Theory and applications of generalized Taylor series in signal processing

This research deals with an alternative expansion for functions called the Bu rmann expansion. This expansion, based on the generalized Taylor series expansion, provides a different class of orthonormal basis functions. Fourier series is an example of an expansion in which a periodic function is represented using sines and cosines. There are other expansions which use different classes of functions for their representations. For example, Taylor series utilizes polynomials for its expansion. The central notion investigated in this work is whether it is possible to expand a function using just about any other function. The answer to the aforementioned question lies within the theory of generalized Taylor series (Chapter 2). Several new results and examples are given showing the capabilities of this class of expansion for applications in parametric estimation and exponential approximation. This expansion is also shown to be useful in representing dilated and translated versions of a signal. After a discussion of time-frequency decomposition in Chapter 3, the interest is then turned to a new application of linear random search and linear prediction coding to a speech signal. This result will illustrate a new method of signal compression. Chapter 4 contains mathematical foundations needed to use the linear random search algorithm. In addition, the linear random search algorithm and the generalized Taylor series are applied to a voice signal and the results are presented (Chapter 5). The results are then compared to a model obtained by the linear prediction coding. Chapter 6 presents a summary of results and some concluding remarks. The results of this study indicate that the linear prediction coding scheme is capable of representing speech signals better than the generalized Taylor series. The generalized Taylor series, however, is preferred to the ordinary Taylor series since the former allows a selection of the root function.