The structure of state covariances and its relation to the power spectrum of the input

Tryphon T. Georgiou

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68 Scopus citations


We study the relationship between power spectra of stationary stochastic inputs to a linear filter and the corresponding state covariances, and identify the structure of positive-semidefinite matrices that qualify as state covariances of the filter. This structure is best revealed by a rank condition pertaining to the solvability of a linear equation involving the state covariance and the system matrices. We then characterize all input power spectra consistent with any specific state covariance. The parametrization of input spectra is achieved through a relation to solutions of an analytic interpolation problem which is analogous, but not equivalent, to a matricial Nehari problem.

Original languageEnglish (US)
Pages (from-to)1056-1066
Number of pages11
JournalIEEE Transactions on Automatic Control
Issue number7
StatePublished - Jul 2002

Bibliographical note

Funding Information:
Manuscript received February 26, 2001; revised January 15, 2002. Recommended by Associate Editor F. Legland. This work was supported in part by the Air Force Office of Scientific Research, the National Science Foundation. The author is with the Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455 USA (e-mail: Publisher Item Identifier 10.1109/TAC.2002.800643.

Copyright 2012 Elsevier B.V., All rights reserved.


  • Linear filters
  • Spectral analysis

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