Abstract
The approximation of a given function by a rational function has been considered extensively by mathematicians. A particular result has been stated by Walsh that the best approximation of a given analytical function is one which interpolates the given function at several properly chosen points. In this paper, transfer functions of multidimensional digital filters with separable denominator are used for the approximation of given multivariate functions. It is shown that the result of Walsh can be generalized in a straightforward manner. By an example it is illustrated how the new result can be applied to, e.g., the order reduction of a higher-order system. In the conclusion we state the usefulness and the limitation of the result.
This is a preview of subscription content, log in via an institution to check access.
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
T. Tzafestas,Multidimensional Systems: Techniques and Applications, New York: Marcel Dekker, 1986.
N.K. Bose,Multidimensional Systems: Progress, Direction and Open Problems, Dordrecht, Holland: 1986.
T. Hinamoto, and S. Maekawa, Design of 2-D Separable Denominator Filters using Canonical Local State Space Models,”IEEE Transactions on Circuits and Systems, vol. CAS-33, 1986, pp. 922–926.
S. Kung, B.C. Levy, M. Morf, and T. Kailath, “New Results in 2-D Systems Theory, II,”Proceedings of the IEEE, vol. 65, pp. 945–961.
G. Lashgari, L.M. Silverman, and J.F. Abramatic, “Approximation of 2-D Separable in Denominator Filters,”IEEE Transactions on Circuits and Systems, vol. CAS-30, 1983, pp. 107–121.
P.K. Rajan, and M.N.S. Swamy, “Quadrantal Symmetric Associated with Two-Dimensional Digital Transfer Functions,”IEEE Transactions on Circuits and Systems, vol. CAS-25, 1978, pp. 340–343.
M. Reinhardt, “Entwurf von 2-D Digitalfiltern mit separablem Nenner,” Diploma thesis at Lehrstuhl für Allgemeine und Theoretische Elektrotechnik, University of Erlangen-Nürnberg.
M.Y. Dabbagh, and W.E. Alexander, “Multiprocessor Implementation of 2-D Denominator-Separable Digital Filters for Real Time Processing,”IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. ASSP-37, 1989, pp. 872–881.
M. Kawamata, and T. Higuchi, “Synthesis of 2-D Separable Denominator Digital Filters with Minimum Roundoff Noise and No Overflow Oscillation,”IEEE Transactions on Circuits and Systems, vol. CAS-33, 1986, pp. 365–372.
D. Raghuramireddy, and R. Unbehauen, “A New Realization Structure for Multiprocessor Implementation of 2-D Denominator-Separable Filters,” inProceedings of IEEE Conference ISCAS-91, I, 1991, pp. 472–475.
J.L. Walsh,Interpolation and Approximation by Rational Functions in the Complex Domain, Providence, RI: American Mathematical Society.
W. Rudin,Function Theory in Polydiscs, New York: Benjamin, 1969.
W. Rudin,Real and Complex Analysis, New York: McGraw-Hill, 1966.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Nie, X., Unbehauen, R. Approximation of multivariable functions by M-D transfer functions with separable denominator. Multidim Syst Sign Process 5, 281–288 (1994). https://doi.org/10.1007/BF00980710
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00980710