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Simple state-space formulations of 2-D frequency transformation and double bilinear transformation

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Abstract

In this paper, an explicit relationship between the two-dimensional (2-D) frequency transformation and the theory of linear fractional transformation (LFT) representation is shown. Based on this relationship, a simple alternative state-space formulation of 2-D frequency transformation for 2-D digital filters is derived by utilizing the well-known Redheffer star product of LFT representations. The proposed formulation is then utilized to establish a simple relationship between the state-space representations of a 2-D continuous system and a 2-D discrete system which are related by the double bilinear transformation. Moreover, the inherent relations among the proposed formulations and the existing results are discussed. It turns out that all the existing results given in the literature can be unified as special or equivalent cases by the new state-space formulation of 2-D frequency transformation in a very concise and elegant form. Numerical examples are given to illustrate the effectiveness of the proposed formulations.

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References

  • Agathoklis P. (1993) The double bilinear transformation for 2-D systems in state-space description. IEEE Transactions on Signal Processing 41(2): 994–996

    Article  Google Scholar 

  • Agathoklis, P., & Kanellakis, A. (1992). Complex domain transformations for 2-D systems in state-space description. In Proceedings of the 1992 ISCAS, Vol. 2 (pp. 710–713). San Diego, CA.

  • Bose N. K. (1982) Applied multidimensional system theory. Van Nostrand Reinhold, New York

    Google Scholar 

  • Bose N. K., Jury E. I. (1974) Positivity and stability tests for multidimensional filters (discrete–continuous). IEEE Transactions on Acoustics, Speech, and Signal Processing ASSP-22(3): 174–180

    Article  MathSciNet  Google Scholar 

  • Chakrabarti S., Mitra S. K. (1977) Design of two-dimensional digital filters via spectral transformations. Proceedings of the IEEE 65(6): 905–914

    Article  Google Scholar 

  • Cockburn, J. C. (2000). Multidimensional realizations of systems with parametric uncertainty. In Proceedings of MTNS, Perpignan, France, June 2000, Session si20a, 6 pages.

  • Cockburn J. C., Morton B. G. (1997) Linear fractional representations of uncertain systems. Automatica 33: 1263–1271

    Article  MATH  MathSciNet  Google Scholar 

  • Constantinides A. G. (1970) Spectral transformations for digital filters. Proceedings of the Institute Electrical Engineers 117(8): 1585–1590

    Article  Google Scholar 

  • Doyle, J., Packard, A., & Zhou, K. (1991). Review of LFTs, LMI, and μ. In Proceedings of the 30th IEEE conference on decision and control, Vol. 2 (pp. 1227–1232).

  • Golub G. H., Van Loan C. F. (1996) Matrix computations (3rd ed.). Johns Hopkins University Press, Baltimore

    MATH  Google Scholar 

  • Hecker, S. (2007). Generation of low order LFT representations for robust control applications. Fortschrittberichte VDI, series 8, no. 1114. Available at: http://mediatum2.ub.tum.de/doc/601652/601652.pdf.

  • Hecker S., Varga A. (2004) Generalized LFT-based representation of parametric uncertain models. European Journal of Control 4: 326–337

    Article  MathSciNet  Google Scholar 

  • Hecker S., Varga A. (2006) Symbolic manipulation techniques for low order LFT-based parametric uncertainty modelling. International Journal of Control 79(11): 1485–1494

    Article  MATH  MathSciNet  Google Scholar 

  • Hinamoto T., Yokoyama S., Inoue T., Zeng W., Lu W.-S. (2002) Analysis and minimization of L 2-sensitivity for linear systems and two-dimensional state-space filters using general controllability and observability gramians. IEEE Transactions on Circuits and Systems CAS-I-49(9): 1279–1289

    MathSciNet  Google Scholar 

  • Jury E. I., Ruridant T. Y. (1986) On the evaluation of double square integral in the (s 1, s 2) complex biplane. IEEE Transactions on Acoustics, Speech, and Signal Processing ASSP-34(3): 630–632

    Article  MathSciNet  Google Scholar 

  • Kailath T. (1980) Linear systems. Prentice-Hall, USA

    MATH  Google Scholar 

  • Koshita S., Kawamata M. (2004) State-space formulation of frequency transformation for 2-D digital filters. IEEE Signal Processing Letters 11(10): 784–787

