Towards a framework for continuous and discrete multidimensional systems
Continuous multidimensional systems described by partial differential equations can be represented by discrete systems in a number of ways. However, the relations between the various forms of continuous, semi-continuous, and discrete multidimensional systems do not fit into an established framework like in the case of one-dimensional systems. This paper contributes to the development of such a framework in the case of multidimensional systems. First, different forms of partial differential equations of physics-based systems are presented. Secondly, it is shown how the different forms of continuous multidimensional systems lead to certain discrete models in current use (finite-difference models, multidimensional wave digital filters, transfer function models). The links between these discrete models are established on the basis of the respective continuous descriptions. The presentation is based on three examples of physical systems (heat flow, transmission of electrical signals, acoustic wave propagation).
- Churchill R.V. (1972): Operational Mathematics. -New York: McGraw-Hill, (3rd Ed).
- Courant R. and Hilbert D. (1968): Methoden der Mathematischen Physik II. -Berlin: Springer.
- Fettweis A. (1986): Wave digital filters: Theory and practice. - Proc. IEEE, Vol. 74, No. 2, pp. 270-327.
- Fettweis A. (1994): Multidimensional wave-digital principles: From filtering to numerical integration. - Proc. Int. Conf. Acoustics, Speech, and Signal Processing, ICASSP 94, Adelaide, pp. VI-173-VI-181.
- Fettweis A. (1999): New results in numerically integrating PDEs by the wave digital approach. - Proc. Int. Symp. Circuits and Systems, ISCAS 99, Orlando, pp. V-17-V-20.
- Fettweis A. and Nitsche G. (1991): Numerical integration of partial differential equations using principles of multidimensional wave digital filters. - J. VLSI Signal Process., Vol. 3, pp. 7-24.
- Gregor J. (1998): The Cauchy problem for partial difference equations. - Acta Applicandae Mathematicae, Vol. 53, pp. 247-263.
- Kowalczuk Z. (1993): Discrete approximation of continuous-time systems: A survey. - IEE Proc.-G, Vol. 140, No. 4, pp. 264-278.
- Korner T.W. (1988): Fourier Analysis. -Cambridge: Cambridge University Press.
- Kraus H. (1996): Simulation of coupled transmission lines by multidimensional wave digital filters. - Proc. Int. Conf. Acoustics, Speech, and Signal Processing, ICASSP 96, Atltanta, pp. III-1747-III-1750.
- Rabenstein R. (1998): Transfer function models for multidimensional systems with bounded spatial domains. - Math. Comput. Modell. Dynam. Syst., Vol. 5, No. 3, pp. 259-278.
- Rabenstein R. and Trautmann L. (1999): Solution of vector partial differential equations by transfer function models. - Proc. Int. Symp. Circuits and Systems, ISCAS 99, Orlando, pp. V-21-V-24.
- Rabenstein R. and Zayati A. (2000): Sound field simulation by computational acoustics, Part I: Simulation algorithm. - Int. J. Adapt. Contr. Signal Process., Vol. 14, pp. 663-680.
- Roesser R.P. (1975): A discrete state-space model for linear image processing. - IEEE Trans. Automat. Contr., Vol. AC-20, No. 1, pp. 1-75.
- Rogers E., Gałkowski K. and Owens D.H. (1997): Control systems theory for linear repetitive processes. - Appl. Math. Comp. Sci., Vol. 7, No. 4, pp. 737-774.
- Schetelig Th. and Rabenstein R. (1998): Simulation of three-dimensional sound propagation with multidimensional wave digital filters. - Proc. Int. Conf. Acoustics, Speech, and Signal Processing, ICASSP 98, Seattle, pp. 3537-3540.
- Tveito A. and Winther R. (1998): Introduction to Partial Differential Equations. -New-York: Springer.
- Veit J. (1996): Boundary value problems for partial difference equations. - Multidim. Syst. Signal Process., Vol. 7, pp. 113-134.