Advertisement

Quantization and Roundoff Errors

  • Robert King
  • Majid Ahmadi
  • Raouf Gorgui-Naguib
  • Alan Kwabwe
  • Mahmood Azimi-Sadjadi

Abstract

A one-dimensional (1-D) digital filter, as noted in Section 1.3, is generally defined by
$${y_n} = \sum\limits_{i = 0}^M {{a_i}{u_{n - i}}} - \sum\limits_{i = 1}^N {{b_i}{y_{n - i}}} $$
(5.1)
where {u n } is the input sequence, {y n } is the output sequence, and a i , and b i are some constants.

Keywords

Digital Filter Roundoff Error Input Quantization Recursive Digital Filter Finite Impulse Response Digital Filter 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    L. R. Rabiner and B. Gold, Theory and Application of Digital Signal Processing, Prentice-Hall, Englewood Cliffs, NJ (1975).Google Scholar
  2. 2.
    L. B. Jackson, An Analysis of Roundoff Noise in Digital Filters, ScD Thesis, Stevens Institute of Technology, Hoboken, New Jersey (1969).Google Scholar
  3. 3.
    B. Liu, Effect of finite word length on the accuracy of digital filters, IEEE Trans. Circuit Theory CT-18, 670–677 (1971).CrossRefGoogle Scholar
  4. 4.
    V. B. Lawrence, Use of Orthogonal Functions in the Design of Digital Filters, PhD Thesis, London University (1972).Google Scholar
  5. 5.
    B. Liu and T. Kaneko, Error analysis of digital filters realized with floating-point arithmetic, Proc. IEEE 57, 1735–1747 (1969).CrossRefGoogle Scholar
  6. 6.
    J. K. Aggarwal, Input Quantization and Arithmetic Roundoff in Digital Filters—A Review, Proc. NATO Advanced Study Institute on Network and Signal Theory, PPL Conf. Pubi. No. 12, 315–343 (1972).Google Scholar
  7. 7.
    W. R. Bennett, Spectra of quantized signals, Bell Syst. Tech. J. 27, 446–472 (1948).Google Scholar
  8. 8.
    A. Papoulis, Probability, Random Variables and Stochastic Processes, McGraw-Hill, New York (1965).zbMATHGoogle Scholar
  9. 9.
    L. Jackson, On the interaction of roundoff noise and dynamic range in digital filters, Bell Syst. Tech. J. 49, 159–184 (1970).zbMATHGoogle Scholar
  10. 10.
    B. Liu and M. E. Van Valkenburg, On roundoff error of fixed-point digital filters using sign-magnitude truncation, IEEE Trans. Circuit Theory CT-19, 536–537 (1972).CrossRefGoogle Scholar
  11. 11.
    I. W. Sandberg, Floating-point-roundoff accumulation in digital-filter realizations, Bell Syst. Tech. J. 46, 1775–1791 (1967).zbMATHGoogle Scholar
  12. 12.
    T. Kaneko and B. Liu, Effect of coefficient rounding in floating-point digital filters, IEEE Trans. Aerosp. Electron. Syst. AES-7, 995–1003 (1971).CrossRefGoogle Scholar
  13. 13.
    E. P. F. Kan and J. K. Aggarwal, Error analysis of digital filters employing floating-point arithmetic, IEEE Trans. Circuit Theory CT-18, 678–686 (1971).CrossRefGoogle Scholar
  14. 14.
    A. Fettweis, Some Statistical Properties of Floating-Point Roundoff Noise, Proc. 16th Midwest Symp. Circuit Theory, II.2.1-II.2.10 (1973).Google Scholar
  15. 15.
    D. S. K. Chan, Roundoff Noise in Cascade Realization of Finite Impulse Response Digital Filters, SB and SM Thesis, Department of Electrical Engineering, Massachusetts Institute of Technology, Cambridge, Mass. (1972).Google Scholar
  16. 16.
    L. B. Jackson, Roundoff-noise analysis for fixed-point digital filters realized in cascade or parallel form, IEEE Trans. Audio Electroacoust. AU-18, 107–122 (1970).CrossRefGoogle Scholar
  17. 17.
