Abstract
We present the theory leading to a theorem that describes all the rational solutions of the lossless inverse scattering problem (LIS-problem) for lossless networks. They are parametrized by a set of points in the closed unit disc of the complex plane. Quite a few classical problems in estimation theory and network theory may be viewed as a special case of the LIS problem. We present a global method to construct LIS solutions using reproducing kernel Hilbert space methods. Finally, we give connections with applications and with some classical interpolation problems and relate the results to maximum entropy approximation theory.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
J. Schur, “Ueber Potenzreihen, die im Innern des Einheitskreises beschränkt sind”, J. für dei Reine und Angewandte Mathematik, vol. 147, (Berlin), pp. 205–232, 1917.
E.A. Guillemin, “Synthesis of Passive Networks”, J. Wiley, New York, 1957.
V. Belevitch, “Classical Network Theory”, Holden-Day, San Francisco, 1968.
G. Pick, “Ueber die Beschränkungen Analytische Funktionen, welche durch vorgegebene Funktionwerte bewirkt werden”, Math. Ann. 77, 7–23, (1916).
R. Nevanlinna, “Ueber beschrankte Analytische Funktionen”, Ann. Acad.-Sci. Fenn., A32, 1–75 (1929).
C. Carathéodory and L. Fejer, “Ueber den Zusammenhang der Extremen von harmonischen Funktionen mit ihren Koeffizienten und über Picard-Landauschen Satz”, Rend. Circ., Mat. Palermo 32 (1911), 218–239.
G. Szegö, “Orthogonal Polynomials”, New York, 1939.
N.I. Akhiezer, “The Classical Moment Problem”, Oliver and Boyd, Edinburgh, 1965.
L.Ya. Geronimus, “Orthogonal Polynomials”, Consultants Bureau, New York, 1961.
N. Leyinson, “The Wiener rms error criterion in filter design and prediction”, J. Math. Phys., Vol. 25, pp. 261–278, Jan. 1947.
P. Dewilde, A.C. Vieira and T. Kailath, “On a Generalized Szegö-Levinson Realization Algorithm for Optimal Linear Predictors Based on a Network Synthesis Approach”, IEEE Trans. on CAS, Vol. CAS-25, No. 9, Sept. 1978.
S. Darlington, “Synthesis of Reactance 4-poles which produce prescribed insertion loss characteristics”, J. Math. Phys. 18, 257–355 (1939).
P.I. Richards, “A special class of functions with positive real part in a half-plane”, Duke Math. J., vol. 14, no. 3, pp. 777–786, sept. 1947.
O. Brune, “Synthesis of a finite two-terminal network whose driving point impedance is a prescribed function of frequency”, J. Math. Phys. 10 (3), 191–236, (Aug. 1931).
V. Belevitch, “Four dimensional transformations of 4-pole matrices with applications to the synthesis of reactance 4-poles”, I.R.E. Trans, on Circuit Theory, Vol. CT-3, no. 2, pp. 105–111, June 1956.
J. Neirynck and Ph. van Bastelaer, “La Synthese des filtres par factorisation de la matrice de transfert”, Revue MBLE, vol. X, No. 1, pp. 5–32.
P. Dewilde, “Cascade Scattering Matrix Synthesis”, Techn. Rep. Stanford Info. Syst. Lab., June 1970.
P. Dewilde, “Input-Output Description of Roomy Systems”, SIAM J. Opt. and Contr. 14, 712–736, (1976).
V.P. Potapov, “The Multiplicative Structure of J-contractive Matrix Functions”, Amer. Math. Soc. Transl. 15, 131–243 (260).
D. Sarason, “Generalized Interpolants in H∞”, Trans. Amer. Math. Soc, 127, (1967), pp. 179–203.
B. Sz.-Nagy and C. Foias, “Harmonic Analysis of Operators on Hilbert Space”, North Holland, Amsterdam, 1970.
B. Sz.-Nagy and C. Foias, “Sur les contractions de l’espace de Hilbert”, Acta Sci. Math., 19–27.
V.M. Adamyan, D.Z. Arov and M.G. Krein, “Infinite Hankel Matrices and Generalized Caratheodory-Fejer and F. Riesz Problems”, Funkcioncal Anal. i. Prilozen. 2 (1969), no. 1, 1–19= Funct. Anal. Appl. 2 (1968), 1–18.
