Acta Applicandae Mathematicae 61: 317–331, 2000. © 2000 Kluwer Academic Publishers. Printed in the Netherlands. 317 On an Alternative Parameterization of the Solutions of the Partial Realization Problem ⋆ TONY VAN GESTEL 1 , MARC VAN BAREL 2 and BART DE MOOR 1 1 Dept. of Electrical Engineering – ESAT/SISTA, K.U. Leuven, Kard. Mercierlaan 94, B-3001 Leuven, Belgium. e-mail: {tony.vangestel, bart.demoor}@esat.kuleuven.ac.be 2 Dept. of Computer Acience – NALAG, K. U. Leuven, Celestijnenlaan 200A, B-3001 Leuven, Belgium. e-mail: marc.vanbarel@cs.kuleuven.ac.be (Received: 30 June 1999) Abstract. The solutions of the partial realization problem have to satisfy a finite number of inter- polation conditions at ∞. The minimal degree of an interpolating deterministic system is called the algebraic degree or McMillan degree of the partial covariance sequence and is easy to compute. The solutions of the partial stochastic realization problem have to satisfy the same interpolation conditions and have to fulfill a positive realness constraint. The minimal degree of a stochastic realization is called the positive degree. In the literature, solutions of the partial realization problem are para- meterized by the Kimura–Georgiou parameterization. Solutions of the partial stochastic realization problem are then obtained by checking the positive realness constraint for the interpolating solutions of the corresponding partial realization problem. In this paper, an alternative parameterization is developed for the solutions of the partial realization problems. Both the solutions of the partial and partial stochastic realization problem are analyzed in this parameterization, while the main concerns are the minimality and the uniqueness of the solutions. Based on the structure of the parameterization, a lower bound for the positive degree is derived. Mathematics Subject Classifications (2000): 93E12, 41A20, 41A05. Key words: rational covariance extension, minimal partial realization, partial stochastic realization, stochastic modeling. 1. Introduction In signal processing, speech processing and system identification, signals can often be modeled as a stationary random sequence generated by passing white noise through a filter or system with a stable transfer function and letting the system come to a statistical steady state [5, 7, 17, 24, 25]. However, before using the filter and the corresponding spectral density e.g. in design processes, the basic inverse problem has to be solved, that is estimating a filter from given data. In practice, ⋆ This work is partially supported by: the Flemish Government: GOA-MIPS; the FWO- Vlaanderen: projects G.0292.95, G.0256.97, G.0278.97, res. comm. ICCoS; the IWT: ITA/GBO/T23, ISIS, EUREKA 1562; the Belgian Fed. Government: IUAP P4-02, IUAP P4-24, Project MD/01/24; the European Commission: TMR Network, Alapedes, SYSIDENT-KIT124.