Super-Resolution from Short-Time Fourier Transform Measurements

While spike trains are obviously not band-limited, the theory of super-resolution tells us that perfect recovery of unknown spike locations and weights from low-pass Fourier transform measurements is possible provided that the minimum spacing, $\Delta$, between spikes is not too small. Specifically, for a cutoff frequency of $f_c$, Donoho [2] shows that exact recovery is possible if $\Delta > 1/f_c$, but does not specify a corresponding recovery method. On the other hand, Cand\`es and Fernandez-Granda [3] provide a recovery method based on convex optimization, which provably succeeds as long as $\Delta > 2/f_c$. In practical applications one often has access to windowed Fourier transform measurements, i.e., short-time Fourier transform (STFT) measurements, only. In this paper, we develop a theory of super-resolution from STFT measurements, and we propose a method that provably succeeds in recovering spike trains from STFT measurements provided that $\Delta > 1/f_c$.Comment: IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), May 2014, to appea

Multiple and changing cycles of active stars I. Methods of analysis and application to the solar cycles

Long-term observational data have information on the magnetic cycles of active stars and that of the Sun. The changes in the activity of our central star have basic effects on Earth, like variations in the global climate. Therefore understanding the nature of these variations is extremely important. The observed variations related to magnetic activity cannot be treated as stationary periodic variations, therefore methods like Fourier transform or different versions of periodogramms give only partial information on the nature of the light variability. We demonstrate that time-frequency distributions provide useful tools for analyzing the observations of active stars. With test data we demonstrate that the observational noise has practically no effect on the determination in the the long-term changes of time-series observations of active stars. The rotational signal may modify the determined cycles, therefore it is advisable to remove it from the data. Wavelets are less powerful in recovering complex long-term changes than other distributions which are discussed. Applying our technique to the sunspot data we find a complicated, multi-scale evolution in the solar activity.Comment: Accepted to Astronomy and Astrophysic