Affiliations: Istituto per le Applicazioni del Calcolo, CNR, Viale del Policlinico 137, 00161 Roma, Italy
Abstract: The aim of this paper is to review several methods developed in the last ten years to solve problems which can be considered as spectral analysis problems in a general sense. A common denominator of these methods can be found in the framework of Padé approximation theory. It turns out that the asymptotic distribution in the complex plane of the poles of the Padé approximants to the Z-transform of the noisy data provides the key information to solve the original spectral analysis problem. The main features of the distribution of the Padé poles are related to the asymptotic behaviour of a set of polynomials orthogonal in a generalized sense with respect to the unknown spectrum. The novelty of this approach consists in exploiting the asymptotic nature of the relevant information, as expressed by the distribution of the Padé poles in the complex plane, in a statistical fashion. Looking at a suitable average behaviour it is then possible to provide regularized solution to the usually very ill-conditioned spectral analysis problems.
Keywords: Padé approximants, signal processing, superresolution, matrix pencils, orthogonal polynomials, singular integral equations, Cauchy integrals, Riemaim surfaces
