PhD defense by Emil Solsbæk Ottosen
13.04.2018 kl. 13.00 - 17.00
Sparse Nonstationary Gabor Expansions With Applications to Music Signals
In this PhD thesis we consider sparseness properties of certain adaptive time-frequency representations. Time-frequency representations are two dimensional signal representa-tions containing information about the frequencies of the signal occurring at any given time point. Traditionally, such representations are obtained by dividing the signal into shorter segments and then applying the Fourier transform on each segment. The segments are obtained by multiplying the signal with a smooth window function and the resulting time-frequency resolution depends on the length of this window function. Mathematically speaking, the procedure described above corresponds to a stationary Gabor expansion obtained by decomposing the signal into a convergent sum of time-frequency localized atoms.
A straightforward generalization of the theory is to apply multiple window functions in the expansion resulting in so-called nonstationary Gabor expansions. Both stationary and nonstationary Gabor expansions have shown great potential in relation to music signals as they tend to produce sparse time-frequency representations. Sparseness of a time-frequency representation is often desirable as the particular characteristics of the signals become easier to identify. The sparseness property also allows for efficient approximations of the signal by thresholding the expansion coefficients.
In this thesis we use a very general class of smoothness spaces known as decomposition spaces to characterize signals with sparse expansions relative to certain nonstationary Gabor frames. Nonstationary Gabor frames can be implemented in both the time and the frequency domain and we provide a separate characterization for each case. As a practical application of the theory, we construct a new time-stretching algorithm based on nonstationary Gabor frames in the time domain. Time-stretching is the application of modifying the length of a signal without affecting its frequencies.
Professor Horia Cornean, Aalborg Universitet
Lektor Jakob Lemvig, Danmarks Tekniske Universitet
Professor Hans G- Feichtinger, University of Vienna
Institutleder Søren Højsgaard
Professor Morten Nielsen, Aalborg Universitet
After PhD defense the department will host a small reception at Skjernvej 4A in room 5.123.
Institut for Matematiske Fag
Skjernvej 4A, aud. 5.018
03.04.2018 kl. 12.00