Multidimensional Systems and Signal Processing

, Volume 28, Issue 1, pp 305–314 | Cite as

On minimal realizations of first-degree 3D systems with separable denominators

  • Thi Loan Nguyen
  • Li Xu
  • Zhiping Lin
  • David B. H. Tay


In this paper, we focus on first-degree three-dimensional (3D) causal systems which have separable denominators. Gröbner basis is applied to prove that not all first-degree 3D systems with separable denominators have minimal realizations (of order 3). This is in contrast to 2D systems with separable denominators which always admit absolutely minimal realizations. Two illustrative examples are presented.


Multidimensional systems State space realization Minimal realization  Gröbner basis 



We wish to acknowledge the funding support for this project from Nanyang Technological University under the Undergraduate Research Experience on Campus (URECA) program.


  1. Antoniou, G. E., Varoufakis, S. J., & Parakevopoulos, P. N. (1988). Minimal state space realization of factorable 2-D systems. IEEE Transaction on Circuits System, 35, 1055–1058.CrossRefMathSciNetGoogle Scholar
  2. Belcastro, C. M. (1994) Uncertainty modeling of real parameter variations for robust control applications. Ph.D. Thesis, University of Drexel, US, December.Google Scholar
  3. Bose, N. K. (1982). Applied multidimensional systems theory. New York: Van Nostrand Reinhold.MATHGoogle Scholar
  4. Bose, N. K. (2003). Multidimensional systems theory and applications. New York: Springer.Google Scholar
  5. Buchberger, B. (1985). Gröbner bases: an algorithmic method in polynomial ideal theory. In N. K. Bose (Ed.), Multidimensional systems theory: Progress, directions and open problems (pp. 184–232). Dordrecht: Reidel.CrossRefGoogle Scholar
  6. Buchberger, B. (2001). Gröbner bases and systems theory. Multidimensional Systems and Signal Processing, 12(3–4), 223–251.CrossRefMATHMathSciNetGoogle Scholar
  7. Chen, C. T. (1970). Introduction to linear system theory. New York: Holt, Rinehart and Winston.Google Scholar
  8. Decker, W., Greuel, G. M., Pfister, G., & Schönemann, H. (2015). Singular 4-0-2—A computer algebra system for polynomial computations.
  9. Doan, M. L., Nguyen, T. T., Lin, Z., & Xu, L. (2015). Notes on minimal realization of multidimensional systems. Multidimensional Systems and Signal Processing, Special Issue on Symbolic Methods in Multidimensional Systems Theory, 26(2), 519–553.CrossRefMATHMathSciNetGoogle Scholar
  10. Galkowski, K. (2001). State-space realizations of linear 2-D systems with extensions to the general \(n\)-D (\(n>\)2) case. LNCIS: Springer.Google Scholar
  11. Hinamoto, T., Doi, A., & Lu, W. S. (2012). Realization of 3-D separable denominator digital filters with low \(l_2\)-sensitivity. IEEE Transaction on Signal Processing, 60(12), 6282–6293.CrossRefMathSciNetGoogle Scholar
  12. Hinamoto, T., & Fairman, F. W. (1981). Separable-denominator state-space realization of two-dimensional filters using a canonic form. IEEE Transaction on Acoustics Speech, Signal Processing, 29, 846–853.CrossRefMathSciNetGoogle Scholar
  13. Kailath, T. (1980). Linear systems. Upper Saddle River: Prentice-Hall, Inc.MATHGoogle Scholar
  14. Lin, Z., Xu, L., & Bose, N. K. (2008). A tutorial on Gröbner bases with applications in signals and systems. IEEE Transactions on Circuits and Systems I: Regular Papers, 55, 445–461.CrossRefGoogle Scholar
  15. Lu, W. S., & Antoniou, A. (1992). Two-dimensional digital filters. New York City: Marcel Dekker, Inc.MATHGoogle Scholar
  16. Magni, J. F. (2006). Linear fractional representation toolbox for use with matlab.
  17. Pommaret, J.-F. (2015). Relative parametrization of linear multidimensional. Multidimensional Systems and Signal Processing, Special Issue on symbolic methods in multidimensional systems theory, 26(2), 405–437.CrossRefMATHMathSciNetGoogle Scholar
  18. Xu, L., Lin, Z., & Anazawa, Y. (2006). Symbolic computation approach for minimal realization of a class of \(n\)D systems. In Proceedings of ICEMS (2006). Nagasaki, Japan. November, pp. 20–23.Google Scholar
  19. Xu, L., Fan, H., Lin, Z., & Bose, N. K. (2008). A direct-construction approach to multidimensional realization and LFR uncertainty modeling. Multidimensional Systems and Signal Processing, 19(3–4), 323–359.CrossRefMATHMathSciNetGoogle Scholar
  20. Xu, L., Yan, S., Lin, Z., & Matsushita, S. (2012). A new elementary operation approach to multidimensional realization and LFR uncertainty modeling: the MIMO case. IEEE Transaction on Circuits and Systems I, 59(3), 638–651.CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Thi Loan Nguyen
    • 1
  • Li Xu
    • 2
  • Zhiping Lin
    • 3
  • David B. H. Tay
    • 4
  1. 1.School of Physics and Mathematics SciencesNanyang Technological UniversitySingaporeSingapore
  2. 2.Department of Electronics and Information SystemsAkita Prefectural UniversityAkitaJapan
  3. 3.School of EEENanyang Technological UniversitySingaporeSingapore
  4. 4.Department of EngineeringLaTrobe UniversityBundooraAustralia

Personalised recommendations