The multi-resolution time domain (MRTD) technique for electromagnetic field equations was proposed by Krumpholz, Katehi et al., using Battle-Lemarie wavelets. The basis principle behind the MRTD is the wavelet-Galerkin time domain (WGTD) approach. Despite its effectiveness in space discretization, the complexity ofthe MRTD makes it unpopular. Recently, the WGTD was significantly simplified by Cheong et al. based on the approximate sampling property ofthe shifted versions ofthe Daubechies compactly supported wavelets. In this paper, we provide a rigorous analysis ofthe MRTD, employing positive sampling functions and their biorthogonal dual. We call our approach as the sampling biorthogonal time-domain (SBTD) technique. The introduced sampling and dual functions are both originated from Daubechies scaling functions of order 2 (referred as to D2), and form a biorthonormal system. This biorthonormal system has exact interpolation properties and demonstrates superiority over the FDTD in terms ofmemory and speed. Numerical examples and comparisons with the traditional FDTD results are provided.
2. Yee, K.S., "Numerical solution ofinitial boundary value problems involving Maxwell's equation in isotropic media," IEEE Trans. Antennas Propagat., Vol. 14, 302-307, 1996.
3. Cheong, Y. W., Y. M. Less, K. H. Ra, J. G. Kang, and C. C. Shin, "Wavelet-Galerkin scheme oftime-dep endent inhomogeneous electromagnetic problems," IEEE Microwave Guided Wave Lett., Vol. 9, 297-299, 1999.
4. Daubechies, I., Ten Lectures on Wavelets, SIAM, 1992.
5. Sweldens, W. and R. Piessens, Wavelet sampling techniques, Proc. Statistical Computing Section, 20-29, 1993.
6. Walter, G. G., Wavelets and Other Orthogonal Systems with Applications, CRC Press, 1994.
7. Tentzeris, E. M., R. L. Robertson, M. Krumpholz, and L. P. B. Katehi, Application ofMR TD to printed transmission lines, Proc. Microwave Theory Tech. Soc., 573-576, 1996.
8. Sheen, D. M., S. M. Ali, M. D. Abouzahra, and J. A. Kong, "Application ofthe three-dimensional finite-difference timedomain method to the analysis ofplanar microstrip circuits," IEEE Trans. Microwave Theory Tech., Vol. 38, 849-857, 1990.
9. Tentzeris, E. M., R. L. Robertson, J. F. Harvey, and L. P. B. Katehi, "PML absorbing boundary conditions for the characterization ofop en microwave circuits using multiresolution time-domain techniques (MRTD)," IEEE Trans. Antenna Propagat., Vol. 47, 1709-1715, 1999.