On the Factorization of Rational Discrete-Time Spectral Densities

Abstract

In this paper, we consider an arbitrary matrix-valued, rational spectral density P(z) . We show with a constructive proof that P(z) admits a factorization of the form P(z)=W'(z^{-1}) W(z) , where W(z) is stochastically minimal. Moreover, W(z) and its right inverse are analytic in regions that may be selected with the only constraint that they satisfy some symplectic-type conditions. By suitably selecting the analyticity regions, this extremely general result particularizes into a corollary that may be viewed as the discrete-time counterpart of the matrix factorization method devised by Youla in his celebrated work

Parametrization of Minimal Spectral Factors of Discrete-Time Rational Spectral Densities

“…In this paper we provide a parametrization of the set of minimal spectral factors of a discretetime spectral density in terms of the all-pass divisors of an all-pass function (the conjugate phase function). Remarkably, our main theorem applies to general spectral densities and gives an answer to a conjecture of [4]. Moreover, this result is particularly interesting in the light of the recent work [13].…”

“…In particular, our result applies to spectral densities that are rank-deficient and/or possess zeros/poles on the unit circle and/or are improper. The basis of our parametrization is the conjugate phase function, an all-pass function that can be explicitly computed from the minimum-phase spectral factor and the maximum phase unstable spectral factor: these two "extremal spectral factors" can, in turn, be explicitly calculated as discussed in [4] for the input-output representation and in the works by Oarȃ and co-workers [23], [25] for the state-space representation. Therefore, thanks to these contributions, our abstract theoretical parametrization result may indeed be used to explicitly provide all the minimal spectral factors of a given spectral density and hence all the minimal representations of the corresponding process {y(t)}.…”

On the State Space and Dynamics Selection in Linear Stochastic Models: A Spectral Factorization Approach

“…2) There is a one to one correspondence between symmetric solutions of (7) and A-invariant subspaces which is defined by the map assigning to each solution Q the Ainvariant subspace ker(Q). 3 Any solution P can actually be seen as the difference say X − X 0 of two arbitrary solutions of an equivalent Riccati equation parametrizing the minimal spectral factors which is defined directly in terms of a realization of Φ and does not involve a reference spectral factor, see [11,Sect. 16.5].…”

“…see [2], [3] and references therein. 3) Fix a minimal realization W (z) = C(zI − A) −1 B + D to provide a parametrization of the model (1).…”

On Minimal Spectral Factors With Zeroes and Poles Lying on Prescribed Regions

“…and was left open as a conjecture in [2]. Our proof makes use of a very elegant and profound parametrization of rational all-pass functions established by Alpay and Gohberg in [1].…”