Long abstract:
Gabor Analysis is the branch of Time-Frequency Analysis which works with series expansions of signals, using time-frequency shifts of a so-called Gabor atom (typically a Gaussian function) along some lattice in the TF-plane.

Description of content...

Gabor Analysis is one of the central topics of Time-Frequency Analysis. It is closely related to the sampling problem for the Short-time Fourier Transform or sliding-window Fourier transfrom (which is also in the background of the MP3 compression). Given a window (which might be a smooth, plateau-like function or a stretched Gaussian function, for example) and a sampling lattice, Gabor analysis provides tools to recover in a simple way the signal from the sampled STFT.

Both theoretical estimates and numerical simulations (e.g. using a toolbox developed in the NuHAG group over many years, or the LTFAT toolbox hosted nowadays at ARI, OEAW, both in Vienna) suggest that one has to find a balance between reasonable low redudandancy and numerical stability of the recovery process and overall good TF-locality of the recovery process. We will demonstrate how MATLAB simulations help to find the best match between a given window (e.g. a generalized Gaussian, which may have an elliptic concentration in the TF-plane) and the sampling lattice, typically via good covering properties.

The results of these simulations provide guide practical guidelines, but also encourage further theoretical studies and suggest possible claims about the structure of general Gabor frames.

Let us finally comment that a derivative of Gabor Analysis are the so-called Wilson bases, which have been used in the signal analysis part of the detection of gravitational waves. The huge LIGO team has obtained the Physics Nobel Prize in 2017. Another new application is the so-called Gaborator, which allows to view spectrograms and even upload private WAV-files. It is based on similar ideas, adapted to audio signals and human hearing. The underlying theory has been also derived by the ARI team.