Abstract
This paper discusses the invariance of the second-order modes of 2-D separable denominator digital filters under frequency transformation. This paper first derives a state-space description for 2-D digital filters obtained by 2-D frequency transformation and then represents the controllability Gramians and the observability Gramians of the transformed 2-D digital filters. This description proves that the second-order modes of 2-D separable denominator digital filters are not invariant under all frequency transformations, but invariant under specific frequency transformations of which transfer functions are strictly proper. These frequency transformations include lowpass-bandpass and lowpass-bandstop frequency transformations keeping the same bandwidth as that of the prototype lowpass filter. It is also shown that the horizontal second-order modes are invariant when only horizontal transformation is applied, and the vertical second-order modes are invariant when only vertical transformation is applied. This paper further remarks significance brought by the invariance of the second-order modes under frequency transformation.
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References
C.T. Mullis R.A. Roberts (December 1976) ArticleTitle“Roundoff Noise in Digital Filters: Frequency Transformations and Invariants” IEEE Trans. Acoust., Speech, Signal Processing ASSP-24 IssueID6 538–550
M. Kawamata T. Higuchi (April 1986) ArticleTitle“Synthesis of 2-D Separable Denominator Digital Filters with Minimum Roundoff Noise and no Overflow Oscillations” IEEE Transactions Circuits and Systems CAS-33 IssueID4 365–372
M. Kawamata, T. Lin, and T. Higuchi, “Minimization of Sensitivity of 2-D State-space Digital Filters and its Relation to 2-D Balanced Realizations,” Proceedings of IEEE International Symposium on Circuits and Systems, 1987, pp. 710–713.
T. Lin M. Kawamata T. Higuchi (July 1986) ArticleTitle“A Unified Study on the Roundoff Noise in 2-D State Space Digital Filters” IEEE Transactions Circuits and Systems 33 IssueIDCAS-7 724–730
W.-S. Lu A. Antoniou (October 1986) ArticleTitle“Synthesis of 2-D State-Space Fixed-point Digital Filter Structures with Minimum Roundoff noise” IEEE Transactions Circuits and Systems CAS-33 IssueID10 965–973
S.Y. Hwang (August 1977) ArticleTitle“Minimum Uncorrelated Unit Noise in State-space Digital Filtering” IEEE Transactions Acoust., Speech, Signal Processing ASSP-25 IssueID4 273–281
T. Lin M. Kawamata T. Higuchi (February 1987) ArticleTitle“Design of 2-D Digital Filters with an Arbitrary Response and no Overflow Oscillations Based on a New Stability Condition” IEEE Transactions Circuits and Systems CAS-34 IssueID2 113–126
W.-S. Lu A. Antoniou (1992) Two-Dimensional Digital Filters Marcel Dekker Newyork
M. Kawamata T. Higuchi (1985) ArticleTitle“A Unified Approach to the Optimal Synthesis of Fixed-point State-space Digital Filters” IEEE Transactions Acoust., Speech, Signal Processing ASSP-33 911–920
T. Lin M. Kawamata T. Higuchi (October 1987) ArticleTitle“Minimization of sensitivity of 2-D Systems and its Relation to 2-D Balanced Realizations,” The Transactions of the Institute of Electronics, Information and Communication Engineers E70 IssueID10 938–944
B.C. Moore (February 1981) ArticleTitle“Principal Component Analysis in Linear Systems: Controllability, Observability, and Model Reduction” IEEE Transactions Automatic Control AC-26 IssueID1 17–32
L. Pernebo L.M. Silverman (April 1982) ArticleTitle“Model Reduction via Balanced State Space Representations” IEEE Transactions Automatic Control AC-27 IssueID2 382–387
B. Lashgari L.M. Silverman J.F. Abramatic (February 1983) ArticleTitle“Approximation of 2-D Separable in Denominator Filters” IEEE Transanctions Circuits and Systems CAS-30 IssueID2 107–121
H. Luo W.-S. Lu A. Antoniou (1995) ArticleTitle“A Weighted Balanced Approximation for 2-D Discrete Systems and its Application to Model Reduction” IEEE Transactions Circuits and Systems-I Fundamentals, Theory and Applications CAS-I-42 419–429
G. Obinata B. Anderson (2000) Model Reduction for Control System Design Springer Verlag Berlin
T. Hinamoto S. Maekawa (1982) ArticleTitle“Canonic Form State-space Realization of Two Dimensional Transfer Functions Having Separable Denominator” Inter Journal Systems Science 13 1083–1091
T. Hinamoto S. Yokoyama T. Inoue W. Zeng W.-S. Lu (September 2002) ArticleTitle“Analysis and Minimization of L2-sensitivity for Linear Systems and Two-dimensional State-space Filters using General Controllability and Observability Gramians” IEEE Transactions Circuits and Systems CAS-I-49 IssueID9 1279–1289 Occurrence HandleMR1930661
A.G. Constantinides (1970) ArticleTitle“Spectral Transformations for Digital Filters” Proceedings of IEE 117 IssueID8 1585–1590
N.A. Pendergrass S.K. Mitra E.I. Jury (January 1976) ArticleTitle“Spectral transformations for two-dimensional digital filters” IEEE Transactions Circuits and Systems CAS-23 IssueID1 26–35
D. Dudgeon R.M. Mersereau (1976) Multidimensional Digital Signal Processing Prentice-Hall NJ
Editor-in Chief D. Zwillinger, Standard Mathematical Tables and Formulae, Boca Raton: CRC Press, 1996.
H. Matsukawa M. Kawamata (August 2001) ArticleTitle“Design of Variable Digital Filters Based on State-space Realizations” IEICE Transactions Fundamentals of Electronics, Communications and Computer Sciences E84-A IssueID8 1822–1830
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First Online Version Published in July, 2005
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Koshita, S., Kawamata, M. Invariance of Second-Order Modes under Frequency Transformation in 2-D Separable Denominator Digital Filters. Multidim Syst Sign Process 16, 305–333 (2005). https://doi.org/10.1007/s11045-004-1679-5
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DOI: https://doi.org/10.1007/s11045-004-1679-5