As studied by Jaulent in 1982, the inverse problem of lossy electric transmission lines is closely related to the inverse scattering of Zakharov-Shabat equations with two potential functions. Focusing on the numerical solution of this inverse scattering problem, we develop a fast one-shot algorithm based on the Gelfand-Levitan-Marchenko equations and on some differential equations derived from the Zakharov-Shabat equations. Compared to existing results, this new algorithm is computationally more efficient. It is then applied to the synthesis of non uniform lossy electric transmission lines.
2. Lundstedt, J. and M. NorgrenLin, "Comparison between frequency domain and time domain methods for parameter reconstruction on nonuniform dispersive transmission lines," Progress In Electromagnetics Research, Vol. 43, 1-37, 2003.
3. Sheen, D. and D. Shepelsky, "Uniqueness in the simultaneous reconstruction of multiparameters of a transmission line," Progress In Electromagnetics Research, Vol. 21, 153-172, 1999.
4. Bilotti, F., L. Vegni, and A. Toscano, "A new stripline high pass filter layout," Journal of Electromagnetic Waves and Applications, Vol. 14, No. 3, 423-439, 2000.
5. Costanzo, S., "Synthesis of multi-step coplanar waveguide-to-microstrip transition," Progress In Electromagnetics Research, Vol. 113, 111-126, 2011.
6. Monti, G., R. De Paolis, and L. Tarricone, "Design of a 3-state reconfigurable CRLH transmission line based on mems switches," Progress In Electromagnetics Research, Vol. 95, 283-297, 2009.
7. Mao, J. F., O. Wing, and F. Y. Chang, "Synthesis of coupled transmission lines," IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, Vol. 44, No. 4, 327-337, 1997.
8. Jaulent, M., "The inverse scattering problem for LCRG transmission lines," Journal of Mathematical Physics, Vol. 23, No. 12, 2286-2290, 1982.
9. Frangos, P. V. and D. L. Jaggard, "Analytical and numerical solution to the two-potential Zakharov-Shabat inverse scattering problem," IEEE Transactions on Antennas and Propagation, Vol. 40, No. 4, 399-404, 1992.
10. Xiao, G. B. and K. Yashiro, "An improved algorithm of constructing potentials from cauchy data and its application in synthesis of nonuniform transmission lines," IEEE Transactions on Microwave Theory and Techniques, Vol. 50, No. 8, 2025-2028, 2002.
11. Gelfand, I. M. and B. M. Levitan, "On the determination of a differential equation by its spearal function," American Mathematical Society Translations, Vol. 1, 253-304, 1955.
12. Marchenko, V. A., "Reconstruction of the potential energy from the phase of scattered waves," Dokl. Akad. Nauk. SSR, Vol. 104, 635-698, 1955.
13. Frangos, P. V., "One-dimensional inverse scattering: Exact methods and applications," Ph.D. Thesis, University of Pennsylvania, 1986.
14. Frangos, P. V. and D. L. Jaggard, "Inverse scattering: Solution of coupled Geland-Levitan-Marchenko integral equations using successive kernel approximations," IEEE Transactions on Antennas and Propagation, Vol. 43, No. 6, 547-552, 1995.
15. Song, G. H. and S. Y. Shin, "Design of corrugated waveguide filters by the Gelfand-Levitan-Marchenko inverse-scattering method," Journal of the Optical Society of America. A, Optics, Image Science, and Vision, Vol. 2, 1905-1915, 1985.
16. Modelski, J. and A. Synyavskyy, "A new numerical method for Zakharov-Shabat's inverse scattering problem solution," International Conference on Microwaves, Radar and Wireless Communications, Krakow, 2006.
17. Frangos, P. V. and D. L. Jaggard, "A numerical solution to the Zakharov-Shabat inverse scattering problem," IEEE Transactions on Antennas and Propagation, Vol. 39, No. 1, 74-79, 1991.
18. Xiao, G. B. and K. Yashiro, "An e±cient algorithm for solving Zakharov-Shabat inverse scattering problem," IEEE Transactions on Antennas and Propagation, Vol. 50, No. 6, 807-811, 2002.
19. Zhang, Q. H., M. Sorine, and M. Admane, "Inverse scattering for soft fault diagnosis in electric transmission lines," IEEE Transactions on Antennas and Propagation, Vol. 59, No. 1, 141-148, 2011.
20. Tang, H. B. and Q. H. Zhang, "Inverse scattering for lossy electric transmission line soft fault diagnosis," IEEE Antennas and Propagation Society International Symposium, Toronto, 2010.
21. Eckhaus, W. and A. V. Harten, The Inverse Scattering Transformation and the Theory of Solitons: An Introduction, Elsevier, 1981.