This summary of the doctoral thesis 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 to be 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. 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.Comment: 29 pages, 8 figure