Transformation methods in signal processing

This dissertation is concerned with the application of the theory of rational functions in signal processing. The PhD thesis summarizes the corresponding results of the author?s research. Since the systems of rational functions are defined by the collection of inverse poles with multiplicities, the following parameters should be determined: the number, the positions and the multiplicities of the inverse poles. Therefore, we develop the hyperbolic variant of the so-called Nelder?Mead and the particle swarm optimization algorithm. In addition, the latter one is integrated into a more general multi-dimensional framework. Furthermore, we perform a detailed stability and error analysis of these methods. We propose an electrocardiogram signal generator based on spline interpolation. It turns to be an efficient tool for testing and evaluating signal models, filtering techniques, etc. In this thesis, the synthesized heartbeats are used to test the diagnostic distortion of the rational representation. In addition, we introduce a mathematical model for the multi-channel QRS complex. We construct a new, generalized time-frequency distribution of signals based on different types of rational functions. This method is applied for the problem of epileptic seizure detection in electroencephalography. The effectiveness of our algorithm is compared to several state-of-the-art feature extraction methods. Numerical methods for computing rational series expansions of signals are also presented in a Matlab toolbox. We provide a fast algorithm to construct discrete rational orthogonal functions. Then, these systems are used to represent discrete-time series which are compared with usual signal compression techniques. The main material of the dissertation has been published in 4 journal and 5 conference papers, and presented at 10 international conferences. The conference papers have appeared in IEEE proceedings.