TELECOMMUNICATIONS AND RADIO ENGINEERING - 2011 Vol. 70,
No 10 		 

 

 

 

PROJECTION APPROXIMATIONS TO THE MATRIX SCATTERING OPERATORS AND RELATIVE CONVERGENCE PHENOMENON



I.V. Petrusenko & Yu.K. Sirenko
A. Usikov Institute of Radio Physics and Electronics,
National Academy of Sciences of Ukraine
12, Academician Proskura St., Kharkiv 61085, Ukraine
E-mail: yks@ire.kharkov.ua

Abstract
A matrix-operator model of the mode-matching technique is examined for a scalar problem of mode diffraction on abrupt discontinuity in the waveguide. The convergence of approximations, which are found via a truncation procedure, to the matrix operators of mode reflection and transmission is studied analytically. Various estimates for these projection approximations are obtained. On the basis of these results, the unconditional and relative convergence phenomena are analyzed. The impact of the “Mittra rule” upon the rate of convergence is validated analytically. The uniform-convergence condition for scattering operator approximations is stated. Properties of the condition number for the truncated matrix-operator equation are discussed. The results thus obtained may well be used to substantiate a highly practically class of mathematical models related to the mode-matching technique.
KEY WORDS: mode-matching technique, truncation procedure, Cayley transform

References

  1. Petrusenko, I.V., (2004), Analytic – numerical analysis of waveguide bends, Electromagnetics, 24:237-254.
  2. Petrusenko, I.V., (2004), Matrix operator technique for analysis of wave transformers, Proc. 10th  Int. Conf. on Mathematical Methods in Electromagnetic Theory (MMET’04), Dnipropetrovs’k,
    pp. 118-120.
  3. Petrusenko, I.V. and Sirenko, Yu.K., (2011), Fresnel formulae for scattering operators, Telecommunications and Radio Engineering, 70(9):749-758.
  4. Petrusenko, I.V. and Sirenko, Yu.K., (2009), Generalization of the power conservation law for scalar mode-diffraction problems, Telecommunications and Radio Engineering, 68(16):1399-1410.
  5. Mittra, R. and Lee, S.W., (1971), Analytical techniques in the theory of guided waves, New York, The Macmillan Company, - 327 p.
  6. Gohberg, I.C. and Fel’dman, I.A., (2006), Convolution equations and projection methods for their solution (Translations of mathematical monographs), Providence, American Mathematical Society,
    - 261 p. (Translated from Russian).
  7. Trenogin, V.A., (2002), Functional analysis, Phizmathlit, Moscow: 488 p. (in Russian).
  8. Weyl, H., (1997), The classical groups: Their invariants and representations, Chichester, Princeton University Press, - 316 p.
  9. Richtmyer, R.D., (1978), Principles of advanced mathematical physics. Vol. 1,New York - Heidelberg - Berlin, Springer-Verlag, - 422 p.
  10. Petrusenko, I.V. and Dmitryuk, S.G., (1986), The method of solution of the generalized waveguide bifurcation problem, Padiotekhnika i Elektronika, 31(7):1285-1293 (in Russian).
  11. Gribanov, Yu.I., (1959), On the complete continuity of matrix operators in vector spaces, Doklady Akademii Nauk SSSR, 129(5):975-978 (in Russian).


pages 843-856

Back