Skip to main content
SpringerLink
Log in

Gröbner Bases and Multidimensional FIR Multirate Systems

  • Published:
Multidimensional Systems and Signal Processing Aims and scope Submit manuscript

Abstract

The polyphase representation with respect to sampling lattices in multidimensional (M-D) multirate signal processing allows us to identify perfect reconstruction (PR) filter banks with unimodular Laurent polynomial matrices, and various problems in the design and analysis of invertible MD multirate systems can be algebraically formulated with the aid of this representation. While the resulting algebraic problems can be solved in one dimension (1-D) by the Euclidean Division Algorithm, we show that Gröbner bases offers an effective solution to them in the M-D case.

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. T. Kalker, H. Park, and M. Vetterli, “Groebner Bases Techniques in Multidimensional Multirate Systems,” Proceedings of ICASSP 95, 1995.

  2. M. Vetterli and J. Kovačević, Wavelets and Subband Coding, Prentice Hall Signal Processing Series: Prentice Hall, 1995.

  3. M. Vetterli, “Filter banks allowing perfect reconstruction,” IEEE Transactions on Signal Processing, vol. 10, 1986, pp. 219–244.

    Google Scholar 

  4. A. J. E. M. Janssen, “Note on a linear system occurring in perfect reconstruction,” IEEE Transactions on Signal Processing, vol. 18, no. 1, 1989, pp. 109–114.

    Google Scholar 

  5. M. Vetterli and C. Herley, “Wavelets and filter banks: Theory and design,” IEEE Transactions on ASSP, vol. 40, 1992, pp. 2207–2232.

    Google Scholar 

  6. P. P. Vaidyanathan, Multirate Systems and Filter Banks, Prentice Hall Signal Processing Series: Prentice Hall, 1993.

  7. B. Buchberger, “Gräbner bases—an algorithmic method in polynomial ideal theory,” in Multidimensional systems theory(N. K. Bose, ed.), 1985, pp. 184–232. Dordrecht: D. Reidel.

    Google Scholar 

  8. D. Cox, J. Little, and D. O'shea, Ideals, Varieties, and Algorithms, Undergraduate Texts in Mathematics: Springer-Verlag, 1992.

  9. B. Mishra, Algorithmic Algebra, Texts and Monographs in Computer Science: Springer-Verlag, 1993.

  10. W. Adams and P. Loustaunau, An Introduction to Gröbner Bases, vol. 3 of Graduate Studies in Mathematics. American Mathematical Society, 1994.

  11. T. Becker and V. Weispfenning, Gröbner Bases, vol. 141 of Graduate Texts in Mathematics. Springer-Verlag, 1993.

  12. D. Eisenbud, Introduction to Commutative Algebra with a View Towards Algebraic Geometry, Graduate Texts in Mathematics: Springer-Verlag, 1995.

  13. X. Chen, I. Reed, T. Helleseth, and T. K. Truong, “Use of Gröbner bases to Decode Binary Cyclic Codes up to the True Minimum Distance,” IEEE Transactions on Information Theory, vol. 40, 1994, pp. 1654–1660.

    Google Scholar 

  14. D. Bayer and M. Stillman, “A theorem on redefining division orders by the reverse lexicographic orders,” Duke J. of Math., vol. 55, 1987, pp. 321–328.

    Google Scholar 

  15. D. Bayer and M. Stillman, “On the complexity of computing syzygies,” J. Symb. Comp., vol. 6, 1988, pp. 135–147.

    Google Scholar 

  16. Gruson, Lazarsfeld, and Peskine, “On a theorem of Castelnuovo, and the equations defining space curves,” Invent. Math., vol. 72, 1983, pp. 491–506.

    Google Scholar 

  17. Winkler, “On the complexity of the Gröbner basis algorithm over k[x; y; z],” in Springer Lect. Notes in Computer Sci.174 (J. Fitch, ed.), Springer-Verlag, 1984.

  18. A. Logar and B. Sturmfels, “Algorithms for the Quillen-Suslin theorem,” Journal of Algebra, vol. 145, 1992, pp. 231–239.

    Google Scholar 

  19. H. Park, A Computational Theory of Laurent Polynomial Rings and Multidimensional FIR Systems. PhD thesis, University of California at Berkeley, 1995.

  20. P. P. Vaidyanathan and T. Chen, “Role of anticausal inverses in multirate filter banks—part i: Systemtheoretic fundamentals,” IEEE Transactions on Signal processing, vol. 43, 1995, pp. 1090–1102.

    Google Scholar 

  21. G. Greuel, G. Pfister, and H. Schoenemann, Singular User Manual, University of Kaiserslautern, version 0.90 ed., 1995. (Available by anonymous ftp from helios.mathematik.uni-kl.de.)

  22. H. Park and C. Woodburn, “An Algorithmic Proof of Suslin's Stability Theorem for Polynomial Rings,” Journal of Algebra, vol. 178, pp. 277–298, 1995.

    Google Scholar 

  23. S. Basu and H. M. Choi, “Multidimensional causal, stable, perfect reconstruction filter banks,” Proceedings of the ICIP-94, vol. 1, 1994, pp. 805–809.

    Google Scholar 

  24. Z. Cvetković and M. Vetterli, “Oversampled Filtered Banks,” IEEE Transactions on Signal Processing, submitted.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Oakland University, Rochester, MI, 48309

    Hyungju Park

  2. Philips Research Laboratories, Prof. Holstlaan 4, 5656 AA, Eindhoven, The Netherlands

    Ton Kalker

  3. Department of Electrical Engineering and Computer Sciences, University of California at Berkeley, Berkeley, CA, 94720

    Martin Vetterli

Authors
  1. Hyungju Park
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ton Kalker
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Martin Vetterli
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Park, H., Kalker, T. & Vetterli, M. Gröbner Bases and Multidimensional FIR Multirate Systems. Multidimensional Systems and Signal Processing 8, 11–30 (1997). https://doi.org/10.1023/A:1008299221759

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1008299221759

Search

Navigation