V.M. Adamyan, D.Z. Arov and M.G. Krein, “Bounded Operators that Commute with a Contraction of Class C of Unit Rank of Nonunitarity” Funkcional Anal. i. Prilozen. 3 (1969), no. 3, 86–87 = Functional Anal. Appl. 3 (1969), 242–243.
V.M. Adamyan and D.Z. Arov, “On Unitary Couplings of Semi-unitary Operators”, Amer. Math. Soc. Transl. (2), Vol. 95, 1970.
V.M. Adamyan, D.Z. Arov and M.G. Krein, “Analytic Properties of Schmidt Pairs for a Hankel Operator and the Generalized Schur-Takagi Problem”, Math. USSR. Sbornik, Vol. 15 (1971), No. 1.
J.W. Helton, “Non-Euclidean Functional Analysis and Electronics”, Bull, of the AMS, Vol. 7, No. 1, July 1982.
J.W. Helton, “Orbit structure of the Möbins transformation semigroup action on H∞ (broadband matching)”, Adv. in Math. Suppl. Studies, vol. 3, Academic Press, New York, 1978, pp. 129–197.
J.W. Helton, “Systems with infinite dimensional state space: the Hilbert Space Approach”, Proc. IEEE 64 (1976), 145–160.
Ph. Delsarte, Y. Genin and Y. Kamp, “Orthogonal Polynomial Matrices on the Unit Circle”, IEEE Trans, on CAS, Vol. 25, pp. 149–160, 1978.
Ph. Delsarte, Y. Genin and Y. Kamp, “Schur Parametrization of Positive-Definite Block-Toeplitz Systems”, SIAM J. on Appl. Math. Vol. 36, pp. 33–46, 1979.
ph. Delsarte, Y. Genin and Y. Kamp, “The Nevanlinna-Pick problem for Matrix-valued functions”, SIAM J. on Appl. Math., Vol. 36, pp. 47–61, 1979.
I.C. Gohberg, “The Problem of Factorization of Operator Functions”, Izvestija Akad. Nauk SSR, Series Math. 28 (1965), 1055–1082, (Russian).
H. Bart, I.C. Gohberg and M.A. Kaashoek, “Minimal Factorization of Matrix and Operator Functions”, Birkhäuser, 1979.
T. Kailath, A. Vieira and M. Morf, “Inverse of Toeplitz Operators, Innovations and Orthogonal Polynomials”, SIAM Rev., Vol. 20, no. 1, pp. 106–119, Jan. 1978.
T. Kailath, “Lectures on Wiener and Kaiman Filtering”, Springer-Verlag, CISM series, 1981.
L. Ljung, T. Kailath and B. Friedlander, “Scattering Theory and Linear Least Squares Estimation”, Part. 1: continuous-time problems”, Proc. IEEE, 64(1), 131–139.
H. Lev-Ari and T. Kailath, “Parametrization and Modeling of Non-stationary Processes”, Techn. Rep. Stanford Univ., 1982.
Y. Genin and P. van Dooren, “On Σ-lossless Transfer Functions and Related Questions”, Rept. R447, Philips Res. Labs., Brussels, Belgium, Nov. 1980.
P. Dewilde and H. Dym, “Schur Recursions, Error Formulas, and Convergence of Rational Estimators for Stationary Stochastic Sequences”, IEEE Trans, on Info. Theory, Vol. IT-27, No. 4, July 1981.
P. Dewilde and H. Dym, “Lossless Chain Scattering Matrices and Optimum Linear Prediction: the Vector Case”, Circuit Theory and Appl., Vol. 9, pp. 135–175 (1981).
P. Dewilde and H. Dym, “Lossless Inverse Scattering with Rational Networks: Theory and Applications”, Techn. Rept., Delft Univ. of Technology, 1982.
T. Kailath, “Lectures on Wiener and Kaiman Filtering”, Lecture Notes CISM, Springer Verlag, 1981.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1984 Springer Basel AG
About this chapter
Cite this chapter
Dewilde, P.M. (1984). The Lossless Inverse Scattering Problem in the Network-Theory Context. In: Dym, H., Gohberg, I. (eds) Topics in Operator Theory Systems and Networks. OT 12: Operator Theory: Advances and Applications, vol 12. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5425-2_4
Download citation
DOI: https://doi.org/10.1007/978-3-0348-5425-2_4
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-5427-6
Online ISBN: 978-3-0348-5425-2
eBook Packages: Springer Book Archive