# A homotopy approach to rational covariance extension with degree constraint

International Journal of Applied Mathematics and Computer Science (2001)

- Volume: 11, Issue: 5, page 1173-1201
- ISSN: 1641-876X

## Access Full Article

top## Abstract

top## How to cite

topEnqvist, Per. "A homotopy approach to rational covariance extension with degree constraint." International Journal of Applied Mathematics and Computer Science 11.5 (2001): 1173-1201. <http://eudml.org/doc/207550>.

@article{Enqvist2001,

abstract = {The solutions to the Rational Covariance Extension Problem (RCEP) are parameterized by the spectral zeros. The rational filter with a specified numerator solving the RCEP can be determined from a known convex optimization problem. However, this optimization problem may become ill-conditioned for some parameter values. A modification of the optimization problem to avoid the ill-conditioning is proposed and the modified problem is solved efficiently by a continuation method.},

author = {Enqvist, Per},

journal = {International Journal of Applied Mathematics and Computer Science},

keywords = {stochastic realization theory; continuation method; ARMA model design; optimization; rational covariance extension problem; ARMA-model design; degree constraint; ill-conditioning},

language = {eng},

number = {5},

pages = {1173-1201},

title = {A homotopy approach to rational covariance extension with degree constraint},

url = {http://eudml.org/doc/207550},

volume = {11},

year = {2001},

}

TY - JOUR

AU - Enqvist, Per

TI - A homotopy approach to rational covariance extension with degree constraint

JO - International Journal of Applied Mathematics and Computer Science

PY - 2001

VL - 11

IS - 5

SP - 1173

EP - 1201

AB - The solutions to the Rational Covariance Extension Problem (RCEP) are parameterized by the spectral zeros. The rational filter with a specified numerator solving the RCEP can be determined from a known convex optimization problem. However, this optimization problem may become ill-conditioned for some parameter values. A modification of the optimization problem to avoid the ill-conditioning is proposed and the modified problem is solved efficiently by a continuation method.

LA - eng

KW - stochastic realization theory; continuation method; ARMA model design; optimization; rational covariance extension problem; ARMA-model design; degree constraint; ill-conditioning

UR - http://eudml.org/doc/207550

ER -

## References

top- Allgower E.L. and Georg K. (1990): Numerical Continuation Methods. — Berlin, New York: Springer.
- Allgower E.L. and Georg K. (1993): Continuation and path following. — Acta Numerica, Vol.2, pp.1–64. Zbl0792.65034
- Arnold V.I. (1983): Geometrical Methods in the Theory of Ordinary Differential Equations. — New York, Berlin: Springer. Zbl0507.34003
- Bauer F.L. (1955): Ein direktes iterationverfahren zur Hurwitz-zerlegung eines polynoms. — Arch. Elek. Ubertragung, Vol.9, pp.285–290.
- Byrnes C.I., Enqvist P. and Lindquist A. (2001): Cepstral coefficients, covariance lags and pole-zero models for finite data strings. — IEEE Trans. Sign. Process, Vol.49, No.4.
- Byrnes C.I., Gusev S.V. and Lindquist A. (1999): A convex optimization approach to the rational covariance extension problem. — SIAM J. Contr. Optim., Vol.37, No.1, pp.211– 229. Zbl0947.30027
- Byrnes C.I., Lindquist A., Gusev S.V. and Matveev A.S. (1995): A complete parametrization of all positive rational extensions of a covariance sequence. — IEEE Trans. Automat. Contr., Vol.40, No.11, pp.1841–1857. Zbl0847.93008
- Caines P.E. (1987): Linear Stochastic Systems. — New York: Wiley.
- Chui C.K. and Chen G. (1992): Signal Processing and Systems Theory. — Berlin: Springer. Zbl0824.93002
- Davidenko D. (1953): On a new method of numerically integrating a system of nonlinear equations. — Dokl. Akad. Nauk SSSR, Vol.88, pp.601–604 (in Russian).
- Den Heijer C. and Rheinboldt W.C. (1981): On steplength algorithms for a class of continuation methods. — SIAM J. Numer. Anal., Vol.18, No.5, pp.925–948. Zbl0472.65042
- Georgiou T.T. (1983): Partial Realization of Covariance Sequences. — Ph.D. Thesis, University of Florida.
- Georgiou T.T. (1987): Realization of power spectra from partial covariance sequences. — IEEE Trans. Acoust. Speech Sign. Process., Vol.ASSP–35, No.4, pp.438–449. Zbl0653.93060
- Goodman T., Michelli C., Rodriguez G. and Seatzu S. (1997): Spectral factorization of Laurent polynomials. — Adv. Comp. Math., Vol.7, No.4, pp.429–454. Zbl0886.65013
- Kalman R.E. (1981): Realization of covariance sequences. — Toeplitz Memorial Conference, Tel Aviv, Israel, pp.331–342.
- Luenberger D.G. (1984): Linear and Nonlinear Programming. — Reading, Mass.: Addison Wesley. Zbl0571.90051
- Markel J.D. and Gray Jr. A.H. (1976): Linear Prediction of Speech. — New York: Springer. Zbl0443.94002
- Nash S.G. and Sofer A. (1996): Linear and Nonlinear Programming. — New York: McGrawHill.
- Ortega J.M. and Rheinboldt W.C. (1970): Iterative Solution of Nonlinear Equations in Several Variables. — New York: Academic Press. Zbl0241.65046
- Porat B. (1994): Digital Processing of Random Signals, Theory & Methods. — Englewood Cliffs. NJ.: Prentice Hall.
- Rudin W. (1976): Principles of Mathematical Analysis. — New York: McGraw Hill. Zbl0346.26002
- Wilson G. (1969): Factorization of the covariance generating function of a pure moving average process. — SIAM J. Numer. Anal., Vol.6, pp.1–7. Zbl0176.46401
- Wu S-P., Boyd S. and Vandenberghe L. (1997): FIR filter design via spectral factorization and convex optimization, In: Applied Computational Control, Signal and Communications (Biswa Datta, Ed.) — Boston: Birkhäuser, pp.215–245. Zbl0963.93026

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.