Abstract
This paper presents the theoretical foundations of a new method for the discrete simulation of multidimensional systems, which are described by linear partial differential equations with constant coefficients. It is based on methods customary in linear systems theory and digital signal processing and uses a frequency-domain representation of the continuous system to be simulated. The selection of appropriate functional transformations for each variable yields an exact treatment of initial and boundary conditions. The heat-flow equation is treated as an example. For this case, a realizing structure for the simulating discrete system is given along with simulation examples.
Similar content being viewed by others
References
Th. Meis and U. Marcowitz,Numerical Solution of Partial Differential Equations, New York: Springer Verlag, 1978.
A. Fettweis and G. Nitsche, “Transformation Approach to Numerically Integrating PDEs by Means of WDF Principles,”Multidimensional Systems and Signal Processing, vol. 2, 1991, pp. 127–159.
A. Fettweis and G. Nitsche, “Numerical Integration of Partial Differential Equations Using Principles of Multidimensional Wave Digital Filters,”Journal of VLSI Signal Processing, vol. 3, 1991, pp. 7–24.
C.-C.J. Kuo and B.C. Levy “Discretization and Solution of Elliptic PDEs — A Digital Signal Processing Approach,”Proc. IEEE, vol. 78, no. 12, December 1990, pp. 1808–1842.
R. Rabenstein, “A Signal Processing Approach to the Digital Simulation of Multi-Dimensional Continuous Systems,” inProc. Eur. Signal Processing Conf., The Hague, Netherlands, 1986, pp. 665–668.
R. Rabenstein, “Simulation of Distributed Linear Systems,” inProc. Eurosim Simulation Congress, Capri, Italy, September/October 1992.
H. W. Schüssler, A Signalprocessing Approach to Simulation,Frequenz, vol. 35, 1981, pp. 174–184.
R. Rabenstein, “Diskrete Simulation linearer mehrdimensionaler kontinuierlicher Systeme,” Ph.D. dissertation (in German), University Erlangen-Nürnberg, Germany, 1991.
G. Doetsch, “Integration von Differentialgleichungen vermittels der endlichen Fourier transformation,”Math. Annalen, vol. 112, 1936, pp. 52–68.
F. Oberhettinger and L. Badii,Tables of Laplace-Transforms, Berlin and New York: Springer Verlag, 1973.
A. Papoulis,Signal Analysis, New York: McGraw-Hill, 1977.
J. Gosse,Technical Guide to Thermal Processes, Cambridge: Cambridge University Press, 1986.
R. Rabenstein, “Design of FIR Digital Filters with Flatness Constraints for the Error Functions,”Circuits, Systems and Signal Processing, to appear.
R. Rabenstein, “A Signal Processing Approach to the Numerical Solution of Parabolic Differential Equations,” inProc. Second European Conf. Multigrid Methods, GMD-Study, no. 110, October 1985, pp. 133–144.
A.V. Oppenheim and A.S. Willsky,Signals and Systems, Englewood Cliffs, NJ: Prentice-Hall, 1983.
R. Sauer and I. Szabó,Mathematische Hilfsmittel des Ingenieurs Teil I, Berlin: Springer-Verlag, 1967.
R. Bracewell,The Fourier-Transform and its Applications, New York: McGraw-Hill, 1978.
A. Papoulis,The Fourier-Integral and its Applications, New York: McGraw-Hill, 1962.
H.W. Schüssler,Netzwerke, Signale und Systeme 2, Berlin and New York: Springer-Verlag, 3rd edition, 1991.
E.I. Jury,Theory and Application of the z-Transform-Method, New York: Wiley, 1964.
I.N. Sneddon,The Use of Integral Transforms, New York: McGraw-Hill, 1972.
R. Rabenstein, “Simulation of Linear Continuous Systems with Distributed Parameters,”Simulation Practice and Theory, to appear.
H. Krauss, R. Rabenstein, “Numerical Solution of the Telegraph Equation by Multidimensional Filters,” in A.G. Constantides et al. (eds.)Proc. DSP & CAES Conf., Nicosia, Cyprus, 1993, pp. 170–175.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Rabenstein, R. Discrete simulation of linear multidimensional continuous systems. Multidim Syst Sign Process 5, 7–40 (1994). https://doi.org/10.1007/BF00985861
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00985861