Lokale Frequenzanalyse mittels Hilbert- und Wavelettransformation

The topic of this thesis is the assignment of local frequencies to a given signal. Two different methods are examined, the assignment through local maxima of the wavelet transform and the assignment through the analytic signal/Hilbert transform approach, i.e. the instantaneous frequency. Especially in the case of the analytic signal it is not clear when the assignment of the frequencies coincides with the physical interpretation. Therefore it is analyzed under which conditions the analytic signal delivers local frequencies in a physical sense. Additionally the weaknesses of the assignment of local frequencies via analytical signal are showed. In particular gives the analytical signal only one local frequency, wherefore a signal decomposition is needed to work on each component seperately. Without such a decomposition the analytical signal approach yields just a mean frequency. Because of the importance for the assignment of local frequencies one expects from components of a decomposition of a signal that the wavelet transform of the components is also a decomposition of the wavelet transform of the original signal. Additionally the sum of all components has to equal the original signal. To fulfill both requirements the definition of components uses the Morlet reconstruction formula. In the occuring integrals will be integrated not over all frequencies (which would deliver the original signal) but over intervalls, which are generated out of a decomposition of the wavelet transform of the original signal. This decomposition is generated out of a decomposition of the time frequency space (R x R+) such that these sets are connected (depending on the wavelet transform of the signal). In this case the first requirement that the sum coincides with the original signal is fulfilled automatically (Morlet reconstruction formula, s. a.). That the wavelet transform of the components is also a decomposition of the wavelet transform of the original signal seems at first obvious but is much harder to proove. The thesis is structured in the following way: Chapter 1: Introduction Chapter 2: Preliminaries, basic definitions and propositions Chapter 3: Historical origin of the concept of the analytical signal. Analysis of the conincidence of local frequencies via analytical signal/Hilbert transform, i.e. instantaneous frequency with physical interpretation. The analysis showed that inner functions (special functions out of Hardy space) play an important role in the cases where the instantaneous frequency coincides with the physical interpretation. Chapter 4: Enhancement of the Morlet reconstruction formula to weak growing functions (a function which divided by a polynome is integrable), enhancement of the analytical signal/Hilbert transform to weak growing functions. Chapter 5: Decompositions of signals. Derivation according to an easy example followed by the general definition and the proof of the desired requirements. Chapter 6: In this chapter it is shown that the local maxima of the wavelet transform delivers local frequencies in the physical sense and when these coincides with those of the analytical singal.