Applying Hyperbolic Wavelets in Frequency Domain Identification Alexandros Soumelidis 1 ,J´ ozsef Bokor 1 and Ferenc Schipp 2 1 Systems and Control Laboratory, Computer and Automation Research Institute, Budapest, Hungary 2 Department of Numerical Analysis, E¨ otv¨ os Lor´ and University, Budapest, Hungary Keywords: System Identification, Discrete–time Systems, Frequency–domain Representations, Wavelets, Hyperbolic Geometry. Abstract: The paper elaborates a hyperbolic wavelet construction for representing signals in the Hardy space H 2 on the unit disc. An efficient computing scheme based on the matrix form of the representation is worked out. The wavelet coefficients can be computed on the basis of discrete time–domain measurements. This wavelet is used to reconstruct poles of functions in H 2 as the basis of nonparametric frequency–domain identification of discrete–time signals and systems. 1 INTRODUCTION Representations of discrete-time signals and systems in the frequency domain are used in many fields of science and technology, e.g. in detection and changes in systems, system identification, and control design. The stable representations of signals and systems of finite energy result in complex analytic functions de- fined on the unit disc of the Hardy space H 2 . The identification of H 2 signals is usually based on phys- ical measurements in the time–domain. Convenient methods for system identification can be obtained in the case when an orthogonal basis of the space H 2 is used. A well-known orthonormal basis in H 2 is the trigonometric system that forms the basis of classi- cal Fourier–transform representations and associated identification methods. Orthogonal bases can also be generated by rational functions and this concept leads to rational orthogonal bases (ROBs) that have gained great significance besides H 2 also in H ∞ system iden- tification (Heuberger et al., 2005). Application of ROBs requires a priori information on the locations of system poles. This paper elaborates a method to obtain representations of H 2 functions that does not use strict a priori assumptions. A promising opportu- nity to realize this arises from some wavelet-type con- struction that utilize the hyperbolic geometry gener- ated by the so-called Blaschke functions. The goal is to apply hyperbolic wavelet methods to identify poles of functions in H 2 . 2 RATIONAL ORTHOGONAL BASES The Blaschke function in H 2 (D) is defined as B b (z) := z − b 1 − bz (z ∈ C, b ∈ D), where b is called the parameter of the Blaschke- function. The parameter b is identical to the zero and b ∗ = 1/ b is the pole of B b . The most important feature of the Blaschke func- tion is that B b : T → T and B b D → D are bijections, as a consequence the Blaschke functions to be inner functions in the space H 2 (D). The discrete Laguerre-system is complete or- thonormed system in H 2 (D) defined by φ n (z)=  1 −|b| 2 1 − bz B n b (z), (n = 0, 1,...). If the pole locations of the system are exactly known one obtains finite rational representations (Soumelidis et al., 2002b). Rational orthogonal bases have intensively been discussed in the context of H 2 and H ∞ identification of systems (Heuberger et al., 2005), and efficient methods have been elaborated that solved the identification problem in the case when — at least approximately — the pole locations are known. Special attention paid on the problems of pole selection and validation (Bokor et al., 1999; e Silva, 2005) as well as methods have been found to refine the pole locations starting from an approximate place- ment (Soumelidis et al., 2002a), however the general problem identifying poles has not been solved so far. 532 Soumelidis A., Bokor J. and Schipp F.. Applying Hyperbolic Wavelets in Frequency Domain Identification. DOI: 10.5220/0004043705320535 In Proceedings of the 9th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2012), pages 532-535 ISBN: 978-989-8565-21-1 Copyright c  2012 SCITEPRESS (Science and Technology Publications, Lda.)