    Article  Google Scholar 

  • Koshita S., Kawamata M. (2005) Invariance of second-order modes under frequency transformation in 2-D separable denominator digital filters. Multidimensional Systems and Signal Processing 16(3): 305–333

    Article  MATH  MathSciNet  Google Scholar 

  • Lancaster P., Tismenetsky M. (1985) The theory of matrices (2nd ed.). Academic Press, New York

    MATH  Google Scholar 

  • Lodge J. H., Fahmy M. M. (1982) The bilinear transformation of two-dimensional state-space systems. IEEE Transactions on Acoustics, Speech, and Signal Processing ASSP-30(3): 500–502

    Article  Google Scholar 

  • Lu W.-S., Antoniou A. (1992) Two-dimensional digital filters. Marcel Dekker, New York

    MATH  Google Scholar 

  • Magni, J. F. (2005). User manual of the linear fractional representation toolbox. Version 2, Technical Report. France, October 2005 (Revised Feb 2006). (http://www.cert.fr/dcsd/idco/perso/Magni/toolboxes.html).

  • Matsukawa H., Kawamata M. (2001) Design of variable digital filters based on state-space realizations. IEICE Transactions Fundamentals of Electronics, Communications and Computer Sciences E84-A(8): 1822–1830

    Google Scholar 

  • Mullis C. T., Roberts R. A. (1976) Roundoff noise in digital filters: Frequency transformations and invariants. IEEE Transactions on Acoustics, Speech, and Signal Processing ASSP-24(6): 538–550

    Article  MathSciNet  Google Scholar 

  • Pendergrass N. A., Mitra S. K., Jury E. I. (1976) Spectral transformations for two-dimensional digital filters. IEEE Transactions on Circuits and Systems CAS-23(1): 26–35

    Article  MathSciNet  Google Scholar 

  • Redheffer R. M. (1960) On a certain linear fractional transformation. Journal of Mathematics and Physics 39: 269–286

    MathSciNet  Google Scholar 

  • Roesser R. P. (1975) A discrete state-space model for linear image processing. IEEE Transactions on Automatic Control 20(1): 1–10

    Article  MATH  MathSciNet  Google Scholar 

  • 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

    Article  MathSciNet  Google Scholar 

  • Xu S., Lam J., Zou Y., Lin Z., Paszke W. (2005) Robust H-infinity filtering for uncertain 2-D continuous systems. IEEE Transaction on Signal Processing 53(5): 1731–1738

    Article  MathSciNet  Google Scholar 

  • Yan S., Xu L., Anazawa Y. (2007) A two-stage approach to the establishment of state-space formulation of 2-D frequency transformation. IEEE Signal Processing Letters 14(12): 960–963

    Article  Google Scholar 

  • Zerz E. (1999) LFT representations of parametrized polynomial systems. IEEE Transactions on Circuits and Systems 46(3): 410–416

    Article  MATH  MathSciNet  Google Scholar 

  • Zerz, E. (2000). Topics in multidimensional linear systems theory. Lecture Notes in Control and Information Sciences 256. London: Springer.

  • Zhang F. (2005) The Schur complement and its applications. Springer, New York

    Book  MATH  Google Scholar 

  • Zhou K., Doyle J. C., Glover K. (1996) Robust and optimal control. Prentice Hall, New Jersey

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Electronics and Information Systems, Akita Prefectural University, Akita, 015-0055, Japan

    Shi Yan, Natsuko Shiratori & Li Xu

Authors
  1. Shi Yan
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  2. Natsuko Shiratori
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  3. Li Xu
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Corresponding author

Correspondence to Li Xu.

Additional information

This work was partially supported by JSPS.KAKENHI 19560048.

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Yan, S., Shiratori, N. & Xu, L. Simple state-space formulations of 2-D frequency transformation and double bilinear transformation. Multidim Syst Sign Process 21, 3–23 (2010). https://doi.org/10.1007/s11045-009-0092-5

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  • DOI: https://doi.org/10.1007/s11045-009-0092-5

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