Spectral Factorization

Patrick Dewilde and Alle-Jan van der Veen
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Patrick Dewilde: Delft University of Technology, DIMES
Alle-Jan van der Veen: Delft University of Technology, DIMES

Chapter 13 in Time-Varying Systems and Computations, 1998, pp 363-381 from Springer

Abstract: 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.

Keywords: Riccati Equation; Lyapunov Equation; Embedding Problem; Spectral Factorization; Nest Algebra (search for similar items in EconPapers)
Date: 1998
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4757-2817-0_13

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DOI: 10.1007/978-1-4757-2817-0_13

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