46, 1–22 (2007). https://doi.org/10.1007/s11075-007-9123-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-007-9123-1

Keywords

Mathematics Subject Classifications (2000)

Search

Navigation