Extended Fourier analysis of signals

This summary of the doctoral thesis [8] is created to emphasize the close connection of the proposed spectral analysis method with the Discrete Fourier Transform (DFT), the most extensively studied and frequently used approach in the history of signal processing. It is shown that in a typical application case, where uniform data readings are transformed to the same number of uniformly spaced frequencies, the results of the classical DFT and proposed approach coincide. The difference in performance appears when the length of the DFT is selected greater than the length of the data. The DFT solves the unknown data problem by padding readings with zeros up to the length of the DFT, while the proposed Extended DFT (EDFT) deals with this situation in a different way, it uses the Fourier integral transform as a target and optimizes the transform basis in the extended frequency range without putting such restrictions on the time domain. Consequently, the Inverse DFT (IDFT) applied to the result of EDFT returns not only known readings, but also the extrapolated data, where classical DFT is able to give back just zeros, and higher resolution are achieved at frequencies where the data has been successfully extended. It has been demonstrated that EDFT able to process data with missing readings or gaps inside or even nonuniformly distributed data. Thus, EDFT significantly extends the usability of the DFT based methods, where previously these approaches have been considered as not applicable [10-38]. The EDFT founds the solution in an iterative way and requires repeated calculations to get the adaptive basis, and this makes it numerical complexity much higher compared to DFT. This disadvantage was a serious problem in the 1990s, when the method has been proposed. Fortunately, since then the power of computers has increased so much that nowadays EDFT application could be considered as a real alternative.