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Spectral Factorization

In: Time-Varying Systems and Computations

Author

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  • Patrick Dewilde

    (Delft University of Technology, DIMES)

  • Alle-Jan van der Veen

    (Delft University of Technology, DIMES)

Abstract

In this chapter we give a simple and straightforward treatment of the spectral factorization problem of a positive operator Ω∈X into Ω = W*W, where Wi ∈ U is outer. We only consider the case where Ω is a strictly positive operator and where its causal part is bounded and has a u.e. stable realization. This leads to a recursive Riccati equation with time-varying coefficients for which the minimal positive definite solution leads to the outer factor. The theory also includes a formulation of a time-varying (strictly-) positive real lemma. In addition, we provide connections with related problems discussed in previous chapters in which Riccati equations appear as well, such as inner-outer factorization and orthogonal embedding. The results can no doubt be formulated in a more general way where strict positivity is not assumed, but we consider these extensions as laying outside the scope of the book.

Suggested Citation

  • Patrick Dewilde & Alle-Jan van der Veen, 1998. "Spectral Factorization," Springer Books, in: Time-Varying Systems and Computations, chapter 13, pages 363-381, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4757-2817-0_13
    DOI: 10.1007/978-1-4757-2817-0_13
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