On time-frequency analysis and time-limitedness

We study two classical problems, namely the concentration of energy problem and the truncation problem. The first problem deals with time-limited signals that have maximal energy in a certain frequency band. The second problem is about estimating the spectrum of a signal, if this signal is only known at a certain interval. Solutions of the first problem can be used to obtain good solutions for the second one by means of a preprocessing algorithm, called tapering. The truncation problem and the tapering algorithm are also studied for time-scale and time-frequency analysis, using the continuous wavelet transform and the Wigner-Ville representation

Integral representations of affine transformations in phase space with an application to energy localization problems

Applying the fractional Fourier transform and the Wigner distribution on a signal in a cascade fashion is equivalent with a rotation of the time and frequency parameters of the Wigner distribution. This report presents a formula for all unitary operators that are related to energy preserving transformations on the parameters of the Wigner distribution by means of such a cascade of operators. Furthermore, such operators are used to solve certain type of energy localization problems via the Weyl correspondence

Automatic phase detection in seismic data using the discrete wavelet transform

Seismic data consist of traces, which contain information about a seismic event, but in some period of time the traces may be just noise. A trace which c ontains seismic information, is called a seismic signal. Seismic signals consist of several typically short energy bursts, called phases, exhibiting several patterns in terms of dominant frequency, amplitude and polarisation. Amongst others, a significant phase is the S-phase. We present a fast algorithm to detect the S-phase in a three-component seismic signal. This new approach is a combination of traditional S-phase detection methods from seismology and the discrete wavele t transform. Stability and correctness of the algorithm will be proved and results will be presented to demonstrate the algorithm