Adaptive Rational Transformations in Biomedical Signal Processing Gerg˝ o Bognár, Sándor Fridli, Péter Kovács, and Ferenc Schipp Abstract In this paper we provide a summary on our recent research activity in the field of biomedical signal processing by means of adaptive transformation methods using rational systems. We have dealt with several questions that can be efficiently treated by using such mathematical modeling techniques. In our constructions the emphasis is on the adaptivity. We have found that a transformation method that is adapted to the specific problem and the signals themselves can perform better than a transformation of general nature. This approach generates several mathematical challenges and questions. These are approximation, representation, optimization, and parameter extraction problems among others. In this paper we give an overview about how these challenges can be properly addressed. We take ECG processing problems as a model to demonstrate them. 1 Introduction Mathematical transformation methods have a long history in signal processing, here we mention only the trigonometric Fourier-system, the wavelets, and other orthogonal systems, like the Hermite or Walsh-system. Although these methods perform generally well, their flexibility and adaptivity is usually limited. Our focus is on the adaptive transformations, where the underlying function systems have free parameters that can be adapted to the specific problem and to the signals themselves. We expect such an adapted method to provide a simpler and more concise representation for the signals that still captures the relevant behavior of them. Our approach is to perform an adaptive transformation by means of rational functions [8]. The rational systems are especially flexible and adaptive, we have G. Bognár () · S. Fridli · P. Kovács · F. Schipp Department of Numerical Analysis, Faculty of Informatics, ELTE Eötvös Loránd University, Budapest, Hungary e-mail: bognargergo@caesar.elte.hu; fridli@inf.elte.hu; kovika@inf.elte.hu; schipp@numanal.inf.elte.hu © Springer Nature Switzerland AG 2019 I. Faragó et al. (eds.), Progress in Industrial Mathematics at ECMI 2018, Mathematics in Industry 30, https://doi.org/10.1007/978-3-030-27550-1_30 239