    D. S. K. Chan and L. R. Rabiner, Theory of roundoff noise in cascade realizations of finite impulse response digital filters, Bell Syst. Tech. J. 52, 329–345 (1973).zbMATHGoogle Scholar
  18. 18.
    L. B. Jackson, An Analysis of Limit Cycles due to Multiplication Rounding in Recursive Digital (Sub) Filters, Proc. 7th Annual Allerton Conference on Circuit and System Theory, 69-78 (October 1969).Google Scholar
  19. 19.
    P. M. Ebert, J. E. Mazo, and M. G. Taylor, Overflow oscillations in digital filters, Bell Syst. Tech. J. 48, 2990–3020 (1969).Google Scholar
  20. 20.
    J. F. Kaiser, Some Practical Considerations in the Realization of Linear Digital Filters, Proc. 3rd Allerton Annual Conference on Circuit and System Theory, 621-633 (October 1965).Google Scholar
  21. 21.
    S. R. Parker and S. F. Hess, Limit cycle oscillations in digital filters, IEEE Trans. Circuit Theory CT-18, 687–697 (1971).CrossRefGoogle Scholar
  22. 22.
    K. P. Prasad and P. S. Reddy, Limit cycles in second-order digital filters, J. Inst. Electron. Telecommun. Eng. (India) 26, 85–86 (1980).Google Scholar
  23. 23.
    V. B. Lawrence and K. V. Mina, A new and interesting class of limit cycles in recursive digital filters, Bell Syst. Tech. J. 58, 379–408 (1979).Google Scholar
  24. 24.
    D. C. McLernon and R. A. King, Additional properties of one-dimensional limit cycles, IEE Proc. 133, Part G, No. 3, 140–144 (1986).MathSciNetGoogle Scholar
  25. 25.
    T. A. C. M. Claasen, W. F. G. Mecklenbrauker, and J. B. H. Peek, Some remarks on the classification of limit cycles in digital filters, Philips Res. Rep. 28, 297–305 (1973).Google Scholar
  26. 26.
    D. C. Munson, Jr., J. H. Strickland, Jr., and T. P. Walker, Maximum amplitude zero-input limit cycles in digital filters, IEEE Trans. Circuits Syst. CAS-31, 266–275 (1984).CrossRefGoogle Scholar
  27. 27.
    N. G. Kingsbury and P. J. W. Rayner, Digital filtering using logarithmic arithmetic, Electron. Lett. 7, 56–58 (1971).CrossRefGoogle Scholar
  28. 28.
    E. E. Swartzlander, Jr. and A. G. Alexopoulos, The Sign/logarithm number system, IEEE Trans. Comput. C-24, 1238–1242 (1975).MathSciNetCrossRefGoogle Scholar
  29. 29.
    T. Kurokawa, Error Analysis of Digital Filters with Logarithmic Number System, Ph.D. Dissertation, Univ. Oklahoma, Norman (1978).Google Scholar
  30. 30.
    T. Kurokawa, J. A. Payne, and S. C. Lee, Error analysis of recursive digital filters implemented with logarithmic number systems, IEEE Trans. Acoust., Speech, Signal Process. ASSP-28, 706–715 (1980).CrossRefGoogle Scholar
  31. 31.
    F. J. Taylor, Logarithmic Arithmetic Unit for Signal Processing, IEEE Int. Conf. on Acoustics, Speech and Signal Process., Vol. 3, 44.10/1–4 (1984).Google Scholar
  32. 32.
    R. V. Hogg and T. Craig, Introduction to Mathematical Statistics, The MacMillan Company, London (1970).Google Scholar

Copyright information

© Springer Science+Business Media New York 1989

Authors and Affiliations

  • Robert King
    • 1
  • Majid Ahmadi
    • 2
  • Raouf Gorgui-Naguib
    • 3
  • Alan Kwabwe
    • 4
  • Mahmood Azimi-Sadjadi
    • 5
  1. 1.Imperial CollegeLondonEngland
  2. 2.University of WindsorWindsorCanada
  3. 3.University of Newcastle upon TyneNewcastle upon TyneEngland
  4. 4.Imperial College and Bankers Trust CompanyLondonEngland
  5. 5.Colorado State UniversityFort CollinsUSA

Personalised recommendations