Approximation based on orthogonal and almost orthogonal functions |
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| Authors: | Dragan Anti? Bratislav Dankovi? Sa?a Nikoli? Marko Milojkovi? Zoran Jovanovi? |
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| Institution: | 1. NASA Langley Research Center, Hampton, VA 23681, USA;2. INRIA Bordeaux Sud-Ouest and Institut de Mathématiques de Bordeaux, 33405 Talence Cedex, France;3. National Institute of Aerospace, Hampton, VA 23666, USA;1. Max Planck Institute for Mathematics in the Sciences, Germany;2. Wesleyan University, United States of America;3. Department of Mathematics, University of Ljubljana, Slovenia;1. The Center of Advanced Medical Engineering and Informatics, Osaka University, 560-8531 Osaka, Japan;2. Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, 560-0043 Osaka, Japan;3. Department of Mechanical Science and Bioengineering Graduate School of Engineering Science, Osaka University, 560-8531 Osaka, Japan |
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| Abstract: | In this paper, we define a class of almost orthogonal rational functions of Legendre type in a new manner. Relations of these functions with classical exponentional functions orthogonal over interval (0, ∞), as well as classical polynomials orthogonal over (0, 1) are explained. Defining relations of these functions can be used for designing almost orthogonal filters. These filters are generators of orthogonal signals and can be successfully applied in finding the best signal approximation in the sense of the mean square error. The filters orthogonal property enables building of physical (in this case electrical) models of dynamical systems (the sources of signals to be approximated) either with less components for the same model accuracy or higher accuracy for the same number of components than the other known models. New filters represent further improvement of previously designed filters, by the same authors, in the sense of simplicity, higher accuracy, lesser approximation time and even a possibility to approximate signals generated by systems with built-in imperfections. Series of experiments were performed to analyze the dependence of approximation accuracy and the number of filters sections. |
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