Computational Aspects of Time-Frequency and Gabor Analysis
Hans G. Feichtinger
given at Malostranske namesti (CS building of the MFF) , Prague, Computer Science (24.01.19 16:15)
Gabor Analysis is a part of Time-Frequency Analysis. It is concerned with localized Fourier expansions of a given signal. In the one-dimensional case one is carrying out Fourier Analysis (via the discrete/fast Fourier transform, the DFT/FFT) of segments of an audio signal. It can be viewed as a kind of inversion of the process of producing music from a score. Sometimes the pictures obtained by this transform which are called spectrogram look like a graphical composition. The method is also at the basis for the MP3 compression algorithm for audio data.
In two dimensions one can compare the approach with the foundation of JPEG compression of images, the current standard for image compression. But instead of a decomposing an image into disjoint 8x8 blocks one has overlapping blocks with smooth transitions.
While the foundations of this theory go back to a paper by D. Gabor from 1946 the
mathematical analysis and parallel to it the computational realization of this theory has started only in the late 80th of the last century. The talk will illustrate the applications, describe the mathematical structure of the problem, and how this analysis has helped to overcome the computational questions involved in this problem. The demonstration found at www.gaborator.com provides a convincing illustration of the subject using audio signals.