A survey of spectral factorization methods

Numerical Linear Algebra with Applications

Volumn 8, Issue 6-7, 2001, Pages 467-496

  Sayed, A H ^a   Kailath, T ^b  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b Stanford University   (United States)

Author keywords

Bauer; Causal; Kalman; Levinson; Riccati equation; Schur; Spectral factorization; Wiener Hopf

Indexed keywords

EID: 0035630383     PISSN: 10705325     EISSN: None     Source Type: Journal    
DOI: 10.1002/nla.250     Document Type: Article

References (36)

1. Wiener N, Hopf E. On a class of singular integral equations. Proceedings of the Prussian Acad. Math. - Phys. Ser. 1931; 696.
2. Wiener N. Extrapolation, Interpolation, and Smoothing of Stationary Time Series. Technology Press and Wiley: NY, 1949. [Originally published in 1942 as a classified Nat. Defense Res. Council Report. Also published under the title Time Series Analysis by MIT Press, Cambridge, MA]
3. Doob JL. Stochastic Processes. Wiley: New York, 1953.
4. Grenander U, Szegö G. Toeplitz Forms and their Applications. University of California Press: Berkley, CA, 1958.
5. Bauer FL. Ein direktes iterations verfahren zur Hurwitz-zerlegung eines polynoms. Archiv Elektrische Uebertragung 1955; 9:285-290.
6. Bauer FL. Beiträge zur entwicklung numerischer verfahren für programmgesteuerte rechenanlagen, II. Direkte faktorisierung eines polynoms. Sitzung Bericht Bayerischen Akademische Wissenschaften 1956; 163-203.
7. Youla DC, Kazanjian NN. Bauer-type factorization of positive matrices and the theory of matrix polynomials orthogonal on the unit circle. IEEE Transactions on Circuits and Systems 1978; CAS-25: 57-69.
8. Rissanen J, Kailath T. Partial realization of random systems. Automation 1972; 8:389-396.
9. Schur I. Über potenzreihen die im Inneren des Einheitskreises beschränkt sind. Journal für die Reine und Angewandte Mathematik 1917; 147:205-232. [English translation in Operator Theory: Advances and Applications 1986; 18:31-88, I. Gohberg (ed), Birkhäuser: Boston.]
10. Schur I. Über potenzreihen die im Inneren des Einheitskreises beschränkt sind. Journal für die Reine und Angewandte Mathematik 1917; 147:205-232. [English translation in Operator Theory: Advances and Applications 1986; 18:31-88, I. Gohberg (ed), Birkhäuser: Boston.]
11. Levinson N. The Wiener r.m.s. (root-mean-square) error criterion in filter design and prediction. Journal of Mathematical Physics 1947; 25:261-278.
12. Durbin J. The fitting of time series models. Revue de L'Institute Internationale de Statistique 1960; 28:233-244.
13. Kailath T. Signal processing applications of some moment problems. In Moments in Mathematics, Vol. 37. Landau H (ed), American Mathematical Society, 1987; 71-109.
14. Dewilde P, Dym H. Schur recursions, error formulas and convergence of rational estimators for stationary stochastic sequences. IEEE Transactions on Information Theory 1981; IT-27(4):446-461.
15. Dewilde P, Dym H. Lossless chain scattering matrices and optimum linear prediction: the vector case. Circuit Theory and Applications 1981; 9:135-175.
16. Georgiou TT, Khargonekar PP. Spectral factorization and Nevanlinna-Pick interpolation. SIAM Journal on Control and Optimization 1987; 25:754-766.
17. Georgiou TT, Khargonekar PP. Spectral factorization and matrix valued functions using interpolation theory. IEEE Transactions on Circuits and Systems 1989; 36:568-574.
18. Constantinescu T. Schur Parameters, Factorization, and Dilation Problems. Birkhäuser: Berlin, Germany, 1996.
19. Lev-Ari H, Kailath T. Triangular factorization of structured Hermitian matrices. In Operator Theory: Advances and Applications, Vol. 18. Gohberg I (ed), Birkhäuser: Boston 1986; 301-324.
20. Kailath T., Sayed AH. Displacement structure: theory and applications. SIAM Review 1995; 37:297-386.
21. Kailath T, Sayed AH (eds). Fast Reliable Algorithms for Matrices with Structure. SIAM: PA, 1999.
22. Szegö G. Orthogonal Polynomials. American Mathematical Society Colloquium Publishing, Vol. 23. Providence, RI, 1939.
23. Geronimus LY. Orthogonal Polynomials. Transl. Consultants Bureau, NY, 1961. [Original in Russian, 1958.]
24. Kailath T, Sayed AH, Hassibi B. Linear Estimation. Prentice-Hall: Englewood Cliffs, NJ, 2000.
25. Kailath T. Linear Systems. Prentice-Hall: Englewood Cliffs, NJ, 1980.
26. Anderson BDO, Hitz KL, Diem ND. Recursive algorithm for spectral factorization. IEEE Transactions on Circuits and Systems 1974; 21(6):742-750.
27. Pappas T, Laub AJ, Sandell NR. On the numerical solution of the discrete-time algebraic Riccati equation. IEEE Transactions on Automatic Control 1980; 25:631-641.
28. Bittanti S, Laub AJ, Willems JC. The Riccati Equation. Springer: Berlin, 1991.
29. Patel RV, Laub AJ, Van Dooren P (eds). Numerical Linear Algebra Techniques for Systems and Control. IEEE Press reprint volume. IEEE: NY, 1994.
30. Lancaster P, Rodman L. Algebraic Riccati Equations. Oxford University Press: NY, 1995.
31. Sayed AH, Lev-Ari H, Kailath T. Lattice filter interpretations of the Chandrasekhar recursions for estimation and spectral factorization. In Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing. San Francisco, CA, 1992; 4:549-552.
32. Sayed AH, Kailath T, Lev-Ari H. Generalized Chandrasekhar recursions from the generalized Schur algorithm. IEEE Transactions on Automatic Control 1994: 39:2265-2269.
33. Chang C, Georgiou TT. On a Schur-algorithm based approach to spectral factorization: connection with the Riccati equation. Linear Algebra and its Applications 1992; 171:233-247.
34. ∞ Theories. SIAM: PA, 1999.
35. Wilson G. Factorization of the covariance generating function of a pure moving average process. SIAM Journal on Numerical Analysis 1969; 6:1-7.
36. Goodman TNT, Micchelli CA, Rodriguez G, Seatzu S. Spectral factorization of Laurent polynomials. Advances in Computational Mathematics 1997; 7:429-445.